Finite Group Databasebeta

← #46846307

Subgroups — #46846307

Order 34688.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D271C211yesmaximal, characteristic
2C2×#22576118C211yesmaximal, characteristic
3#31774241C212yesmaximal, complemented
4#31774237C212yesmaximal, complemented
5#31774233C211yesmaximal, characteristic, complemented
6#31774230C211yesmaximal, characteristic, complemented
7#31774227C211yesmaximal, characteristic, complemented
8#31773936C212yesmaximal, complemented
9#31773904C212yesmaximal, complemented
10#31773859C211yesmaximal, characteristic
11#483×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271C2211yescharacteristic
13C22×C4×D271C2211yescharacteristic
14C2×#16864637C2212yes—
15#22576140C2211yescharacteristic
16#22576139C2211yescharacteristic
17#22576138—21no—
18#22576137—21no—
19#22576136C2211yescharacteristic
20#22576118—21no—
21#22576118—21no—
22#22576118C2212yes—
23#22576118C2212yes—
24C2×C4⋊Dic271C2211yescharacteristic
25C4⋊C4×D271—21no—
26C4⋊C4×D271C2211yescharacteristic
27C2×(C2×D271)⋊C4—21no—
28C22⋊C4×D271—21no—
29C2×C4×Dic271—21no—
30#22576094C2212yescomplemented
31#22576093C2212yescomplemented
32#22576091C2212yescomplemented
33#22576085C2212yescomplemented
34#22576084C2212yescomplemented
35#22576083C2212yescomplemented
36C22⋊(C271⋊C8)C2211yescharacteristic
37(C2×D271)⋊C8C2212yes—
38C22⋊Q8×C271C2211yescharacteristic
39C2×C4⋊C4×C271C2211yescharacteristic
40#106×C271C2212yescomplemented
41#103×C271C2212yescomplemented
42C22⋊C8×C271C2211yescharacteristic
43C23×D271D411yescharacteristic
44C2×Dic542—21no—
45C2×Dic542C2311yescharacteristic
46Q8×D271—24no—
47C22×Dic271—21no—
48C22×Dic271D411yescharacteristic
49C2×C4×D271—24no—
50C2×C4×D271—21no—
51C2×C4×D271—21no—
52C2×C4×D271D411yescharacteristic
53C2×C4×D271D411yescharacteristic
54C2×C4×D271C2311yescharacteristic
55C2×C4×D271C2311yescharacteristic
56#16864637—24no—
57C2×C271⋊C8C2312yes—
58C4⋊Dic271—21no—
59C4⋊Dic271—21no—
60C4⋊Dic271C2311yescharacteristic
61C4⋊Dic271C2312yes—
62Dic271⋊C4—21no—
63Dic271⋊C4—21no—
64Dic271⋊C4—21no—
65C22⋊Dic271—21no—
66(C2×D271)⋊C4—21no—
67(C2×D271)⋊C4—21no—
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4—22no—
70C4×Dic271—22no—
71C4×Dic271—22no—
72C2×Q8×C271C2311yescharacteristic
73C22×C4×C271—21no—
74C22×C4×C271C2311yescharacteristic
75C2×C8×C271C2312yes—
76C4⋊C4×C271—21no—
77C4⋊C4×C271—21no—
78C4⋊C4×C271C2311yescharacteristic
79C4⋊C4×C271C2312yes—
80C22⋊C4×C271—21no—
81C22×D271—22no—
82C22×D271—22no—
83C22×D271—41no—
84C22×D271C2×D411yescharacteristic
85C22×D271C2×D411yescharacteristic
86Dic542—22no—
87Dic542—42no—
88C2×Dic271—22no—
89C2×Dic271—22no—
90C2×Dic271—21no—
91C2×Dic271—21no—
92C2×Dic271—42no—
93C2×Dic271C2×D411yescharacteristic
94C2×Dic271C2×D411yescharacteristic
95C4×D271—22no—
96C4×D271—22no—
97C4×D271—22no—
98C4×D271—42no—
99C4×D271—42no—
100C4×D271#4114yes—
101C271⋊C8—22no—
102C23×C271C2×D411yescharacteristic
103Q8×C271—22no—
104C2×C4×C271—22no—
105C2×C4×C271—21no—
106C2×C4×C271—21no—
107C2×C4×C271—22no—
108C2×C4×C271—42no—
109C2×C4×C271C2×D411yescharacteristic
110C2×C4×C271C2411yescharacteristic
111C8×C271—22no—
112C22×C271—21no—
113C22×C271—21no—
114C22×C271#13712yes—
115C22×C271C22×D411yescharacteristic
116C2×D271—21no—
117C2×D271—21no—
118C2×D271—21no—
119C2×D271—21no—
120C2×D271—21no—
121C2×D271—44no—
122C2×D271#12312yes—
123C4×C271—21no—
124C4×C271—41no—
125C4×C271—42no—
126C4×C271—41no—
127C4×C271C2×#4112yes—
128Dic271—21no—
129Dic271—41no—
130Dic271—41no—
131Dic271—42no—
132Dic271#12312yes—
133C2×C271—22no—
134C2×C271C2×#12311yescharacteristic
135C2×C271C2×#13711yescharacteristic
136C2×C271#57511yescharacteristic
137D271—22no—
138D271—41no—
139C271#309511yescharacteristic, complemented, Sylow
140#3095—2711nomaximal, Sylow
141C2×C22⋊Q8—2711no—
142C2×#117—2711no—
143#485—2712no—
144#485—2712no—
145#484—2711no—
146#484—2711no—
147#483—2711no—
148#483D27111yescharacteristic, complemented
149#428—2712no—
150#419—2712no—
151#407—2711no—
152C22×Q8—2711no—
153C23×C4—2711no—
154C2×M16—2712no—
155C22⋊Q8—2711no—
156C22⋊Q8—2711no—
157C22⋊Q8—2711no—
158C22⋊Q8—5421no—
159C22⋊Q8—5421no—
160C22⋊Q8C2×D27111yescharacteristic
161#117—2712no—
162#117—2712no—
163#117—5421no—
164#117—5421no—
165C2×C4⋊C4—2711no—
166C2×C4⋊C4—2711no—
167C2×C4⋊C4—5421no—
168C2×C4⋊C4C2×D27111yescharacteristic
169C2×C22⋊C4—5421no—
170C2×C22⋊C4—5421no—
171C2×C42—5421no—
172#107—2712no—
173#107—2712no—
174#106—2712no—
175#106C2×D27112yescomplemented
176#103—2712no—
177#103—2712no—
178#103—2712no—
179#103C2×D27112yescomplemented
180C22⋊C8—2711no—
181C22⋊C8—2712no—
182C22⋊C8C2×D27111yescharacteristic
183C24—2711no—
184C2×Q8—2711no—
185C2×Q8—5421no—
186C2×Q8—5424no—
187C2×Q8C22×D27111yescharacteristic
188C22×C4—21no—
189C22×C4—2711no—
190C22×C4—2711no—
191C22×C4—2711no—
192C22×C4—2711no—
193C22×C4—2711no—
194C22×C4—5424no—
195C22×C4—5421no—
196C22×C4—5421no—
197C22×C4—5421no—
198C22×C4C22×D27111yescharacteristic
199M16—5424no—
200C2×C8—2712no—
201C2×C8C22×D27112yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—2712no—
205C4⋊C4—2711no—
206C4⋊C4—5421no—
207C4⋊C4—5421no—
208C4⋊C4—5421no—
209C4⋊C4—5421no—
210C4⋊C4—5421no—
211C4⋊C4C22×D27112yes—
212C4⋊C4C22×D27111yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—5422no—
215C22⋊C4—5422no—
216C22⋊C4—5421no—
217C22⋊C4—5421no—
218C22⋊C4—5421no—
219C42—5422no—
220C42—5422no—
221C23—2711no—
222C23—2711no—
223C23—5422no—
224C23—5422no—
225C23—10841no—
226C23D4×D27111yescharacteristic
227Q8—22no—
228Q8—5422no—
229Q8—10842no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—2711no—
236C2×C4—2714no—
237C2×C4—2711no—
238C2×C4—5422no—
239C2×C4—5422no—
240C2×C4—5422no—
241C2×C4—5422no—
242C2×C4—5422no—
243C2×C4—5421no—
244C2×C4—5421no—
245C2×C4—10842no—
246C2×C4—10842no—
247C2×C4—10842no—
248C2×C4D4×D27111yescharacteristic
249C2×C4C23×D27111yescharacteristic
250C8—22no—
251C8—5422no—
252C22—21no—
253C22—21no—
254C22—2712no—
255C22—5421no—
256C22—5421no—
257C22—5421no—
258C22—5421no—
259C22—5421no—
260C22—10844no—
261C22#2257618912yes—
262C22C2×D4×D27111yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—2712no—
268C4—5421no—
269C4—10841no—
270C4—10842no—
271C4—10841no—
272C4#41×D27112yes—
273C2—22no—
274C2—5422no—
275C2—10841no—
276C2#123×D27111yescharacteristic
277C2C2×#2257618911yescharacteristic
278C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#483×C271C2
socleC22×C271C22×D4
prime radicalC23×D271D4
hypercenter#483D271
hypocenterC271#3095
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #31774241C2semidirect
4. #31774237C2semidirect
5. #31774233C2semidirect
6. #31774230C2semidirect
7. #31774227C2semidirect
8. #31773936C2semidirect
9. #31773904C2semidirect
11. #483×C271C2semidirect
30. #22576094C22semidirect
31. #22576093C22semidirect
32. #22576091C22semidirect
33. #22576085C22semidirect
34. #22576084C22semidirect
35. #22576083C22semidirect
40. #106×C271C22semidirect
41. #103×C271C22semidirect
139. C271#3095semidirect
148. #483D271semidirect
175. #106C2×D271semidirect
179. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2712
C2×#225761182
#317742412
#317742372
#317742332
#317742302
#317742272
#317739362
#317739042
#317738592
#483×C2712
#3095271