Finite Group Databasebeta

← #46848412

Subgroups — #46848412

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#31774585C211yesmaximal, characteristic
2Q8×D1084C211yesmaximal, characteristic
3#31774267C211yesmaximal, characteristic, complemented
4#31774240C211yesmaximal, characteristic, complemented
5#31774229C211yesmaximal, characteristic, complemented
6#31774156C211yesmaximal, characteristic, complemented
7#31774139C211yesmaximal, characteristic, complemented
8#31774108C211yesmaximal, characteristic, complemented
9#31774054C211yesmaximal, characteristic, complemented
10#31774054C211yesmaximal, characteristic, complemented
11#31774002C211yesmaximal, characteristic, complemented
12#31773967C211yesmaximal, characteristic
13#31773904C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#479×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×Dic1084C2211yescharacteristic
19C2×Dic1084C2211yescharacteristic
20C2×#16864644C2211yescharacteristic
21C2×#16864637C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic
23#22576160—21no—
24#22576160C2211yescharacteristic
25#22576155—21no—
26#22576151—21no—
27#22576145—21no—
28#22576140—21no—
29#22576140—21no—
30#22576140C2211yescharacteristic
31#22576140C2211yescharacteristic
32#22576140C2211yescharacteristic
33#22576140C2211yescharacteristic
34C4×Dic542C2211yescharacteristic
35C4×D1084—21no—
36C4×D1084C2211yescharacteristic
37#22576118—21no—
38#22576118C2211yescharacteristic
39C4⋊C4×D271C2211yescharacteristic
40#22576091C2211yescharacteristic
41#22576085C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576083C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576083C2211yescharacteristic
48#22576083C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576071C2211yescharacteristic
52#125×C271C2211yescharacteristic
53C4×Q8×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic
55#106×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542—21no—
61C2×Dic542C2311yescharacteristic
62C2×Dic542C2311yescharacteristic
63C2×Dic542C2311yescharacteristic
64Q8×D271—22no—
65Q8×D271—22no—
66Q8×D271—41no—
67Q8×D271—41no—
68C2×D1084D411yescharacteristic
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271C2311yescharacteristic
73C2×C4×D271C2311yescharacteristic
74Dic1084—41no—
75Dic1084—41no—
76#16864644—22no—
77#16864637—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—41no—
81C4⋊Dic271—21no—
82C4⋊Dic271—21no—
83C4⋊Dic271—21no—
84C4⋊Dic271—21no—
85C4⋊Dic271—21no—
86C4⋊Dic271—21no—
87C4⋊Dic271D411yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89Dic271⋊C4—21no—
90Dic271⋊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)⋊C4D411yescharacteristic
97(C2×D271)⋊C4D411yescharacteristic
98C4×Dic271—21no—
99C4×Dic271—21no—
100C2×Q8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C4⋊C4×C271—21no—
104C4⋊C4×C271—21no—
105C4⋊C4×C271—21no—
106C4⋊C4×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C42×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271C2×D411yescharacteristic
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—21no—
116Dic542—21no—
117Dic542—21no—
118Dic542—21no—
119Dic542—42no—
120Dic542—42no—
121D1084—22no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C4×D271—42no—
134C4×D271—42no—
135C271⋊C8—21no—
136C271⋊C8—21no—
137Q8×C271—21no—
138Q8×C271#4112yes—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271—21no—
148C22×C271C22×D411yescharacteristic
149C2×D271—21no—
150C2×D271—21no—
151C2×D271—21no—
152C2×D271—21no—
153C2×D271—21no—
154C2×D271#13712yes—
155C4×C271—22no—
156C4×C271—21no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271C2×#4111yescharacteristic
162C4×C271H(5,𝔽2)11yescharacteristic
163Dic271—21no—
164Dic271—21no—
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168Dic271—41no—
169C2×C271#54611yescharacteristic
170C2×C271C2×#13711yescharacteristic
171C2×C271#57811yescharacteristic
172D271—22no—
173D271—41no—
174C271#331211yescharacteristic, complemented, Sylow
175#3312—2711nomaximal, Sylow
176#552—2711no—
177D4×Q8—2711no—
178#491—2711no—
179#485—2711no—
180#483—2711no—
181#479D27111yescharacteristic, complemented
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×Q8—2711no—
192C22×Q8—2711no—
193C2×Q16—2711no—
194C2×Q16—2711no—
195C2×Q16—2711no—
196C2×M16—2711no—
197C22×C8—2711no—
198C4⋊Q8—2711no—
199C4⋊Q8—5421no—
200#126—5421no—
201#125—5421no—
202#125C2×D27111yescharacteristic
203#123—5421no—
204C22⋊Q8—2711no—
205C22⋊Q8—2711no—
206C22⋊Q8—2711no—
207C22⋊Q8—2711no—
208C22⋊Q8—5421no—
209C22⋊Q8—5421no—
210C4×Q8—2711no—
211C4×Q8C2×D27111yescharacteristic
212C4×D4—2711no—
213C4×D4—5421no—
214#117—2711no—
215#117—5421no—
216C2×C4⋊C4—2711no—
217#107C2×D27111yescharacteristic
218#106—2711no—
219#106C2×D27111yescharacteristic
220C4⋊C8C2×D27111yescharacteristic
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#103C2×D27111yescharacteristic
230#103C2×D27111yescharacteristic
231C22⋊C8—2711no—
232C22⋊C8—2711no—
233#97—2711no—
234C2×Q8—2711no—
235C2×Q8—2711no—
236C2×Q8—2711no—
237C2×Q8—5421no—
238C2×Q8—5421no—
239C2×Q8—5422no—
240C2×Q8—5422no—
241C2×Q8—10841no—
242C2×Q8—10841no—
243C2×Q8C22×D27111yescharacteristic
244C2×D4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—5421no—
248C22×C4—5421no—
249C22×C4—5421no—
250Q16—5422no—
251Q16—10841no—
252Q16—10841no—
253M16—10841no—
254C2×C8—2711no—
255C2×C8—2711no—
256C2×C8—10841no—
257C2×C8C22×D27111yescharacteristic
258C2×C8C22×D27111yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—2711no—
263C4⋊C4—2711no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4C22×D27111yescharacteristic
273C4⋊C4C22×D27111yescharacteristic
274C4⋊C4C22×D27111yescharacteristic
275C22⋊C4—2711no—
276C22⋊C4—2711no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C42—21no—
283C42—5421no—
284C42—5421no—
285C42C22×D27111yescharacteristic
286C23—2711no—
287C23—2711no—
288Q8—21no—
289Q8—5421no—
290Q8—5421no—
291Q8—5421no—
292Q8—5421no—
293Q8—5421no—
294Q8—5421no—
295Q8—10842no—
296Q8—10842no—
297Q8#1686465512yes—
298D4—5422no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2711no—
304C2×C4—2711no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—5421no—
312C2×C4—5421no—
313C2×C4—10842no—
314C2×C4—10842no—
315C2×C4—10842no—
316C2×C4D4×D27111yescharacteristic
317C2×C4D4×D27111yescharacteristic
318C2×C4C23×D27111yescharacteristic
319C8—21no—
320C8—21no—
321C8—5421no—
322C8—5421no—
323C22—2712no—
324C22—5421no—
325C22—5421no—
326C22—5421no—
327C22—5421no—
328C22—5421no—
329C22C2×D4×D27111yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—5421no—
337C4—5421no—
338C4—10841no—
339C4—10841no—
340C4—10841no—
341C4—10841no—
342C4C2×#1686465511yescharacteristic
343C4#2257620611yescharacteristic
344C2—5422no—
345C2—10841no—
346C2#3177453811yescharacteristic
347C2#137×D27111yescharacteristic
348C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#479×C271C2
socleC22×C271C22×D4
prime radicalC2×D1084D4
hypercenter#479D271
hypocenterC271#3312
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
3. #31774267C2semidirect
4. #31774240C2semidirect
5. #31774229C2semidirect
6. #31774156C2semidirect
7. #31774139C2semidirect
8. #31774108C2semidirect
9. #31774054C2semidirect
10. #31774054C2semidirect
11. #31774002C2semidirect
13. #31773904C2semidirect
14. #103×D271C2semidirect
15. #479×C271C2semidirect
174. C271#3312semidirect
181. #479D271semidirect

Maximal subgroups

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

subgroupindex
#317745852
Q8×D10842
#317742672
#317742402
#317742292
#317741562
#317741392
#317741082
#317740542
#317740022
#317739672
#317739042
#103×D2712
#479×C2712
#3312271