Finite Group Databasebeta

← #30615865

Subgroups — #30615865

Order 15616.

790 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462462C211yesmaximal, characteristic
2#21462384C211yesmaximal, characteristic
3#21461887C212yesmaximal, complemented
4#21461886C212yesmaximal, complemented
5#21461885C212yesmaximal, complemented
6#21461884C212yesmaximal, complemented
7C2×#16840638C211yesmaximal, characteristic
8#21461499C211yesmaximal, characteristic
9#21461499C211yesmaximal, characteristic
10#21461484C211yesmaximal, characteristic
11#427×D61C211yesmaximal, characteristic
12C22×#11758213C2211yescharacteristic
13C22×#11758213C2211yescharacteristic
14C2×D61⋊M16C2211yescharacteristic
15C2×#14519856C2211yescharacteristic
16#16840769C2212yes—
17#16840769C2212yes—
18#16840768—22no—
19#16840767—21no—
20#16840766—21no—
21#16840727C2212yes—
22#16840727C2212yes—
23#16840726—22no—
24C4×#11758213—21no—
25C4×#11758213—21no—
26#16840638C2212yes—
27#16840638C2212yes—
28#16840632C2212yes—
29#16840632C2212yes—
30#16840632C2212yes—
31#16840632C2212yes—
32(C2×#6923)⋊C8C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C22×C8×D61C2211yescharacteristic
35#117×D61C2211yescharacteristic
36#117×D61C2211yescharacteristic
37#16839800C411yescharacteristic, complemented
38#16839799C2211yescharacteristic
39#16839799C2211yescharacteristic
40#16839798C411yescharacteristic, complemented
41#16839798C411yescharacteristic, complemented
42#16839797C2211yescharacteristic
43#16839797C2211yescharacteristic
44#107×D61C411yescharacteristic, complemented
45#107×D61C411yescharacteristic, complemented
46#106×D61C411yescharacteristic, complemented
47#106×D61C411yescharacteristic, complemented
48C61×#427C411yescharacteristic, complemented
49C2×#11758213—21no—
50C2×#11758213—22no—
51C2×#11758213—22no—
52C2×#11758213—22no—
53C2×#11758213—21no—
54C2×#11758213D411yescharacteristic
55C2×#11758213D411yescharacteristic
56C2×#11758213Q814yes—
57C2×#11758213C2312yes—
58C2×#11758213C2312yes—
59D61⋊M16—22no—
60D61⋊M16—22no—
61C2×#11758217D412yes—
62C2×D61⋊C8C2312yes—
63C22×#92420—21no—
64C22×#92420—21no—
65#14519856—22no—
66#14519856—22no—
67C2×C4⋊#6923—21no—
68C2×C22⋊#6923—21no—
69C2×C4×#6923C2312yes—
70#14519844—22no—
71#14519844—22no—
72C4⋊#92420—22no—
73C4⋊#92420—22no—
74#14519832—22no—
75#14519832—22no—
76C4×#92420—22no—
77C4×#92420—22no—
78C22×C4×D61C2311yescharacteristic
79C22×#92418C2×C411yescharacteristic
80C2×C8×D61—22no—
81C2×C8×D61—22no—
82C2×C8×D61C2×C411yescharacteristic
83C2×C8×D61C2×C411yescharacteristic
84C2×C8×D61C2311yescharacteristic
85C2×C8×D61C2311yescharacteristic
86#14519736C2×C411yescharacteristic
87#14519736C2×C411yescharacteristic
88#14519735C2311yescharacteristic
89#14519735C2311yescharacteristic
90#14519735C2311yescharacteristic
91#14519735C2311yescharacteristic
92#14519734—21no—
93#14519734—21no—
94#14519733—21no—
95#14519733—21no—
96C4⋊C4×D61C2×C411yescharacteristic
97C4⋊C4×D61C2×C411yescharacteristic
98C4⋊C4×D61C2×C411yescharacteristic
99C4⋊C4×D61C2×C411yescharacteristic
100C22⋊C4×D61—21no—
101C22⋊C4×D61—21no—
102C42×D61—21no—
103C42×D61—21no—
104#14519711C2×C411yescharacteristic
105#14519711C2×C411yescharacteristic
106#14519710C2×C411yescharacteristic
107#14519710C2×C411yescharacteristic
108#14519710C2×C411yescharacteristic
109#14519710C2×C411yescharacteristic
110#14519709C2×C411yescharacteristic
111#14519709C2×C411yescharacteristic
112#14519708C2×C411yescharacteristic
113#14519708C2×C411yescharacteristic
114#14519708C2×C411yescharacteristic
115#14519708C2×C411yescharacteristic
116C22×C8×C61C2×C411yescharacteristic
117#117×C61C2×C411yescharacteristic
118#117×C61C2×C411yescharacteristic
119#107×C61C2×C411yescharacteristic
120#107×C61C2×C411yescharacteristic
121#106×C61C2×C411yescharacteristic
122#106×C61C2×C411yescharacteristic
123C23×D61#4111yescharacteristic
124C22×#6923—22no—
125C22×C4×C61C22×C411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61C22×C411yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×C4×D61C2×Q812yes—
134C2×C4×D61#4112yes—
135C2×C4×D61#4112yes—
136C2×C4×D61#4112yes—
137C2×C4×D61#4111yescharacteristic
138C2×C4×D61C2411yescharacteristic
139C22⋊#6923—42no—
140#11758217—24no—
141C2×#92420—24no—
142C2×#92420—22no—
143C2×#92420—22no—
144C2×#92420—42no—
145C2×#92420C2×D412yes—
146C2×#92420C2×D412yes—
147C4⋊#6923—42no—
148C4×#6923—22no—
149C4×#6923—22no—
150#11758213—24no—
151#11758213—22no—
152#11758213—22no—
153#11758213—22no—
154#11758213—22no—
155#11758213—22no—
156#11758213—42no—
157#11758213C2×Q814yes—
158D61⋊C8—22no—
159D61⋊C8—22no—
160C2×C8×C61—22no—
161C2×C8×C61C22×C411yescharacteristic
162C2×C8×C61C22×C411yescharacteristic
163C4⋊C4×C61C22×C411yescharacteristic
164C4⋊C4×C61C22×C411yescharacteristic
165C4⋊C4×C61C22×C411yescharacteristic
166C4⋊C4×C61C22×C411yescharacteristic
167C22⋊C4×C61—21no—
168C22⋊C4×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C22⋊Dic61—21no—
172C22⋊Dic61—21no—
173#11758198—21no—
174#11758198—21no—
175#11758198—21no—
176#11758198—21no—
177C4⋊Dic61C22×C411yescharacteristic
178C4⋊Dic61C22×C411yescharacteristic
179C4⋊Dic61C22×C411yescharacteristic
180C4⋊Dic61C22×C411yescharacteristic
181#11758196—21no—
182#11758196—21no—
183#11758196—21no—
184#11758196—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C2×#92418—22no—
192C2×#92418C22×C411yescharacteristic
193C2×#92418C22×C411yescharacteristic
194C8×D61—22no—
195C8×D61—22no—
196C8×D61—21no—
197C8×D61—21no—
198C8×D61—21no—
199C8×D61—21no—
200C23×C61C4×D411yescharacteristic
201C22×D61—41no—
202C22×D61—41no—
203C22×D61C4×D411yescharacteristic
204C22×D61#12512yes—
205C22×D61#12612yes—
206C22×D61C2×#4111yescharacteristic
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61#11712yes—
215C2×C4×C61C4×D411yescharacteristic
216C2×C4×C61C4×Q812yes—
217C2×C4×C61C23×C411yescharacteristic
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61C4×D411yescharacteristic
223C2×Dic61C22⋊Q812yes—
224C2×Dic61#12312yes—
225C2×Dic61C22×D411yescharacteristic
226C4×D61—81no—
227C4×D61—81no—
228C4×D61—81no—
229C4×D61—81no—
230C4×D61#11712yes—
231C4×D61C4×Q812yes—
232C4×D61C22⋊Q812yes—
233C4×D61#12312yes—
234C4×D61#12512yes—
235C4×D61#12612yes—
236C4×D61C22×Q811yescharacteristic
237C4×D61C2×#4111yescharacteristic
238C4×D61C2×#4111yescharacteristic
239C4×D61C2×#4111yescharacteristic
240#92420—24no—
241#92420—22no—
242#92420—22no—
243#92420—42no—
244C8×C61—21no—
245C8×C61—21no—
246#92418—21no—
247#92418—21no—
248#21043—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61#38712yes—
252C22×C61C2×C4×D411yescharacteristic
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61#38712yes—
260C2×D61C2×#12511yescharacteristic
261C2×D61C2×#12611yescharacteristic
262#6923—82no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61#38712yes—
268C4×C61C2×#11711yescharacteristic
269C4×C61C2×C4×Q811yescharacteristic
270Dic61—41no—
271Dic61—41no—
272Dic61—41no—
273Dic61—41no—
274Dic61#38712yes—
275Dic61C2×C22⋊Q811yescharacteristic
276Dic61C2×#12311yescharacteristic
277C2×#427—611no—
278#2905—611no—
279#2873—611no—
280#1922—612no—
281#1922—612no—
282#1922—612no—
283#1922—612no—
284C2×#341—611no—
285#1733—611no—
286#1733—611no—
287#1729—611no—
288C2×C61—22no—
289C2×C61C2×#38711yescharacteristic
290C2×C61#295711yescharacteristic
291C2×C61#295711yescharacteristic
292D61—21no—
293D61—21no—
294D61—22no—
295C22×M16—611no—
296C22×M16—611no—
297C22×M16—611no—
298C23×C8—611no—
299C2×#117—611no—
300C2×#117—611no—
301C2×#117—611no—
302#427—611no—
303#427—611no—
304#427—611no—
305#427—611no—
306#427—611no—
307#427—611no—
308#427—611no—
309#427#692311yescharacteristic, complemented
310C2×#107—611no—
311C2×#107—611no—
312C2×#106—611no—
313C2×#106—611no—
314#424—612no—
315#424—612no—
316#424—1222no—
317#423—1221no—
318#423—1221no—
319#405—612no—
320#405—612no—
321#405—1222no—
322C4×M16—1221no—
323C4×M16—1221no—
324#341—612no—
325#341—612no—
326#339—612no—
327#339—612no—
328#339—612no—
329#339—612no—
330#336—612no—
331#336—612no—
332C61#2104311yescharacteristic, complemented, Sylow
333C23×C4—611no—
334C2×M16—612no—
335C2×M16—611no—
336C2×M16—611no—
337C2×M16—614no—
338C2×M16—612no—
339C2×M16—612no—
340C2×M16—1222no—
341C2×M16—1222no—
342C2×M16—1222no—
343C2×M16—1221no—
344C2×M16—1221no—
345C2×M16—1222no—
346C2×M16—1222no—
347C22×C8—612no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8—611no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8—1222no—
354C22×C8—1222no—
355C22×C8—1221no—
356C22×C8—1221no—
357C22×C8C2×#692311yescharacteristic
358#117—611no—
359#117—611no—
360#117—611no—
361#117—611no—
362#117—611no—
363#117—611no—
364#117—1221no—
365#117—1221no—
366#117—1221no—
367#117—1221no—
368#117—1222no—
369#117—1222no—
370#117C2×#692311yescharacteristic
371#117C2×#692311yescharacteristic
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C42—612no—
381C2×C42—1221no—
382C2×C42—1221no—
383#107—611no—
384#107—611no—
385#107—611no—
386#107—611no—
387#107—611no—
388#107—611no—
389#107C2×#692311yescharacteristic
390#107C2×#692311yescharacteristic
391#106—611no—
392#106—611no—
393#106—611no—
394#106—611no—
395#106—611no—
396#106—611no—
397#106C2×#692311yescharacteristic
398#106C2×#692311yescharacteristic
399C4⋊C8—1222no—
400C4⋊C8—1222no—
401C4⋊C8—1222no—
402C4⋊C8—1222no—
403#97—1222no—
404#97—1222no—
405C4×C8—1222no—
406C4×C8—1222no—
407C24—611no—
408C22×C4—611no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1222no—
422C22×C4C22×#692311yescharacteristic
423M16—614no—
424M16—1222no—
425M16—1222no—
426M16—1224no—
427M16—1222no—
428M16—1224no—
429M16—1222no—
430M16—1222no—
431M16—2442no—
432C2×C8—22no—
433C2×C8—612no—
434C2×C8—612no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C2×C8—1224no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—2442no—
450C2×C8C22×#692311yescharacteristic
451C2×C8C22×#692311yescharacteristic
452C4⋊C4—611no—
453C4⋊C4—611no—
454C4⋊C4—611no—
455C4⋊C4—611no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—2442no—
461C4⋊C4C22×#692311yescharacteristic
462C4⋊C4C22×#692311yescharacteristic
463C4⋊C4C22×#692311yescharacteristic
464C4⋊C4C22×#692311yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—1221no—
468C22⋊C4—1221no—
469C22⋊C4—1221no—
470C22⋊C4—1221no—
471C22⋊C4—1221no—
472C22⋊C4—1221no—
473C22⋊C4—2442no—
474C42—21no—
475C42—21no—
476C42—1221no—
477C42—1221no—
478C42—1221no—
479C42—1221no—
480C42—1222no—
481C42—1222no—
482C42—1221no—
483C42—1221no—
484C23—611no—
485C23—611no—
486C23—612no—
487C23—612no—
488C23—2441no—
489C23—2441no—
490C23D4×#692311yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—611no—
498C2×C4—611no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—612no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—4881no—
514C2×C4—4881no—
515C2×C4—4881no—
516C2×C4—4881no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4#1451985612yes—
521C2×C4D4×#692311yescharacteristic
522C2×C4Q8×#692312yes—
523C2×C4C23×#692311yescharacteristic
524C8—21no—
525C8—21no—
526C8—1222no—
527C8—1222no—
528C8—1224no—
529C8—1221no—
530C8—1221no—
531C8—2442no—
532C22—21no—
533C22—21no—
534C22—611no—
535C22—611no—
536C22—612no—
537C22—1221no—
538C22—1221no—
539C22—1221no—
540C22—1221no—
541C22—2442no—
542C22—2442no—
543C22#1684071112yes—
544C22C2×D4×#692311yescharacteristic
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—611no—
550C4—611no—
551C4—612no—
552C4—4882no—
553C4—2441no—
554C4—2441no—
555C4—2441no—
556C4—2441no—
557C4#1684071112yes—
558C4C2×#1451985611yescharacteristic
559C4C2×Q8×#692311yescharacteristic
560C2—22no—
561C2—1221no—
562C2—1221no—
563C2—1222no—
564C2C2×#1684071111yescharacteristic
565C2#2146259611yescharacteristic
566C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#427C4
socleC22×C61C2×C4×D4
prime radicalC23×D61#41
cosocleC2×C4×D61C24
hypercenter#427#6923
hypocenterC61#21043
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #21461887C2semidirect
4. #21461886C2semidirect
5. #21461885C2semidirect
6. #21461884C2semidirect
37. #16839800C4semidirect
40. #16839798C4semidirect
41. #16839798C4semidirect
44. #107×D61C4semidirect
45. #107×D61C4semidirect
46. #106×D61C4semidirect
47. #106×D61C4semidirect
48. C61×#427C4semidirect
309. #427#6923semidirect
332. C61#21043semidirect

Maximal subgroups

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

subgroupindex
#214624622
#214623842
#214618872
#214618862
#214618852
#214618842
C2×#168406382
#214614992
#214614842
#427×D612
#2104361