Finite Group Databasebeta

← #29699462

Subgroups — #29699462

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic61C211yesmaximal, characteristic
2#21454481C211yesmaximal, characteristic, complemented
3#21454385C211yesmaximal, characteristic, complemented
4#21446533C211yesmaximal, characteristic, complemented
5#21446527C211yesmaximal, characteristic, complemented
6#21446419C211yesmaximal, characteristic, complemented
7#21446417C211yesmaximal, characteristic, complemented
8#21446148C211yesmaximal, characteristic
9#21445955C211yesmaximal, characteristic, complemented
10#21445837C211yesmaximal, characteristic, complemented
11#21445837C211yesmaximal, characteristic, complemented
12#21445685C211yesmaximal, characteristic, complemented
13#21445664C211yesmaximal, characteristic, complemented
14#21445661C211yesmaximal, characteristic, complemented
15C61×#1876C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#41×Dic61C2211yescharacteristic
18#41×Dic61C2211yescharacteristic
19Q16×Dic61C411yescharacteristic, complemented
20Q16×Dic61C411yescharacteristic, complemented
21SD16×Dic61C411yescharacteristic, complemented
22SD16×Dic61C411yescharacteristic, complemented
23SD16×Dic61C411yescharacteristic, complemented
24SD16×Dic61C411yescharacteristic, complemented
25D8×Dic61C411yescharacteristic, complemented
26D8×Dic61C411yescharacteristic, complemented
27#16839820—21no—
28#16839820—21no—
29#16839816—21no—
30#16839812—21no—
31#16839800C2211yescharacteristic
32#16839790—21no—
33#16839790C2211yescharacteristic
34#16839788—21no—
35#16839788C2211yescharacteristic
36#16839785—21no—
37#16839785C2211yescharacteristic
38#16839785C2211yescharacteristic
39C2×#14519706—21no—
40C2×#14519706C2211yescharacteristic
41C2×#14519704C2211yescharacteristic
42C2×#14519704C2211yescharacteristic
43#16839758C2211yescharacteristic
44#16839758C2211yescharacteristic
45#16839729C2211yescharacteristic
46#16839720C2211yescharacteristic
47#16839717C2211yescharacteristic
48C2×#14519688C2211yescharacteristic
49C2×#14519687C2211yescharacteristic
50C2×C8×Dic61C2211yescharacteristic
51C4×#11758196C2211yescharacteristic
52#16839527C2211yescharacteristic
53#16839526C2211yescharacteristic
54#16839526C2211yescharacteristic
55#16839523C2211yescharacteristic
56#16839523C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839516C2211yescharacteristic
60C2×#135×C61C2211yescharacteristic
61C2×#131×C61C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#408C2211yescharacteristic
65C61×#341C2211yescharacteristic
66C61×#336C2211yescharacteristic
67#14519784—22no—
68#14519784—41no—
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C22×#92418—21no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75Q8×Dic61C2×C411yescharacteristic
76Q8×Dic61C2×C411yescharacteristic
77Q8×Dic61C2×C411yescharacteristic
78Q8×Dic61C2×C411yescharacteristic
79D4×Dic61—21no—
80D4×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61—21no—
83D4×Dic61C2×C411yescharacteristic
84D4×Dic61C2×C411yescharacteristic
85D4×Dic61C2×C411yescharacteristic
86D4×Dic61C2×C411yescharacteristic
87#14519736—21no—
88#14519736—21no—
89#14519736C2311yescharacteristic
90#14519736C2311yescharacteristic
91C2×#11758196—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519711C2×C411yescharacteristic
97#14519711C2×C411yescharacteristic
98#14519709C2×C411yescharacteristic
99#14519709C2×C411yescharacteristic
100#14519706—21no—
101#14519706—21no—
102#14519706—21no—
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706D411yescharacteristic
109#14519706D411yescharacteristic
110#14519706D411yescharacteristic
111#14519706D411yescharacteristic
112#14519704—22no—
113#14519704—41no—
114#14519701C2×C411yescharacteristic
115#14519701C2×C411yescharacteristic
116#14519701C2×C411yescharacteristic
117#14519701C2×C411yescharacteristic
118#14519698C2×C411yescharacteristic
119#14519698C2×C411yescharacteristic
120#14519698C2×C411yescharacteristic
121#14519698C2×C411yescharacteristic
122C22⋊#92418—21no—
123C22⋊#92418—21no—
124#14519688—41no—
125#14519687—21no—
126#14519687—21no—
127#14519687—41no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C8×Dic61C2×C411yescharacteristic
131C8×Dic61C2×C411yescharacteristic
132C8×Dic61C2×C411yescharacteristic
133C8×Dic61C2×C411yescharacteristic
134#14519684—21no—
135C2×#41×C61C2311yescharacteristic
136C2×#41×C61C2311yescharacteristic
137#135×C61—22no—
138#135×C61—22no—
139C2×Q16×C61C2×C411yescharacteristic
140C2×SD16×C61C2×C411yescharacteristic
141C2×SD16×C61C2×C411yescharacteristic
142C2×D8×C61C2×C411yescharacteristic
143#131×C61—22no—
144#131×C61—41no—
145C2×M16×C61—21no—
146C2×M16×C61C2311yescharacteristic
147C2×M16×C61C2311yescharacteristic
148C22×C8×C61—21no—
149C22×C8×C61C2311yescharacteristic
150C22×C8×C61C2311yescharacteristic
151C2×C42×C61C2311yescharacteristic
152#104×C61—22no—
153#104×C61—41no—
154C22⋊C8×C61—21no—
155C22⋊C8×C61—21no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61—21no—
159#41×C61—21no—
160#41×C61—21no—
161#41×C61—21no—
162#41×C61#4112yes—
163#41×C61#4112yes—
164C2×Q8×C61C22×C411yescharacteristic
165C2×Q8×C61C22×C411yescharacteristic
166C2×D4×C61—21no—
167C2×D4×C61—21no—
168C2×D4×C61C22×C411yescharacteristic
169C2×D4×C61C22×C411yescharacteristic
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61C2411yescharacteristic
174C22×Dic61—21no—
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61—21no—
178C22×Dic61—21no—
179C22×Dic61#4111yescharacteristic
180C22×Dic61#4111yescharacteristic
181Q16×C61—22no—
182SD16×C61—22no—
183SD16×C61—22no—
184D8×C61—22no—
185M16×C61—21no—
186M16×C61—21no—
187M16×C61—21no—
188M16×C61—21no—
189M16×C61—42no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—42no—
197C2×C8×C61C22×C411yescharacteristic
198C2×C8×C61C22×C411yescharacteristic
199C2×C8×C61C2×D411yescharacteristic
200C2×C8×C61C2×D411yescharacteristic
201C42×C61—21no—
202C42×C61—21no—
203C22⋊Dic61—21no—
204C22⋊Dic61—21no—
205C22⋊Dic61—21no—
206C22⋊Dic61—21no—
207C22⋊Dic61—21no—
208C22⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61C22×C411yescharacteristic
214C4⋊Dic61C22×C411yescharacteristic
215C4⋊Dic61C22×C411yescharacteristic
216C4⋊Dic61C22×C411yescharacteristic
217#11758196—42no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230#11758194—21no—
231#11758194—21no—
232#11758194—21no—
233#11758194—21no—
234#11758194—42no—
235C2×#92418—21no—
236C2×#92418—21no—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418—42no—
242C2×#92418C22×C411yescharacteristic
243C2×#92418C22×C411yescharacteristic
244C2×#92418C2×D411yescharacteristic
245C2×#92418C2×D411yescharacteristic
246C23×C61—21no—
247C23×C61—21no—
248C23×C61C2×#4111yescharacteristic
249Q8×C61—21no—
250Q8×C61—21no—
251Q8×C61—21no—
252Q8×C61#11712yes—
253D4×C61—22no—
254D4×C61—22no—
255D4×C61—21no—
256D4×C61—21no—
257D4×C61—21no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61#11712yes—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—21no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61C23×C411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×C4×C612−1+411yescharacteristic
276C2×Dic61—22no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—41no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—81no—
288C2×Dic61—81no—
289C2×Dic61C4×D411yescharacteristic
290C2×Dic61C4×D411yescharacteristic
291C2×Dic61C4×D411yescharacteristic
292C2×Dic61C4×D411yescharacteristic
293C8×C61—21no—
294C8×C61—21no—
295C8×C61—21no—
296C8×C61—21no—
297C8×C61—41no—
298C8×C61—41no—
299#92418—21no—
300#92418—21no—
301#92418—21no—
302#92418—21no—
303#92418—41no—
304#92418—41no—
305#22782—611nomaximal, Sylow
306C22×C61—21no—
307C22×C61—21no—
308C22×C61—22no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61C2×C4×D411yescharacteristic
315C22×C61#55211yescharacteristic
316C22×C61#55311yescharacteristic
317C4×C61—22no—
318C4×C61—41no—
319C4×C61—81no—
320C4×C61C2×#11711yescharacteristic
321C4×C61#52011yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#54811yescharacteristic
324Dic61—21no—
325Dic61—21no—
326Dic61—21no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
331Dic61—81no—
332Dic61#44412yes—
333C4×#135—611no—
334#2925—611no—
335#2906—611no—
336#1938—611no—
337#1938—611no—
338#1922—611no—
339#1922—611no—
340#1876—611no—
341#1876D6111yescharacteristic
342#1837—611no—
343#1806—611no—
344#1806—611no—
345#1770—611no—
346#1765—611no—
347#1765—611no—
348C2×C61—21no—
349C2×C61—21no—
350C2×C61—42no—
351C2×C61—81no—
352C2×C61#235211yescharacteristic
353C2×C61C2×#44411yescharacteristic
354C2×C61#295711yescharacteristic
355C2×#135C2×D6111yescharacteristic
356C2×#131—611no—
357C2×#131C2×D6111yescharacteristic
358C4×#41—611no—
359C4×#41—611no—
360C4×Q16—611no—
361C4×Q16—611no—
362C4×SD16—611no—
363C4×SD16—611no—
364C4×SD16—611no—
365C4×SD16—611no—
366C4×D8—611no—
367C4×D8—611no—
368#433—1221no—
369#433—1221no—
370#432—1221no—
371#431—1221no—
372#427—611no—
373#424—611no—
374#424—611no—
375#424—1221no—
376#424—1221no—
377#423—611no—
378#423—611no—
379#423—1221no—
380C2×C4⋊C8—611no—
381C2×C4⋊C8—1221no—
382C2×#104—611no—
383C2×#104—611no—
384C2×#104C2×D6111yescharacteristic
385C2×#104C2×D6111yescharacteristic
386#416—611no—
387#416—611no—
388#408—611no—
389#408C2×D6111yescharacteristic
390#405—611no—
391#405—611no—
392C2×#97—611no—
393C2×#97—611no—
394C2×C4×C8—611no—
395C4×C4⋊C4—611no—
396#341—611no—
397#341—611no—
398#341—611no—
399#341C2×D6111yescharacteristic
400#339—611no—
401#339—611no—
402#339—611no—
403#339—611no—
404#336—611no—
405#336C2×D6111yescharacteristic
406C61#2278211yescharacteristic, complemented, Sylow
407C2×#41C22×D6111yescharacteristic
408C2×#41C22×D6111yescharacteristic
409#135—22no—
410#135—22no—
411C2×Q16C4×D6111yescharacteristic
412C2×SD16C4×D6111yescharacteristic
413C2×SD16C4×D6111yescharacteristic
414C2×D8C4×D6111yescharacteristic
415#131—22no—
416#131—41no—
417#131—1222no—
418#131—2441no—
419C2×M16—21no—
420C2×M16—611no—
421C2×M16—611no—
422C2×M16—1221no—
423C2×M16C22×D6111yescharacteristic
424C2×M16C22×D6111yescharacteristic
425C22×C8—21no—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8—1221no—
429C22×C8C22×D6111yescharacteristic
430C22×C8C22×D6111yescharacteristic
431C4×Q8—611no—
432C4×Q8—611no—
433C4×Q8—611no—
434C4×Q8—611no—
435C4×D4—611no—
436C4×D4—611no—
437C4×D4—611no—
438C4×D4—611no—
439C4×D4—1221no—
440C4×D4—1221no—
441C4×D4—1221no—
442C4×D4—1221no—
443#117—611no—
444#117—611no—
445#117—1221no—
446#117—1221no—
447C2×C4⋊C4—1221no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453#107—611no—
454#107—611no—
455#106—611no—
456#106—611no—
457C4⋊C8—611no—
458C4⋊C8—611no—
459C4⋊C8—611no—
460C4⋊C8—611no—
461C4⋊C8—1221no—
462C4⋊C8—1221no—
463C4⋊C8—1221no—
464C4⋊C8—1221no—
465C4⋊C8—1221no—
466C4⋊C8—1221no—
467C4⋊C8—1221no—
468C4⋊C8—1221no—
469#104—22no—
470#104—41no—
471#104—1222no—
472#104—2441no—
473#103—611no—
474#103—611no—
475#103—611no—
476#103—611no—
477#102—611no—
478#102—611no—
479#102—611no—
480#102—611no—
481C22⋊C8—21no—
482C22⋊C8—21no—
483C22⋊C8—1221no—
484C22⋊C8—1221no—
485#97—1221no—
486#97—1221no—
487#97—2441no—
488#97—2441no—
489C4×C8—611no—
490C4×C8—611no—
491C4×C8—611no—
492C4×C8—611no—
493C4×C8—1221no—
494C4×C8—1221no—
495#95—1221no—
496#41—21no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—21no—
501#41—21no—
502#41#1175822412yes—
503#41#1175822412yes—
504C2×Q8C2×C4×D6111yescharacteristic
505C2×Q8C2×C4×D6111yescharacteristic
506C2×D4—21no—
507C2×D4—21no—
508C2×D4C2×C4×D6111yescharacteristic
509C2×D4C2×C4×D6111yescharacteristic
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4C23×D6111yescharacteristic
521Q16—22no—
522SD16—22no—
523SD16—22no—
524D8—22no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—21no—
529M16—42no—
530M16—1221no—
531M16—1221no—
532M16—1221no—
533M16—1221no—
534M16—2442no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—42no—
542C2×C8—611no—
543C2×C8—611no—
544C2×C8—611no—
545C2×C8—611no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—2442no—
553C2×C8C2×C4×D6111yescharacteristic
554C2×C8C2×C4×D6111yescharacteristic
555C2×C8D4×D6111yescharacteristic
556C2×C8D4×D6111yescharacteristic
557C4⋊C4—611no—
558C4⋊C4—611no—
559C4⋊C4—611no—
560C4⋊C4—611no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—2442no—
566C22⋊C4—1221no—
567C22⋊C4—1221no—
568C22⋊C4—1221no—
569C22⋊C4—1221no—
570C22⋊C4—1221no—
571C22⋊C4—1221no—
572C42—21no—
573C42—21no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—1221no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—1221no—
583C42—1221no—
584C42—1221no—
585C42—1221no—
586C23—21no—
587C23—21no—
588C23#41×D6111yescharacteristic
589Q8—21no—
590Q8—21no—
591Q8—21no—
592Q8#1451973312yes—
593D4—22no—
594D4—22no—
595D4—21no—
596D4—21no—
597D4—21no—
598D4—41no—
599D4—41no—
600D4#1451973312yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—611no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—4881no—
620C2×C4—4881no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4C22×C4×D6111yescharacteristic
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C2×#1175822411yescharacteristic
630C2×C4C2×#1175822411yescharacteristic
631C2×C4#1451982211yescharacteristic
632C2×C4#1451982411yescharacteristic
633C8—21no—
634C8—21no—
635C8—21no—
636C8—21no—
637C8—41no—
638C8—41no—
639C8—1221no—
640C8—1221no—
641C8—1221no—
642C8—1221no—
643C8—2441no—
644C8—2441no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22C4×D4×D6111yescharacteristic
654C22#1684044911yescharacteristic
655C22#1684045511yescharacteristic
656C4—22no—
657C4—41no—
658C4—81no—
659C4—612no—
660C4—1221no—
661C4—1221no—
662C4—1221no—
663C4—4881no—
664C4—2441no—
665C4—2441no—
666C4—2441no—
667C4—2441no—
668C4C2×#1451973311yescharacteristic
669C4#1684024111yescharacteristic
670C4#1684039911yescharacteristic
671C4#1684041811yescharacteristic
672C2—21no—
673C2—21no—
674C2—42no—
675C2—81no—
676C2#2144874611yescharacteristic
677C2#444×D6111yescharacteristic
678C2#2145473111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1876C2
socleC22×C61C2×C4×D4
prime radicalC2×#135×C61C22
cosocleC22×C4×C61C24
hypercenter#1876D61
hypocenterC61#22782
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1876

Complements

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

NH ≅ G/Ntype
2. #21454481C2semidirect
3. #21454385C2semidirect
4. #21446533C2semidirect
5. #21446527C2semidirect
6. #21446419C2semidirect
7. #21446417C2semidirect
9. #21445955C2semidirect
10. #21445837C2semidirect
11. #21445837C2semidirect
12. #21445685C2semidirect
13. #21445664C2semidirect
14. #21445661C2semidirect
19. Q16×Dic61C4semidirect
20. Q16×Dic61C4semidirect
21. SD16×Dic61C4semidirect
22. SD16×Dic61C4semidirect
23. SD16×Dic61C4semidirect
24. SD16×Dic61C4semidirect
25. D8×Dic61C4semidirect
26. D8×Dic61C4semidirect
406. C61#22782semidirect

Maximal subgroups

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

subgroupindex
#135×Dic612
#214544812
#214543852
#214465332
#214465272
#214464192
#214464172
#214461482
#214459552
#214458372
#214456852
#214456642
#214456612
C61×#18762
#2278261