Finite Group Databasebeta

← #30620440

Subgroups — #30620440

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#446C211yesmaximal, characteristic
2#21462338C211yesmaximal, characteristic
3#21462003C212yesmaximal
4#21462000C211yesmaximal, characteristic
5#21461998C212yesmaximal
6#21461997C211yesmaximal, characteristic
7#21461690C212yesmaximal
8#21461688C211yesmaximal, characteristic
9#21461591C212yesmaximal
10C4×#14519840C211yesmaximal, characteristic
11Q82×D61C211yesmaximal, characteristic
12#16840814—22no—
13#16840813—21no—
14#16840812—21no—
15D61⋊(C4⋊C8)C2211yescharacteristic
16D61⋊(C4⋊C8)C2212yes—
17C2×#14519840C2211yescharacteristic
18C2×#14519840C2212yes—
19C2×#14519840C2211yescharacteristic
20#16840721C2212yes—
21C4×D61⋊C8C2211yescharacteristic
22(C2×#6923)⋊Q8—21no—
23#16840711—22no—
24#16840708C2211yescharacteristic
25#16840706C2212yes—
26C4⋊C4×#6923—21no—
27C4×C4⋊#6923C2211yescharacteristic
28C4×C4⋊#6923C2212yes—
29#16840635C2211yescharacteristic
30#16840635C2211yescharacteristic
31#16840635C2212yes—
32#16840635C2212yes—
33#16840635C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
36Q8×Dic122—21no—
37Q8×Dic122C411yescharacteristic, complemented
38Q8×Dic122C412yescomplemented
39Q8×Dic122C411yescharacteristic, complemented
40#16840492—21no—
41#16840492—22no—
42#16840492C411yescharacteristic, complemented
43#16840492C412yescomplemented
44C4⋊Q8×D61—22no—
45C4⋊Q8×D61—21no—
46C4⋊Q8×D61C2211yescharacteristic
47C4⋊Q8×D61C2212yes—
48C4×Q8×D61—21no—
49C4×Q8×D61C2211yescharacteristic
50C4×Q8×D61C2211yescharacteristic
51C4×Q8×D61C2212yes—
52Q82×C61C411yescharacteristic, complemented
53C2×D61⋊C8C2311yescharacteristic
54C2×D61⋊C8C2312yes—
55C2×D61⋊C8C2311yescharacteristic
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923C2311yescharacteristic
59C2×C4⋊#6923C2312yes—
60C2×C4⋊#6923C2311yescharacteristic
61C2×C4×#6923—21no—
62C2×C4×#6923—22no—
63#14519844—22no—
64#14519844—21no—
65C4⋊#92420—21no—
66C4⋊#92420—22no—
67#14519840—22no—
68#14519840—24no—
69#14519840Q814yescomplemented
70#14519832—22no—
71#14519832—22no—
72C4×#92420—21no—
73C4×#92420—21no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—21no—
77(C2×#6923)⋊C4—22no—
78C2×Q8×D61—22no—
79C2×Q8×D61—21no—
80C2×Q8×D61D411yescharacteristic
81C2×Q8×D61C2311yescharacteristic
82#14519777—22no—
83#14519777—21no—
84#14519777C2×C411yescharacteristic
85#14519777C2×C412yes—
86#14519776—22no—
87#14519776—21no—
88#14519776—22no—
89#14519776—21no—
90#14519776—22no—
91#14519776—22no—
92#14519776—22no—
93#14519776—21no—
94#14519776—22no—
95#14519776C2×C412yes—
96#14519776C2×C412yes—
97#14519776C2×C412yes—
98#14519775—22no—
99#14519775—22no—
100#14519775—22no—
101#14519775—21no—
102#14519775—22no—
103Q8×Dic61—21no—
104Q8×Dic61C2×C411yescharacteristic
105Q8×Dic61C2×C412yes—
106Q8×Dic61D411yescharacteristic
107C4×Dic122—21no—
108C4×Dic122—22no—
109C4×Dic122—21no—
110C4×Dic122—22no—
111C4×Dic122C2×C411yescharacteristic
112C4×Dic122C2×C412yes—
113C4×Dic122C2×C411yescharacteristic
114C4×Dic122C2×C412yes—
115#14519742—22no—
116#14519742—21no—
117#14519742—21no—
118#14519742—22no—
119#14519742C2×C411yescharacteristic
120#14519742C2×C412yes—
121#14519742D411yescharacteristic
122#14519742D412yes—
123C4⋊C4×D61—22no—
124C4⋊C4×D61—21no—
125C4⋊C4×D61—22no—
126C4⋊C4×D61—21no—
127C4⋊C4×D61D412yes—
128C4⋊C4×D61D411yescharacteristic
129C4⋊C4×D61C2311yescharacteristic
130C4⋊C4×D61C2312yes—
131C42×D61—22no—
132C42×D61—21no—
133C42×D61C2312yes—
134C42×D61C2311yescharacteristic
135C4⋊Q8×C61—22no—
136C4⋊Q8×C61—21no—
137C4⋊Q8×C61C2×C411yescharacteristic
138C4⋊Q8×C61C2×C412yes—
139C4×Q8×C61—21no—
140C4×Q8×C61C2×C411yescharacteristic
141C4×Q8×C61C2×C412yes—
142C4×Q8×C61C2×C411yescharacteristic
143C22×#6923—21no—
144C22×#6923—21no—
145C22×#6923—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61—22no—
148C2×Q8×C61C22⋊C411yescharacteristic
149C2×Q8×C61C22×C411yescharacteristic
150Q8×D61—22no—
151Q8×D61—22no—
152Q8×D61—21no—
153Q8×D61—21no—
154Q8×D61—82no—
155Q8×D61—81no—
156Q8×D61—82no—
157Q8×D61—81no—
158Q8×D61C2×Q812yes—
159Q8×D61#4112yes—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—22no—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122C22⋊C411yescharacteristic
175C2×Dic122C22⋊C412yes—
176C2×Dic122C22×C411yescharacteristic
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—22no—
180C4⋊#6923—21no—
181C4⋊#6923—22no—
182C4⋊#6923—21no—
183C4⋊#6923—21no—
184C4⋊#6923—22no—
185C4⋊#6923—42no—
186C4⋊#6923—81no—
187C4⋊#6923C2×Q812yes—
188C4×#6923—42no—
189C4×#6923—82no—
190D61⋊C8—21no—
191D61⋊C8—21no—
192D61⋊C8—22no—
193D61⋊C8—21no—
194D61⋊C8—22no—
195D61⋊C8C2×Q812yes—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61C22⋊C412yes—
201C4⋊C4×C61C22⋊C411yescharacteristic
202C4⋊C4×C61C22×C411yescharacteristic
203C4⋊C4×C61C22×C412yes—
204C42×C61—22no—
205C42×C61—21no—
206C42×C61C22×C412yes—
207C42×C61C22×C411yescharacteristic
208C4⋊Dic61—22no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—22no—
212C4⋊Dic61C22⋊C412yes—
213C4⋊Dic61C22⋊C411yescharacteristic
214C4⋊Dic61C22×C411yescharacteristic
215C4⋊Dic61C22×C412yes—
216#11758196—21no—
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220#11758196—22no—
221#11758196—21no—
222#11758196—22no—
223#11758196—21no—
224#11758196—22no—
225#11758196C22⋊C412yes—
226#11758196C22⋊C412yes—
227#11758196C22⋊C412yes—
228C4×Dic61—22no—
229C4×Dic61—21no—
230C4×Dic61—22no—
231C4×Dic61—21no—
232C4×Dic61—22no—
233C4×Dic61—21no—
234C4×Dic61C22×C412yes—
235C4×Dic61C22×C411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238C4×Dic61C2×D412yes—
239C4×Dic61C2×D412yes—
240C22×D61C22×D411yescharacteristic
241C2×#6923—22no—
242C2×#6923—41no—
243C2×#6923—41no—
244C2×#6923—41no—
245C2×#6923—42no—
246C2×#6923—41no—
247C2×#6923—42no—
248C2×#6923—42no—
249C2×#6923—41no—
250Q8×C61—21no—
251Q8×C61—22no—
252Q8×C61—82no—
253Q8×C61—81no—
254Q8×C61C4×Q812yes—
255C2×C4×C61—21no—
256C2×C4×C61—22no—
257C2×C4×C61—21no—
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C412yes—
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C411yescharacteristic
270C2×Dic61C22×D411yescharacteristic
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—21no—
277C4×D61—22no—
278C4×D61—21no—
279C4×D61—41no—
280C4×D61—42no—
281C4×D61—41no—
282C4×D61—42no—
283C4×D61#13712yes—
284C4×D61#13712yes—
285C4×D61C22×Q811yescharacteristic
286C4×D61C2×#4111yescharacteristic
287C4×D612−1+411yescharacteristic
288C4×D612−1+411yescharacteristic
289Dic122—21no—
290Dic122—22no—
291Dic122—22no—
292Dic122—22no—
293Dic122—24no—
294Dic122—82no—
295Dic122—82no—
296Dic122—82no—
297Dic122—82no—
298Dic122—81no—
299Dic122C4×Q812yes—
300#92420—22no—
301#92420—41no—
302#92420—42no—
303#22525—611nomaximal, Sylow
304C22×C61C22×C22⋊C411yescharacteristic
305C2×D61#54811yescharacteristic
306C2×D61D4×Q811yescharacteristic
307C2×D61C2×#13711yescharacteristic
308C2×D61C2×#13711yescharacteristic
309C2×D61#57611yescharacteristic
310C2×D61#57711yescharacteristic
311#6923—42no—
312#6923—81no—
313#6923—82no—
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—21no—
318C4×C61—41no—
319C4×C61—42no—
320C4×C61#41912yes—
321C4×C61C2×C4×Q811yescharacteristic
322C4×C61#51911yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—22no—
326Dic61—21no—
327Dic61—41no—
328Dic61—42no—
329Dic61#41912yes—
330Dic61#54811yescharacteristic
331Dic61D4×Q811yescharacteristic
332C2×Q82—611no—
333C2×#446—611no—
334#2355—611no—
335#1993—612no—
336#1990—611no—
337#1988—612no—
338#1987—611no—
339#1805—612no—
340#1803—611no—
341#1763—612no—
342C4×#103—611no—
343C2×C61Q8×C22⋊C411yescharacteristic
344C2×C61C2×#41911yescharacteristic
345C2×C61#289711yescharacteristic
346D61#336911yescharacteristic
347D61#337511yescharacteristic
348D61#338511yescharacteristic
349D61#338711yescharacteristic
350Q82—612no—
351Q82—611no—
352Q82—612no—
353Q82—611no—
354Q82—611no—
355Q82—1221no—
356Q82—1221no—
357Q82—1222no—
358Q82#692311yescharacteristic, complemented
359C2×C4⋊Q8—611no—
360C2×C4⋊Q8—612no—
361C2×C4⋊Q8—1221no—
362C2×C4⋊Q8—1222no—
363C2×C4×Q8—611no—
364C2×C4×Q8—611no—
365C2×C4×Q8—612no—
366C2×C4×Q8—1221no—
367#446—1221no—
368#446—1221no—
369#446—1222no—
370C2×C4⋊C8—611no—
371C2×C4⋊C8—612no—
372C2×#103—611no—
373C2×#103—612no—
374C2×#103—611no—
375C2×#97—612no—
376C2×C4×C8—611no—
377#391—1221no—
378#387—1222no—
379#384—611no—
380#382—612no—
381C4×C4⋊C4—611no—
382C4×C4⋊C4—612no—
383C4×C4⋊C4—1221no—
384#340—612no—
385#340—612no—
386#340—612no—
387#340—612no—
388#340—611no—
389#340—612no—
390#340—611no—
391C61#2252511yescharacteristic, complemented, Sylow
392C22×Q8—611no—
393C22×Q8—611no—
394C22×Q8—1222no—
395C22×Q8—1221no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—612no—
399C4⋊Q8—22no—
400C4⋊Q8—21no—
401C4⋊Q8—612no—
402C4⋊Q8—612no—
403C4⋊Q8—612no—
404C4⋊Q8—611no—
405C4⋊Q8—612no—
406C4⋊Q8—1222no—
407C4⋊Q8—1221no—
408C4⋊Q8—1221no—
409C4⋊Q8—1222no—
410C4⋊Q8—1221no—
411C4⋊Q8—1222no—
412C4⋊Q8—1222no—
413C4⋊Q8—1222no—
414C4⋊Q8—1222no—
415C4⋊Q8—1222no—
416C4⋊Q8—1221no—
417C4⋊Q8—1222no—
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420C4⋊Q8—1222no—
421C4⋊Q8—1221no—
422C4⋊Q8C2×#692311yescharacteristic
423C4⋊Q8C2×#692312yes—
424C4×Q8—21no—
425C4×Q8—611no—
426C4×Q8—611no—
427C4×Q8—612no—
428C4×Q8—612no—
429C4×Q8—611no—
430C4×Q8—611no—
431C4×Q8—612no—
432C4×Q8—611no—
433C4×Q8—612no—
434C4×Q8—611no—
435C4×Q8—612no—
436C4×Q8—1221no—
437C4×Q8—1222no—
438C4×Q8—1222no—
439C4×Q8—1221no—
440C4×Q8—1221no—
441C4×Q8—1222no—
442C4×Q8—1222no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×Q8C2×#692311yescharacteristic
446C4×Q8C2×#692312yes—
447C4×Q8C2×#692311yescharacteristic
448C2×C4⋊C4—611no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—612no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—1222no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1222no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—1221no—
461C2×C42—612no—
462C2×C42—611no—
463C2×C42—1221no—
464C2×C42—1222no—
465C2×C42—1222no—
466C2×C42—1221no—
467C4⋊C8—1221no—
468C4⋊C8—1222no—
469C4⋊C8—1221no—
470C4⋊C8—1222no—
471#103—614no—
472#103—1224no—
473#103—1222no—
474#97—1222no—
475#97—1222no—
476C4×C8—1221no—
477C4×C8—1221no—
478#95—1221no—
479#95—1222no—
480#95—1222no—
481#95—1222no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—612no—
485C2×Q8—611no—
486C2×Q8—612no—
487C2×Q8—612no—
488C2×Q8—611no—
489C2×Q8—1221no—
490C2×Q8—1222no—
491C2×Q8—1222no—
492C2×Q8—1222no—
493C2×Q8—1222no—
494C2×Q8—1222no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1222no—
499C2×Q8—4882no—
500C2×Q8—4881no—
501C2×Q8—4882no—
502C2×Q8—4881no—
503C2×Q8C22⋊#692311yescharacteristic
504C2×Q8C22×#692311yescharacteristic
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—612no—
508C22×C4—612no—
509C22×C4—611no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1222no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1222no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1222no—
521C2×C8—1221no—
522C2×C8—1222no—
523C2×C8—1221no—
524C2×C8—1222no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—611no—
534C4⋊C4—612no—
535C4⋊C4—611no—
536C4⋊C4—612no—
537C4⋊C4—1221no—
538C4⋊C4—1222no—
539C4⋊C4—1221no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1221no—
545C4⋊C4—1222no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1222no—
549C4⋊C4—1221no—
550C4⋊C4—1222no—
551C4⋊C4—1221no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—1221no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C4⋊C4C22⋊#692312yes—
558C4⋊C4C22⋊#692311yescharacteristic
559C4⋊C4C22×#692311yescharacteristic
560C4⋊C4C22×#692312yes—
561C42—22no—
562C42—21no—
563C42—611no—
564C42—611no—
565C42—612no—
566C42—612no—
567C42—611no—
568C42—612no—
569C42—1221no—
570C42—1222no—
571C42—1221no—
572C42—1222no—
573C42—1222no—
574C42—1221no—
575C42—4882no—
576C42—2442no—
577C42C22×#692312yes—
578C42C22×#692311yescharacteristic
579C23—611no—
580Q8—22no—
581Q8—21no—
582Q8—82no—
583Q8—81no—
584Q8—612no—
585Q8—1222no—
586Q8—1222no—
587Q8—1224no—
588Q8—1222no—
589Q8—1221no—
590Q8—4882no—
591Q8—4881no—
592Q8—4882no—
593Q8—4882no—
594Q8—4882no—
595Q8Q8×#692312yes—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—611no—
600C2×C4—611no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—611no—
607C2×C4—612no—
608C2×C4—611no—
609C2×C4—612no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1221no—
619C2×C4—1222no—
620C2×C4—1221no—
621C2×C4—1222no—
622C2×C4—2441no—
623C2×C4—2442no—
624C2×C4—2441no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4C2×C22⋊#692312yes—
635C2×C4C2×C22⋊#692311yescharacteristic
636C2×C4C2×C22⋊#692312yes—
637C2×C4C2×C22⋊#692311yescharacteristic
638C2×C4C23×#692311yescharacteristic
639C8—1222no—
640C8—2441no—
641C8—2442no—
642C22—611no—
643C22—611no—
644C22—611no—
645C22—611no—
646C22—611no—
647C22—611no—
648C22C22×C22⋊#692311yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—21no—
653C4—42no—
654C4—41no—
655C4—611no—
656C4—611no—
657C4—612no—
658C4—1222no—
659C4—1222no—
660C4—1222no—
661C4—1221no—
662C4—4882no—
663C4—4881no—
664C4—2442no—
665C4—2441no—
666C4—2442no—
667C4#1684075512yes—
668C4C2×Q8×#692311yescharacteristic
669C4#1684083011yescharacteristic
670C2—611no—
671C2—611no—
672C2—611no—
673C2—611no—
674C2Q8×C22⋊#692311yescharacteristic
675C2C2×#1684075511yescharacteristic
676C2#2146244011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingQ82×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×D61C22×D4
cosocleC2×C4×D61C24
hypercenterQ82#6923
hypocenterC61#22525
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q82

Complements

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

NH ≅ G/Ntype
37. Q8×Dic122C4semidirect
38. Q8×Dic122C4semidirect
39. Q8×Dic122C4semidirect
42. #16840492C4semidirect
43. #16840492C4semidirect
52. Q82×C61C4semidirect
69. #14519840Q8semidirect
358. Q82#6923semidirect
391. C61#22525semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4462
#214623382
#214620032
#214620002
#214619982
#214619972
#214616902
#214616882
#214615912
C4×#145198402
Q82×D612
#2252561