Finite Group Databasebeta

← #46848415

Subgroups — #46848415

Order 34688.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774587C211yesmaximal, characteristic
2#31774265C211yesmaximal, characteristic, complemented
3#31774235C211yesmaximal, characteristic, complemented
4#31774225C211yesmaximal, characteristic, complemented
5#31774204C211yesmaximal, characteristic, complemented
6#31774153C211yesmaximal, characteristic, complemented
7#31774136C211yesmaximal, characteristic, complemented
8#31774104C211yesmaximal, characteristic, complemented
9#31774053C211yesmaximal, characteristic, complemented
10#31774053C211yesmaximal, characteristic, complemented
11#31774002C211yesmaximal, characteristic, complemented
12#31773969C211yesmaximal, characteristic
13#31773906C211yesmaximal, characteristic, complemented
14C2×#22576084C211yesmaximal, characteristic, complemented
15D4×Q8×C271C211yesmaximal, characteristic
16C2×#16864644C2211yescharacteristic
17C2×#16864644C2211yescharacteristic
18C2×#16864644C2211yescharacteristic
19C2×#16864638C2211yescharacteristic
20C22×C271⋊C8C2211yescharacteristic
21#22576153—21no—
22#22576152C2211yescharacteristic
23#22576150—21no—
24#22576141—21no—
25#22576139C2211yescharacteristic
26#22576139C2211yescharacteristic
27C4×Dic542C2211yescharacteristic
28#22576119C2211yescharacteristic
29#22576116—21no—
30C2×C4⋊Dic271C2211yescharacteristic
31#22576093C2211yescharacteristic
32#22576091C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576084C2211yescharacteristic
36#22576084C2211yescharacteristic
37#22576084C2211yescharacteristic
38#22576084C2211yescharacteristic
39#22576084C2211yescharacteristic
40#22576084C2211yescharacteristic
41#22576084C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576083C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic
47#22576071C2211yescharacteristic
48C22×Q8×C271C2211yescharacteristic
49C22×Q8×C271C2211yescharacteristic
50C4⋊Q8×C271—21no—
51C4⋊Q8×C271C2211yescharacteristic
52C22⋊Q8×C271—21no—
53C22⋊Q8×C271—21no—
54C22⋊Q8×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56C4×Q8×C271C2211yescharacteristic
57C4×D4×C271—21no—
58C4×D4×C271C2211yescharacteristic
59C2×Dic542C2311yescharacteristic
60C22×Dic271—21no—
61#16864644—22no—
62#16864644—41no—
63#16864644—41no—
64#16864638—41no—
65C2×C271⋊C8—41no—
66C2×C271⋊C8C2311yescharacteristic
67C2×C271⋊C8C2311yescharacteristic
68C2×C271⋊C8C2311yescharacteristic
69C2×C271⋊C8C2311yescharacteristic
70C4⋊Dic271—21no—
71C4⋊Dic271C2311yescharacteristic
72C4⋊Dic271C2311yescharacteristic
73C4⋊Dic271C2311yescharacteristic
74Dic271⋊C4—21no—
75Dic271⋊C4—21no—
76Dic271⋊C4—21no—
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79C22⋊Dic271—21no—
80C22⋊Dic271—21no—
81C22⋊Dic271—21no—
82C4×Dic271—21no—
83C4×Dic271—21no—
84C2×Q8×C271—21no—
85C2×Q8×C271—21no—
86C2×Q8×C271—22no—
87C2×Q8×C271—22no—
88C2×Q8×C271—41no—
89C2×Q8×C271—41no—
90C2×Q8×C271C2311yescharacteristic
91C2×Q8×C271C2311yescharacteristic
92C2×Q8×C271C2311yescharacteristic
93C2×D4×C271D411yescharacteristic
94C22×C4×C271—21no—
95C22×C4×C271—21no—
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C4⋊C4×C271—21no—
99C4⋊C4×C271—21no—
100C4⋊C4×C271—21no—
101C4⋊C4×C271—21no—
102C4⋊C4×C271—21no—
103C4⋊C4×C271D411yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C22⋊C4×C271D411yescharacteristic
108C22⋊C4×C271D411yescharacteristic
109C42×C271—21no—
110C42×C271C2311yescharacteristic
111Dic542—21no—
112Dic542—21no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—42no—
118C271⋊C8—21no—
119C271⋊C8—21no—
120C271⋊C8—21no—
121C271⋊C8—21no—
122C23×C271C2×D411yescharacteristic
123C23×C271C2×D411yescharacteristic
124Q8×C271—21no—
125Q8×C271—21no—
126Q8×C271—21no—
127Q8×C271—21no—
128Q8×C271—21no—
129Q8×C271—42no—
130Q8×C271—42no—
131Q8×C271#4112yes—
132D4×C271—22no—
133C2×C4×C271—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—42no—
142C2×C4×C271—42no—
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C22×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271—21no—
153C22×C271#13712yes—
154C22×C271C22×D411yescharacteristic
155C4×C271—21no—
156C4×C271—21no—
157C4×C271—22no—
158C4×C271—21no—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271C2×#4111yescharacteristic
164C4×C271H(5,𝔽2)11yescharacteristic
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168Dic271—41no—
169C2×C271—22no—
170C2×C271—41no—
171C2×C271#54611yescharacteristic
172C2×C271C2×#13711yescharacteristic
173C2×C271#57811yescharacteristic
174C271#331211yescharacteristic, complemented, Sylow
175#3312—2711nomaximal, Sylow
176#552—2711no—
177D4×Q8D27111yescharacteristic
178#491—2711no—
179#485—2711no—
180#483—2711no—
181#479—2711no—
182#470—2711no—
183#467—2711no—
184#462—2711no—
185#451—2711no—
186#451—2711no—
187#441—2711no—
188#435—2711no—
189#419—2711no—
190C2×#103—2711no—
191C22×Q8C2×D27111yescharacteristic
192C22×Q8C2×D27111yescharacteristic
193C2×Q16—2711no—
194C2×Q16—2711no—
195C2×Q16—2711no—
196C2×M16—2711no—
197C22×C8—2711no—
198C4⋊Q8—21no—
199C4⋊Q8C2×D27111yescharacteristic
200#126—5421no—
201#125—2711no—
202#125—5421no—
203#123—5421no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—2711no—
207C22⋊Q8—2711no—
208C22⋊Q8C2×D27111yescharacteristic
209C22⋊Q8C2×D27111yescharacteristic
210C4×Q8—2711no—
211C4×Q8C2×D27111yescharacteristic
212C4×D4—21no—
213C4×D4C2×D27111yescharacteristic
214#117—2711no—
215#117—5421no—
216C2×C4⋊C4—2711no—
217#107—2711no—
218#106—2711no—
219#106—2711no—
220C4⋊C8—2711no—
221#103—2711no—
222#103—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103—2711no—
231C22⋊C8—2711no—
232C22⋊C8—2711no—
233#97—2711no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—2711no—
241C2×Q8C22×D27111yescharacteristic
242C2×Q8C22×D27111yescharacteristic
243C2×Q8C22×D27111yescharacteristic
244C2×D4C22⋊D27111yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—5421no—
248C22×C4C22×D27111yescharacteristic
249C22×C4C22×D27111yescharacteristic
250Q16—5422no—
251Q16—10841no—
252Q16—10841no—
253M16—10841no—
254C2×C8—2711no—
255C2×C8—2711no—
256C2×C8—2711no—
257C2×C8—2711no—
258C2×C8—10841no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—2711no—
265C4⋊C4—2711no—
266C4⋊C4—2711no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4C22⋊D27111yescharacteristic
274C4⋊C4C22×D27111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4C22⋊D27111yescharacteristic
281C22⋊C4C22⋊D27111yescharacteristic
282C42—21no—
283C42—5421no—
284C42—5421no—
285C42C22×D27111yescharacteristic
286C23C2×C22⋊D27111yescharacteristic
287C23C2×C22⋊D27111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—42no—
295Q8—5421no—
296Q8—5421no—
297Q8#1686465312yes—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—5421no—
312C2×C4—5421no—
313C2×C4—10842no—
314C2×C4C2×C22⋊D27111yescharacteristic
315C2×C4C2×C22⋊D27111yescharacteristic
316C2×C4C2×C22⋊D27111yescharacteristic
317C2×C4C2×C22⋊D27111yescharacteristic
318C2×C4C23×D27111yescharacteristic
319C8—5421no—
320C8—5421no—
321C8—5421no—
322C8—5421no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#2257619112yes—
329C22C22×C22⋊D27111yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—10841no—
339C4—10841no—
340C4—10841no—
341C4—10841no—
342C4C2×#1686465311yescharacteristic
343C4#2257620511yescharacteristic
344C2—22no—
345C2—41no—
346C2#3177454111yescharacteristic
347C2C2×#2257619111yescharacteristic
348C2#3177471211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingD4×Q8×C271C2
socleC22×C271C22×D4
prime radicalC2×D4×C271D4
hypercenterD4×Q8D271
hypocenterC271#3312
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #31774265C2semidirect
3. #31774235C2semidirect
4. #31774225C2semidirect
5. #31774204C2semidirect
6. #31774153C2semidirect
7. #31774136C2semidirect
8. #31774104C2semidirect
9. #31774053C2semidirect
10. #31774053C2semidirect
11. #31774002C2semidirect
13. #31773906C2semidirect
14. C2×#22576084C2semidirect
174. C271#3312semidirect

Maximal subgroups

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

subgroupindex
#317745872
#317742652
#317742352
#317742252
#317742042
#317741532
#317741362
#317741042
#317740532
#317740022
#317739692
#317739062
C2×#225760842
D4×Q8×C2712
#3312271