Finite Group Databasebeta

← #46848211

Subgroups — #46848211

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#31774600C211yesmaximal, characteristic, complemented
2#31774542C211yesmaximal, characteristic, complemented
3#31774254C211yesmaximal, characteristic, complemented
4#31774234C211yesmaximal, characteristic, complemented
5#31774223C211yesmaximal, characteristic, complemented
6#31774134C211yesmaximal, characteristic, complemented
7#31774127C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774043C211yesmaximal, characteristic, complemented
10#31774032C211yesmaximal, characteristic, complemented
11C4×#16864639C211yesmaximal, characteristic, complemented
12#31773962C211yesmaximal, characteristic, complemented
13#31773890C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#478×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864643C2211yescharacteristic, complemented
19C2×D2168C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21C2×C8×D271C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic, complemented
23#22576155—21no—
24#22576148C2211yescharacteristic
25#22576145—21no—
26#22576144—21no—
27#22576140C2211yescharacteristic
28#22576135—21no—
29#22576135C2211yescharacteristic
30#22576135C2211yescharacteristic
31#22576135C2211yescharacteristic, complemented
32#22576134—21no—
33#22576130—21no—
34C4×D1084C2211yescharacteristic
35C4×D1084C2211yescharacteristic, complemented
36#22576122—21no—
37#22576118C2211yescharacteristic, complemented
38C4⋊C4×D271C2211yescharacteristic
39C2×(C2×D271)⋊C4—21no—
40#22576093C2211yescharacteristic, complemented
41#22576085C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic, complemented
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C4×C271⋊C8C2211yescharacteristic, complemented
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—
60#16864653—22no—
61#16864653—41no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084D411yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71D4×D271—22no—
72D4×D271—41no—
73C22×Dic271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77C2×C4×D271C2311yescharacteristic
78#16864643—41no—
79D2168—41no—
80#16864639—22no—
81C2×C271⋊C8C2311yescharacteristic
82C2×C271⋊C8C2311yescharacteristic
83C8×D271—41no—
84C8×D271—41no—
85C4⋊Dic271—21no—
86C4⋊Dic271D411yescharacteristic
87C4⋊Dic271C2311yescharacteristic
88Dic271⋊C4—21no—
89(C2×D271)⋊C4—21no—
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)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4D411yescharacteristic
98(C2×D271)⋊C4D411yescharacteristic
99C4×Dic271—21no—
100C2×D4×C271C2311yescharacteristic
101C22×C4×C271—21no—
102C2×C8×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C4⋊C4×C271—21no—
105C4⋊C4×C271—21no—
106C4⋊C4×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116C22×D271C2×D411yescharacteristic
117C22×D271C2×D411yescharacteristic
118Dic542—21no—
119Dic542—21no—
120D1084—22no—
121D1084—21no—
122D1084—21no—
123D1084—21no—
124D1084—21no—
125D1084—42no—
126D1084—81no—
127D1084—81no—
128C22⋊D271—42no—
129C22⋊D271—42no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—42no—
133C2×Dic271C2×D411yescharacteristic
134C2×Dic271C2×D411yescharacteristic
135C4×D271—21no—
136C4×D271—21no—
137C4×D271—21no—
138C4×D271—21no—
139C4×D271—81no—
140C4×D271—81no—
141C271⋊C8—21no—
142C271⋊C8—21no—
143C23×C271—21no—
144D4×C271—21no—
145D4×C271#4112yes—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—81no—
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—22no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—84no—
171C2×D271#13512yes—
172C4×C271—21no—
173C4×C271—21no—
174C4×C271—41no—
175C4×C271—41no—
176C4×C271—41no—
177C4×C271C2×#4111yescharacteristic
178C4×C271H(5,𝔽2)11yescharacteristic
179Dic271—21no—
180Dic271—21no—
181Dic271—41no—
182Dic271—41no—
183C2×C271—42no—
184C2×C271#54611yescharacteristic
185C2×C271C2×#13511yescharacteristic
186C2×C271#57611yescharacteristic
187D271—22no—
188D271—41no—
189D271—81no—
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#478D27111yescharacteristic, complemented
199#466—2711no—
200#465—2711no—
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×SD16—2711no—
211C2×D8—2711no—
212C2×D8—2711no—
213C22×C8—2711no—
214C22×C8—2711no—
215#126—5421no—
216#125C2×D27111yescharacteristic, complemented
217#124—2711no—
218#123—5421no—
219#123—5421no—
220C22⋊Q8—2711no—
221#121—2711no—
222#121—2711no—
223#121—2711no—
224#121—5421no—
225#121—5421no—
226#120—5421no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—5421no—
230C4×D4C2×D27111yescharacteristic
231#117—2711no—
232C2×C4⋊C4—2711no—
233C2×C22⋊C4—5421no—
234#107—2711no—
235#107C2×D27111yescharacteristic, complemented
236#106C2×D27111yescharacteristic, complemented
237C4⋊C8C2×D27111yescharacteristic, complemented
238#103—2711no—
239#103—2711no—
240#103—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102C2×D27111yescharacteristic
247#102C2×D27111yescharacteristic
248C22⋊C8—2711no—
249C22⋊C8—2711no—
250C4×C8—2711no—
251C24—5421no—
252#41—5422no—
253#41—10841no—
254C2×Q8—2711no—
255C2×D4—2711no—
256C2×D4—2711no—
257C2×D4—2711no—
258C2×D4—5421no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5422no—
263C2×D4—5421no—
264C2×D4—10841no—
265C2×D4C22×D27111yescharacteristic
266C22×C4—21no—
267C22×C4—2711no—
268C22×C4—2711no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4—5421no—
272SD16—10841no—
273D8—5422no—
274D8—10841no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8—10841no—
278C2×C8—10841no—
279C2×C8C22×D27111yescharacteristic
280C2×C8C22×D27111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4C22×D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C4⋊C4C22×D27111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—2711no—
292C22⋊C4—2711no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C42—5421no—
302C42C22×D27111yescharacteristic
303C23—21no—
304C23—2711no—
305C23—2711no—
306C23—5421no—
307C23—5421no—
308C23—10842no—
309C23—10842no—
310C23—21681no—
311Q8—5421no—
312Q8—5421no—
313D4—21no—
314D4—5422no—
315D4—5421no—
316D4—5421no—
317D4—5421no—
318D4—5421no—
319D4—10842no—
320D4—10842no—
321D4—10842no—
322D4—21681no—
323D4—21681no—
324D4#1686465512yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—81no—
329C2×C4—2711no—
330C2×C4—2711no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—10842no—
338C2×C4—21681no—
339C2×C4—21681no—
340C2×C4D4×D27111yescharacteristic
341C2×C4D4×D27111yescharacteristic
342C2×C4C23×D27111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5421no—
346C8—5421no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—2712no—
351C22—5421no—
352C22—5421no—
353C22—5421no—
354C22—5421no—
355C22—5421no—
356C22—10841no—
357C22—10841no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22—10841no—
362C22—21684no—
363C22C2×D4×D27111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—5421no—
370C4—5421no—
371C4—10841no—
372C4—10841no—
373C4C2×#1686465511yescharacteristic
374C4#2257620611yescharacteristic
375C2—42no—
376C2—5422no—
377C2—10841no—
378C2—21681no—
379C2—21681no—
380C2#3177453811yescharacteristic
381C2#135×D27111yescharacteristic
382C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#478×C271C2
socleC22×C271C22×D4
hypercenter#478D271
hypocenterC271#3297
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. #31774600C2semidirect
2. #31774542C2semidirect
3. #31774254C2semidirect
4. #31774234C2semidirect
5. #31774223C2semidirect
6. #31774134C2semidirect
7. #31774127C2semidirect
8. #31774113C2semidirect
9. #31774043C2semidirect
10. #31774032C2semidirect
11. C4×#16864639C2semidirect
12. #31773962C2semidirect
13. #31773890C2semidirect
14. #102×D271C2semidirect
15. #478×C271C2semidirect
18. C2×#16864643C22semidirect
22. C2×C8×D271C22semidirect
31. #22576135C22semidirect
35. C4×D1084C22semidirect
37. #22576118C22semidirect
40. #22576093C22semidirect
41. #22576085C22semidirect
42. #22576083C22semidirect
43. #22576083C22semidirect
48. #22576080C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C4×C271⋊C8C22semidirect
52. #125×C271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
56. C4⋊C8×C271C22semidirect
191. C271#3297semidirect
198. #478D271semidirect
216. #125C2×D271semidirect
235. #107C2×D271semidirect
236. #106C2×D271semidirect
237. C4⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746002
#317745422
#317742542
#317742342
#317742232
#317741342
#317741272
#317741132
#317740432
#317740322
C4×#168646392
#317739622
#317738902
#102×D2712
#478×C2712
#3297271