Finite Group Databasebeta

← #46848274

Subgroups — #46848274

Order 34688.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774618C211yesmaximal, characteristic, complemented
2#31774546C211yesmaximal, characteristic
3#31774246C211yesmaximal, characteristic, complemented
4#31774239C211yesmaximal, characteristic, complemented
5#31774236C211yesmaximal, characteristic, complemented
6#31774175C211yesmaximal, characteristic, complemented
7#31774136C211yesmaximal, characteristic, complemented
8#31774135C211yesmaximal, characteristic, complemented
9#31774105C211yesmaximal, characteristic, complemented
10#31774056C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#452×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×#16864644C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576159C2211yescharacteristic
21#22576158C2211yescharacteristic
22#22576155—21no—
23#22576153—21no—
24#22576146—21no—
25#22576143—21no—
26#22576141—21no—
27#22576139C2211yescharacteristic
28#22576136C2211yescharacteristic
29#22576136C2211yescharacteristic
30#22576132—21no—
31Q8×Dic271C2211yescharacteristic
32#22576125—21no—
33D4×Dic271C2211yescharacteristic
34C4×C22⋊D271C2211yescharacteristic
35#22576119C2211yescharacteristic
36#22576118C2211yescharacteristic
37C2×Dic271⋊C4—21no—
38#22576094C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576092C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576083C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×Q16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864653—22no—
60#16864653—41no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271D411yescharacteristic
65C22×Dic271—21no—
66C22×Dic271—21no—
67C2×C4×D271C2311yescharacteristic
68#16864644—41no—
69C2×C271⋊C8—41no—
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C8×D271—41no—
73C4⋊Dic271—21no—
74C4⋊Dic271C2311yescharacteristic
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271D411yescharacteristic
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4D411yescharacteristic
92C4×Dic271—21no—
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271C2311yescharacteristic
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104Q16×C271—41no—
105SD16×C271—22no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271C2×D411yescharacteristic
112Dic542—21no—
113Dic542—21no—
114Dic542—42no—
115C22⋊D271—22no—
116C22⋊D271—42no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—42no—
122C2×Dic271—42no—
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C4×D271—21no—
127C4×D271—21no—
128C271⋊C8—21no—
129C271⋊C8—21no—
130C23×C271—21no—
131C23×C271C2×D411yescharacteristic
132Q8×C271—21no—
133Q8×C271—21no—
134Q8×C271—21no—
135Q8×C271—21no—
136D4×C271—21no—
137D4×C271—42no—
138D4×C271#4112yes—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—42no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2411yescharacteristic
147C8×C271—21no—
148C8×C271—21no—
149C22×C271—21no—
150C22×C271—22no—
151C22×C271—21no—
152C22×C271—21no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271C22×D411yescharacteristic
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—21no—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271C2×#4111yescharacteristic
164C4×C271H(5,𝔽2)11yescharacteristic
165Dic271—21no—
166Dic271—21no—
167Dic271—41no—
168Dic271—41no—
169Dic271—41no—
170Dic271—41no—
171Dic271#13512yes—
172C2×C271—41no—
173C2×C271—42no—
174C2×C271#54611yescharacteristic
175C2×C271C2×#13511yescharacteristic
176C2×C271#57811yescharacteristic
177D271—41no—
178C271#330211yescharacteristic, complemented, Sylow
179#3302—2711nomaximal, Sylow
180#556—2711no—
181#547—2711no—
182#487—2711no—
183#485—2711no—
184#485—2711no—
185#474—2711no—
186#467—2711no—
187#467—2711no—
188#462—2711no—
189#452—2711no—
190#452D27111yescharacteristic, complemented
191C4×SD16—2711no—
192C8×D4—2711no—
193#416—2711no—
194#416—2711no—
195C2×#41—2711no—
196C2×#41C2×D27111yescharacteristic
197C2×Q16—2711no—
198C2×Q16C2×D27111yescharacteristic
199C2×SD16C2×D27111yescharacteristic
200C22×C8—2711no—
201C22×C8—2711no—
202C4⋊Q8—2711no—
203C4⋊Q8—2711no—
204#126—5421no—
205#126—5421no—
206#124—5421no—
207#123—5421no—
208#123—5421no—
209C22⋊Q8—2711no—
210C22⋊Q8—2711no—
211C22⋊Q8—2711no—
212C22⋊Q8C2×D27111yescharacteristic
213#121—5421no—
214C4×Q8—2711no—
215C4×Q8—5421no—
216C4×D4—2711no—
217C4×D4—2711no—
218#117—2711no—
219#117—2711no—
220C2×C4⋊C4—5421no—
221#107—2711no—
222#107—2711no—
223#106—2711no—
224C4⋊C8—2711no—
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103—2711no—
231#103C2×D27111yescharacteristic
232#102—2711no—
233#102—2711no—
234#102C2×D27111yescharacteristic
235C22⋊C8—2711no—
236C22⋊C8C2×D27111yescharacteristic
237C4×C8—2711no—
238#41—22no—
239#41—41no—
240#41—5422no—
241#41—10841no—
242C2×Q8—2711no—
243C2×Q8—5421no—
244C2×Q8C22×D27111yescharacteristic
245C2×Q8C22×D27111yescharacteristic
246C2×D4—21no—
247C2×D4—2711no—
248C2×D4—5421no—
249C2×D4C22×D27111yescharacteristic
250C22×C4—21no—
251C22×C4—2711no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4C22×D27111yescharacteristic
255Q16—41no—
256Q16—10841no—
257SD16—22no—
258C2×C8—2711no—
259C2×C8—2711no—
260C2×C8—10841no—
261C2×C8—10841no—
262C2×C8C22×D27111yescharacteristic
263C2×C8C22×D27111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—2711no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2711no—
279C22⋊C4—2711no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—2711no—
287C42—5421no—
288C42—5421no—
289C42—5421no—
290C23—21no—
291C23—2711no—
292C23D4×D27111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—5421no—
298Q8—5421no—
299Q8—10842no—
300D4—21no—
301D4—42no—
302D4—5422no—
303D4—10842no—
304D4#1686465312yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—2711no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—10842no—
321C2×C4—10842no—
322C2×C4D4×D27111yescharacteristic
323C2×C4C23×D27111yescharacteristic
324C8—21no—
325C8—21no—
326C8—5421no—
327C8—5421no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—5421no—
335C22—5421no—
336C22—5421no—
337C22C2×D4×D27111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—2712no—
343C4—5421no—
344C4—5421no—
345C4—10841no—
346C4—10841no—
347C4—10841no—
348C4—10841no—
349C4C2×#1686465311yescharacteristic
350C4#2257620511yescharacteristic
351C2—41no—
352C2—42no—
353C2—10841no—
354C2#3177453411yescharacteristic
355C2#135×D27111yescharacteristic
356C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#452×C271C2
socleC22×C271C22×D4
prime radical#31774546C2
hypercenter#452D271
hypocenterC271#3302
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #31774618C2semidirect
3. #31774246C2semidirect
4. #31774239C2semidirect
5. #31774236C2semidirect
6. #31774175C2semidirect
7. #31774136C2semidirect
8. #31774135C2semidirect
9. #31774105C2semidirect
10. #31774056C2semidirect
11. SD16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773893C2semidirect
14. #31773890C2semidirect
15. #452×C271C2semidirect
178. C271#3302semidirect
190. #452D271semidirect

Maximal subgroups

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

subgroupindex
#317746182
#317745462
#317742462
#317742392
#317742362
#317741752
#317741362
#317741352
#317741052
#317740562
SD16×Dic2712
(C22⋊D271)⋊C82
#317738932
#317738902
#452×C2712
#3302271