Finite Group Databasebeta

← #30616764

Subgroups — #30616764

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C211yesmaximal, characteristic
2C2×C4×#11758217C211yesmaximal, characteristic
3#21462011C212yesmaximal, complemented
4#21461962C212yesmaximal, complemented
5#21461846C214yesmaximal
6#21461572C214yesmaximal, complemented
7C2×#16840314C211yesmaximal, characteristic
8C22×#11758217C2212yes—
9#16840783—21no—
10#16840782—21no—
11#16840779—21no—
12#16840778—21no—
13#16840765C2212yes—
14C2×#14519838C2218yes—
15#16840732C2212yes—
16C4×#11758217—21no—
17C4×#11758217—21no—
18C4×#11758217C2212yes—
19C4×#11758217C2212yes—
20C4×#11758217C2212yes—
21C2×#14519832—21no—
22C2×C4×#92420—21no—
23#16840642C2218yes—
24#16840314—21no—
25#16840314—21no—
26#16840314—21no—
27#16840314—21no—
28#16840314—21no—
29#16840314—21no—
30#16840314—21no—
31#16840314—21no—
32C2×#14519776C2211yescharacteristic
33C2×#14519775C2211yescharacteristic
34C2×#14519767C412yes—
35C2×#14519756C411yescharacteristic
36C2×#14519754C2212yes—
37C2×Q8×Dic61C411yescharacteristic
38C2×#14519742C411yescharacteristic
39C2×#14519742C2212yes—
40C2×#14519733C412yes—
41C22×C4×Dic61C2211yescharacteristic
42C2×C22⋊Q8×C61C411yescharacteristic
43C2×#11758217—22no—
44C2×#11758217—28no—
45C2×#11758217C2314yes—
46C2×#11758217C2314yes—
47C22×#92420—22no—
48#14519844—22no—
49C4⋊#92420—22no—
50#14519838—216no—
51C22⋊#92420—22no—
52C22⋊#92420—22no—
53#14519832—22no—
54#14519832—22no—
55#14519832—22no—
56C4×#92420—22no—
57C4×#92420—22no—
58C4×#92420—22no—
59C22×Dic122C2×C411yescharacteristic
60C22×Dic122C2312yes—
61C23×Dic61C2×C411yescharacteristic
62C23×Dic61C2311yescharacteristic
63#14519776—21no—
64#14519776—21no—
65#14519776—21no—
66#14519776—21no—
67#14519775—21no—
68#14519775—21no—
69#14519775—21no—
70#14519775—21no—
71#14519767—22no—
72#14519767—22no—
73#14519767—22no—
74#14519767—22no—
75#14519756—21no—
76#14519756—21no—
77#14519756—21no—
78#14519756—21no—
79#14519754—22no—
80#14519754—22no—
81#14519754—22no—
82#14519754—22no—
83Q8×Dic61—21no—
84Q8×Dic61—21no—
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87#14519742—21no—
88#14519742—21no—
89#14519742—21no—
90#14519742—21no—
91#14519742—22no—
92#14519742—22no—
93#14519742—22no—
94#14519742—22no—
95#14519733—22no—
96#14519733—22no—
97#14519733—22no—
98#14519733—22no—
99C2×C4⋊Dic61C2×C412yes—
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758196C2×C412yes—
102C2×#11758196C2×C412yes—
103C2×#11758196C2×C412yes—
104C2×#11758196C2×C412yes—
105C2×#11758196C2312yes—
106C2×C22⋊Dic61C2×C412yes—
107C2×C4×Dic61C2×C412yes—
108C2×C4×Dic61C2×C412yes—
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61D411yescharacteristic
117C2×C4×Dic61D411yescharacteristic
118C2×C4×Dic61D411yescharacteristic
119C2×C4×Dic61C2311yescharacteristic
120C2×C4×Dic61C2311yescharacteristic
121C22×Q8×C61C2×C411yescharacteristic
122C23×C4×C61C2×C411yescharacteristic
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—21no—
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C4⋊C4×C61C2×C412yes—
129C2×C22⋊C4×C61C2×C412yes—
130C24×C61C22×C411yescharacteristic
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22×C412yes—
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C412yes—
142C22×C4×C61C22×C411yescharacteristic
143C22×Dic61—24no—
144C22×Dic61—24no—
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22×C412yes—
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C412yes—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—28no—
164C2×Dic122—42no—
165#11758217—28no—
166#11758217—24no—
167#11758217—28no—
168#11758217#4118yes—
169C2×#92420—28no—
170C2×#92420—24no—
171C2×#92420—24no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊Dic61—22no—
179C22⋊Dic61—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61C22⋊C411yescharacteristic
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—41no—
217C23×C61C2×C22⋊C411yescharacteristic
218C23×C61C2×C22⋊C411yescharacteristic
219C23×C61C23×C411yescharacteristic
220Q8×C61—42no—
221Q8×C61—42no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—42no—
250C2×Dic61#10114yescomplemented
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C22×D411yescharacteristic
258C2×Dic61C22×D411yescharacteristic
259C2×Dic61C2×#4114yes—
260Dic122—42no—
261Dic122—42no—
262Dic122—44no—
263Dic122—44no—
264#92420—28no—
265#92420—44no—
266#21425—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—44no—
275C22×C61C2×#10112yes—
276C22×C61C22×C22⋊C411yescharacteristic
277C22×C61C4×#4112yes—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61#41312yes—
284Dic61—21no—
285Dic61—21no—
286Dic61—21no—
287Dic61—21no—
288Dic61—42no—
289Dic61—42no—
290Dic61C2×#10112yes—
291Dic61C2×#10112yes—
292Dic61#52512yes—
293Dic61#53212yes—
294C2×#533—611no—
295#2920—611no—
296C2×C4×M16—611no—
297#1997—612no—
298#1966—612no—
299#1901—614no—
300C4×#101—614no—
301C2×C61—22no—
302C2×C61—22no—
303C2×C61—41no—
304C2×C61#229711yescharacteristic
305C2×C61C22×#10111yescharacteristic
306C2×C61C2×#41311yescharacteristic
307C22×M16—612no—
308#533—1221no—
309#533—1221no—
310#533—1221no—
311#533—1221no—
312#533—1221no—
313#533—1221no—
314#533—1221no—
315#533—1221no—
316C2×C4⋊Q8—611no—
317C2×C4⋊Q8—611no—
318C2×#125—612no—
319C2×C22⋊Q8—611no—
320C2×C22⋊Q8—612no—
321C2×C22⋊Q8#692311yescharacteristic
322C2×C4×Q8—611no—
323C2×C4×Q8—611no—
324C2×C4×Q8—612no—
325C2×#117—612no—
326C22×C42—611no—
327#432—1221no—
328#432—1221no—
329#431—1221no—
330#431—1221no—
331#423—612no—
332C2×#101—618no—
333#407—612no—
334C4×M16—612no—
335C4×M16—612no—
336C4×M16—612no—
337C4×M16—1221no—
338C4×M16—1221no—
339C2×#97—1221no—
340C2×C4×C8—1221no—
341#343—618no—
342C61#2142511yescharacteristic, complemented, Sylow
343C22×Q8—612no—
344C22×Q8—611no—
345C22×Q8C2×#692311yescharacteristic
346C23×C4—611no—
347C23×C4—611no—
348C23×C4C2×#692311yescharacteristic
349C2×M16—614no—
350C2×M16—614no—
351C2×M16—1228no—
352C2×M16—1222no—
353C22×C8—1222no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362#125—1222no—
363#125—1222no—
364#125—1222no—
365#125—1222no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377C22⋊Q8—1221no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382C4×Q8—1221no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×Q8—1221no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390#117—1222no—
391#117—1222no—
392#117—1222no—
393#117—1222no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4C2×#692312yes—
402C2×C4⋊C4C2×#692311yescharacteristic
403C2×C22⋊C4—612no—
404C2×C22⋊C4C2×#692312yes—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—612no—
418C2×C42—612no—
419C4⋊C8—1222no—
420C4⋊C8—1222no—
421#101—12216no—
422C22⋊C8—1222no—
423C22⋊C8—1222no—
424#97—1222no—
425#97—1222no—
426#97—1222no—
427C4×C8—1222no—
428C4×C8—1222no—
429C4×C8—1222no—
430C24C22×#692311yescharacteristic
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—42no—
434C2×Q8—1228no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—2442no—
440C22×C4—42no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—1224no—
456C22×C4—1224no—
457C22×C4C22⋊#692311yescharacteristic
458C22×C4C22⋊#692311yescharacteristic
459C22×C4C22⋊#692311yescharacteristic
460C22×C4C22⋊#692311yescharacteristic
461C22×C4C22×#692312yes—
462C22×C4C22×#692311yescharacteristic
463C22×C4C22×#692312yes—
464C22×C4C22×#692311yescharacteristic
465M16—618no—
466M16—1224no—
467M16—1228no—
468M16—1228no—
469C2×C8—1224no—
470C2×C8—1228no—
471C2×C8—1224no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—1222no—
511C42—1222no—
512C42—1222no—
513C42—1222no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23C2×C22⋊#692311yescharacteristic
518C23C2×C22⋊#692311yescharacteristic
519C23C23×#692311yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—2442no—
523Q8—2444no—
524Q8—2444no—
525Q8—2442no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—614no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—614no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1224no—
559C2×C4—2442no—
560C2×C4C2×C22⋊#692311yescharacteristic
561C2×C4C2×C22⋊#692311yescharacteristic
562C2×C4C2×C22⋊#692311yescharacteristic
563C2×C4C2×C22⋊#692311yescharacteristic
564C8—1228no—
565C8—2444no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—44no—
574C22D61⋊#10112yes—
575C22C22×C22⋊#692311yescharacteristic
576C22#41×#692312yes—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—612no—
583C4—612no—
584C4—612no—
585C4—612no—
586C4—1221no—
587C4—1221no—
588C4—1221no—
589C4—1221no—
590C4—2442no—
591C4—2442no—
592C4#1684074912yes—
593C2—22no—
594C2—22no—
595C2—41no—
596C2#2146228011yescharacteristic
597C2C2×D61⋊#10111yescharacteristic
598C2C2×#1684074911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#21425
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21462011C2semidirect
4. #21461962C2semidirect
6. #21461572C2semidirect
250. C2×Dic61#101semidirect
342. C61#21425semidirect

Maximal subgroups

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

subgroupindex
#214624952
C2×C4×#117582172
#214620112
#214619622
#214618462
#214615722
C2×#168403142
#2142561