Finite Group Databasebeta

← #46848255

Subgroups — #46848255

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#31774589C211yesmaximal, characteristic, complemented
2#31774558C211yesmaximal, characteristic
3#31774253C211yesmaximal, characteristic, complemented
4#31774237C211yesmaximal, characteristic, complemented
5#31774211C211yesmaximal, characteristic, complemented
6#31774206C211yesmaximal, characteristic, complemented
7#31774135C211yesmaximal, characteristic, complemented
8#31774128C211yesmaximal, characteristic, complemented
9#31774121C211yesmaximal, characteristic, complemented
10#31774058C211yesmaximal, characteristic, complemented
11Q16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14C2×#22576084C211yesmaximal, characteristic, complemented
15#451×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576155—21no—
21#22576151—21no—
22#22576151C2211yescharacteristic
23#22576149—21no—
24#22576149C2211yescharacteristic
25#22576144—21no—
26#22576136—21no—
27#22576136C2211yescharacteristic
28#22576136C2211yescharacteristic
29#22576134—21no—
30#22576134C2211yescharacteristic
31Q8×Dic271C2211yescharacteristic
32Q8×Dic271C2211yescharacteristic
33C4×C22⋊D271—21no—
34C4×C22⋊D271C2211yescharacteristic
35#22576118C2211yescharacteristic
36#22576116—21no—
37C2×C4⋊Dic271C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576082C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C22×Q8×C271C2211yescharacteristic
53C2×Q16×C271C2211yescharacteristic
54C2×Q16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—41no—
61C2×D1084—21no—
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271D411yescharacteristic
65C22×Dic271—21no—
66C2×C4×D271—21no—
67C2×C4×D271C2311yescharacteristic
68#16864641—41no—
69C2×C271⋊C8—41no—
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C8×D271—41no—
73C4⋊Dic271—21no—
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271D411yescharacteristic
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
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
96C2×Q8×C271—21no—
97C2×Q8×C271—22no—
98C2×Q8×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×Q8×C271C2311yescharacteristic
101C2×Q8×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104Q16×C271—22no—
105Q16×C271—41no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113D1084—21no—
114D1084—21no—
115D1084—42no—
116C22⋊D271—22no—
117C22⋊D271—42no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—42no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—21no—
126C4×D271—21no—
127C4×D271—42no—
128C271⋊C8—21no—
129C271⋊C8—21no—
130C23×C271C2×D411yescharacteristic
131Q8×C271—21no—
132Q8×C271—21no—
133Q8×C271—21no—
134Q8×C271—21no—
135Q8×C271—21no—
136Q8×C271—42no—
137Q8×C271#4112yes—
138C2×C4×C271—21no—
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—21no—
151C22×C271#13512yes—
152C22×C271C22×D411yescharacteristic
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C4×C271—21no—
160C4×C271—22no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271—41no—
164C4×C271C2×#4111yescharacteristic
165C4×C271H(5,𝔽2)11yescharacteristic
166Dic271—21no—
167Dic271—21no—
168Dic271—21no—
169Dic271—41no—
170Dic271—41no—
171Dic271—41no—
172C2×C271—22no—
173C2×C271#54611yescharacteristic
174C2×C271C2×#13511yescharacteristic
175C2×C271#57811yescharacteristic
176D271—41no—
177D271—81no—
178C271#330011yescharacteristic, complemented, Sylow
179#3300—2711nomaximal, Sylow
180#552—2711no—
181#548—2711no—
182#488—2711no—
183#485—2711no—
184#480—2711no—
185#479—2711no—
186#467—2711no—
187#465—2711no—
188#464—2711no—
189#452—2711no—
190#451D27111yescharacteristic, complemented
191C4×Q16—2711no—
192C8×D4—2711no—
193#416—2711no—
194C2×#103—2711no—
195C2×#41—2711no—
196C22×Q8C2×D27111yescharacteristic
197C2×Q16C2×D27111yescharacteristic
198C2×Q16C2×D27111yescharacteristic
199C2×SD16—2711no—
200C22×C8—2711no—
201C22×C8—2711no—
202#126—5421no—
203#125—2711no—
204#125—5421no—
205#124—2711no—
206#124—5421no—
207#123—5421no—
208C22⋊Q8—2711no—
209C22⋊Q8—2711no—
210C22⋊Q8—5421no—
211C22⋊Q8C2×D27111yescharacteristic
212#121—2711no—
213#121—5421no—
214C4×Q8—2711no—
215C4×Q8—2711no—
216C4×D4—2711no—
217C4×D4—5421no—
218#117—2711no—
219#117—5421no—
220C2×C4⋊C4—2711no—
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#103C2×D27111yescharacteristic
231#103C2×D27111yescharacteristic
232#102—2711no—
233#102—2711no—
234#102—2711no—
235C22⋊C8—2711no—
236C22⋊C8C2×D27111yescharacteristic
237C4×C8—2711no—
238#41—5422no—
239#41—10841no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8C22×D27111yescharacteristic
244C2×Q8C22×D27111yescharacteristic
245C2×Q8C22×D27111yescharacteristic
246C2×D4—2711no—
247C2×D4—2711no—
248C2×D4—5421no—
249C2×D4—5421no—
250C22×C4—21no—
251C22×C4—2711no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4C22×D27111yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—10841no—
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—2711no—
291C23—5421no—
292C23D4×D27111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8#1686465312yes—
300D4—5422no—
301D4—5421no—
302D4—5421no—
303D4—10842no—
304D4—10842no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—2711no—
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—21no—
330C22—5421no—
331C22—5421no—
332C22—5421no—
333C22—10841no—
334C22—10841no—
335C22—10841no—
336C22#2257618312yes—
337C22C2×D4×D27111yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—5421no—
344C4—5421no—
345C4—5421no—
346C4—10841no—
347C4—10841no—
348C4—10841no—
349C4C2×#1686465311yescharacteristic
350C4#2257620511yescharacteristic
351C2—22no—
352C2—10841no—
353C2—21681no—
354C2#3177453411yescharacteristic
355C2C2×#2257618311yescharacteristic
356C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#451×C271C2
socleC22×C271C22×D4
prime radical#31774558C2
hypercenter#451D271
hypocenterC271#3300
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #31774589C2semidirect
3. #31774253C2semidirect
4. #31774237C2semidirect
5. #31774211C2semidirect
6. #31774206C2semidirect
7. #31774135C2semidirect
8. #31774128C2semidirect
9. #31774121C2semidirect
10. #31774058C2semidirect
11. Q16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773892C2semidirect
14. C2×#22576084C2semidirect
15. #451×C271C2semidirect
178. C271#3300semidirect
190. #451D271semidirect

Maximal subgroups

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

subgroupindex
#317745892
#317745582
#317742532
#317742372
#317742112
#317742062
#317741352
#317741282
#317741212
#317740582
Q16×Dic2712
(C22⋊D271)⋊C82
#317738922
C2×#225760842
#451×C2712
#3300271