Finite Group Databasebeta

← #30617329

Subgroups — #30617329

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462608C211yesmaximal, characteristic, complemented
2C4×#14519864C211yesmaximal, characteristic
3#21461994C211yesmaximal, characteristic, complemented
4#21461934C211yesmaximal, characteristic, complemented
5#21461886C212yesmaximal, complemented
6#21461788C212yesmaximal
7#21461691C211yesmaximal, characteristic
8#21461622C212yesmaximal, complemented
9#21461612C211yesmaximal, characteristic, complemented
10#21461594C212yesmaximal
11#21455608C211yesmaximal, characteristic
12C2×#14519864C2211yescharacteristic
13C2×#14519864C2211yescharacteristic
14#16840813—21no—
15#16840812—21no—
16#16840811—21no—
17Q8×#92420—21no—
18C2×#14519847C2212yes—
19#16840769C2212yes—
20#16840766C2211yescharacteristic
21C2×#14519841C2212yes—
22#16840727C2212yes—
23#16840727C2211yescharacteristic
24#16840727C2211yescharacteristic
25#16840726—21no—
26#16840726—21no—
27C4×#11758213C2211yescharacteristic
28C4×#11758217—21no—
29C4×D61⋊C8C2211yescharacteristic
30C2×C4×#92420—21no—
31C2×C8×#6923C2212yes—
32C4×C4⋊#6923C2211yescharacteristic
33#16840638C2212yes—
34#16840632C2212yes—
35#16840632C2212yes—
36#16840632C2212yes—
37(C2×#6923)⋊C8C2212yes—
38#16840317C2211yescharacteristic
39C4×#11758224C2211yescharacteristic
40#16840072C412yescomplemented
41#16840064C412yescomplemented
42#16840054C411yescharacteristic, complemented
43#16840048C411yescharacteristic, complemented
44#16840042C411yescharacteristic, complemented
45#16839797C2212yes—
46C4⋊C8×D61C2211yescharacteristic
47#16839755C2212yes—
48C61×#476C411yescharacteristic, complemented
49#14519864—22no—
50#14519864—22no—
51#14519864—22no—
52#14519864Q814yescomplemented
53C2×#11758213C2312yes—
54C2×#11758213C2311yescharacteristic
55C2×#11758213C2311yescharacteristic
56C2×#11758217—21no—
57C2×#11758217—21no—
58C2×D61⋊C8C2311yescharacteristic
59C2×D61⋊C8C2311yescharacteristic
60C22×#92420—21no—
61C22×#92420—21no—
62C2×C4⋊#6923—21no—
63C2×C4×#6923C2312yes—
64#14519847—24no—
65#14519844—22no—
66C4⋊#92420—22no—
67C4⋊#92420—21no—
68C4⋊#92420—21no—
69#14519841—24no—
70#14519832—21no—
71#14519832—21no—
72#14519832—22no—
73#14519832—21no—
74C4×#92420—21no—
75C4×#92420—21no—
76C4×#92420—22no—
77C4×#92420—21no—
78C4×#92420D411yescharacteristic
79C4×#92420D411yescharacteristic
80C4×#92420D411yescharacteristic
81C4×#92420D411yescharacteristic
82C8×#6923—24no—
83(C2×#6923)⋊C4—21no—
84C2×#11758224C2311yescharacteristic
85C2×C8×D61C2312yes—
86#14519777C2×C411yescharacteristic
87#14519768C2×C412yes—
88#14519757—21no—
89#14519753—21no—
90Q8×Dic61—21no—
91C4×Dic122C2×C411yescharacteristic
92#14519742—21no—
93#14519742C2×C411yescharacteristic
94D4×Dic61C2×C411yescharacteristic
95C4×C22⋊D61—21no—
96#14519737—21no—
97#14519735C2311yescharacteristic
98#14519735C2312yes—
99#14519733—21no—
100C2×C4×Dic61—21no—
101C42×D61C2311yescharacteristic
102#14519710C2×C412yes—
103#14519710C2×C412yes—
104#14519709C2×C412yes—
105C4⋊#92418C2×C411yescharacteristic
106#14519706C2×C412yes—
107#14519702C2×C412yes—
108#14519700C2×C412yes—
109#14519698C2×C412yes—
110C4⋊Q8×C61C2×C411yescharacteristic
111C4×D4×C61C2×C411yescharacteristic
112#106×C61C2×C412yes—
113C4⋊C8×C61C2×C411yescharacteristic
114#102×C61C2×C412yes—
115C22×#6923—22no—
116C2×Q8×C61—21no—
117C2×D4×C61C22×C411yescharacteristic
118C22×C4×C61—21no—
119C2×C22⋊D61—21no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122#11758224—21no—
123#11758224—21no—
124#11758224C2×Q812yes—
125#11758224#4112yes—
126C2×C4×D61#4111yescharacteristic
127C2×C4×D61#4111yescharacteristic
128C2×C4×D61C2411yescharacteristic
129C2×Dic122—21no—
130C2×Dic122C22×C411yescharacteristic
131#11758217—24no—
132#11758217—24no—
133C2×#92420—24no—
134C2×#92420—21no—
135C2×#92420—21no—
136C2×#92420—24no—
137C2×#92420—22no—
138C2×#92420C2×D411yescharacteristic
139C2×#92420C2×D411yescharacteristic
140C2×#92420C2×D411yescharacteristic
141C2×#92420C2×D411yescharacteristic
142C4⋊#6923—24no—
143C4×#6923—22no—
144C4×#6923—22no—
145#11758213—21no—
146#11758213—21no—
147#11758213—22no—
148#11758213—22no—
149#11758213C2×Q812yes—
150#11758213#4112yes—
151D61⋊C8—21no—
152D61⋊C8—21no—
153D61⋊C8C2×Q812yes—
154D61⋊C8#4112yes—
155C2×C8×C61C22×C412yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61C22×C412yes—
158C4⋊C4×C61C22×C411yescharacteristic
159C22⋊C4×C61—21no—
160C42×C61C22×C411yescharacteristic
161C22⋊Dic61—21no—
162#11758198—21no—
163#11758198—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61C22×C412yes—
167C4⋊Dic61C22×C411yescharacteristic
168#11758196—21no—
169#11758196—21no—
170#11758196—22no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—22no—
175C4×Dic61C22×C411yescharacteristic
176C4×Dic61C2×D411yescharacteristic
177C4×Dic61C2×D411yescharacteristic
178C2×#92418C22×C412yes—
179C8×D61—22no—
180C8×D61—22no—
181C23×C61—21no—
182C22×D61C2×#4111yescharacteristic
183C2×#6923—24no—
184C2×#6923—42no—
185C2×#6923—42no—
186Q8×C61—42no—
187D4×C61—21no—
188D4×C61C4×Q812yes—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61C4×D411yescharacteristic
193C2×C4×C61C4×D411yescharacteristic
194C2×C4×C61C23×C411yescharacteristic
195C22⋊D61—22no—
196C22⋊D61—22no—
197C2×Dic61—21no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—21no—
201C2×Dic61—22no—
202C2×Dic61—42no—
203C2×Dic61C4×D411yescharacteristic
204C2×Dic61C4×D411yescharacteristic
205C2×Dic61C22×D411yescharacteristic
206C4×D61—22no—
207C4×D61—41no—
208C4×D61C22×Q811yescharacteristic
209C4×D61C2×#4111yescharacteristic
210C4×D61C2×#4111yescharacteristic
211C4×D61C2×#4111yescharacteristic
212Dic122—21no—
213Dic122—42no—
214Dic122C4×Q812yes—
215#92420—22no—
216#92420—21no—
217#92420—21no—
218#92420—22no—
219#92420—42no—
220#92420C22⋊Q812yes—
221#92420C22⋊Q812yes—
222C8×C61—22no—
223#92418—22no—
224#21627—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61C2×C4×D411yescharacteristic
229C2×D61—21no—
230C2×D61—21no—
231C2×D61#52911yescharacteristic
232C2×D61#53311yescharacteristic
233#6923—44no—
234C4×C61—21no—
235C4×C61—41no—
236C4×C61—42no—
237C4×C61#44312yes—
238C4×C61C2×C4×Q811yescharacteristic
239C4×C61C4×#4111yescharacteristic
240Dic61—21no—
241Dic61—21no—
242Dic61—22no—
243Dic61—42no—
244Dic61—41no—
245Dic61C2×C22⋊Q811yescharacteristic
246Dic61#52511yescharacteristic
247#3076—611no—
248#2960—611no—
249C4×#131—611no—
250#1986—611no—
251#1950—611no—
252#1922—612no—
253#1866—612no—
254#1806—611no—
255#1772—612no—
256C8×C4⋊C4—611no—
257#1764—612no—
258C2×C61—42no—
259C2×C61C4×C22⋊Q811yescharacteristic
260C2×C61C2×#44311yescharacteristic
261C2×C61#295511yescharacteristic
262D61—21no—
263D61—21no—
264C2×#131—611no—
265C2×#131—611no—
266#533—611no—
267C4×#41—611no—
268#479—612no—
269#478—612no—
270#477—611no—
271#476#692311yescharacteristic, complemented
272#475—611no—
273#474—611no—
274#446—1221no—
275#446—1221no—
276C8×Q8—1221no—
277C8×Q8—1221no—
278C2×#108—612no—
279#427—612no—
280#424—612no—
281#423—611no—
282C2×C4⋊C8—611no—
283C2×#104—612no—
284#416—612no—
285#405—611no—
286#405—611no—
287#405—612no—
288#405—1221no—
289#405—1221no—
290C4×M16—611no—
291C4×M16—1221no—
292C2×C4×C8—611no—
293C2×C4×C8—612no—
294C2×C4×C8—1221no—
295C4×C4⋊C4—611no—
296#341—612no—
297#339—612no—
298#339—612no—
299#339—612no—
300#336—612no—
301C61#2162711yescharacteristic, complemented, Sylow
302C2×#41—611no—
303#131—614no—
304#131—1222no—
305#131—1222no—
306#131—1222no—
307C2×M16—611no—
308C2×M16—611no—
309C2×M16—612no—
310C2×M16—1221no—
311C2×M16—1221no—
312C22×C8—611no—
313C22×C8—611no—
314C22×C8—612no—
315C22×C8—1221no—
316C22×C8—1221no—
317C4⋊Q8—611no—
318C4⋊Q8C2×#692311yescharacteristic
319#125—612no—
320C22⋊Q8—1221no—
321C22⋊Q8—1221no—
322C4×Q8—611no—
323C4×Q8—611no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×D4—611no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4C2×#692311yescharacteristic
330#117—611no—
331#117—612no—
332#117—1221no—
333C2×C4⋊C4—1221no—
334C2×C42—611no—
335C2×C42—612no—
336C2×C42—1221no—
337#108—1224no—
338#107—612no—
339#107—612no—
340#106—612no—
341#106C2×#692312yes—
342C4⋊C8—612no—
343C4⋊C8—611no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8—1222no—
347C4⋊C8—1222no—
348C4⋊C8C2×#692311yescharacteristic
349#104—1224no—
350#103—612no—
351#103—612no—
352#102—612no—
353#102C2×#692312yes—
354#97—1221no—
355#97—1221no—
356#97—1221no—
357#97—1222no—
358C4×C8—611no—
359C4×C8—611no—
360C4×C8—611no—
361C4×C8—611no—
362C4×C8—1221no—
363C4×C8—1224no—
364C4×C8—1221no—
365C4×C8—1221no—
366C4×C8—1222no—
367#95—1221no—
368#41—612no—
369#41—612no—
370#41—1221no—
371#41—1221no—
372C2×Q8—21no—
373C2×Q8—611no—
374C2×Q8—1221no—
375C2×D4—1221no—
376C2×D4C22×#692311yescharacteristic
377C22×C4—21no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1222no—
384M16—612no—
385M16—612no—
386M16—1224no—
387M16—1221no—
388M16—1221no—
389M16—1224no—
390M16—1222no—
391M16—1222no—
392C2×C8—611no—
393C2×C8—611no—
394C2×C8—612no—
395C2×C8—612no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—612no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1224no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1221no—
406C2×C8—1224no—
407C2×C8—1222no—
408C2×C8C22×#692312yes—
409C4⋊C4—21no—
410C4⋊C4—612no—
411C4⋊C4—611no—
412C4⋊C4—1221no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1224no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4C22×#692312yes—
419C4⋊C4C22×#692311yescharacteristic
420C22⋊C4—21no—
421C22⋊C4—1221no—
422C22⋊C4—1222no—
423C22⋊C4—1221no—
424C42—611no—
425C42—611no—
426C42—611no—
427C42—1221no—
428C42—1222no—
429C42—1221no—
430C42—1222no—
431C42—1222no—
432C42—1221no—
433C42C22×#692311yescharacteristic
434C23—21no—
435C23—611no—
436Q8—42no—
437Q8—612no—
438Q8—1221no—
439Q8—2442no—
440D4—21no—
441D4—1222no—
442D4—1222no—
443D4Q8×#692312yes—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—611no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—1221no—
457C2×C4—1222no—
458C2×C4—1224no—
459C2×C4—1221no—
460C2×C4—1222no—
461C2×C4—2441no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4D4×#692311yescharacteristic
466C2×C4D4×#692311yescharacteristic
467C2×C4C23×#692311yescharacteristic
468C8—22no—
469C8—612no—
470C8—612no—
471C8—1221no—
472C8—1221no—
473C8—1222no—
474C8—1222no—
475C8—1222no—
476C8—2442no—
477C22—22no—
478C22—41no—
479C22—41no—
480C22—611no—
481C22—611no—
482C22—1221no—
483C22—1221no—
484C22C2×D4×#692311yescharacteristic
485C4—21no—
486C4—41no—
487C4—42no—
488C4—611no—
489C4—611no—
490C4—1222no—
491C4—1221no—
492C4—1221no—
493C4—2444no—
494C4—2442no—
495C4—2441no—
496C4#1684080512yes—
497C4C2×Q8×#692311yescharacteristic
498C4#41×#692311yescharacteristic
499C2—42no—
500C2—1221no—
501C2—1221no—
502C2C22⋊Q8×#692311yescharacteristic
503C2C2×#1684080511yescharacteristic
504C2#2146259111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#476C4
socleC22×C61C2×C4×D4
prime radicalC2×#11758224C23
cosocleC2×C4×D61C24
hypercenter#476#6923
hypocenterC61#21627
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #21462608C2semidirect
3. #21461994C2semidirect
4. #21461934C2semidirect
5. #21461886C2semidirect
8. #21461622C2semidirect
9. #21461612C2semidirect
40. #16840072C4semidirect
41. #16840064C4semidirect
42. #16840054C4semidirect
43. #16840048C4semidirect
44. #16840042C4semidirect
48. C61×#476C4semidirect
52. #14519864Q8semidirect
271. #476#6923semidirect
301. C61#21627semidirect

Maximal subgroups

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

subgroupindex
#214626082
C4×#145198642
#214619942
#214619342
#214618862
#214617882
#214616912
#214616222
#214616122
#214615942
#214556082
#2162761