Finite Group Databasebeta

← #29686944

Subgroups — #29686944

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840024C211yesmaximal, characteristic
2#21454637C211yesmaximal, characteristic, complemented
3#21454623C211yesmaximal, characteristic
4C2×#16839859C211yesmaximal, characteristic, complemented
5#21454246C211yesmaximal, characteristic, complemented
6#21446725C211yesmaximal, characteristic, complemented
7#21446601C211yesmaximal, characteristic, complemented
8#21446474C211yesmaximal, characteristic, complemented
9#21446333C211yesmaximal, characteristic, complemented
10#21446330C211yesmaximal, characteristic
11#21445882C211yesmaximal, characteristic, complemented
12#21445724C211yesmaximal, characteristic, complemented
13#21445614C211yesmaximal, characteristic, complemented
14#100×Dic61C211yesmaximal, characteristic, complemented
15C61×#1931C211yesmaximal, characteristic, complemented
16C2×#14519810C2211yescharacteristic
17C2×#14519803C2211yescharacteristic
18C22×#11758200C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20C2×#14519783C2211yescharacteristic
21#16840230C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24#16840024C418yescomplemented
25#16839871—21no—
26#16839868—21no—
27#16839861—21no—
28#16839859—21no—
29#16839859—21no—
30#16839859—21no—
31#16839859—21no—
32#16839859—21no—
33#16839856—21no—
34#16839853—21no—
35#16839849—21no—
36#16839842—21no—
37C2×#14519712C2211yescharacteristic
38C2×#14519712C2211yescharacteristic
39#16839805C2211yescharacteristic
40C2×#14519710C2211yescharacteristic
41#16839770C2211yescharacteristic
42#16839764C2211yescharacteristic
43#16839761C2211yescharacteristic
44C2×#14519702C2211yescharacteristic
45C2×#14519695C2211yescharacteristic
46C2×#14519695C2211yescharacteristic
47C2×#14519694C2211yescharacteristic
48#16839719—21no—
49#16839718C2211yescharacteristic
50#16839716—21no—
51M16×Dic61C2211yescharacteristic
52M16×Dic61C2211yescharacteristic
53M16×Dic61C2211yescharacteristic
54C4×#11758189—21no—
55C2×#14519687C2211yescharacteristic
56#97×D61—21no—
57#16839531C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839527C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C61×#418C2211yescharacteristic
63C2×#102×C61C2211yescharacteristic
64C2×#100×C61C2211yescharacteristic
65C61×#405C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#341C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C23×Dic61—21no—
71#14519810—22no—
72#14519810C2×C414yescomplemented
73#14519803—22no—
74#14519803C2×C414yes—
75C2×Dic244—21no—
76C2×#11758190—21no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×#11758200—21no—
80C2×#11758200—22no—
81C2×#11758200C2×C414yescomplemented
82C2×#11758199—21no—
83#14519785—22no—
84#14519785C2×C414yescomplemented
85#14519783—22no—
86#14519783—22no—
87C2×#11758194C2311yescharacteristic
88C2×#11758194C2311yescharacteristic
89C2×#11758194C2311yescharacteristic
90C2×#11758189—21no—
91C2×#11758189—21no—
92C22×#92418C2311yescharacteristic
93C2×C8×D61—21no—
94C2×C8×D61—21no—
95#14519742—21no—
96#14519742—21no—
97#14519742—21no—
98#14519742—21no—
99#14519742D411yescharacteristic
100#14519742D411yescharacteristic
101D4×Dic61—21no—
102D4×Dic61—21no—
103D4×Dic61—21no—
104D4×Dic61—21no—
105#14519739D411yescharacteristic
106#14519739D411yescharacteristic
107#14519737—21no—
108#14519737—21no—
109#14519734—21no—
110C2×C4⋊Dic61C2311yescharacteristic
111C2×#11758196—21no—
112C2×C22⋊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#14519695—22no—
131#14519695C2×C414yescomplemented
132#14519694—22no—
133#14519688—21no—
134#14519687—21no—
135#14519687—21no—
136#14519687—21no—
137#14519687D411yescharacteristic
138#14519687D411yescharacteristic
139#14519687D411yescharacteristic
140#14519687D411yescharacteristic
141C4×#92418—21no—
142C8×Dic61—21no—
143C8×Dic61—21no—
144C8×Dic61—21no—
145C22×D4×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#102×C61—21no—
155#102×C61—21no—
156#102×C61—21no—
157#102×C61—21no—
158#100×C61C2×C414yescomplemented
159#97×C61—21no—
160C4×C8×C61—21no—
161C24×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61C22×C412yes—
168C22×C4×C61—21no—
169C22×C4×C61C2411yescharacteristic
170C2×C22⋊D61—21no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—41no—
174C22×Dic61—41no—
175C22×Dic61—82no—
176C22×Dic61#4111yescharacteristic
177C22×Dic61#4111yescharacteristic
178#11758222—21no—
179#11758222—21no—
180#11758222C22×C412yes—
181#11758222#4112yes—
182C2×D244C2×D411yescharacteristic
183C2×C4×D61—21no—
184C2×C4×D61—21no—
185C2×C4×D61—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122C22×C412yes—
192C2×Dic122C2×D411yescharacteristic
193M16×C61—21no—
194M16×C61—21no—
195M16×C61—21no—
196M16×C61—21no—
197M16×C61C22×C412yes—
198M16×C61C22×C412yes—
199C2×C8×C61—21no—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C2×C8×C61C2×D411yescharacteristic
205C2×C8×C61C2×D411yescharacteristic
206C4⋊C4×C61—21no—
207C4⋊C4×C61—21no—
208C4⋊C4×C61C2×D411yescharacteristic
209C4⋊C4×C61C2×D411yescharacteristic
210C22⋊C4×C61—21no—
211C42×C61—21no—
212C22⋊Dic61—41no—
213C22⋊Dic61—41no—
214#11758200—24no—
215#11758200—24no—
216#11758200—24no—
217#11758199—24no—
218#11758198—21no—
219#11758198—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222#11758196—21no—
223#11758196—21no—
224#11758196—41no—
225#11758196—41no—
226C4×Dic61—21no—
227C4×Dic61—21no—
228C4×Dic61—41no—
229C4×Dic61—41no—
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234#11758194—21no—
235#11758194—21no—
236#11758194—21no—
237#11758194—21no—
238#11758194C22×C412yes—
239#11758194#4112yes—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418—21no—
243C2×#92418C22×C412yes—
244C2×#92418C2×D411yescharacteristic
245C2×#92418C2×D411yescharacteristic
246C2×#92418#4112yes—
247Dic244—24no—
248#11758190—24no—
249#11758189—24no—
250#11758189—24no—
251C8×D61—24no—
252C8×D61—24no—
253C23×C61—22no—
254C23×C61—41no—
255C23×C61—41no—
256C23×C61—81no—
257C23×C61—81no—
258C23×C61C2×#4111yescharacteristic
259C22×D61—21no—
260D4×C61—22no—
261D4×C61—22no—
262D4×C61—41no—
263D4×C61—41no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—41no—
267C2×C4×C61—81no—
268C2×C4×C61C23×C411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C22⋊D61—22no—
275C22⋊D61—22no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—82no—
284C2×Dic61—82no—
285C2×Dic61—81no—
286C2×Dic61#12111yescharacteristic
287C2×Dic61#12111yescharacteristic
288C2×Dic61#12111yescharacteristic
289C2×Dic61#12111yescharacteristic
290D244—21no—
291D244C4×D412yes—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—81no—
295C4×D61—81no—
296Dic122—21no—
297Dic122—22no—
298Dic122—22no—
299Dic122—41no—
300Dic122—41no—
301Dic122C4×D412yes—
302C8×C61—22no—
303C8×C61—22no—
304C8×C61—21no—
305C8×C61—21no—
306C8×C61—41no—
307#92418—22no—
308#92418—22no—
309#92418—41no—
310#92418C4×D412yes—
311#92418C4×D412yes—
312#21557—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—42no—
316C22×C61—82no—
317C22×C61—82no—
318C22×C61—81no—
319C22×C61—81no—
320C22×C61C4×#4111yescharacteristic
321C22×C61C2×#12111yescharacteristic
322C22×C61#52911yescharacteristic
323C2×D61—22no—
324C2×D61—41no—
325C2×D61—41no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—81no—
329C4×C61C2×C4×D411yescharacteristic
330C4×C61C2×C4×D411yescharacteristic
331C4×C61#52511yescharacteristic
332C4×C61#53211yescharacteristic
333Dic61—22no—
334Dic61—21no—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—42no—
339Dic61—81no—
340C2×#471—611no—
341C4×#137—611no—
342C4×AGL(1,ℤ/8)—611no—
343C2×#440—611no—
344#2875—611no—
345#1977—611no—
346#1952—611no—
347#1931D6111yescharacteristic, complemented
348#1930—611no—
349#1908—611no—
350#1907—611no—
351#1816—611no—
352#1778—611no—
353C4×#101—611no—
354C4×#100—611no—
355C2×C61—21no—
356C2×C61—21no—
357C2×C61—82no—
358C2×C61C4×#12111yescharacteristic
359C2×C61#306811yescharacteristic
360C2×C61#306911yescharacteristic
361D61—42no—
362C2×#137—611no—
363C2×AGL(1,ℤ/8)—611no—
364C22×SD16—611no—
365C2×#131—611no—
366C2×#131—611no—
367C4×#41—611no—
368C2×C4×Q8—611no—
369C2×C4×D4—611no—
370#471—618no—
371#442—1221no—
372#441—1221no—
373#440—1221no—
374#440—1221no—
375#440—1221no—
376#440—1221no—
377#440—1221no—
378#440—1221no—
379C4×Q16—1221no—
380C4×SD16—1221no—
381C4×SD16—1221no—
382C4×D8—1221no—
383C2×#108—611no—
384C2×#108—611no—
385#428—611no—
386#428C2×D6111yescharacteristic
387C2×#107—611no—
388#419—611no—
389#418C2×D6111yescharacteristic
390#417—611no—
391#417—611no—
392C2×#103—611no—
393C2×#102C2×D6111yescharacteristic
394C2×#101—611no—
395C2×#101—611no—
396C2×#100—611no—
397C2×#100C2×D6111yescharacteristic
398#405—611no—
399#405—1221no—
400#405—1221no—
401#405C2×D6111yescharacteristic
402C4×M16—611no—
403C4×M16—611no—
404C4×M16—611no—
405C4×M16—1221no—
406C2×#97—611no—
407C2×#97—1221no—
408#343—611no—
409#343—611no—
410#343—611no—
411#343C2×D6111yescharacteristic
412#341—611no—
413#341C2×D6111yescharacteristic
414C61#2155711yescharacteristic, complemented, Sylow
415C2×#41—611no—
416C22×Q8—611no—
417C22×D4C22×D6111yescharacteristic
418C23×C4—1221no—
419#137—614no—
420#137—1222no—
421AGL(1,ℤ/8)—614no—
422AGL(1,ℤ/8)—1222no—
423C2×Q16—1221no—
424C2×SD16—614no—
425C2×SD16—1221no—
426C2×SD16—1221no—
427C2×SD16—1221no—
428C2×SD16—1222no—
429C2×SD16—1221no—
430C2×D8—1221no—
431#131—614no—
432#131—1222no—
433#131—1222no—
434#131—1222no—
435C2×M16—611no—
436C2×M16—611no—
437C2×M16—611no—
438C2×M16—1221no—
439C2×M16—1221no—
440C2×M16C22×D6111yescharacteristic
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C22×C8—611no—
444C22×C8—1221no—
445C22×C8—1221no—
446C22×C8C22×D6111yescharacteristic
447C4×Q8—611no—
448C4×Q8—611no—
449C4×Q8—1221no—
450C4×Q8—1221no—
451C4×Q8—1221no—
452C4×Q8—1221no—
453C4×D4—611no—
454C4×D4—611no—
455C4×D4—1221no—
456C4×D4—1221no—
457C4×D4—1221no—
458C4×D4—1221no—
459C4×D4—1221no—
460C4×D4—1221no—
461#117—1221no—
462#117C22×D6111yescharacteristic
463C2×C4⋊C4—611no—
464C2×C4⋊C4—1221no—
465C2×C4⋊C4C22×D6111yescharacteristic
466C2×C22⋊C4—1221no—
467C2×C42—611no—
468C2×C42—1221no—
469C2×C42—1221no—
470#108—614no—
471#108—1222no—
472#107—21no—
473#107—1221no—
474#107—1221no—
475#107—1221no—
476#106—21no—
477#106—1221no—
478#103—1221no—
479#103—1221no—
480#103—1221no—
481#103—1221no—
482#103—1221no—
483#103—1221no—
484#102—21no—
485#102—21no—
486#102—21no—
487#102—21no—
488#102—1221no—
489#102—1221no—
490#101—614no—
491#101—1222no—
492#100—1222no—
493#100C4×D6114yescomplemented
494#97—21no—
495#97—611no—
496#97—611no—
497#97—611no—
498#97—611no—
499#97—1221no—
500#97—1221no—
501#97—1221no—
502#97—1221no—
503C4×C8—21no—
504C4×C8—1221no—
505C4×C8—1221no—
506C4×C8—1221no—
507C4×C8—1221no—
508C24—21no—
509#41—612no—
510#41—612no—
511#41—1221no—
512#41—1221no—
513C2×Q8—612no—
514C2×Q8—611no—
515C2×Q8—1221no—
516C2×Q8—1221no—
517C2×Q8—1221no—
518C2×Q8—1221no—
519C2×Q8—1221no—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—611no—
526C2×D4—1221no—
527C2×D4C2×C4×D6112yes—
528C22×C4—21no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—4882no—
537C22×C4—2441no—
538C22×C4—2441no—
539C22×C4C23×D6111yescharacteristic
540Q16—1224no—
541SD16—1224no—
542SD16—1224no—
543SD16—1224no—
544SD16—1224no—
545D8—1224no—
546M16—21no—
547M16—21no—
548M16—21no—
549M16—21no—
550M16—612no—
551M16—612no—
552M16—1224no—
553M16—1224no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16C2×C4×D6112yes—
559M16C2×C4×D6112yes—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—21no—
565C2×C8—612no—
566C2×C8—611no—
567C2×C8—611no—
568C2×C8—612no—
569C2×C8—1224no—
570C2×C8—1224no—
571C2×C8—1221no—
572C2×C8—1221no—
573C2×C8—1221no—
574C2×C8D4×D6111yescharacteristic
575C2×C8D4×D6111yescharacteristic
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—1221no—
579C4⋊C4—1221no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—2441no—
583C4⋊C4—2441no—
584C4⋊C4D4×D6111yescharacteristic
585C4⋊C4D4×D6111yescharacteristic
586C22⋊C4—21no—
587C22⋊C4—1221no—
588C22⋊C4—1221no—
589C22⋊C4—2441no—
590C22⋊C4—2441no—
591C42—21no—
592C42—611no—
593C42—611no—
594C42—611no—
595C42—611no—
596C42—1221no—
597C42—1221no—
598C42—2441no—
599C42—2441no—
600C23—22no—
601C23—41no—
602C23—41no—
603C23—81no—
604C23—81no—
605C23—1221no—
606C23#41×D6111yescharacteristic
607Q8—612no—
608Q8—1221no—
609Q8—1222no—
610Q8—1222no—
611Q8—2441no—
612Q8—2441no—
613D4—22no—
614D4—22no—
615D4—41no—
616D4—41no—
617D4—612no—
618D4—1222no—
619D4—1222no—
620D4—1221no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—41no—
624C2×C4—81no—
625C2×C4—611no—
626C2×C4—611no—
627C2×C4—611no—
628C2×C4—611no—
629C2×C4—1221no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—4881no—
634C2×C4—4881no—
635C2×C4—4882no—
636C2×C4—4882no—
637C2×C4—4881no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4C22×C4×D6111yescharacteristic
644C2×C4C2×D4×D6111yescharacteristic
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4#41×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4#41×D6111yescharacteristic
649C8—22no—
650C8—22no—
651C8—21no—
652C8—21no—
653C8—41no—
654C8—612no—
655C8—612no—
656C8—1222no—
657C8—1222no—
658C8—2441no—
659C22—21no—
660C22—21no—
661C22—42no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—1222no—
667C22—2441no—
668C22—2441no—
669C22#1684023011yescharacteristic
670C22#121×D6111yescharacteristic
671C22#1684029511yescharacteristic
672C4—41no—
673C4—41no—
674C4—81no—
675C4—1222no—
676C4—1221no—
677C4—1221no—
678C4—1221no—
679C4—1221no—
680C4—4881no—
681C4—2442no—
682C4C4×D4×D6111yescharacteristic
683C4C4×D4×D6111yescharacteristic
684C4#1684027011yescharacteristic
685C4#1684031111yescharacteristic
686C2—21no—
687C2—21no—
688C2—82no—
689C2—2442no—
690C2#2144833111yescharacteristic
691C2#2145554511yescharacteristic
692C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1931C2
socleC22×C61C2×#121
prime radicalC2×#14519803C22
cosocleC22×C4×C61C24
hypercenter#1931D61
hypocenterC61#21557
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1931

Complements

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

NH ≅ G/Ntype
2. #21454637C2semidirect
4. C2×#16839859C2semidirect
5. #21454246C2semidirect
6. #21446725C2semidirect
7. #21446601C2semidirect
8. #21446474C2semidirect
9. #21446333C2semidirect
11. #21445882C2semidirect
12. #21445724C2semidirect
13. #21445614C2semidirect
14. #100×Dic61C2semidirect
15. C61×#1931C2semidirect
24. #16840024C4semidirect
72. #14519810C2×C4semidirect
81. C2×#11758200C2×C4semidirect
84. #14519785C2×C4semidirect
117. #14519712C2×C4semidirect
131. #14519695C2×C4semidirect
158. #100×C61C2×C4semidirect
347. #1931D61semidirect
414. C61#21557semidirect
493. #100C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400242
#214546372
#214546232
C2×#168398592
#214542462
#214467252
#214466012
#214464742
#214463332
#214463302
#214458822
#214457242
#214456142
#100×Dic612
C61×#19312
#2155761