Finite Group Databasebeta

← #30622667

Subgroups — #30622667

Order 15616.

750 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#21462353C211yesmaximal, characteristic
2#21462348C211yesmaximal, characteristic
3#21461999C211yesmaximal, characteristic
4#21461997C211yesmaximal, characteristic
5#21461924C211yesmaximal, characteristic
6#21461922C211yesmaximal, characteristic
7#21461883C211yesmaximal, characteristic
8#21461882C211yesmaximal, characteristic
9#21461882C211yesmaximal, characteristic
10#21461880C211yesmaximal, characteristic
11#21461635C211yesmaximal, characteristic
12#21461633C211yesmaximal, characteristic
13#107×#6923C211yesmaximal, characteristic
14#21461617C211yesmaximal, characteristic
15#501×D61C211yesmaximal, characteristic
16C2×Q8×#6923—21no—
17C2×#14519846C2211yescharacteristic
18C2×#14519845C2211yescharacteristic
19D61⋊(C4⋊C8)C2211yescharacteristic
20C2×#14519831C2211yescharacteristic
21C2×C8×#6923C2211yescharacteristic
22(C2×#6923)⋊Q8—21no—
23#16840713—21no—
24#16840713C2211yescharacteristic
25#16840713C2211yescharacteristic
26#16840713C2211yescharacteristic
27#16840713C2211yescharacteristic
28#16840711—21no—
29#16840711—21no—
30#16840711—21no—
31#16840711—21no—
32C4⋊C4×#6923—21no—
33C4⋊C4×#6923C2211yescharacteristic
34C4⋊C4×#6923C2211yescharacteristic
35C4⋊C4×#6923C2211yescharacteristic
36C4⋊C4×#6923C2211yescharacteristic
37C4×C4⋊#6923C2211yescharacteristic
38C4×C4⋊#6923C2211yescharacteristic
39C4×C4⋊#6923C2211yescharacteristic
40#16840636C2211yescharacteristic
41#16840636C2211yescharacteristic
42#16840635C2211yescharacteristic
43#16840635C2211yescharacteristic
44#16840635C2211yescharacteristic
45#16840635C2211yescharacteristic
46#16840635C2211yescharacteristic
47#16840635C2211yescharacteristic
48#16840635C2211yescharacteristic
49#16840635C2211yescharacteristic
50(C2×#6923)⋊C8C2211yescharacteristic
51(C2×#6923)⋊C8C2211yescharacteristic
52C4⋊Q8×D61C2211yescharacteristic
53#125×D61C2211yescharacteristic
54#16840172C411yescharacteristic, complemented
55#16840171C411yescharacteristic, complemented
56#16840170C411yescharacteristic, complemented
57#16840170C411yescharacteristic, complemented
58#16840169C411yescharacteristic, complemented
59#16840169C411yescharacteristic, complemented
60#16840168C411yescharacteristic, complemented
61#107×D61C2211yescharacteristic
62#107×D61C2211yescharacteristic
63#106×D61C2211yescharacteristic
64#106×D61C2211yescharacteristic
65#97×D61C2211yescharacteristic
66C61×#501C411yescharacteristic, complemented
67C2×D61⋊C8C2311yescharacteristic
68C2×D61⋊C8C2311yescharacteristic
69Q8×#6923—44no—
70C2×C4⋊#6923—21no—
71C2×C4⋊#6923—21no—
72C2×C4⋊#6923—21no—
73C2×C4⋊#6923—21no—
74C2×C4⋊#6923D411yescharacteristic
75C2×C4⋊#6923D411yescharacteristic
76C2×C4⋊#6923C2311yescharacteristic
77C2×C4⋊#6923C2311yescharacteristic
78C2×C4⋊#6923C2311yescharacteristic
79C2×C4⋊#6923C2311yescharacteristic
80C2×C4×#6923—21no—
81C2×C4×#6923—21no—
82C2×C4×#6923—21no—
83C2×C4×#6923—21no—
84C2×C4×#6923—21no—
85C2×C4×#6923—21no—
86C2×C4×#6923C2311yescharacteristic
87C2×C4×#6923C2311yescharacteristic
88#14519846Q814yescomplemented
89#14519845—22no—
90#14519844—21no—
91C4⋊#92420—21no—
92#14519831—22no—
93C8×#6923—22no—
94(C2×#6923)⋊C4—21no—
95(C2×#6923)⋊C4—21no—
96(C2×#6923)⋊C4—21no—
97(C2×#6923)⋊C4—21no—
98(C2×#6923)⋊C4—21no—
99(C2×#6923)⋊C4—21no—
100(C2×#6923)⋊C4—21no—
101(C2×#6923)⋊C4—21no—
102(C2×#6923)⋊C4—21no—
103(C2×#6923)⋊C4—21no—
104(C2×#6923)⋊C4D411yescharacteristic
105(C2×#6923)⋊C4D411yescharacteristic
106C2×Q8×D61—21no—
107C2×C8×D61C2311yescharacteristic
108C2×C8×D61C2311yescharacteristic
109#14519777C2×C411yescharacteristic
110#14519776—21no—
111#14519776C2×C411yescharacteristic
112#14519776C2×C411yescharacteristic
113#14519775—21no—
114#14519769C2×C411yescharacteristic
115#14519768C2×C411yescharacteristic
116#14519768C2×C411yescharacteristic
117#14519767—21no—
118#14519767—21no—
119C4⋊C4×D61—21no—
120C4⋊C4×D61—21no—
121C4⋊C4×D61—21no—
122C4⋊C4×D61C2311yescharacteristic
123C4⋊C4×D61C2311yescharacteristic
124C4⋊C4×D61C2311yescharacteristic
125C4⋊C4×D61C2311yescharacteristic
126C42×D61C2311yescharacteristic
127#14519711C2×C411yescharacteristic
128#14519711C2×C411yescharacteristic
129#14519710C2×C411yescharacteristic
130#14519710C2×C411yescharacteristic
131#14519710C2×C411yescharacteristic
132#14519710C2×C411yescharacteristic
133#14519709C2×C411yescharacteristic
134#14519709C2×C411yescharacteristic
135#14519708C2×C411yescharacteristic
136#14519708C2×C411yescharacteristic
137#14519708C2×C411yescharacteristic
138#14519708C2×C411yescharacteristic
139#14519688C2×C411yescharacteristic
140#14519687C2×C411yescharacteristic
141#14519687C2×C411yescharacteristic
142C4⋊Q8×C61C2×C411yescharacteristic
143#125×C61C2×C411yescharacteristic
144#107×C61C2×C411yescharacteristic
145#107×C61C2×C411yescharacteristic
146#106×C61C2×C411yescharacteristic
147#106×C61C2×C411yescharacteristic
148#97×C61C2×C411yescharacteristic
149C22×#6923—21no—
150C22×#6923—21no—
151C22×#6923—21no—
152C22×#6923—21no—
153C22×#6923C2×D411yescharacteristic
154C22×#6923C2×D411yescharacteristic
155C22×#6923C2×D411yescharacteristic
156C22×#6923C2×D411yescharacteristic
157C2×Q8×C61—21no—
158Q8×D61—42no—
159Q8×D61—42no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2411yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C4⋊#6923—22no—
173C4⋊#6923—21no—
174C4⋊#6923—21no—
175C4⋊#6923—21no—
176C4⋊#6923—21no—
177C4⋊#6923—21no—
178C4⋊#6923—21no—
179C4⋊#6923—41no—
180C4⋊#6923—42no—
181C4⋊#6923—42no—
182C4⋊#6923—42no—
183C4⋊#6923—81no—
184C4⋊#6923C2×Q812yes—
185C4⋊#6923C2×Q812yes—
186C4×#6923—21no—
187C4×#6923—21no—
188C4×#6923—21no—
189C4×#6923—21no—
190C4×#6923—42no—
191C4×#6923—42no—
192C4×#6923—42no—
193C4×#6923—42no—
194C4×#6923—81no—
195C4×#6923—81no—
196D61⋊C8—21no—
197D61⋊C8—21no—
198D61⋊C8—21no—
199D61⋊C8—21no—
200C2×C8×C61C22×C411yescharacteristic
201C2×C8×C61C22×C411yescharacteristic
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61C22×C411yescharacteristic
206C4⋊C4×C61C22×C411yescharacteristic
207C4⋊C4×C61C22×C411yescharacteristic
208C4⋊C4×C61C22×C411yescharacteristic
209C42×C61C22×C411yescharacteristic
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61C22×C411yescharacteristic
214C4⋊Dic61C22×C411yescharacteristic
215C4⋊Dic61C22×C411yescharacteristic
216C4⋊Dic61C22×C411yescharacteristic
217#11758196—21no—
218#11758196—21no—
219#11758196—21no—
220#11758196—21no—
221#11758196—21no—
222#11758196—21no—
223#11758196—21no—
224#11758196—21no—
225#11758196—21no—
226#11758196—21no—
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61#4111yescharacteristic
229C4×Dic61#4111yescharacteristic
230C2×#92418C22×C411yescharacteristic
231C2×#92418C22×C411yescharacteristic
232C8×D61—21no—
233C8×D61—21no—
234C8×D61C2×Q812yes—
235C8×D61#4112yes—
236C22×D61C22×D411yescharacteristic
237C2×#6923—21no—
238C2×#6923—21no—
239C2×#6923—21no—
240C2×#6923—21no—
241C2×#6923—21no—
242C2×#6923—22no—
243C2×#6923—22no—
244C2×#6923—21no—
245C2×#6923—21no—
246C2×#6923—21no—
247C2×#6923—21no—
248C2×#6923—21no—
249C2×#6923—21no—
250C2×#6923—41no—
251C2×#6923—41no—
252C2×#6923—41no—
253C2×#6923—41no—
254C2×#6923—41no—
255C2×#6923—41no—
256C2×#6923—41no—
257C2×#6923—41no—
258C2×#6923—41no—
259C2×#6923—41no—
260C2×#6923AGL(1,ℤ/8)12yes—
261Q8×C61—42no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61C4×D411yescharacteristic
267C2×C4×C61C4×D411yescharacteristic
268C2×C4×C61C23×C411yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61C4×D411yescharacteristic
274C2×Dic61C4×D411yescharacteristic
275C2×Dic61C2×#4111yescharacteristic
276C4×D61—21no—
277C4×D61—21no—
278C4×D61—21no—
279C4×D61—21no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—41no—
288C4×D61C22×Q811yescharacteristic
289C4×D61C2×#4111yescharacteristic
290C4×D612−1+411yescharacteristic
291C4×D612−1+411yescharacteristic
292Dic122—42no—
293Dic122—42no—
294Dic122—42no—
295#92420—41no—
296#92420—41no—
297C8×C61—21no—
298C8×C61C4×Q812yescomplemented
299#92418—21no—
300#92418C4×Q812yescomplemented
301#23166—611nomaximal, Sylow
302C22×C61C2×C4×D411yescharacteristic
303C2×D61#54711yescharacteristic
304C2×D61D4×Q811yescharacteristic
305C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
306C2×D61#57511yescharacteristic
307C2×D61#57711yescharacteristic
308C2×D61#57811yescharacteristic
309#6923—22no—
310#6923—41no—
311#6923—41no—
312#6923—41no—
313#6923—42no—
314#6923—42no—
315#6923—81no—
316#6923—81no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61C2×C4×Q811yescharacteristic
324C4×C61#51911yescharacteristic
325Dic61—21no—
326Dic61—21no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
331Dic61#55611yescharacteristic
332Dic61#55711yescharacteristic
333C2×#501—611no—
334#2370—611no—
335#2365—611no—
336#1989—611no—
337#1987—611no—
338#1943—611no—
339#1941—611no—
340#1921—611no—
341#1920—611no—
342#1920—611no—
343#1918—611no—
344#1776—611no—
345#1776—611no—
346C4×#107—611no—
347#1771—611no—
348C2×C61#236511yescharacteristic
349C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
350C2×C61#294711yescharacteristic
351D61#334311yescharacteristic
352D61#335011yescharacteristic
353D61#338411yescharacteristic
354D61#339011yescharacteristic
355C2×C4⋊Q8—611no—
356C2×#125—611no—
357C2×C4×Q8—1221no—
358#501—611no—
359#501—611no—
360#501—611no—
361#501—611no—
362#501—611no—
363#501—611no—
364#501—611no—
365#501#692311yescharacteristic, complemented
366C2×#107—611no—
367C2×#107—611no—
368C2×#107—611no—
369C2×#106—611no—
370C2×#106—611no—
371C2×#106—611no—
372C2×C4⋊C8—611no—
373C2×#97—611no—
374C2×#97—611no—
375C2×C4×C8—611no—
376#391—1221no—
377#389—611no—
378#389—611no—
379#389—611no—
380#389—611no—
381#389—1221no—
382#387—1221no—
383#387—1221no—
384#387—1221no—
385#387—1221no—
386C4×C4⋊C4—611no—
387C4×C4⋊C4—611no—
388C4×C4⋊C4—611no—
389C4×C4⋊C4—611no—
390C4×C4⋊C4—611no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—611no—
393C4×C4⋊C4—1221no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#336—611no—
405#336—611no—
406C61#2316611yescharacteristic, complemented, Sylow
407C22×Q8—1221no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C4⋊Q8—611no—
413C4⋊Q8—611no—
414C4⋊Q8—611no—
415C4⋊Q8—1221no—
416C4⋊Q8—1221no—
417C4⋊Q8C2×#692311yescharacteristic
418#125—611no—
419#125—611no—
420#125—611no—
421#125—1221no—
422#125—1221no—
423#125C2×#692311yescharacteristic
424C4×Q8—2444no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42—1221no—
451#107—614no—
452#107—611no—
453#107—611no—
454#107—611no—
455#107—611no—
456#107—611no—
457#107—611no—
458#107C2×#692311yescharacteristic
459#107C2×#692311yescharacteristic
460#106—611no—
461#106—611no—
462#106—611no—
463#106—611no—
464#106—611no—
465#106—611no—
466#106—1222no—
467#106C2×#692311yescharacteristic
468#106C2×#692311yescharacteristic
469C4⋊C8—1221no—
470C4⋊C8—1221no—
471#97—611no—
472#97—611no—
473#97—611no—
474#97—1222no—
475#97C2×#692311yescharacteristic
476C4×C8—1222no—
477#95—611no—
478#95—611no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489C2×Q8—21no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—2442no—
494C2×Q8—2442no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C2×C8—611no—
511C2×C8—612no—
512C2×C8—612no—
513C2×C8—611no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8C22×#692311yescharacteristic
523C2×C8C22×#692311yescharacteristic
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—612no—
528C4⋊C4—612no—
529C4⋊C4—611no—
530C4⋊C4—611no—
531C4⋊C4—611no—
532C4⋊C4—611no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1222no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—4881no—
554C4⋊C4—2442no—
555C4⋊C4—2441no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4C22×#692311yescharacteristic
559C4⋊C4C22×#692311yescharacteristic
560C4⋊C4C22×#692311yescharacteristic
561C4⋊C4C22×#692311yescharacteristic
562C42—611no—
563C42—611no—
564C42—611no—
565C42—1221no—
566C42—1221no—
567C42—1221no—
568C42—1221no—
569C42—4881no—
570C42—4881no—
571C42—2442no—
572C42—2442no—
573C42—2442no—
574C42—2442no—
575C42C22×#692311yescharacteristic
576C23—611no—
577Q8—42no—
578Q8—2442no—
579Q8—2442no—
580Q8—2442no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—612no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4D4×#692311yescharacteristic
633C2×C4D4×#692311yescharacteristic
634C2×C4C23×#692311yescharacteristic
635C8—21no—
636C8—612no—
637C8—1221no—
638C8—2441no—
639C8—2441no—
640C8Q8×#692312yescomplemented
641C22—611no—
642C22—611no—
643C22—611no—
644C22—611no—
645C22—611no—
646C22—611no—
647C22C2×D4×#692311yescharacteristic
648C4—21no—
649C4—21no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—611no—
655C4—611no—
656C4—1222no—
657C4—1221no—
658C4—1221no—
659C4—4881no—
660C4—4881no—
661C4—2441no—
662C4—2441no—
663C4—2441no—
664C4—2442no—
665C4—2442no—
666C4—2441no—
667C4—2441no—
668C4—2441no—
669C4—2441no—
670C4C2×Q8×#692311yescharacteristic
671C4#1684083011yescharacteristic
672C2—611no—
673C2—611no—
674C2—611no—
675C2—611no—
676C2#2146234811yescharacteristic
677C2AGL(1,ℤ/8)×#692311yescharacteristic
678C2#2146257711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#501C4
socleC22×C61C2×C4×D4
prime radicalC22×D61C22×D4
cosocleC2×C4×D61C24
hypercenter#501#6923
hypocenterC61#23166
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
54. #16840172C4semidirect
55. #16840171C4semidirect
56. #16840170C4semidirect
57. #16840170C4semidirect
58. #16840169C4semidirect
59. #16840169C4semidirect
60. #16840168C4semidirect
66. C61×#501C4semidirect
88. #14519846Q8semidirect
298. C8×C61C4×Q8semidirect
300. #92418C4×Q8semidirect
365. #501#6923semidirect
406. C61#23166semidirect
640. C8Q8×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623532
#214623482
#214619992
#214619972
#214619242
#214619222
#214618832
#214618822
#214618802
#214616352
#214616332
#107×#69232
#214616172
#501×D612
#2316661