Finite Group Databasebeta

← #30617336

Subgroups — #30617336

Order 15616.

738 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×#14519865C211yesmaximal, characteristic
3#21461994C211yesmaximal, characteristic, complemented
4#21461934C211yesmaximal, characteristic, complemented
5#21461884C212yesmaximal, complemented
6#21461789C212yesmaximal
7#21461692C211yesmaximal, characteristic
8#21461626C212yesmaximal, complemented
9#21461616C211yesmaximal, characteristic, complemented
10#21461600C212yesmaximal
11#21455610C211yesmaximal, characteristic
12C2×#14519865C2211yescharacteristic
13C2×#14519865C2211yescharacteristic
14#16840813—21no—
15#16840812—21no—
16#16840811—21no—
17Q8×#92420—21no—
18C2×#14519847C2212yes—
19#16840769C2212yes—
20#16840766C2211yescharacteristic
21C2×#14519842C2212yes—
22#16840727—21no—
23#16840727—21no—
24#16840727C2212yes—
25#16840727C2211yescharacteristic
26#16840727C2211yescharacteristic
27C4×#11758213—21no—
28C4×#11758213C2211yescharacteristic
29C2×#14519831C2212yes—
30C4×D61⋊C8—21no—
31C4×D61⋊C8C2211yescharacteristic
32C4×C4⋊#6923C2211yescharacteristic
33#16840638C2212yes—
34#16840632C2212yes—
35#16840632C2212yes—
36#16840632C2212yes—
37(C2×#6923)⋊C8C2212yes—
38#16840317C2211yescharacteristic
39C4×#11758226C2211yescharacteristic
40#16840071C412yescomplemented
41#16840065C412yescomplemented
42#16840049C411yescharacteristic, complemented
43#16840047C411yescharacteristic, complemented
44#16840043C411yescharacteristic, complemented
45#16839799C2212yes—
46C4⋊C8×D61C2211yescharacteristic
47#16839757C2212yes—
48C61×#477C411yescharacteristic, complemented
49#14519865—22no—
50#14519865—22no—
51#14519865—22no—
52#14519865Q814yescomplemented
53C2×#11758213—21no—
54C2×#11758213—21no—
55C2×#11758213C2312yes—
56C2×#11758213C2311yescharacteristic
57C2×#11758213C2311yescharacteristic
58C2×D61⋊C8—21no—
59C2×D61⋊C8—21no—
60C2×D61⋊C8C2311yescharacteristic
61C2×D61⋊C8C2311yescharacteristic
62C2×C4⋊#6923—21no—
63C2×C4×#6923C2312yes—
64#14519847—24no—
65#14519844—22no—
66C4⋊#92420—22no—
67C4⋊#92420—21no—
68C4⋊#92420—21no—
69#14519842—24no—
70#14519832—21no—
71#14519832—21no—
72#14519832—22no—
73#14519832—21no—
74#14519831—24no—
75C4×#92420—21no—
76C4×#92420—21no—
77C4×#92420—22no—
78C4×#92420—21no—
79C4×#92420D411yescharacteristic
80C4×#92420D411yescharacteristic
81C4×#92420D411yescharacteristic
82C4×#92420D411yescharacteristic
83(C2×#6923)⋊C4—21no—
84C2×#11758226C2311yescharacteristic
85C2×C8×D61C2312yes—
86#14519777C2×C411yescharacteristic
87#14519768C2×C412yes—
88#14519757—21no—
89#14519753—21no—
90Q8×Dic61—21no—
91Q8×Dic61C2×C411yescharacteristic
92#14519742—21no—
93C4×D244—21no—
94C4×D244C2×C411yescharacteristic
95#14519739—21no—
96#14519739C2×C411yescharacteristic
97#14519735—21no—
98#14519735C2311yescharacteristic
99#14519735C2312yes—
100C42×D61—21no—
101C42×D61C2311yescharacteristic
102#14519711C2×C412yes—
103#14519708C2×C412yes—
104#14519708C2×C412yes—
105C4⋊#92418C2×C411yescharacteristic
106#14519706C2×C412yes—
107#14519701C2×C412yes—
108#14519699C2×C412yes—
109#14519697C2×C412yes—
110C4⋊Q8×C61C2×C411yescharacteristic
111C4×Q8×C61C2×C411yescharacteristic
112#107×C61C2×C412yes—
113C4⋊C8×C61C2×C411yescharacteristic
114#103×C61C2×C412yes—
115C22×#6923—22no—
116C2×Q8×C61—21no—
117C2×Q8×C61C22×C411yescharacteristic
118#11758226—21no—
119#11758226—21no—
120#11758226C2×Q812yes—
121#11758226#4112yes—
122C2×D244—21no—
123C2×D244C22×C411yescharacteristic
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61#4111yescharacteristic
128C2×C4×D61#4111yescharacteristic
129C2×C4×D61C2411yescharacteristic
130C2×Dic122—21no—
131C2×#92420—21no—
132C2×#92420—21no—
133C2×#92420—22no—
134C2×#92420C2×D411yescharacteristic
135C2×#92420C2×D411yescharacteristic
136C2×#92420C2×D411yescharacteristic
137C2×#92420C2×D411yescharacteristic
138C4⋊#6923—42no—
139C4×#6923—22no—
140C4×#6923—22no—
141#11758213—24no—
142#11758213—21no—
143#11758213—21no—
144#11758213—24no—
145#11758213—22no—
146#11758213—22no—
147#11758213C2×Q812yes—
148#11758213#4112yes—
149D61⋊C8—24no—
150D61⋊C8—21no—
151D61⋊C8—21no—
152D61⋊C8—24no—
153D61⋊C8C2×Q812yes—
154D61⋊C8#4112yes—
155C2×C8×C61C22×C412yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C22×C412yes—
159C4⋊C4×C61C22×C411yescharacteristic
160C42×C61—21no—
161C42×C61C22×C411yescharacteristic
162#11758198—21no—
163#11758198—22no—
164#11758198—21no—
165#11758198—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61C22×C412yes—
170C4⋊Dic61C22×C411yescharacteristic
171#11758196—22no—
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—
181C22×D61—21no—
182C22×D61C2×#4111yescharacteristic
183C2×#6923—42no—
184C2×#6923—42no—
185C2×#6923—42no—
186Q8×C61—21no—
187Q8×C61—42no—
188Q8×C61C4×Q812yes—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61C4×D411yescharacteristic
193C2×C4×C61C4×D411yescharacteristic
194C2×C4×C61C23×C411yescharacteristic
195C2×Dic61—21no—
196C2×Dic61—22no—
197C2×Dic61C4×D411yescharacteristic
198C2×Dic61C4×D411yescharacteristic
199C2×Dic61C22×D411yescharacteristic
200D244—22no—
201D244—22no—
202D244—21no—
203D244C4×Q812yes—
204C4×D61—22no—
205C4×D61—22no—
206C4×D61—22no—
207C4×D61—41no—
208C4×D61—42no—
209C4×D61—42no—
210C4×D61C22×Q811yescharacteristic
211C4×D61C2×#4111yescharacteristic
212C4×D61C2×#4111yescharacteristic
213C4×D61C2×#4111yescharacteristic
214Dic122—42no—
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#21628—611nomaximal, Sylow
225C22×C61C2×C4×D411yescharacteristic
226C2×D61—21no—
227C2×D61—21no—
228C2×D61—22no—
229C2×D61—41no—
230C2×D61—41no—
231C2×D61#52911yescharacteristic
232C2×D61#53311yescharacteristic
233#6923—82no—
234C4×C61—21no—
235C4×C61—22no—
236C4×C61—41no—
237C4×C61—42no—
238C4×C61#44412yes—
239C4×C61C2×C4×Q811yescharacteristic
240C4×C61C4×#4111yescharacteristic
241Dic61—21no—
242Dic61—21no—
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#1773—612no—
256#1770—611no—
257#1765—612no—
258C2×C61C4×C22⋊Q811yescharacteristic
259C2×C61C2×#44411yescharacteristic
260C2×C61#295511yescharacteristic
261D61—21no—
262D61—21no—
263D61—42no—
264C2×#131—611no—
265C2×#131—611no—
266#533—611no—
267C4×#41—611no—
268#479—612no—
269#478—612no—
270#477#692311yescharacteristic, complemented
271#476—611no—
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×#97—612no—
293C2×C4×C8—611no—
294C2×C4×C8—1221no—
295C4×C4⋊C4—611no—
296#341—612no—
297#339—612no—
298#339—612no—
299#339—612no—
300#336—612no—
301C61#2162811yescharacteristic, 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—1221no—
324C4×Q8—1221no—
325C4×Q8C2×#692311yescharacteristic
326C4×D4—611no—
327C4×D4—611no—
328C4×D4—1221no—
329C4×D4—1221no—
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#107C2×#692312yes—
340#106—612no—
341#106—612no—
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#103C2×#692312yes—
352#102—612no—
353#102—612no—
354#97—1221no—
355#97—1224no—
356#97—1221no—
357#97—1221no—
358#97—1222no—
359C4×C8—611no—
360C4×C8—611no—
361C4×C8—611no—
362C4×C8—611no—
363C4×C8—1221no—
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—1221no—
374C2×Q8C22×#692311yescharacteristic
375C2×D4—611no—
376C2×D4—1221no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—1221no—
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—21no—
411C4⋊C4—612no—
412C4⋊C4—611no—
413C4⋊C4—1222no—
414C4⋊C4—1221no—
415C4⋊C4—1222no—
416C4⋊C4—1221no—
417C4⋊C4—2442no—
418C4⋊C4C22×#692312yes—
419C4⋊C4C22×#692311yescharacteristic
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—1222no—
423C22⋊C4—1221no—
424C42—21no—
425C42—611no—
426C42—611no—
427C42—611no—
428C42—1221no—
429C42—1222no—
430C42—1221no—
431C42—1222no—
432C42—1222no—
433C42C22×#692311yescharacteristic
434C23—611no—
435C23—1221no—
436Q8—21no—
437Q8—42no—
438Q8—2442no—
439Q8Q8×#692312yes—
440D4—612no—
441D4—1222no—
442D4—1222no—
443D4—1221no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
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—1222no—
457C2×C4—1221no—
458C2×C4—1222no—
459C2×C4—2441no—
460C2×C4—2442no—
461C2×C4—2442no—
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—611no—
478C22—611no—
479C22—1221no—
480C22—1221no—
481C22—1222no—
482C22—2441no—
483C22—2441no—
484C22C2×D4×#692311yescharacteristic
485C4—22no—
486C4—21no—
487C4—41no—
488C4—42no—
489C4—611no—
490C4—611no—
491C4—1221no—
492C4—1221no—
493C4—4882no—
494C4—2441no—
495C4—2442no—
496C4#1684080812yes—
497C4C2×Q8×#692311yescharacteristic
498C4#41×#692311yescharacteristic
499C2—1221no—
500C2—1221no—
501C2—2442no—
502C2C22⋊Q8×#692311yescharacteristic
503C2C2×#1684080811yescharacteristic
504C2#2146259111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#477C4
socleC22×C61C2×C4×D4
prime radicalC2×#11758226C23
cosocleC2×C4×D61C24
hypercenter#477#6923
hypocenterC61#21628
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #21462608C2semidirect
3. #21461994C2semidirect
4. #21461934C2semidirect
5. #21461884C2semidirect
8. #21461626C2semidirect
9. #21461616C2semidirect
40. #16840071C4semidirect
41. #16840065C4semidirect
42. #16840049C4semidirect
43. #16840047C4semidirect
44. #16840043C4semidirect
48. C61×#477C4semidirect
52. #14519865Q8semidirect
270. #477#6923semidirect
301. C61#21628semidirect

Maximal subgroups

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

subgroupindex
#214626082
C4×#145198652
#214619942
#214619342
#214618842
#214617892
#214616922
#214616262
#214616162
#214616002
#214556102
#2162861