Finite Group Databasebeta

← #46848183

Subgroups — #46848183

Order 34688.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774605C211yesmaximal, characteristic, complemented
2#31774531C211yesmaximal, characteristic
3#31774242C211yesmaximal, characteristic, complemented
4#31774217C211yesmaximal, characteristic, complemented
5#31774213C211yesmaximal, characteristic, complemented
6#31774186C211yesmaximal, characteristic, complemented
7#31774129C211yesmaximal, characteristic, complemented
8#31774095C211yesmaximal, characteristic, complemented
9#31774047C211yesmaximal, characteristic, complemented
10#31774046C211yesmaximal, characteristic, complemented
11#31773982C211yesmaximal, characteristic, complemented
12C8×C22⋊D271C211yesmaximal, characteristic, complemented
13C2×#22576082C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C2×D4×D271—21no—
18C2×D4×D271C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×#16864641C2211yescharacteristic, complemented
21C2×C8×D271C2211yescharacteristic, complemented
22#22576159C2211yescharacteristic, complemented
23#22576156C2211yescharacteristic
24#22576150—21no—
25#22576140—21no—
26#22576137—21no—
27#22576136C2211yescharacteristic, complemented
28#22576135C2211yescharacteristic
29#22576133—21no—
30#22576130—21no—
31#22576128—21no—
32Q8×Dic271C2211yescharacteristic, complemented
33#22576122C2211yescharacteristic
34C4×C22⋊D271C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36C2×C4⋊Dic271C2211yescharacteristic, complemented
37C4⋊C4×D271C2211yescharacteristic, complemented
38#22576091C2211yescharacteristic, complemented
39#22576091C2211yescharacteristic, complemented
40Dic271⋊C8C2211yescharacteristic, complemented
41#22576082C2211yescharacteristic
42#22576082C2211yescharacteristic
43#22576082C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic, complemented
45#22576081C2211yescharacteristic, complemented
46#22576081C2211yescharacteristic, complemented
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic, complemented
49C22⋊(C271⋊C8)C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic, complemented
53C22×C8×C271C2211yescharacteristic, complemented
54C22⋊Q8×C271C2211yescharacteristic, complemented
55#121×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C23×D271—21no—
61C2×Dic542—21no—
62C2×D1084—22no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—41no—
66C2×D1084C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271D411yescharacteristic
72D4×D271—22no—
73D4×D271—41no—
74D4×D271—44no—
75C22×Dic271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271C2311yescharacteristic
79#16864641—22no—
80#16864641—41no—
81C2×C271⋊C8C2311yescharacteristic
82C2×C271⋊C8C2311yescharacteristic
83C8×D271—41no—
84C4⋊Dic271—21no—
85C4⋊Dic271—21no—
86C4⋊Dic271C2311yescharacteristic
87C4⋊Dic271C2311yescharacteristic
88Dic271⋊C4—21no—
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91Dic271⋊C4D411yescharacteristic
92C22⋊Dic271D411yescharacteristic
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97C4×Dic271—21no—
98C4×Dic271C2311yescharacteristic
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103SD16×C271—41no—
104C2×C8×C271—41no—
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271—21no—
108C4⋊C4×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22⋊C4×C271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—42no—
115C22×D271—42no—
116C22×D271—42no—
117C22×D271—42no—
118C22×D271—81no—
119C22×D271—81no—
120C22×D271C2×D411yescharacteristic
121Dic542—81no—
122D1084—21no—
123D1084—21no—
124D1084—21no—
125D1084—42no—
126D1084—42no—
127D1084#4112yes—
128C22⋊D271—42no—
129C22⋊D271—42no—
130C22⋊D271—81no—
131C22⋊D271SD1614yescomplemented
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—81no—
135C2×Dic271C2×D411yescharacteristic
136C2×Dic271C2×D411yescharacteristic
137C2×Dic271C2×D411yescharacteristic
138C4×D271—21no—
139C4×D271—21no—
140C4×D271—42no—
141C4×D271—81no—
142C271⋊C8—21no—
143C271⋊C8—21no—
144C23×C271—21no—
145C23×C271C2×D411yescharacteristic
146Q8×C271—21no—
147Q8×C271—21no—
148D4×C271—21no—
149D4×C271—21no—
150D4×C271—42no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—21no—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271C2×D411yescharacteristic
157C2×C4×C271C2411yescharacteristic
158C8×C271—21no—
159C8×C271—21no—
160C22×C271—21no—
161C22×C271—21no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271—41no—
165C22×C271C2×SD1612yes—
166C22×C271C22×D411yescharacteristic
167C2×D271—21no—
168C2×D271—21no—
169C2×D271—22no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—41no—
174C2×D271—41no—
175C2×D271—84no—
176C2×D271—84no—
177C2×D271C2×SD1612yes—
178C4×C271—21no—
179C4×C271—41no—
180C4×C271—41no—
181C4×C271—41no—
182C4×C271C2×#4111yescharacteristic
183C4×C271H(5,𝔽2)11yescharacteristic
184Dic271—21no—
185Dic271—21no—
186Dic271—41no—
187Dic271—41no—
188Dic271C2×SD1612yes—
189C2×C271—22no—
190C2×C271—81no—
191C2×C271#54611yescharacteristic
192C2×C271C22×SD1611yescharacteristic
193C2×C271#57611yescharacteristic
194D271—22no—
195D271—42no—
196D271—81no—
197C271#329511yescharacteristic, complemented, Sylow
198#3295—2711nomaximal, Sylow
199#554—2711no—
200D42—2711no—
201#486—2711no—
202#481—2711no—
203#481—2711no—
204#476—2711no—
205#465—2711no—
206#465D27111yescharacteristic, complemented
207#460—2711no—
208#450—2711no—
209#450—2711no—
210C4×SD16—2711no—
211C8×D4—2711no—
212C2×#102—2711no—
213C2×#102—2711no—
214C22×D4—2711no—
215C22×D4—2711no—
216C22×D4—5421no—
217C2×SD16—2711no—
218C2×SD16—2711no—
219C2×SD16C2×D27111yescharacteristic, complemented
220C22×C8—2711no—
221C22×C8C2×D27111yescharacteristic, complemented
222C4⋊Q8—2711no—
223#127—2711no—
224#125—5421no—
225C22⋊Q8—2711no—
226C22⋊Q8—5421no—
227C22⋊Q8—5421no—
228C22⋊Q8C2×D27111yescharacteristic, complemented
229#121—2711no—
230#121—5421no—
231#121C2×D27111yescharacteristic
232#120—5421no—
233#120—5421no—
234C4×Q8—2711no—
235C4×D4—2711no—
236C4×D4—2711no—
237C4×D4—5421no—
238C2×C4⋊C4—2711no—
239C2×C4⋊C4—2711no—
240#106—2711no—
241#106—2711no—
242#106C2×D27111yescharacteristic, complemented
243C4⋊C8—2711no—
244#103C2×D27111yescharacteristic, complemented
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102—2711no—
250#102—2711no—
251#102—2711no—
252#102—2711no—
253#102C2×D27111yescharacteristic, complemented
254C22⋊C8—2711no—
255C22⋊C8—2711no—
256C4×C8—2711no—
257C24—5421no—
258C24—5421no—
259C2×Q8—5421no—
260C2×Q8C22×D27111yescharacteristic
261C2×D4—21no—
262C2×D4—2711no—
263C2×D4—2711no—
264C2×D4—2711no—
265C2×D4—5422no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4—5421no—
269C2×D4—5421no—
270C2×D4—5422no—
271C2×D4—5421no—
272C2×D4—10841no—
273C2×D4—10841no—
274C2×D4—10844no—
275C2×D4C22×D27111yescharacteristic
276C22×C4—2711no—
277C22×C4—5421no—
278C22×C4—5421no—
279C22×C4—5421no—
280C22×C4C22×D27111yescharacteristic
281SD16—41no—
282SD16—5422no—
283SD16—10841no—
284C2×C8—41no—
285C2×C8—2711no—
286C2×C8—2711no—
287C2×C8—10841no—
288C2×C8C22×D27111yescharacteristic
289C2×C8C22×D27111yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2711no—
292C4⋊C4—2711no—
293C4⋊C4—2711no—
294C4⋊C4—5421no—
295C4⋊C4—5421no—
296C4⋊C4—5421no—
297C4⋊C4—5421no—
298C4⋊C4—5421no—
299C4⋊C4C22×D27111yescharacteristic
300C4⋊C4C22×D27111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2711no—
304C22⋊C4—2711no—
305C22⋊C4—5421no—
306C22⋊C4—5421no—
307C22⋊C4—5421no—
308C42—2711no—
309C42—5421no—
310C23—21no—
311C23—2711no—
312C23—5421no—
313C23—5421no—
314C23—10842no—
315C23—10842no—
316C23—10842no—
317C23—10842no—
318C23—21681no—
319C23—21681no—
320C23D4×D27111yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21681no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—2714no—
328D4—2712no—
329D4—5421no—
330D4—5421no—
331D4—5421no—
332D4—10842no—
333D4—10842no—
334D4—10842no—
335D4—10842no—
336D4—21681no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2711no—
343C2×C4—2711no—
344C2×C4—2711no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—10842no—
350C2×C4—21681no—
351C2×C4—21681no—
352C2×C4D4×D27111yescharacteristic
353C2×C4C23×D27111yescharacteristic
354C8—21no—
355C8—21no—
356C8—5421no—
357C8—5421no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2712no—
364C22—5421no—
365C22—5421no—
366C22—5422no—
367C22—10841no—
368C22—10841no—
369C22—10841no—
370C22—10841no—
371C22—10841no—
372C22—21684no—
373C22—21684no—
374C22SD16×D27112yes—
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2712no—
381C4—5421no—
382C4—5421no—
383C4—10841no—
384C4—10841no—
385C4#41×D27111yescharacteristic
386C4#2257620511yescharacteristic
387C2—22no—
388C2—81no—
389C2—5422no—
390C2—10842no—
391C2—21681no—
392C2#3177453611yescharacteristic
393C2C2×SD16×D27111yescharacteristic
394C2#3177469911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
prime radical#31774531C2
hypercenter#465D271
hypocenterC271#3295
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31774605C2semidirect
3. #31774242C2semidirect
4. #31774217C2semidirect
5. #31774213C2semidirect
6. #31774186C2semidirect
7. #31774129C2semidirect
8. #31774095C2semidirect
9. #31774047C2semidirect
10. #31774046C2semidirect
11. #31773982C2semidirect
12. C8×C22⋊D271C2semidirect
13. C2×#22576082C2semidirect
14. #102×D271C2semidirect
15. #465×C271C2semidirect
20. C2×#16864641C22semidirect
21. C2×C8×D271C22semidirect
22. #22576159C22semidirect
27. #22576136C22semidirect
32. Q8×Dic271C22semidirect
36. C2×C4⋊Dic271C22semidirect
37. C4⋊C4×D271C22semidirect
38. #22576091C22semidirect
39. #22576091C22semidirect
40. Dic271⋊C8C22semidirect
43. #22576082C22semidirect
44. #22576082C22semidirect
45. #22576081C22semidirect
46. #22576081C22semidirect
48. #22576080C22semidirect
49. C22⋊(C271⋊C8)C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
52. C2×SD16×C271C22semidirect
53. C22×C8×C271C22semidirect
54. C22⋊Q8×C271C22semidirect
56. #106×C271C22semidirect
57. #103×C271C22semidirect
58. #102×C271C22semidirect
131. C22⋊D271SD16semidirect
197. C271#3295semidirect
206. #465D271semidirect
219. C2×SD16C2×D271semidirect
221. C22×C8C2×D271semidirect
228. C22⋊Q8C2×D271semidirect
242. #106C2×D271semidirect
244. #103C2×D271semidirect
253. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746052
#317745312
#317742422
#317742172
#317742132
#317741862
#317741292
#317740952
#317740472
#317740462
#317739822
C8×C22⋊D2712
C2×#225760822
#102×D2712
#465×C2712
#3295271