Finite Group Databasebeta

← #29686903

Subgroups — #29686903

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840036C211yesmaximal, characteristic
2#21454637C211yesmaximal, characteristic
3#21454634C211yesmaximal, characteristic
4C2×#16839854C211yesmaximal, characteristic, complemented
5#21454241C211yesmaximal, characteristic
6#21446733C211yesmaximal, characteristic, complemented
7#21446601C211yesmaximal, characteristic, complemented
8#21446474C211yesmaximal, characteristic, complemented
9#21446337C211yesmaximal, characteristic
10#21446333C211yesmaximal, characteristic
11#21445882C211yesmaximal, characteristic
12#21445720C211yesmaximal, characteristic, complemented
13#101×Dic61C211yesmaximal, characteristic, complemented
14#21445614C211yesmaximal, characteristic, complemented
15C61×#1930C211yesmaximal, characteristic, complemented
16C2×#14519810C2211yescharacteristic
17C2×#14519808C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20C2×#14519783C2211yescharacteristic
21#16840230C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×#14519742C2211yescharacteristic
24#16840036C418yescomplemented
25#16839868—21no—
26#16839866—21no—
27#16839865—21no—
28#16839861—21no—
29#16839856—21no—
30#16839854—21no—
31#16839854—21no—
32#16839854—21no—
33#16839854—21no—
34#16839854—21no—
35#16839849—21no—
36#16839847—21no—
37C2×#14519712C2211yescharacteristic
38C2×#14519712C2211yescharacteristic
39#16839805C2211yescharacteristic
40C2×#14519710C2211yescharacteristic
41#16839770C2211yescharacteristic
42#16839764C2211yescharacteristic
43#16839761C2211yescharacteristic
44C2×#14519702C2211yescharacteristic
45C2×#14519696C2211yescharacteristic
46C2×#14519695C2211yescharacteristic
47C2×#14519695C2211yescharacteristic
48#16839719—21no—
49#16839718C2211yescharacteristic
50#16839716—21no—
51#16839714—21no—
52M16×Dic61C2211yescharacteristic
53M16×Dic61C2211yescharacteristic
54M16×Dic61C2211yescharacteristic
55C2×C8×Dic61C2211yescharacteristic
56C4×C8×D61—21no—
57#16839531C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839527C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C61×#419C2211yescharacteristic
63C2×#103×C61C2211yescharacteristic
64C2×#101×C61C2211yescharacteristic
65C61×#405C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#341C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70#14519810—22no—
71#14519810C2×C414yes—
72#14519808—22no—
73#14519808C2×C414yes—
74C2×Dic244—21no—
75C2×#11758202—21no—
76C2×#11758202—21no—
77C2×#11758202—21no—
78C2×#11758202—22no—
79C2×#11758202C2×C414yescomplemented
80C2×#11758190—21no—
81C2×#11758201—21no—
82#14519785—22no—
83#14519785C2×C414yes—
84#14519783—22no—
85#14519783—22no—
86C2×#11758194C2311yescharacteristic
87C2×#11758194C2311yescharacteristic
88C2×#11758194C2311yescharacteristic
89C2×#11758189—21no—
90C2×#11758189—21no—
91C22×#92418C2311yescharacteristic
92C2×C8×D61—21no—
93C2×C8×D61—21no—
94Q8×Dic61—21no—
95Q8×Dic61—21no—
96Q8×Dic61—21no—
97Q8×Dic61—21no—
98#14519742—21no—
99#14519742—21no—
100#14519742—21no—
101#14519742—21no—
102#14519742D411yescharacteristic
103#14519742D411yescharacteristic
104#14519739D411yescharacteristic
105#14519739D411yescharacteristic
106#14519737—21no—
107#14519737—21no—
108#14519734—21no—
109C2×C4⋊Dic61—21no—
110C2×C4⋊Dic61C2311yescharacteristic
111C2×#11758196—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—21no—
114C2×C4×Dic61C2311yescharacteristic
115C42×D61—21no—
116#14519712—22no—
117#14519712C2×C414yescomplemented
118#14519710—21no—
119#14519710—21no—
120#14519710—21no—
121#14519708—21no—
122#14519702—21no—
123#14519702—21no—
124#14519702—21no—
125#14519700—21no—
126#14519700—21no—
127#14519700—21no—
128#14519699—21no—
129#14519697—21no—
130#14519696—22no—
131#14519695—22no—
132#14519695C2×C414yescomplemented
133#14519688—21no—
134#14519687—21no—
135#14519687—21no—
136#14519687—21no—
137C4×#92418—21no—
138C8×Dic61—21no—
139C8×Dic61—21no—
140C8×Dic61—21no—
141C8×Dic61D411yescharacteristic
142C8×Dic61D411yescharacteristic
143C8×Dic61D411yescharacteristic
144C8×Dic61D411yescharacteristic
145C22×Q8×C61C2311yescharacteristic
146C2×M16×C61C2311yescharacteristic
147C2×M16×C61C2311yescharacteristic
148C2×M16×C61C2311yescharacteristic
149C22×C8×C61C2311yescharacteristic
150#117×C61C2311yescharacteristic
151C2×C4⋊C4×C61C2311yescharacteristic
152#107×C61—21no—
153#106×C61—21no—
154#103×C61—21no—
155#103×C61—21no—
156#103×C61—21no—
157#103×C61—21no—
158#101×C61C2×C414yescomplemented
159#97×C61—21no—
160C4×C8×C61—21no—
161C2×Q8×C61—21no—
162C2×Q8×C61—21no—
163C2×Q8×C61—21no—
164C2×Q8×C61—21no—
165C2×Q8×C61—21no—
166C2×Q8×C61C22×C412yes—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61C2411yescharacteristic
170C2×C22⋊D61—21no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61#4111yescharacteristic
174C22×Dic61#4111yescharacteristic
175#11758222—21no—
176#11758222—21no—
177#11758222C22×C412yes—
178#11758222#4112yes—
179C2×D244C2×D411yescharacteristic
180C2×C4×D61—21no—
181C2×C4×D61—21no—
182C2×C4×D61—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122C22×C412yes—
189C2×Dic122C2×D411yescharacteristic
190M16×C61—21no—
191M16×C61—21no—
192M16×C61—21no—
193M16×C61—21no—
194M16×C61C22×C412yes—
195M16×C61C22×C412yes—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198C2×C8×C61—21no—
199C2×C8×C61—21no—
200C2×C8×C61—21no—
201C2×C8×C61C2×D411yescharacteristic
202C2×C8×C61C2×D411yescharacteristic
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61C2×D411yescharacteristic
206C4⋊C4×C61C2×D411yescharacteristic
207C22⋊C4×C61—21no—
208C42×C61—21no—
209#11758202—24no—
210#11758202—24no—
211#11758202—24no—
212#11758201—24no—
213#11758198—21no—
214#11758198—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—41no—
218C4⋊Dic61—41no—
219#11758196—21no—
220#11758196—21no—
221#11758196—41no—
222#11758196—41no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—41no—
226C4×Dic61—41no—
227C4×Dic61—41no—
228C4×Dic61—41no—
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233#11758194—21no—
234#11758194—21no—
235#11758194—21no—
236#11758194—21no—
237#11758194C22×C412yes—
238#11758194#4112yes—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418C22×C412yes—
243C2×#92418C2×D411yescharacteristic
244C2×#92418C2×D411yescharacteristic
245C2×#92418#4112yes—
246Dic244—24no—
247#11758190—24no—
248#11758189—24no—
249#11758189—24no—
250C8×D61—24no—
251C8×D61—24no—
252C23×C61C2×#4111yescharacteristic
253C22×D61—21no—
254Q8×C61—22no—
255Q8×C61—22no—
256Q8×C61—41no—
257Q8×C61—41no—
258C2×C4×C61—22no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—81no—
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C22⋊D61—22no—
272C22⋊D61—22no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—81no—
281C2×Dic61#12111yescharacteristic
282C2×Dic61#12111yescharacteristic
283C2×Dic61#12111yescharacteristic
284C2×Dic61#12111yescharacteristic
285D244—21no—
286D244C4×D412yes—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—42no—
290C4×D61—42no—
291Dic122—21no—
292Dic122—22no—
293Dic122—22no—
294Dic122—41no—
295Dic122—41no—
296Dic122C4×D412yes—
297C8×C61—22no—
298C8×C61—22no—
299C8×C61—21no—
300C8×C61—21no—
301C8×C61—41no—
302#92418—22no—
303#92418—22no—
304#92418—41no—
305#92418C4×D412yes—
306#92418C4×D412yes—
307C4×#473—611nomaximal, Sylow
308C22×C61—21no—
309C22×C61—21no—
310C22×C61C4×#4111yescharacteristic
311C22×C61C2×#12111yescharacteristic
312C22×C61#52911yescharacteristic
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—42no—
319C4×C61—81no—
320C4×C61C2×C4×D411yescharacteristic
321C4×C61C2×C4×D411yescharacteristic
322C4×C61#52511yescharacteristic
323C4×C61#53211yescharacteristic
324Dic61—21no—
325Dic61—21no—
326Dic61—21no—
327Dic61—22no—
328Dic61—42no—
329Dic61—81no—
330Dic61#47312yescomplemented
331C2×#473—611no—
332C4×#137—611no—
333C4×#137—611no—
334C2×C4×Q16—611no—
335C4×#131—611no—
336#1979—611no—
337#1952—611no—
338#1930—611no—
339#1930D6111yescharacteristic, complemented
340#1908—611no—
341#1908—611no—
342#1816—611no—
343C4×#108—611no—
344C4×#101—611no—
345C4×#101—611no—
346C2×C61—21no—
347C2×C61—21no—
348C2×C61C4×#12111yescharacteristic
349C2×C61C2×#47311yescharacteristic
350C2×C61#306711yescharacteristic
351D61—42no—
352C2×#137—611no—
353C2×#137—611no—
354C22×Q16—611no—
355C2×#131—611no—
356C2×#131—611no—
357C4×#41—611no—
358C2×C4×Q8—611no—
359C2×C4×Q8—611no—
360#473—618no—
361#441—1221no—
362#441—1221no—
363#440—1221no—
364#440—1221no—
365C4×Q16—1221no—
366C4×Q16—1221no—
367C4×Q16—1221no—
368C4×Q16—1221no—
369C4×Q16—1221no—
370C4×Q16—1221no—
371C4×SD16—1221no—
372C4×SD16—1221no—
373C2×#108—611no—
374C2×#108—611no—
375#428—611no—
376#428C2×D6111yescharacteristic
377C2×#107—611no—
378#419—611no—
379#419C2×D6111yescharacteristic
380#417—611no—
381#417—611no—
382C2×#103—611no—
383C2×#103C2×D6111yescharacteristic
384C2×#101—611no—
385C2×#101—611no—
386C2×#101—611no—
387C2×#101C2×D6111yescharacteristic
388#405—611no—
389#405—1221no—
390#405—1221no—
391#405C2×D6111yescharacteristic
392C4×M16—611no—
393C4×M16—611no—
394C4×M16—611no—
395C4×M16—1221no—
396C2×C4×C8—611no—
397C2×C4×C8—1221no—
398#343—611no—
399#343—611no—
400#343—611no—
401#343C2×D6111yescharacteristic
402#341—611no—
403#341C2×D6111yescharacteristic
404C61C4×#47311yescharacteristic, complemented, Sylow
405C2×#41—611no—
406C22×Q8—611no—
407C22×Q8C22×D6111yescharacteristic
408#137—614no—
409#137—614no—
410#137—1222no—
411#137—1222no—
412C2×Q16—614no—
413C2×Q16—1221no—
414C2×Q16—1221no—
415C2×Q16—1221no—
416C2×Q16—1222no—
417C2×Q16—1221no—
418C2×SD16—1221no—
419C2×SD16—1221no—
420#131—614no—
421#131—1222no—
422#131—1222no—
423#131—1222no—
424C2×M16—611no—
425C2×M16—611no—
426C2×M16—611no—
427C2×M16—1221no—
428C2×M16—1221no—
429C2×M16C22×D6111yescharacteristic
430C2×M16C22×D6111yescharacteristic
431C2×M16C22×D6111yescharacteristic
432C22×C8—611no—
433C22×C8—1221no—
434C22×C8—1221no—
435C22×C8C22×D6111yescharacteristic
436C4×Q8—611no—
437C4×Q8—611no—
438C4×Q8—1221no—
439C4×Q8—1221no—
440C4×Q8—1221no—
441C4×Q8—1221no—
442C4×Q8—1221no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×Q8—1221no—
446C4×D4—611no—
447C4×D4—611no—
448C4×D4—1221no—
449C4×D4—1221no—
450#117—1221no—
451#117C22×D6111yescharacteristic
452C2×C4⋊C4—611no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C42—611no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42—1221no—
460#108—614no—
461#108—1222no—
462#107—21no—
463#107—1221no—
464#107—1221no—
465#107—1221no—
466#106—21no—
467#106—1221no—
468#103—21no—
469#103—21no—
470#103—21no—
471#103—21no—
472#103—1221no—
473#103—1221no—
474#103—1221no—
475#103—1221no—
476#103—1221no—
477#103—1221no—
478#102—1221no—
479#102—1221no—
480#101—614no—
481#101—1222no—
482#101—1222no—
483#101C4×D6114yescomplemented
484#97—21no—
485#97—1221no—
486#97—1221no—
487#97—1221no—
488#97—1221no—
489C4×C8—21no—
490C4×C8—611no—
491C4×C8—611no—
492C4×C8—611no—
493C4×C8—611no—
494C4×C8—1221no—
495C4×C8—1221no—
496C4×C8—1221no—
497C4×C8—1221no—
498#41—612no—
499#41—612no—
500#41—1221no—
501#41—1221no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—612no—
508C2×Q8—611no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8C2×C4×D6112yes—
515C2×D4—611no—
516C2×D4—1221no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4C23×D6111yescharacteristic
527Q16—1224no—
528Q16—1224no—
529Q16—1224no—
530Q16—1224no—
531SD16—1224no—
532SD16—1224no—
533M16—21no—
534M16—21no—
535M16—21no—
536M16—21no—
537M16—612no—
538M16—612no—
539M16—1224no—
540M16—1224no—
541M16—1221no—
542M16—1221no—
543M16—1221no—
544M16—1221no—
545M16C2×C4×D6112yes—
546M16C2×C4×D6112yes—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—612no—
553C2×C8—611no—
554C2×C8—611no—
555C2×C8—612no—
556C2×C8—1224no—
557C2×C8—1224no—
558C2×C8—1221no—
559C2×C8—1221no—
560C2×C8—1221no—
561C2×C8D4×D6111yescharacteristic
562C2×C8D4×D6111yescharacteristic
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—2441no—
570C4⋊C4—2441no—
571C4⋊C4—2441no—
572C4⋊C4—2441no—
573C4⋊C4D4×D6111yescharacteristic
574C4⋊C4D4×D6111yescharacteristic
575C22⋊C4—21no—
576C22⋊C4—1221no—
577C22⋊C4—1221no—
578C42—21no—
579C42—611no—
580C42—611no—
581C42—611no—
582C42—611no—
583C42—1221no—
584C42—1221no—
585C42—2441no—
586C42—2441no—
587C42—2441no—
588C42—2441no—
589C23—1221no—
590C23#41×D6111yescharacteristic
591Q8—22no—
592Q8—22no—
593Q8—41no—
594Q8—41no—
595Q8—612no—
596Q8—1221no—
597Q8—1222no—
598Q8—1222no—
599Q8—2441no—
600Q8—2441no—
601D4—612no—
602D4—1222no—
603D4—1222no—
604D4—1221no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—81no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—4881no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4C22×C4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×D4×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C8—22no—
635C8—22no—
636C8—21no—
637C8—21no—
638C8—41no—
639C8—612no—
640C8—612no—
641C8—1222no—
642C8—1222no—
643C8—2441no—
644C22—21no—
645C22—21no—
646C22—1222no—
647C22—2441no—
648C22—2441no—
649C22#1684023011yescharacteristic
650C22#121×D6111yescharacteristic
651C22#1684029511yescharacteristic
652C4—42no—
653C4—41no—
654C4—41no—
655C4—81no—
656C4—612no—
657C4—1222no—
658C4—1221no—
659C4—1221no—
660C4—1221no—
661C4—4881no—
662C4—2442no—
663C4C4×D4×D6111yescharacteristic
664C4C4×D4×D6111yescharacteristic
665C4#1684027011yescharacteristic
666C4#1684031111yescharacteristic
667C2—21no—
668C2—21no—
669C2—2442no—
670C2#2144833111yescharacteristic
671C2#473×D6111yescharacteristic
672C2#2145553411yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1930C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1930D61
hypocenterC61C4×#473
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1930

Complements

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

NH ≅ G/Ntype
4. C2×#16839854C2semidirect
6. #21446733C2semidirect
7. #21446601C2semidirect
8. #21446474C2semidirect
12. #21445720C2semidirect
13. #101×Dic61C2semidirect
14. #21445614C2semidirect
15. C61×#1930C2semidirect
24. #16840036C4semidirect
79. C2×#11758202C2×C4semidirect
117. #14519712C2×C4semidirect
132. #14519695C2×C4semidirect
158. #101×C61C2×C4semidirect
330. Dic61#473semidirect
339. #1930D61semidirect
404. C61C4×#473semidirect
483. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400362
#214546372
#214546342
C2×#168398542
#214542412
#214467332
#214466012
#214464742
#214463372
#214463332
#214458822
#214457202
#101×Dic612
#214456142
C61×#19302
C4×#47361