Finite Group Databasebeta

← #46848178

Subgroups — #46848178

Order 34688.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774551C211yesmaximal, characteristic, complemented
2D4×D1084C211yesmaximal, characteristic, complemented
3#31774244C211yesmaximal, characteristic, complemented
4#31774211C212yesmaximal, complemented
5#31774178C211yesmaximal, characteristic, complemented
6#31774133C212yesmaximal, complemented
7#31774093C211yesmaximal, characteristic, complemented
8#31774033C212yesmaximal, complemented
9C4×D2168C211yesmaximal, characteristic, complemented
10C2×#22576082C212yesmaximal, complemented
11C8×D4×C271C211yesmaximal, characteristic, complemented
12C2×#16864653—21no—
13C22×D1084C2212yes—
14C2×D4×D271—21no—
15C2×D2168C2211yescharacteristic
16C2×D2168C2212yescomplemented
17#22576160C2211yescharacteristic, complemented
18#22576157C2211yescharacteristic
19#22576144—22no—
20#22576140—21no—
21#22576135—21no—
22#22576134C2212yes—
23#22576134C2212yescomplemented
24#22576130—22no—
25C4×D1084C2211yescharacteristic
26C4×D1084C2211yescharacteristic, complemented
27C2×C4⋊Dic271C2212yescomplemented
28#22576094C2211yescharacteristic, complemented
29#22576094C2212yescomplemented
30#22576082C2211yescharacteristic
31#22576082C2212yes—
32#22576082C2212yescomplemented
33#22576082C2212yescomplemented
34#22576082C2211yescharacteristic, complemented
35#22576082C2212yescomplemented
36C22×C8×C271C2212yescomplemented
37C4×D4×C271C2211yescharacteristic
38C4⋊C8×C271C2211yescharacteristic, complemented
39C22⋊C8×C271C2212yescomplemented
40C4×C8×C271C2211yescharacteristic, complemented
41C23×D271—22no—
42#16864653—44no—
43C2×Dic542—21no—
44C2×D1084—24no—
45C2×D1084—22no—
46C2×D1084—21no—
47C2×D1084—21no—
48C2×D1084—42no—
49C2×D1084C2312yes—
50C2×D1084C2311yescharacteristic
51C2×D1084C2311yescharacteristic
52C2×C22⋊D271—22no—
53C2×C22⋊D271—22no—
54D4×D271—44no—
55C22×Dic271—22no—
56C2×C4×D271—21no—
57C2×C4×D271—21no—
58D2168—22no—
59D2168—42no—
60C4⋊Dic271—22no—
61C4⋊Dic271—21no—
62C4⋊Dic271C2311yescharacteristic
63C4⋊Dic271C2312yes—
64C4⋊Dic271C2311yescharacteristic
65(C2×D271)⋊C4—21no—
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4—21no—
69C2×D4×C271D411yescharacteristic
70C22×C4×C271C2312yes—
71C2×C8×C271—42no—
72C2×C8×C271C2311yescharacteristic
73C2×C8×C271C2312yes—
74C2×C8×C271C2311yescharacteristic
75C4⋊C4×C271D411yescharacteristic
76C22⋊C4×C271D412yes—
77C42×C271C2311yescharacteristic
78C22×D271—21no—
79C22×D271—21no—
80C22×D271—22no—
81C22×D271—44no—
82C22×D271—44no—
83C22×D271—82no—
84Dic542—42no—
85D1084—21no—
86D1084—22no—
87D1084—21no—
88D1084—21no—
89D1084—22no—
90D1084—44no—
91D1084—42no—
92D1084—81no—
93D1084#4112yescomplemented
94C22⋊D271—44no—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C2×Dic271—21no—
98C2×Dic271—22no—
99C2×Dic271—44no—
100C4×D271—42no—
101C4×D271—42no—
102C23×C271C2×D412yes—
103D4×C271D814yescomplemented
104C2×C4×C271—22no—
105C2×C4×C271—22no—
106C2×C4×C271C2×D411yescharacteristic
107C2×C4×C271C2×D412yes—
108C2×C4×C271C2×D411yescharacteristic
109C2×C4×C271C2411yescharacteristic
110C8×C271—21no—
111C8×C271—21no—
112C8×C271—22no—
113C22×C271—22no—
114C22×C271—22no—
115C22×C271C2×D814yes—
116C22×C271C22×D411yescharacteristic
117C2×D271—22no—
118C2×D271—41no—
119C2×D271—41no—
120C2×D271—42no—
121C2×D271—41no—
122C2×D271—41no—
123C2×D271—42no—
124C2×D271—42no—
125C2×D271—41no—
126C2×D271—88no—
127C4×C271—22no—
128C4×C271—21no—
129C4×C271C2×D812yes—
130C4×C271C2×#4111yescharacteristic
131C4×C271H(5,𝔽2)11yescharacteristic
132Dic271—41no—
133Dic271—41no—
134Dic271—42no—
135C2×C271—24no—
136C2×C271#54611yescharacteristic
137C2×C271C22×D811yescharacteristic
138C2×C271#57711yescharacteristic
139D271—42no—
140D271—81no—
141D271—82no—
142C271#329411yescharacteristic, complemented, Sylow
143#3294—2711nomaximal, Sylow
144#547—2711no—
145D42—2711no—
146#486—2711no—
147#480—2712no—
148#474—2711no—
149#466—2712no—
150#459—2711no—
151#447—2712no—
152C4×D8—2711no—
153C8×D4D27111yescharacteristic, complemented
154C2×#102—2712no—
155C2×#41—5421no—
156C22×D4—2712no—
157C22×D4—5421no—
158C2×D8—2711no—
159C2×D8—2712no—
160C22×C8C2×D27112yescomplemented
161C4⋊Q8—2711no—
162#127—2711no—
163#123—5422no—
164C22⋊Q8—5421no—
165#121—2712no—
166#121—2712no—
167#121—5421no—
168#120—5422no—
169C4×D4—2711no—
170C4×D4—2711no—
171C4×D4C2×D27111yescharacteristic
172C2×C4⋊C4—2712no—
173#107—2711no—
174#107—2712no—
175C4⋊C8C2×D27111yescharacteristic, complemented
176#102—2711no—
177#102—2712no—
178#102—2712no—
179#102—2712no—
180#102—2711no—
181#102—2712no—
182C22⋊C8C2×D27112yescomplemented
183C4×C8C2×D27111yescharacteristic, complemented
184C24—5422no—
185#41—10844no—
186C2×Q8—5421no—
187C2×D4—2711no—
188C2×D4—2712no—
189C2×D4—2711no—
190C2×D4—5424no—
191C2×D4—5422no—
192C2×D4—5422no—
193C2×D4—5422no—
194C2×D4—5421no—
195C2×D4—5421no—
196C2×D4—10842no—
197C2×D4—10844no—
198C2×D4D108411yescharacteristic
199C22×C4—5422no—
200C22×C4—5421no—
201C22×C4—5421no—
202C22×C4C22×D27112yes—
203D8—5422no—
204D8—10842no—
205C2×C8—42no—
206C2×C8C22×D27111yescharacteristic
207C2×C8C22×D27111yescharacteristic
208C2×C8C22×D27112yes—
209C4⋊C4—2711no—
210C4⋊C4—2712no—
211C4⋊C4—2711no—
212C4⋊C4—5422no—
213C4⋊C4—5421no—
214C4⋊C4D108411yescharacteristic
215C22⋊C4—5421no—
216C22⋊C4—5421no—
217C22⋊C4—5422no—
218C22⋊C4—5422no—
219C22⋊C4D108412yes—
220C42C22×D27111yescharacteristic
221C23—5421no—
222C23—5421no—
223C23—5422no—
224C23—10844no—
225C23—10844no—
226C23—21682no—
227C23C2×D108412yes—
228Q8—10842no—
229D4—2712no—
230D4—5421no—
231D4—5422no—
232D4—5421no—
233D4—5421no—
234D4—5422no—
235D4—10844no—
236D4—10844no—
237D4—10844no—
238D4—10842no—
239D4—21681no—
240D4D216814yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—5421no—
244C2×C4—5421no—
245C2×C4—5422no—
246C2×C4—10844no—
247C2×C4—10842no—
248C2×C4—10842no—
249C2×C4C2×D108411yescharacteristic
250C2×C4C2×D108412yes—
251C2×C4C2×D108411yescharacteristic
252C2×C4C23×D27111yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—5422no—
259C22—10841no—
260C22—10841no—
261C22—10842no—
262C22—10841no—
263C22—10841no—
264C22—10842no—
265C22—10842no—
266C22—10841no—
267C22—21688no—
268C22C2×D216814yes—
269C22C22×D108411yescharacteristic
270C4—22no—
271C4—21no—
272C4—10841no—
273C4—10841no—
274C4—10842no—
275C4C2×D216812yes—
276C4#41×D27111yescharacteristic
277C4#2257620511yescharacteristic
278C2—24no—
279C2—10842no—
280C2—21682no—
281C2—21681no—
282C2#3177454211yescharacteristic
283C2C22×D216811yescharacteristic
284C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC8×D4×C271C2
socleC22×C271C22×D4
hypercenterC8×D4D271
hypocenterC271#3294
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #31774551C2semidirect
2. D4×D1084C2semidirect
3. #31774244C2semidirect
4. #31774211C2semidirect
5. #31774178C2semidirect
6. #31774133C2semidirect
7. #31774093C2semidirect
8. #31774033C2semidirect
9. C4×D2168C2semidirect
10. C2×#22576082C2semidirect
11. C8×D4×C271C2semidirect
16. C2×D2168C22semidirect
17. #22576160C22semidirect
23. #22576134C22semidirect
26. C4×D1084C22semidirect
27. C2×C4⋊Dic271C22semidirect
28. #22576094C22semidirect
29. #22576094C22semidirect
32. #22576082C22semidirect
33. #22576082C22semidirect
34. #22576082C22semidirect
35. #22576082C22semidirect
36. C22×C8×C271C22semidirect
38. C4⋊C8×C271C22semidirect
39. C22⋊C8×C271C22semidirect
40. C4×C8×C271C22semidirect
93. D1084#41semidirect
103. D4×C271D8semidirect
142. C271#3294semidirect
153. C8×D4D271semidirect
160. C22×C8C2×D271semidirect
175. C4⋊C8C2×D271semidirect
182. C22⋊C8C2×D271semidirect
183. C4×C8C2×D271semidirect
240. D4D2168semidirect

Maximal subgroups

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

subgroupindex
#317745512
D4×D10842
#317742442
#317742112
#317741782
#317741332
#317740932
#317740332
C4×D21682
C2×#225760822
C8×D4×C2712
#3294271