Finite Group Databasebeta

← #30611249

Subgroups — #30611249

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840784C211yesmaximal, characteristic
2#21462266C211yesmaximal, characteristic
3#21461257C214yesmaximal
4#21461241C214yesmaximal, complemented
5#21461213C212yesmaximal, complemented
6#21461211C212yesmaximal, complemented
7#533×D61C211yesmaximal, characteristic
8#16840784—21no—
9#16840784—21no—
10#16840784C2212yes—
11#16840784C2212yes—
12C2×C4⋊#92420—21no—
13C2×#14519834C2212yes—
14C2×#14519832—21no—
15#16840707C2212yes—
16#16840706—21no—
17#16840705—21no—
18#16840704C2212yes—
19C4×C4⋊#6923—21no—
20C4×C22⋊#6923—21no—
21#16840698C2212yes—
22#16840616C2218yes—
23#16840615C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840318—21no—
26#16840317C2211yescharacteristic
27#16840317C2211yescharacteristic
28#16840316—21no—
29#16840316C2212yes—
30#16840315—21no—
31#16840314—21no—
32#16840314C412yescomplemented
33C4⋊Q8×D61C411yescharacteristic, complemented
34C4⋊Q8×D61C411yescharacteristic, complemented
35#125×D61—21no—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61C2212yes—
38C4×Q8×D61—21no—
39C4×Q8×D61C412yescomplemented
40#117×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C22×#92420—22no—
44C2×C4⋊#6923—22no—
45C2×C22⋊#6923—22no—
46C2×C4×#6923—22no—
47C4⋊#92420—22no—
48C4⋊#92420—22no—
49#14519834—22no—
50#14519834C2314yes—
51#14519832—22no—
52#14519832—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4C2314yes—
57C2×Q8×D61—21no—
58C2×Q8×D61C2×C412yes—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519776—21no—
65#14519776—21no—
66#14519776—21no—
67#14519776C2×C412yes—
68#14519775—21no—
69#14519775C2×C412yes—
70#14519769—21no—
71#14519768—21no—
72#14519768—21no—
73#14519767—22no—
74#14519767—22no—
75#14519757—21no—
76#14519757C2312yes—
77#14519756—21no—
78#14519756C2×C412yes—
79#14519755—22no—
80#14519755—22no—
81#14519754—22no—
82#14519754—22no—
83#14519753—21no—
84#14519753C2312yes—
85Q8×Dic61—21no—
86Q8×Dic61C2×C412yes—
87C4×Dic122—21no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90C4×Dic122C2×C412yes—
91#14519742—22no—
92#14519742—22no—
93#14519742—21no—
94#14519742C2×C412yes—
95#14519736—21no—
96#14519735—21no—
97#14519735—21no—
98#14519734—22no—
99#14519733—22no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—22no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61C2×C412yes—
106C22⋊C4×D61—22no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊Q8×C61C2×C411yescharacteristic
120C4⋊Q8×C61C2×C411yescharacteristic
121#125×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61C2×C412yes—
124C4×Q8×C61—21no—
125C4×Q8×C61C2×C412yes—
126#117×C61—21no—
127C2×C42×C61C2×C411yescharacteristic
128C23×D61C2×D411yescharacteristic
129C22×#6923—24no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C22×C4×C61C22⋊C411yescharacteristic
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×Dic61C22⋊C411yescharacteristic
136C22×Dic61C22⋊C411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138Q8×D61—44no—
139Q8×D61—44no—
140C2×C4×D61—24no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22×C411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122C22×C412yes—
161C22⋊#6923—44no—
162C2×#92420—24no—
163C2×#92420—44no—
164C4⋊#6923—44no—
165C4×#6923—44no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61C22×C412yes—
172C22⋊C4×C61—22no—
173C42×C61—22no—
174C42×C61C22⋊C411yescharacteristic
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22×C411yescharacteristic
177C42×C61C22×C411yescharacteristic
178C22⋊Dic61—22no—
179#11758198—22no—
180#11758198—22no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61C22×C412yes—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—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
210C23×C61C2×C22⋊C411yescharacteristic
211C22×D61—41no—
212C22×D61—41no—
213C22×D61C2×C22⋊C411yescharacteristic
214C22×D61#13714yes—
215C22×D61C22×D411yescharacteristic
216C2×#6923—44no—
217C2×#6923—44no—
218C2×#6923—44no—
219Q8×C61—42no—
220Q8×C61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—41no—
237C2×Dic61C2×C22⋊C411yescharacteristic
238C2×Dic61C2×C22⋊C411yescharacteristic
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C22×D411yescharacteristic
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—44no—
256C4×D61—44no—
257Dic122—44no—
258Dic122—42no—
259Dic122—44no—
260Dic122—42no—
261#92420—44no—
262#17596—611nomaximal, Sylow
263C22×C61—21no—
264C22×C61—21no—
265C22×C61#41912yes—
266C22×C61C22×C22⋊C411yescharacteristic
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—44no—
272C2×D61#41912yes—
273C2×D61C2×#13712yes—
274#6923—84no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61#42112yes—
283Dic61—21no—
284Dic61—21no—
285Dic61—21no—
286Dic61—21no—
287Dic61—21no—
288Dic61—42no—
289Dic61—42no—
290Dic61#41012yes—
291C2×#533—611no—
292C2×#432—611no—
293#2283—611no—
294#1520—614no—
295#1515—614no—
296#1506—612no—
297#1506—612no—
298C2×C61—22no—
299C2×C61C2×#41011yescharacteristic
300C2×C61C2×#41911yescharacteristic
301C2×C61C2×#42111yescharacteristic
302D61—22no—
303D61—22no—
304#533—611no—
305#533—611no—
306#533—612no—
307#533—612no—
308#533—611no—
309#533—1221no—
310#533—1221no—
311#533—1221no—
312#533—1221no—
313#533#692311yescharacteristic, complemented
314C2×C4⋊Q8—611no—
315C2×C4⋊Q8—611no—
316C2×#125—1221no—
317C2×C22⋊Q8—612no—
318C2×C22⋊Q8—1221no—
319C2×C4×Q8—612no—
320C2×C4×Q8—1221no—
321C2×#117—1221no—
322C22×C42—611no—
323#432—612no—
324#432—612no—
325#432—1221no—
326#432—1221no—
327C2×C4⋊C8—1221no—
328C2×C22⋊C8—612no—
329C2×#97—1221no—
330#383—612no—
331#382—1221no—
332#381—1221no—
333#380—612no—
334C4×C4⋊C4—1221no—
335C4×C22⋊C4—1221no—
336#376—612no—
337#328—618no—
338#328—618no—
339C61#1759611yescharacteristic, complemented, Sylow
340C22×Q8—612no—
341C22×Q8—1221no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C22×C8—1222no—
346C4⋊Q8—612no—
347C4⋊Q8—612no—
348C4⋊Q8—611no—
349C4⋊Q8—611no—
350C4⋊Q8—1221no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8C2×#692311yescharacteristic
355C4⋊Q8C2×#692311yescharacteristic
356#125—21no—
357#125—1222no—
358#125—1222no—
359#125—1221no—
360#125—1221no—
361#125—1221no—
362C22⋊Q8—21no—
363C22⋊Q8—612no—
364C22⋊Q8—612no—
365C22⋊Q8—612no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372C22⋊Q8—1221no—
373C22⋊Q8C2×#692312yes—
374C4×Q8—21no—
375C4×Q8—612no—
376C4×Q8—612no—
377C4×Q8—612no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×Q8—1221no—
382C4×Q8—1222no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×Q8C2×#692312yes—
386#117—21no—
387#117—1222no—
388#117—1222no—
389#117—1221no—
390#117—1221no—
391#117—1221no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1222no—
413C2×C42—1222no—
414C2×C42C2×#692311yescharacteristic
415C4⋊C8—1222no—
416C4⋊C8—1222no—
417C22⋊C8—614no—
418C22⋊C8—1222no—
419#97—1222no—
420#97—1222no—
421#95—614no—
422#95—1222no—
423#95—1222no—
424#95—1222no—
425C24—611no—
426C2×Q8—21no—
427C2×Q8—612no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1221no—
431C2×Q8—2444no—
432C2×Q8—2444no—
433C2×Q8C22×#692312yes—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
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—1224no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1224no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4C22⋊#692311yescharacteristic
456C22×C4C22⋊#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C2×C8—1224no—
459C2×C8—2444no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—612no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2444no—
480C4⋊C4C22×#692312yes—
481C22⋊C4—22no—
482C22⋊C4—1222no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—2444no—
486C42—22no—
487C42—611no—
488C42—611no—
489C42—611no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—1222no—
500C42—1222no—
501C42—1222no—
502C42—2444no—
503C42C22⋊#692311yescharacteristic
504C42C22⋊#692311yescharacteristic
505C42C22×#692311yescharacteristic
506C42C22×#692311yescharacteristic
507C23—611no—
508C23—611no—
509C23—614no—
510C23—2441no—
511C23—2441no—
512C23C2×C22⋊#692311yescharacteristic
513Q8—42no—
514Q8—42no—
515Q8—2442no—
516Q8—2444no—
517Q8—2444no—
518Q8—2442no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2441no—
549C2×C4—2444no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4C2×C22⋊#692311yescharacteristic
553C2×C4C2×C22⋊#692311yescharacteristic
554C2×C4C2×C22⋊#692311yescharacteristic
555C2×C4C2×C22⋊#692311yescharacteristic
556C2×C4C2×C22⋊#692311yescharacteristic
557C2×C4C23×#692311yescharacteristic
558C8—2444no—
559C22—21no—
560C22—21no—
561C22—612no—
562C22—612no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—1221no—
567C22—2444no—
568C22#1684075512yes—
569C22C22×C22⋊#692311yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—612no—
578C4—1221no—
579C4—1221no—
580C4—1221no—
581C4—1221no—
582C4—1221no—
583C4—4884no—
584C4—2442no—
585C4—2442no—
586C4#1684075912yes—
587C2—22no—
588C2—1222no—
589C2—1222no—
590C2D61⋊#41011yescharacteristic
591C2C2×#1684075511yescharacteristic
592C2C2×#1684075911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#17596
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
4. #21461241C2semidirect
5. #21461213C2semidirect
6. #21461211C2semidirect
24. #16840319C4semidirect
32. #16840314C4semidirect
33. C4⋊Q8×D61C4semidirect
34. C4⋊Q8×D61C4semidirect
39. C4×Q8×D61C4semidirect
42. C61×#533C4semidirect
313. #533#6923semidirect
339. C61#17596semidirect

Maximal subgroups

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

subgroupindex
C2×#168407842
#214622662
#214612572
#214612412
#214612132
#214612112
#533×D612
#1759661