Finite Group Databasebeta

← #29700839

Subgroups — #29700839

Order 15616.

998 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839946C211yesmaximal, characteristic
2#21454660C211yesmaximal, characteristic, complemented
3#21454658C211yesmaximal, characteristic, complemented
4#21454350C211yesmaximal, characteristic
5#21454349C211yesmaximal, characteristic
6#21446620C211yesmaximal, characteristic, complemented
7#21446473C211yesmaximal, characteristic, complemented
8#21446472C211yesmaximal, characteristic, complemented
9#21446335C211yesmaximal, characteristic, complemented
10#21446332C211yesmaximal, characteristic, complemented
11#21446211C211yesmaximal, characteristic, complemented
12#21446044C211yesmaximal, characteristic, complemented
13#21445657C211yesmaximal, characteristic, complemented
14#21445653C211yesmaximal, characteristic, complemented
15C61×#1859C211yesmaximal, characteristic, complemented
16C2×#14519824C2211yescharacteristic
17C2×#14519810C2211yescharacteristic
18C2×#14519807C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20C2×#14519784C2211yescharacteristic
21C2×#14519777C2211yescharacteristic
22#16840243C2211yescharacteristic
23#16840239C2211yescharacteristic
24#16839946C418yescomplemented
25#16839868—21no—
26#16839867—21no—
27#16839862—21no—
28#16839861—21no—
29#16839856—21no—
30C4×#11758202—21no—
31#16839849—21no—
32C4×#11758200—21no—
33#16839802C2211yescharacteristic
34C2×#14519705C2211yescharacteristic
35#16839772C2211yescharacteristic
36#16839770—21no—
37#16839770C2211yescharacteristic
38#16839769—21no—
39#16839761—21no—
40#16839761C2211yescharacteristic
41#16839760—21no—
42#16839758C2211yescharacteristic
43#16839756—21no—
44#16839756C2211yescharacteristic
45#16839755—21no—
46C2×#14519700—21no—
47C2×#14519700C2211yescharacteristic
48C2×#14519700C2211yescharacteristic
49C2×#14519700C2211yescharacteristic
50C2×#14519700C2211yescharacteristic
51#103×D61—21no—
52C2×#14519695C2211yescharacteristic
53C2×#14519695C2211yescharacteristic
54#16839720C2211yescharacteristic
55#16839718C2211yescharacteristic
56C2×C4×#92418C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839526C2211yescharacteristic
59#16839526C2211yescharacteristic
60#16839522C2211yescharacteristic
61C61×#520C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C61×#384C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#339C2211yescharacteristic
68#14519824—22no—
69#14519824—22no—
70#14519824—41no—
71#14519824C2×C414yes—
72C2×#11758222—21no—
73C2×#11758222C2311yescharacteristic
74C2×#11758224—21no—
75C2×#11758224—41no—
76C22×Dic122—21no—
77C22×Dic122C2311yescharacteristic
78C22×Dic122C2311yescharacteristic
79C2×Q8×D61—21no—
80#14519810—22no—
81#14519810C2×C414yescomplemented
82#14519807—22no—
83#14519807C2×C414yescomplemented
84C2×Dic244—21no—
85C2×#11758202—21no—
86C2×#11758190—21no—
87C2×#11758200—21no—
88#14519785—22no—
89#14519785—41no—
90#14519784—22no—
91#14519784—41no—
92C2×#11758194—21no—
93C2×#11758194C2311yescharacteristic
94C2×#11758194C2311yescharacteristic
95C2×#11758189—21no—
96C22×#92418—21no—
97C22×#92418C2311yescharacteristic
98C22×#92418C2311yescharacteristic
99C2×C8×D61—21no—
100#14519777—41no—
101#14519777C2×C414yescomplemented
102C4×Dic122—21no—
103C4×Dic122—21no—
104#14519742—21no—
105#14519742—21no—
106#14519742—21no—
107#14519739—21no—
108C4×C22⋊D61—41no—
109#14519736C2311yescharacteristic
110#14519735—21no—
111#14519733—41no—
112C2×C4⋊Dic61—41no—
113C4⋊C4×D61—21no—
114#14519710—21no—
115#14519708—21no—
116#14519705C2×C414yescomplemented
117#14519702—21no—
118#14519702—21no—
119#14519701D411yescharacteristic
120#14519701D411yescharacteristic
121#14519700—21no—
122#14519700—21no—
123#14519700—21no—
124#14519700—21no—
125#14519700—21no—
126#14519700—21no—
127#14519700—21no—
128#14519700—21no—
129#14519700—21no—
130#14519700—21no—
131#14519700—21no—
132#14519700D411yescharacteristic
133#14519700D411yescharacteristic
134#14519698D411yescharacteristic
135#14519698D411yescharacteristic
136#14519697—21no—
137#14519697D411yescharacteristic
138#14519697D411yescharacteristic
139#14519695—41no—
140#14519695C2×C414yescomplemented
141#14519688—21no—
142#14519687—21no—
143C4×#92418—21no—
144C4×#92418—41no—
145C8×Dic61—21no—
146C2×#41×C61C2311yescharacteristic
147C2×M16×C61C2311yescharacteristic
148C2×M16×C61C2311yescharacteristic
149C4×Q8×C61—21no—
150C4×D4×C61—21no—
151C4×D4×C61—41no—
152#117×C61—21no—
153#117×C61C2311yescharacteristic
154C2×C4⋊C4×C61—21no—
155C2×C4⋊C4×C61—41no—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C4⋊C4×C61C2311yescharacteristic
158C2×C42×C61C2311yescharacteristic
159#107×C61—21no—
160#106×C61—21no—
161#104×C61C2×C414yescomplemented
162#103×C61—21no—
163#102×C61—21no—
164#41×C61—21no—
165#41×C61—21no—
166#41×C61C22×C412yes—
167#41×C61#4112yes—
168C2×Q8×C61C2×D411yescharacteristic
169C2×D4×C61—21no—
170C2×D4×C61C2×D411yescharacteristic
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61—21no—
174C22×C4×C61—21no—
175C22×C4×C61—41no—
176C22×C4×C61C2411yescharacteristic
177C2×C22⋊D61—21no—
178C2×C22⋊D61—41no—
179C22×Dic61—21no—
180C22×Dic61—21no—
181C22×Dic61—41no—
182Q8×D61—24no—
183Q8×D61—42no—
184#11758224—24no—
185#11758224—42no—
186#11758224—44no—
187#11758224—44no—
188#11758222—21no—
189#11758222—21no—
190#11758222—24no—
191#11758222—42no—
192#11758222C22×C412yes—
193#11758222#4112yes—
194C2×D244C2×D411yescharacteristic
195C2×C4×D61—21no—
196C2×C4×D61—41no—
197C2×C4×D61—41no—
198C2×Dic122—21no—
199C2×Dic122—21no—
200C2×Dic122—21no—
201C2×Dic122—21no—
202C2×Dic122—21no—
203C2×Dic122—21no—
204C2×Dic122—21no—
205C2×Dic122—21no—
206C2×Dic122—24no—
207C2×Dic122—21no—
208C2×Dic122—21no—
209C2×Dic122—42no—
210C2×Dic122—41no—
211C2×Dic122C22×C412yes—
212C2×Dic122C22×C412yes—
213C2×Dic122C2×D411yescharacteristic
214M16×C61—21no—
215M16×C61—21no—
216M16×C61—21no—
217M16×C61C22×C412yes—
218C2×C8×C61—21no—
219C2×C8×C61—21no—
220C4⋊C4×C61—21no—
221C4⋊C4×C61—21no—
222C4⋊C4×C61—21no—
223C4⋊C4×C61—21no—
224C4⋊C4×C61—41no—
225C4⋊C4×C61—82no—
226C4⋊C4×C61C2×D411yescharacteristic
227C4⋊C4×C61C2×D411yescharacteristic
228C22⋊C4×C61—21no—
229C22⋊C4×C61—41no—
230C42×C61—21no—
231C42×C61—21no—
232C42×C61—41no—
233C42×C61C22×C412yes—
234C22⋊Dic61—21no—
235#11758202—24no—
236#11758200—24no—
237#11758198—41no—
238C4⋊Dic61—44no—
239C4⋊Dic61C2×D411yescharacteristic
240C4⋊Dic61C2×D411yescharacteristic
241#11758196—41no—
242#11758196—41no—
243#11758196—41no—
244C4×Dic61—21no—
245C4×Dic61—41no—
246C4×Dic61—41no—
247C4×Dic61—41no—
248#11758194—21no—
249#11758194—21no—
250#11758194—21no—
251#11758194—42no—
252#11758194C22×C412yes—
253C2×#92418—21no—
254C2×#92418—21no—
255C2×#92418—21no—
256C2×#92418—21no—
257C2×#92418—21no—
258C2×#92418—21no—
259C2×#92418—42no—
260C2×#92418C2×D411yescharacteristic
261C2×#92418C2×D411yescharacteristic
262C2×#92418C2×D411yescharacteristic
263C2×#92418C2×D411yescharacteristic
264Dic244—24no—
265#11758190—24no—
266#11758189—42no—
267C8×D61—42no—
268C23×C61—21no—
269C23×C61H(5,𝔽2)11yescharacteristic
270C22×D61—21no—
271Q8×C61—21no—
272Q8×C61C4×D412yes—
273D4×C61—22no—
274D4×C61—22no—
275D4×C61—21no—
276D4×C61C4×D412yes—
277C2×C4×C61—22no—
278C2×C4×C61—22no—
279C2×C4×C61—22no—
280C2×C4×C61—21no—
281C2×C4×C61—22no—
282C2×C4×C61—22no—
283C2×C4×C61—21no—
284C2×C4×C61—21no—
285C2×C4×C61—41no—
286C2×C4×C61—41no—
287C2×C4×C61—81no—
288C2×C4×C61—81no—
289C2×C4×C61—161no—
290C2×C4×C61C23×C411yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×C4×C61C22×D411yescharacteristic
293C2×C4×C61C2×#4111yescharacteristic
294C2×C4×C61C2×#4111yescharacteristic
295C2×C4×C61H(5,𝔽2)11yescharacteristic
296C22⋊D61—22no—
297C22⋊D61—22no—
298C22⋊D61—44no—
299C2×Dic61—21no—
300C2×Dic61—22no—
301C2×Dic61—22no—
302C2×Dic61—21no—
303C2×Dic61—21no—
304C2×Dic61—44no—
305C2×Dic61—41no—
306C2×Dic61—41no—
307C2×Dic61—41no—
308C2×Dic61—41no—
309D244—21no—
310D244C4×D412yes—
311C4×D61—22no—
312C4×D61—22no—
313C4×D61—44no—
314C4×D61—161no—
315Dic122—21no—
316Dic122—22no—
317Dic122—22no—
318Dic122—22no—
319Dic122—22no—
320Dic122—44no—
321Dic122—41no—
322Dic122—41no—
323Dic122—41no—
324Dic122—41no—
325Dic122C4×D412yes—
326C8×C61—22no—
327C8×C61—41no—
328#92418—22no—
329#92418—21no—
330#92418—21no—
331#92418—21no—
332#92418—21no—
333#92418—41no—
334#22894—611nomaximal, Sylow
335C22×C61—21no—
336C22×C61—21no—
337C22×C61—22no—
338C22×C61—41no—
339C22×C61—41no—
340C22×C61#51811yescharacteristic
341C22×C61D4211yescharacteristic
342C22×C61#55111yescharacteristic
343C2×D61—22no—
344C2×D61—41no—
345C2×D61—41no—
346C4×C61—24no—
347C4×C61—22no—
348C4×C61—41no—
349C4×C61—41no—
350C4×C61—81no—
351C4×C61—81no—
352C4×C61C2×C4×D411yescharacteristic
353C4×C61C2×C4×D411yescharacteristic
354C4×C61#54611yescharacteristic
355C4×C61#54611yescharacteristic
356Dic61—22no—
357Dic61—42no—
358Dic61—42no—
359Dic61—41no—
360Dic61—41no—
361C2×#456—611no—
362#2947—611no—
363#2947—611no—
364#2898—611no—
365#2898—611no—
366#1956—611no—
367#1930—611no—
368#1930—611no—
369#1908—611no—
370#1908—611no—
371#1886—611no—
372#1859—611no—
373#1859D6111yescharacteristic, complemented
374#1764—611no—
375#1764—611no—
376C2×C61—21no—
377C2×C61—21no—
378C2×C61—42no—
379C2×C61#235811yescharacteristic
380C2×C61#329111yescharacteristic
381C2×C61#329311yescharacteristic
382D61—42no—
383C2×2−1+4—611no—
384C2×#137—611no—
385C2×#137—611no—
386C2×#131—611no—
387C2×#131—611no—
388C2×C4⋊Q8—611no—
389#520—611no—
390#520C2×D6111yescharacteristic
391#519—611no—
392#456—618no—
393#441—1221no—
394#441—1221no—
395#440—1221no—
396#440—1221no—
397C4×Q16—1221no—
398C4×Q16—1221no—
399C4×SD16—1221no—
400C4×SD16—1221no—
401#428—611no—
402#428C2×D6111yescharacteristic
403C2×#104—611no—
404C2×#104C2×D6111yescharacteristic
405#419—611no—
406#419—611no—
407#419—1221no—
408#419—1221no—
409#417—611no—
410#417—1221no—
411#417—1221no—
412#417C2×D6111yescharacteristic
413#416—611no—
414#416—611no—
415#416—1221no—
416#416—1221no—
417C2×#103—611no—
418C2×#103—611no—
419C2×#103—611no—
420C2×#103—611no—
421C2×#103—1221no—
422C2×#103—1221no—
423C2×#101—611no—
424C2×#101—611no—
425#405—611no—
426#405—611no—
427C2×C4×C8—611no—
428#384C2×D6111yescharacteristic
429#343—611no—
430#343C2×D6111yescharacteristic
431#341—611no—
432#341—611no—
433#339—611no—
434#339C2×D6111yescharacteristic
435C61#2289411yescharacteristic, complemented, Sylow
4362−1+4—614no—
4372−1+4—1222no—
4382−1+4—1222no—
4392−1+4—2441no—
440C2×#41—611no—
441C2×#41—1221no—
442C2×#41—1221no—
443C2×#41—2441no—
444C2×#41C22×D6111yescharacteristic
445C22×Q8—611no—
446C22×Q8—611no—
447C22×Q8—1221no—
448C22×Q8—1221no—
449#137—614no—
450#137—614no—
451#137—1222no—
452#137—1222no—
453C2×Q16—1221no—
454C2×Q16—1221no—
455C2×SD16—1221no—
456C2×SD16—1221no—
457#131—1222no—
458#131—1222no—
459#131—2441no—
460#131—2441no—
461C2×M16—611no—
462C2×M16—611no—
463C2×M16—1221no—
464C2×M16—1221no—
465C2×M16C22×D6111yescharacteristic
466C2×M16C22×D6111yescharacteristic
467C22×C8—611no—
468C22×C8—611no—
469C22×C8—1221no—
470C22×C8—1221no—
471C4⋊Q8—614no—
472C4⋊Q8—2441no—
473C4×Q8—21no—
474C4×Q8—1221no—
475C4×Q8—1221no—
476C4×Q8—1221no—
477C4×Q8—1221no—
478C4×Q8—1221no—
479C4×D4—21no—
480C4×D4—41no—
481C4×D4—1221no—
482C4×D4—2441no—
483#117—21no—
484#117—611no—
485#117—1221no—
486#117—2441no—
487#117C22×D6111yescharacteristic
488C2×C4⋊C4—21no—
489C2×C4⋊C4—41no—
490C2×C4⋊C4—1221no—
491C2×C4⋊C4—2441no—
492C2×C4⋊C4C22×D6111yescharacteristic
493C2×C4⋊C4C22×D6111yescharacteristic
494C2×C42C22×D6111yescharacteristic
495#107—21no—
496#107—1221no—
497#106—21no—
498#106—1221no—
499#104—614no—
500#104C4×D6114yescomplemented
501#103—21no—
502#103—611no—
503#103—611no—
504#103—611no—
505#103—611no—
506#103—1221no—
507#103—1221no—
508#103—1221no—
509#103—1221no—
510#103—1221no—
511#103—1221no—
512#103—1221no—
513#103—1221no—
514#103—1221no—
515#103—1221no—
516#103—1221no—
517#103—1221no—
518#103—1221no—
519#102—21no—
520#102—611no—
521#102—611no—
522#102—611no—
523#102—611no—
524#102—1221no—
525#101—614no—
526#101—2441no—
527#97—1221no—
528#97—1221no—
529C4×C8—1221no—
530C4×C8—1221no—
531C4×C8—2441no—
532#41—21no—
533#41—21no—
534#41—612no—
535#41—612no—
536#41—1224no—
537#41—1221no—
538#41—1221no—
539#41—1224no—
540#41—2442no—
541#41—2444no—
542#41—2442no—
543#41—2444no—
544#41C2×C4×D6112yes—
545#41#1175822612yes—
546C2×Q8—612no—
547C2×Q8—612no—
548C2×Q8—611no—
549C2×Q8—1224no—
550C2×Q8—1221no—
551C2×Q8—1221no—
552C2×Q8—1224no—
553C2×Q8—1221no—
554C2×Q8—1221no—
555C2×Q8—1221no—
556C2×Q8—1221no—
557C2×Q8—1221no—
558C2×Q8—1221no—
559C2×Q8—1221no—
560C2×Q8—1221no—
561C2×Q8—2441no—
562C2×Q8—2442no—
563C2×Q8—2442no—
564C2×Q8D4×D6111yescharacteristic
565C2×D4—21no—
566C2×D4—611no—
567C2×D4—1221no—
568C2×D4—2441no—
569C2×D4D4×D6111yescharacteristic
570C22×C4—21no—
571C22×C4—21no—
572C22×C4—21no—
573C22×C4—21no—
574C22×C4—41no—
575C22×C4—1221no—
576C22×C4—1221no—
577C22×C4—1221no—
578C22×C4—2441no—
579C22×C4—2441no—
580C22×C4—2441no—
581C22×C4C23×D6111yescharacteristic
582Q16—1224no—
583Q16—1224no—
584SD16—1224no—
585SD16—1224no—
586M16—21no—
587M16—21no—
588M16—21no—
589M16—612no—
590M16—1221no—
591M16—1221no—
592M16—1221no—
593M16—2442no—
594M16—2442no—
595M16C2×C4×D6112yes—
596C2×C8—21no—
597C2×C8—21no—
598C2×C8—611no—
599C2×C8—611no—
600C2×C8—611no—
601C2×C8—611no—
602C2×C8—1221no—
603C2×C8—1221no—
604C2×C8—1221no—
605C2×C8—1221no—
606C2×C8—1221no—
607C2×C8—1221no—
608C2×C8—2442no—
609C2×C8—2442no—
610C4⋊C4—21no—
611C4⋊C4—21no—
612C4⋊C4—21no—
613C4⋊C4—21no—
614C4⋊C4—41no—
615C4⋊C4—82no—
616C4⋊C4—611no—
617C4⋊C4—611no—
618C4⋊C4—2444no—
619C4⋊C4—2441no—
620C4⋊C4—2441no—
621C4⋊C4—2441no—
622C4⋊C4D4×D6111yescharacteristic
623C4⋊C4D4×D6111yescharacteristic
624C22⋊C4—21no—
625C22⋊C4—41no—
626C22⋊C4—1221no—
627C22⋊C4—2441no—
628C42—21no—
629C42—21no—
630C42—41no—
631C42—1221no—
632C42—2441no—
633C42—2441no—
634C42—2441no—
635C42C2×C4×D6112yes—
636C23—21no—
637C23—1221no—
638C23#1451982211yescharacteristic
639Q8—21no—
640Q8—612no—
641Q8—1221no—
642Q8—1222no—
643Q8—1222no—
644Q8—1222no—
645Q8—1222no—
646Q8—2444no—
647Q8—2441no—
648Q8—2441no—
649Q8—2441no—
650Q8—2441no—
651Q8#1451973912yes—
652D4—22no—
653D4—22no—
654D4—21no—
655D4—612no—
656D4—1222no—
657D4—1222no—
658D4—1221no—
659D4—2444no—
660D4#1451973912yes—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—21no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—161no—
674C2×C4—1221no—
675C2×C4—1222no—
676C2×C4—1222no—
677C2×C4—1222no—
678C2×C4—1222no—
679C2×C4—1221no—
680C2×C4—1221no—
681C2×C4—9761no—
682C2×C4—2444no—
683C2×C4—2444no—
684C2×C4—2441no—
685C2×C4—2441no—
686C2×C4—2441no—
687C2×C4—2441no—
688C2×C4C22×C4×D6111yescharacteristic
689C2×C4C2×D4×D6111yescharacteristic
690C2×C4C2×D4×D6111yescharacteristic
691C2×C4#41×D6111yescharacteristic
692C2×C4C2×#1175822611yescharacteristic
693C2×C4#1451982311yescharacteristic
694C8—22no—
695C8—41no—
696C8—1222no—
697C8—1221no—
698C8—1221no—
699C8—1221no—
700C8—1221no—
701C8—2441no—
702C22—21no—
703C22—21no—
704C22—22no—
705C22—41no—
706C22—41no—
707C22—1222no—
708C22—2441no—
709C22—2441no—
710C22#1684023411yescharacteristic
711C22#1684039411yescharacteristic
712C22#1684043911yescharacteristic
713C4—24no—
714C4—22no—
715C4—41no—
716C4—41no—
717C4—81no—
718C4—81no—
719C4—1222no—
720C4—2442no—
721C4—2442no—
722C4—2441no—
723C4—2441no—
724C4C4×D4×D6111yescharacteristic
725C4C2×#1451973911yescharacteristic
726C4#1684040311yescharacteristic
727C4#1684040411yescharacteristic
728C2—21no—
729C2—21no—
730C2—42no—
731C2—2442no—
732C2#2144879511yescharacteristic
733C2#2145761411yescharacteristic
734C2#2145762011yescharacteristic
subgroupG/H
centerC22#16840394
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1859C2
socleC22×C61D42
prime radicalC2×#14519824C22
cosocleC22×C4×C61C24
hypercenter#1859D61
hypocenterC61#22894
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1859

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454660C2semidirect
3. #21454658C2semidirect
6. #21446620C2semidirect
7. #21446473C2semidirect
8. #21446472C2semidirect
9. #21446335C2semidirect
10. #21446332C2semidirect
11. #21446211C2semidirect
12. #21446044C2semidirect
13. #21445657C2semidirect
14. #21445653C2semidirect
15. C61×#1859C2semidirect
24. #16839946C4semidirect
81. #14519810C2×C4semidirect
83. #14519807C2×C4semidirect
101. #14519777C2×C4semidirect
116. #14519705C2×C4semidirect
140. #14519695C2×C4semidirect
161. #104×C61C2×C4semidirect
373. #1859D61semidirect
435. C61#22894semidirect
500. #104C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399462
#214546602
#214546582
#214543502
#214543492
#214466202
#214464732
#214464722
#214463352
#214463322
#214462112
#214460442
#214456572
#214456532
C61×#18592
#2289461