Finite Group Databasebeta

← #46848220

Subgroups — #46848220

Order 34688.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774599C211yesmaximal, characteristic, complemented
2#31774538C211yesmaximal, characteristic, complemented
3#31774253C211yesmaximal, characteristic, complemented
4#31774230C211yesmaximal, characteristic, complemented
5#31774223C211yesmaximal, characteristic, complemented
6#31774196C211yesmaximal, characteristic, complemented
7#31774132C211yesmaximal, characteristic, complemented
8#31774114C211yesmaximal, characteristic, complemented
9#31774041C211yesmaximal, characteristic, complemented
10#31774030C211yesmaximal, characteristic, complemented
11#31773977C211yesmaximal, characteristic, complemented
12C8×C22⋊D271C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14C2×#22576082C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic, complemented
21#22576153—21no—
22#22576151C2211yescharacteristic, complemented
23#22576149C2211yescharacteristic
24#22576145—21no—
25#22576144—21no—
26#22576135—21no—
27#22576135C2211yescharacteristic
28#22576132—21no—
29#22576131C2211yescharacteristic, complemented
30#22576130—21no—
31D4×Dic271—21no—
32D4×Dic271C2211yescharacteristic, complemented
33#22576122C2211yescharacteristic
34C4×C22⋊D271C2211yescharacteristic
35#22576118C2211yescharacteristic, complemented
36C2×C4⋊Dic271C2211yescharacteristic
37C22⋊C4×D271—21no—
38#22576093C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40Dic271⋊C8C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic, complemented
53C22×C8×C271C2211yescharacteristic
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864655—22no—
61#16864655—41no—
62C2×D1084—22no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084—41no—
67C2×D1084C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271D411yescharacteristic
71C22×Dic271—21no—
72C22×Dic271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76#16864639—22no—
77#16864639—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—41no—
81C4⋊Dic271—21no—
82C4⋊Dic271—21no—
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271D411yescharacteristic
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96C4×Dic271—21no—
97C4×Dic271C2311yescharacteristic
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271—21no—
100C2×D4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102SD16×C271—41no—
103C2×C8×C271—41no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271—21no—
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116C22×D271C2×D411yescharacteristic
117D1084—21no—
118D1084—21no—
119D1084—21no—
120D1084—42no—
121D1084—42no—
122D1084—81no—
123D1084#4112yes—
124C22⋊D271—22no—
125C22⋊D271—81no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—81no—
129C2×Dic271—81no—
130C2×Dic271C2×D411yescharacteristic
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—42no—
136C4×D271—81no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139C23×C271—21no—
140C23×C271C2×D411yescharacteristic
141Q8×C271—21no—
142Q8×C271—21no—
143D4×C271—21no—
144D4×C271—21no—
145D4×C271—42no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271#13512yes—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—22no—
164C2×D271—21no—
165C2×D271—21no—
166C2×D271—41no—
167C2×D271—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—84no—
172C4×C271—21no—
173C4×C271—41no—
174C4×C271—41no—
175C4×C271—41no—
176C4×C271C2×#4111yescharacteristic
177C4×C271H(5,𝔽2)11yescharacteristic
178Dic271—21no—
179Dic271—21no—
180Dic271—21no—
181Dic271—41no—
182Dic271—41no—
183C2×C271—22no—
184C2×C271—81no—
185C2×C271#54611yescharacteristic
186C2×C271C2×#13511yescharacteristic
187C2×C271#57611yescharacteristic
188D271—41no—
189D271—42no—
190D271—81no—
191C271#329711yescharacteristic, complemented, Sylow
192#3297—2711nomaximal, Sylow
193#553—2711no—
194#546—2711no—
195#488—2711no—
196#484—2711no—
197#482—2711no—
198#478—2711no—
199#466—2711no—
200#465D27111yescharacteristic, complemented
201#463—2711no—
202#449—2711no—
203#447—2711no—
204C4×D8—2711no—
205C8×D4—2711no—
206#416—2711no—
207C2×#102—2711no—
208C2×#41—2711no—
209C22×D4—2711no—
210C2×SD16C2×D27111yescharacteristic, complemented
211C2×D8—2711no—
212C2×D8—2711no—
213C22×C8—2711no—
214C22×C8C2×D27111yescharacteristic
215#126—5421no—
216#125—2711no—
217#124—2711no—
218#123—5421no—
219#123—5421no—
220C22⋊Q8C2×D27111yescharacteristic
221#121—2711no—
222#121—2711no—
223#121—5421no—
224#121—5421no—
225#121C2×D27111yescharacteristic
226#120—5421no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—2711no—
230C4×D4—5421no—
231#117—2711no—
232C2×C4⋊C4—2711no—
233C2×C22⋊C4—5421no—
234#107—2711no—
235#107—2711no—
236#106C2×D27111yescharacteristic, complemented
237C4⋊C8—2711no—
238#103—2711no—
239#103—2711no—
240#103C2×D27111yescharacteristic, complemented
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102C2×D27111yescharacteristic, complemented
248C22⋊C8—2711no—
249C22⋊C8—2711no—
250C4×C8—2711no—
251C24—5421no—
252#41—5422no—
253#41—10841no—
254C2×Q8C22×D27111yescharacteristic
255C2×D4—21no—
256C2×D4—2711no—
257C2×D4—2711no—
258C2×D4—2711no—
259C2×D4—5422no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4—10841no—
265C2×D4C22×D27111yescharacteristic
266C22×C4—2711no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4C22×D27111yescharacteristic
272SD16—41no—
273D8—5422no—
274D8—10841no—
275C2×C8—41no—
276C2×C8—2711no—
277C2×C8—2711no—
278C2×C8—10841no—
279C2×C8C22×D27111yescharacteristic
280C2×C8C22×D27111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4C22×D27111yescharacteristic
289C4⋊C4C22×D27111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2711no—
293C22⋊C4—2711no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C42—2711no—
302C42—5421no—
303C23—21no—
304C23—2711no—
305C23—5421no—
306C23—5421no—
307C23—10842no—
308C23—10842no—
309C23—21681no—
310C23D4×D27111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—2712no—
317D4—5422no—
318D4—5421no—
319D4—5421no—
320D4—5421no—
321D4—10842no—
322D4—10842no—
323D4—21681no—
324D4—21681no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2711no—
331C2×C4—2711no—
332C2×C4—2711no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—10842no—
338C2×C4—21681no—
339C2×C4—21681no—
340C2×C4—21681no—
341C2×C4D4×D27111yescharacteristic
342C2×C4C23×D27111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5421no—
346C8—5421no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—5421no—
353C22—5422no—
354C22—5421no—
355C22—5421no—
356C22—10841no—
357C22—10841no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22—21684no—
362C22#2257618112yes—
363C22C2×D4×D27111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—5421no—
369C4—5421no—
370C4—5421no—
371C4—10841no—
372C4—10841no—
373C4#41×D27111yescharacteristic
374C4#2257620511yescharacteristic
375C2—22no—
376C2—81no—
377C2—10841no—
378C2—10842no—
379C2—21681no—
380C2#3177453611yescharacteristic
381C2C2×#2257618111yescharacteristic
382C2#3177469911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
hypercenter#465D271
hypocenterC271#3297
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. #31774599C2semidirect
2. #31774538C2semidirect
3. #31774253C2semidirect
4. #31774230C2semidirect
5. #31774223C2semidirect
6. #31774196C2semidirect
7. #31774132C2semidirect
8. #31774114C2semidirect
9. #31774041C2semidirect
10. #31774030C2semidirect
11. #31773977C2semidirect
12. C8×C22⋊D271C2semidirect
13. #31773892C2semidirect
14. C2×#22576082C2semidirect
15. #465×C271C2semidirect
20. C2×C8×D271C22semidirect
22. #22576151C22semidirect
29. #22576131C22semidirect
32. D4×Dic271C22semidirect
35. #22576118C22semidirect
38. #22576093C22semidirect
39. #22576093C22semidirect
40. Dic271⋊C8C22semidirect
41. #22576084C22semidirect
42. #22576084C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
52. C2×SD16×C271C22semidirect
56. #106×C271C22semidirect
57. #103×C271C22semidirect
58. #102×C271C22semidirect
191. C271#3297semidirect
200. #465D271semidirect
210. C2×SD16C2×D271semidirect
236. #106C2×D271semidirect
240. #103C2×D271semidirect
247. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745992
#317745382
#317742532
#317742302
#317742232
#317741962
#317741322
#317741142
#317740412
#317740302
#317739772
C8×C22⋊D2712
#317738922
C2×#225760822
#465×C2712
#3297271