Finite Group Databasebeta

← #46848955

Subgroups — #46848955

Order 34688.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774610C211yesmaximal, characteristic, complemented
2#31774548C211yesmaximal, characteristic, complemented
3#31774294C211yesmaximal, characteristic, complemented
4#31774224C211yesmaximal, characteristic, complemented
5#31774221C211yesmaximal, characteristic, complemented
6#31774217C211yesmaximal, characteristic, complemented
7#31774215C211yesmaximal, characteristic, complemented
8#31774186C211yesmaximal, characteristic, complemented
9#31774176C211yesmaximal, characteristic, complemented
10#31774151C211yesmaximal, characteristic, complemented
11#31774150C211yesmaximal, characteristic, complemented
12#31773979C211yesmaximal, characteristic, complemented
13#31773973C211yesmaximal, characteristic, complemented
14#31773938C211yesmaximal, characteristic, complemented
15#428×C271C211yesmaximal, characteristic, complemented
16C2×#16864654—21no—
17C2×D2168C2211yescharacteristic
18C2×#16864637C2211yescharacteristic
19#22576159C2211yescharacteristic
20#22576159C2211yescharacteristic
21#22576148—21no—
22#22576146—21no—
23#22576145—21no—
24#22576143—21no—
25#22576138—21no—
26#22576135C2211yescharacteristic
27#22576135C2211yescharacteristic
28#22576134C2211yescharacteristic
29#22576134C2211yescharacteristic
30#22576133—21no—
31#22576122C2211yescharacteristic
32#22576122C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34#22576118C2211yescharacteristic
35#22576117—21no—
36C4⋊C4×D271C2211yescharacteristic
37#22576093C2211yescharacteristic
38#22576092C2211yescharacteristic
39#22576091C2211yescharacteristic
40#22576091C2211yescharacteristic
41#22576091C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576080C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×M16×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic
54C2×C4⋊C4×C271C2211yescharacteristic
55#107×C271C2211yescharacteristic
56#107×C271C2211yescharacteristic
57#106×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic
59#16864654—44no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×D1084—21no—
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271D411yescharacteristic
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271C2311yescharacteristic
73D2168—22no—
74#16864637—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4D411yescharacteristic
83C22⋊Dic271D411yescharacteristic
84(C2×D271)⋊C4—21no—
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×Dic271C2311yescharacteristic
94C22×C4×C271—21no—
95C22×C4×C271C2311yescharacteristic
96M16×C271—22no—
97C2×C8×C271C2311yescharacteristic
98C2×C8×C271C2311yescharacteristic
99C4⋊C4×C271—21no—
100C4⋊C4×C271C2311yescharacteristic
101C4⋊C4×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C22⋊C4×C271—21no—
105C42×C271—21no—
106C22×D271—21no—
107C22×D271—21no—
108C22×D271C2×D411yescharacteristic
109Dic542—42no—
110Dic542—81no—
111D1084—21no—
112D1084—21no—
113D1084—21no—
114D1084—21no—
115D1084—42no—
116D1084—81no—
117C22⋊D271—22no—
118C22⋊D271—42no—
119C22⋊D271—81no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—21no—
126C4×D271—21no—
127C4×D271—42no—
128C4×D271—42no—
129C4×D271—42no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132C23×C271C2×D411yescharacteristic
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—42no—
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2411yescharacteristic
142C8×C271—21no—
143C8×C271#4112yescomplemented
144C22×C271—21no—
145C22×C271—21no—
146C22×C271—21no—
147C22×C271C22×D411yescharacteristic
148C2×D271—21no—
149C2×D271—21no—
150C2×D271—21no—
151C2×D271—41no—
152C2×D271—41no—
153C2×D271—41no—
154C2×D271—41no—
155C2×D271—41no—
156C2×D271—41no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271C2×#4111yescharacteristic
163C4×C2712−1+411yescharacteristic
164Dic271—21no—
165Dic271—21no—
166Dic271—41no—
167Dic271—41no—
168Dic271AGL(1,ℤ/8)12yes—
169C2×C271—41no—
170C2×C271#54711yescharacteristic
171C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C271#57711yescharacteristic
173D271—41no—
174D271—81no—
175D271—81no—
176C271#335411yescharacteristic, complemented, Sylow
177#3354—2711nomaximal, Sylow
178#555—2711no—
179#547—2711no—
180#498—2711no—
181#482—2711no—
182#482—2711no—
183#481—2711no—
184#481—2711no—
185#476—2711no—
186#474—2711no—
187#469—2711no—
188#469—2711no—
189C4×D8—2711no—
190#436—2711no—
191#428—2711no—
192#428D27111yescharacteristic, complemented
193C2×#41—5421no—
194C2×D8—2711no—
195C2×M16—2711no—
196C2×M16C2×D27111yescharacteristic
197C4⋊Q8—2711no—
198C4⋊Q8—2711no—
199#124—5421no—
200#124—5421no—
201#123—5421no—
202#123—5421no—
203C22⋊Q8—5421no—
204#121—2711no—
205#121—2711no—
206#121—2711no—
207#121—2711no—
208#121—5421no—
209C4×D4—2711no—
210C4×D4—2711no—
211C4×D4—2711no—
212#117—2711no—
213#117—5421no—
214#117C2×D27111yescharacteristic
215C2×C4⋊C4—2711no—
216C2×C4⋊C4C2×D27111yescharacteristic
217#107—2711no—
218#107C2×D27111yescharacteristic
219#107C2×D27111yescharacteristic
220#106—2711no—
221#106—2711no—
222#106—2711no—
223#106—2711no—
224#106C2×D27111yescharacteristic
225#106C2×D27111yescharacteristic
226C4⋊C8—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231#102—2711no—
232#102—2711no—
233C22⋊C8—2711no—
234C22⋊C8—2711no—
235C4×C8—2711no—
236#41—10844no—
237C2×Q8—5421no—
238C2×Q8—5421no—
239C2×D4—2711no—
240C2×D4—2711no—
241C2×D4—2711no—
242C2×D4—5421no—
243C2×D4—5421no—
244C2×D4—5421no—
245C2×D4—5421no—
246C22×C4—21no—
247C22×C4—2711no—
248C22×C4—5421no—
249C22×C4—5421no—
250C22×C4—5421no—
251C22×C4C22×D27111yescharacteristic
252D8—5422no—
253M16—22no—
254M16—5422no—
255C2×C8—2711no—
256C2×C8—2711no—
257C2×C8C22×D27111yescharacteristic
258C2×C8C22×D27111yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4C22×D27111yescharacteristic
267C4⋊C4C22×D27111yescharacteristic
268C4⋊C4C22×D27111yescharacteristic
269C4⋊C4C22×D27111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—2711no—
272C22⋊C4—2711no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4—5421no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C42—21no—
281C42—2711no—
282C42—5421no—
283C23—2711no—
284C23—5421no—
285C23—5421no—
286C23D4×D27111yescharacteristic
287Q8—10842no—
288Q8—21681no—
289D4—5422no—
290D4—5421no—
291D4—5421no—
292D4—5421no—
293D4—5421no—
294D4—10842no—
295D4—10842no—
296D4—21681no—
297D4—21681no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—2711no—
306C2×C4—2711no—
307C2×C4—2711no—
308C2×C4—5421no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—5421no—
312C2×C4—10842no—
313C2×C4—10842no—
314C2×C4—10842no—
315C2×C4D4×D27111yescharacteristic
316C2×C4C23×D27111yescharacteristic
317C8—21no—
318C8—5421no—
319C8—5421no—
320C8#1686465412yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—5421no—
325C22—5421no—
326C22—5421no—
327C22—10841no—
328C22—10841no—
329C22—10841no—
330C22—10841no—
331C22—10841no—
332C22—10841no—
333C22C2×D4×D27111yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—2712no—
340C4—5421no—
341C4—5421no—
342C4—10841no—
343C4—10841no—
344C4C2×#1686465411yescharacteristic
345C4#2257620811yescharacteristic
346C2—41no—
347C2—10841no—
348C2—21681no—
349C2—21681no—
350C2#3177454811yescharacteristic
351C2AGL(1,ℤ/8)×D27111yescharacteristic
352C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#428×C271C2
socleC22×C271C22×D4
hypercenter#428D271
hypocenterC271#3354
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #31774610C2semidirect
2. #31774548C2semidirect
3. #31774294C2semidirect
4. #31774224C2semidirect
5. #31774221C2semidirect
6. #31774217C2semidirect
7. #31774215C2semidirect
8. #31774186C2semidirect
9. #31774176C2semidirect
10. #31774151C2semidirect
11. #31774150C2semidirect
12. #31773979C2semidirect
13. #31773973C2semidirect
14. #31773938C2semidirect
15. #428×C271C2semidirect
143. C8×C271#41semidirect
176. C271#3354semidirect
192. #428D271semidirect
320. C8#16864654semidirect

Maximal subgroups

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

subgroupindex
#317746102
#317745482
#317742942
#317742242
#317742212
#317742172
#317742152
#317741862
#317741762
#317741512
#317741502
#317739792
#317739732
#317739382
#428×C2712
#3354271