Finite Group Databasebeta

← #46846159

Subgroups — #46846159

Order 34688.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774453C211yesmaximal, characteristic, complemented
2C4×D4×D271C211yesmaximal, characteristic
3#31774198C211yesmaximal, characteristic, complemented
4#31774197C211yesmaximal, characteristic, complemented
5#31774194C211yesmaximal, characteristic, complemented
6#31774186C211yesmaximal, characteristic, complemented
7#31774185C211yesmaximal, characteristic, complemented
8#31774176C211yesmaximal, characteristic, complemented
9#31774175C211yesmaximal, characteristic, complemented
10#31773938C211yesmaximal, characteristic, complemented
11#31773936C211yesmaximal, characteristic, complemented
12#31773919C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic
14#31773900C211yesmaximal, characteristic
15#478×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C22×C4×D271—21no—
18C2×#16864637C2211yescharacteristic
19C2×#16864637C2211yescharacteristic
20#22576159C2211yescharacteristic, complemented
21#22576159C2211yescharacteristic, complemented
22#22576152C2211yescharacteristic, complemented
23#22576138—21no—
24#22576138—21no—
25#22576125—21no—
26#22576125—21no—
27C4×D1084C2211yescharacteristic
28D4×Dic271C2211yescharacteristic
29#22576122C2211yescharacteristic
30C4×C22⋊D271—21no—
31(C22⋊D271)⋊C4—21no—
32#22576118C2211yescharacteristic
33#22576118C2211yescharacteristic
34C4⋊C4×D271C2211yescharacteristic
35C22⋊C4×D271—21no—
36C42×D271C2211yescharacteristic
37#22576094C2211yescharacteristic, complemented
38#22576093C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41#22576091C2211yescharacteristic, complemented
42#22576091C2211yescharacteristic, complemented
43C4⋊(C271⋊C8)C2211yescharacteristic, complemented
44Dic271⋊C8C2211yescharacteristic, complemented
45Dic271⋊C8C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic
50#22576080C2211yescharacteristic
51#22576080C2211yescharacteristic
52#125×C271C2211yescharacteristic, complemented
53C4×D4×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56C4⋊C8×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C23×D271—21no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64D4×D271—22no—
65D4×D271Q814yescomplemented
66C22×Dic271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—44no—
71C2×C4×D271D411yescharacteristic
72C2×C4×D271D411yescharacteristic
73C2×C4×D271C2311yescharacteristic
74#16864637—22no—
75#16864637—22no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91C4×Dic271—21no—
92C4×Dic271—21no—
93C4×Dic271D411yescharacteristic
94C4×Dic271D411yescharacteristic
95C4×Dic271C2311yescharacteristic
96C2×D4×C271C2311yescharacteristic
97C22×C4×C271—21no—
98C2×C8×C271C2311yescharacteristic
99C2×C8×C271C2311yescharacteristic
100C4⋊C4×C271—21no—
101C4⋊C4×C271—21no—
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C42×C271C2311yescharacteristic
107C22×D271—21no—
108C22×D271—24no—
109C22×D271—42no—
110C22×D271—42no—
111C22×D271C2×D411yescharacteristic
112Dic542—42no—
113Dic542—42no—
114D1084—21no—
115D1084C2×Q812yes—
116C22⋊D271—24no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—42no—
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C4×D271—22no—
125C4×D271—41no—
126C4×D271—42no—
127C4×D271—42no—
128C4×D271—42no—
129C4×D271C2×Q812yes—
130C4×D271#4112yes—
131C271⋊C8—21no—
132C271⋊C8—21no—
133C23×C271—21no—
134D4×C271—21no—
135D4×C271C2×Q812yes—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—42no—
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2411yescharacteristic
143C8×C271—21no—
144C8×C271—21no—
145C22×C271—22no—
146C22×C271—41no—
147C22×C271—41no—
148C22×C271C22×D411yescharacteristic
149C2×D271—21no—
150C2×D271—21no—
151C2×D271—22no—
152C2×D271—48no—
153C2×D271—41no—
154C2×D271—41no—
155C2×D271C22⋊Q812yes—
156C4×C271—21no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271C22×Q811yescharacteristic
162C4×C271C2×#4111yescharacteristic
163Dic271—21no—
164Dic271—41no—
165Dic271—41no—
166Dic271—41no—
167Dic271C22⋊Q812yes—
168Dic271AGL(1,ℤ/8)12yes—
169C2×C271—42no—
170C2×C271C2×C22⋊Q811yescharacteristic
171C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C271#57511yescharacteristic
173D271—22no—
174D271—42no—
175C271#307711yescharacteristic, complemented, Sylow
176#3077—2711nomaximal, Sylow
177#533—2711no—
178C2×C4×D4—2711no—
179#478—2711no—
180#478—2711no—
181#478—2711no—
182#478D27111yescharacteristic, complemented
183#476—2711no—
184#476—2711no—
185#474—2711no—
186#474—2711no—
187#428—2711no—
188#428—2711no—
189#423—2711no—
190#418—2711no—
191#418—2711no—
192C22×D4—2711no—
193C23×C4—5421no—
194C2×M16—2711no—
195C2×M16—2711no—
196C4⋊Q8—2711no—
197C4⋊Q8—2711no—
198#125—2711no—
199#125C2×D27111yescharacteristic, complemented
200C22⋊Q8—5421no—
201C22⋊Q8—5421no—
202C4×Q8—5421no—
203C4×Q8—5421no—
204C4×D4—2711no—
205C4×D4—2711no—
206C4×D4—2711no—
207C4×D4—5421no—
208C4×D4—5421no—
209C4×D4C2×D27111yescharacteristic
210#117—2711no—
211#117—2711no—
212C2×C4⋊C4—2711no—
213C2×C22⋊C4—5421no—
214C2×C42—2711no—
215#107—2711no—
216#107—2711no—
217#107—2711no—
218#107C2×D27111yescharacteristic, complemented
219#106—2711no—
220#106—2711no—
221#106—2711no—
222#106C2×D27111yescharacteristic, complemented
223C4⋊C8—2711no—
224C4⋊C8—2711no—
225C4⋊C8—2711no—
226C4⋊C8C2×D27111yescharacteristic, complemented
227#102—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231#102—2711no—
232#102—2711no—
233#102C2×D27111yescharacteristic
234#102C2×D27111yescharacteristic
235C24—5421no—
236C2×Q8—5421no—
237C2×Q8—5421no—
238C2×D4—2714no—
239C2×D4—2711no—
240C2×D4—5422no—
241C2×D4—5421no—
242C2×D4C22×D27111yescharacteristic
243C22×C4—21no—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—5421no—
248C22×C4—5421no—
249C22×C4—5421no—
250C22×C4—5421no—
251C22×C4—10844no—
252M16—5422no—
253M16—5422no—
254C2×C8—2711no—
255C2×C8—2711no—
256C2×C8C22×D27111yescharacteristic
257C2×C8C22×D27111yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4C22×D27111yescharacteristic
269C4⋊C4C22×D27111yescharacteristic
270C4⋊C4C22×D27111yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4—5421no—
277C42—2711no—
278C42—2711no—
279C42—2711no—
280C42—5421no—
281C42—5421no—
282C42C22×D27111yescharacteristic
283C23—21no—
284C23—2711no—
285C23—5424no—
286C23—5421no—
287C23—10842no—
288C23—10842no—
289Q8—10842no—
290Q8—10842no—
291D4—21no—
292D4—2712no—
293D4—5424no—
294D4—5421no—
295D4Q8×D27112yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—2711no—
301C2×C4—2711no—
302C2×C4—2712no—
303C2×C4—2712no—
304C2×C4—2711no—
305C2×C4—5422no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—10841no—
310C2×C4—10842no—
311C2×C4—10842no—
312C2×C4—10842no—
313C2×C4—10842no—
314C2×C4D4×D27111yescharacteristic
315C2×C4D4×D27111yescharacteristic
316C2×C4C23×D27111yescharacteristic
317C8—21no—
318C8—21no—
319C8—5421no—
320C8—5421no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—2712no—
325C22—5421no—
326C22—5421no—
327C22—5422no—
328C22—10848no—
329C22—10841no—
330C22—10841no—
331C22C2×D4×D27111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2712no—
338C4—2712no—
339C4—5421no—
340C4—10841no—
341C4—10841no—
342C4—10841no—
343C4C2×Q8×D27111yescharacteristic
344C4#41×D27111yescharacteristic
345C2—42no—
346C2—5422no—
347C2—10842no—
348C2C22⋊Q8×D27111yescharacteristic
349C2AGL(1,ℤ/8)×D27111yescharacteristic
350C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#478×C271C2
socleC22×C271C22×D4
prime radicalC2×D4×D271C22
hypercenter#478D271
hypocenterC271#3077
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #31774453C2semidirect
3. #31774198C2semidirect
4. #31774197C2semidirect
5. #31774194C2semidirect
6. #31774186C2semidirect
7. #31774185C2semidirect
8. #31774176C2semidirect
9. #31774175C2semidirect
10. #31773938C2semidirect
11. #31773936C2semidirect
12. #31773919C2semidirect
15. #478×C271C2semidirect
20. #22576159C22semidirect
21. #22576159C22semidirect
22. #22576152C22semidirect
37. #22576094C22semidirect
38. #22576093C22semidirect
39. #22576093C22semidirect
40. #22576092C22semidirect
41. #22576091C22semidirect
42. #22576091C22semidirect
43. C4⋊(C271⋊C8)C22semidirect
44. Dic271⋊C8C22semidirect
45. Dic271⋊C8C22semidirect
52. #125×C271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
56. C4⋊C8×C271C22semidirect
65. D4×D271Q8semidirect
175. C271#3077semidirect
182. #478D271semidirect
199. #125C2×D271semidirect
218. #107C2×D271semidirect
222. #106C2×D271semidirect
226. C4⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744532
C4×D4×D2712
#317741982
#317741972
#317741942
#317741862
#317741852
#317741762
#317741752
#317739382
#317739362
#317739192
#317739002
#478×C2712
#3077271