Finite Group Databasebeta

← #46839724

Subgroups — #46839724

Order 34688.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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, complemented
2C2×#22576138C212yesmaximal, complemented
3C2×#22576122C211yesmaximal, characteristic
4#31773817C211yesmaximal, characteristic, complemented
5#31773810C211yesmaximal, characteristic, complemented
6#31773808C211yesmaximal, characteristic, complemented
7#31773801C212yesmaximal, complemented
8#31773775C212yesmaximal
9#31773768C212yesmaximal, complemented
10#31773756C211yesmaximal, characteristic, complemented
11#395×C271C211yesmaximal, characteristic, complemented
12C22×Dic542C2211yescharacteristic
13C22×Dic542C2212yes—
14C2×Q8×D271—28no—
15C2×Q8×D271—24no—
16C22×D1084C2211yescharacteristic
17C22×C4×D271C2212yes—
18C22×C4×D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20#22576138—28no—
21#22576122—24no—
22C2×Dic271⋊C4C2212yes—
23C2×Dic271⋊C4C2212yes—
24C2×(C2×D271)⋊C4C2212yes—
25C2×(C2×D271)⋊C4C2211yescharacteristic
26C2×(C2×D271)⋊C4C2212yes—
27C2×(C2×D271)⋊C4C2212yes—
28C2×(C2×D271)⋊C4C2211yescharacteristic
29C2×C4×Dic271C2211yescharacteristic
30C271⋊#95C2211yescharacteristic
31C271⋊#95C2212yes—
32C271⋊#95C2212yes—
33C271⋊#95C2212yes—
34C22×Q8×C271C2211yescharacteristic
35C2×C4⋊C4×C271C2212yes—
36C2×C4⋊C4×C271C2211yescharacteristic
37#95×C271C2211yescharacteristic
38#95×C271C2212yes—
39C23×D271C2311yescharacteristic
40C23×D271C2311yescharacteristic
41C2×Dic542—24no—
42C2×Dic542—24no—
43C2×Dic542—24no—
44C2×Dic542—22no—
45C2×Dic542D418yescomplemented
46Q8×D271—232no—
47C2×D1084—22no—
48C2×D1084—22no—
49C2×D1084Q814yescomplemented
50C22×Dic271C2311yescharacteristic
51C22×Dic271C2312yes—
52C22×Dic271C2312yes—
53C22×Dic271C2311yescharacteristic
54C2×C4×D271—24no—
55C2×C4×D271—24no—
56C2×C4×D271—24no—
57C2×C4×D271—24no—
58C2×C4×D271—22no—
59C2×C4×D271—24no—
60C2×C4×D271D414yes—
61Dic271⋊C4—24no—
62Dic271⋊C4—24no—
63(C2×D271)⋊C4—22no—
64(C2×D271)⋊C4—24no—
65(C2×D271)⋊C4—24no—
66(C2×D271)⋊C4—24no—
67(C2×D271)⋊C4—22no—
68C4×Dic271—22no—
69C2×Q8×C271—24no—
70C2×Q8×C271—22no—
71C22×C4×C271C2311yescharacteristic
72C22×C4×C271C2312yes—
73C22×C4×C271C2312yes—
74C22×C4×C271C2311yescharacteristic
75C22×C4×C271C2311yescharacteristic
76C4⋊C4×C271—24no—
77C4⋊C4×C271—22no—
78C22×D271—22no—
79C22×D271—22no—
80C22×D271—22no—
81C22×D271—21no—
82C22×D271—21no—
83C22×D271—22no—
84C22×D271C2×D412yes—
85C22×D271C2×Q812yes—
86C22×D271C2×Q812yescomplemented
87C22×D271#4112yes—
88Dic542—28no—
89Dic542—216no—
90D1084—28no—
91C2×Dic271—22no—
92C2×Dic271—22no—
93C2×Dic271—21no—
94C2×Dic271—22no—
95C2×Dic271—22no—
96C2×Dic271—22no—
97C2×Dic271—22no—
98C2×Dic271C2×D414yes—
99C2×Dic271C2×D412yes—
100C2×Dic271C2×D414yes—
101C4×D271—216no—
102C4×D271—28no—
103C4×D271—44no—
104C23×C271C2411yescharacteristic
105Q8×C271—28no—
106C2×C4×C271—21no—
107C2×C4×C271—22no—
108C2×C4×C271—22no—
109C2×C4×C271—22no—
110C2×C4×C271—21no—
111C2×C4×C271—22no—
112C2×C4×C271—21no—
113C2×C4×C271—22no—
114C2×C4×C271—22no—
115C2×C4×C271—22no—
116C2×C4×C271C2×D414yes—
117C2×C4×C271C2×D412yes—
118C2×C4×C271C2×Q812yescomplemented
119C22×C271C22×D411yescharacteristic
120C22×C271C22×D412yes—
121C22×C271C22×Q811yescharacteristic
122C22×C271C2×#4111yescharacteristic
123C22×C2712−1+412yes—
124C2×D271—22no—
125C2×D271—22no—
126C2×D271—24no—
127C2×D271—24no—
128C2×D271—22no—
129C2×D271—24no—
130C2×D271—41no—
131C2×D271—41no—
132C2×D271—42no—
133C2×D271C22⋊Q814yescomplemented
134C4×C271—24no—
135C4×C271—22no—
136C4×C271—22no—
137C4×C271—41no—
138C4×C271—42no—
139Dic271—24no—
140Dic271—41no—
141Dic271—42no—
142Dic271#12014yes—
143C2×C271C2×#12011yescharacteristic
144C2×C271C2×C22⋊Q811yescharacteristic
145C2×C271#53611yescharacteristic
146C2×C271#54812yes—
147C2×C271D4×Q812yes—
148D271—24no—
149D271—42no—
150C271#240911yescharacteristic, complemented, Sylow
151#2409—2711nomaximal, Sylow
152C23×Q8—2711no—
153C2×C22⋊Q8—2712no—
154C2×C4×D4—2711no—
155#395—2711no—
156#395D27111yescharacteristic, complemented
157#393—2711no—
158#393—2711no—
159#391—2712no—
160#386—2712no—
161#385—2712no—
162#380—2711no—
163C22×Q8—2712no—
164C22×Q8—2711no—
165C22×Q8—5424no—
166C22×Q8—5428no—
167C22×Q8C2×D27111yescharacteristic
168C22×D4—2711no—
169C23×C4—2711no—
170C23×C4—2711no—
171C23×C4—2712no—
172C22⋊Q8—5428no—
173C4×D4—5424no—
174C2×C4⋊C4—2712no—
175C2×C4⋊C4—2712no—
176C2×C4⋊C4C2×D27111yescharacteristic
177C2×C4⋊C4C2×D27112yes—
178C2×C22⋊C4—2711no—
179C2×C22⋊C4—2711no—
180C2×C22⋊C4—2712no—
181C2×C22⋊C4—2712no—
182C2×C22⋊C4—2712no—
183C2×C42—2711no—
184#95—2711no—
185#95—2712no—
186#95—2712no—
187#95—2712no—
188#95C2×D27111yescharacteristic
189#95C2×D27112yes—
190C24—2711no—
191C24—2711no—
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—2718no—
195C2×Q8—5424no—
196C2×Q8—5424no—
197C2×Q8—5424no—
198C2×Q8—54232no—
199C2×Q8—5422no—
200C2×D4—2714no—
201C2×D4—5422no—
202C2×D4—5422no—
203C22×C4—2711no—
204C22×C4—2712no—
205C22×C4—2714no—
206C22×C4—2711no—
207C22×C4—2712no—
208C22×C4—5424no—
209C22×C4—5424no—
210C22×C4—5422no—
211C22×C4—5424no—
212C22×C4—5424no—
213C22×C4—5424no—
214C22×C4C22×D27112yes—
215C22×C4C22×D27111yescharacteristic
216C22×C4C22×D27111yescharacteristic
217C22×C4C22×D27112yes—
218C22×C4C22×D27111yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—5424no—
222C4⋊C4—5424no—
223C22⋊C4—5424no—
224C22⋊C4—5422no—
225C22⋊C4—5424no—
226C22⋊C4—5422no—
227C22⋊C4—5424no—
228C42—5422no—
229C23—2712no—
230C23—2712no—
231C23—2712no—
232C23—2712no—
233C23—5422no—
234C23—5422no—
235C23—5422no—
236C23—5421no—
237C23—5421no—
238C23—5422no—
239C23C23×D27111yescharacteristic
240Q8—28no—
241Q8—54216no—
242Q8—5428no—
243D4—5428no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—2712no—
255C2×C4—2714no—
256C2×C4—2714no—
257C2×C4—5428no—
258C2×C4—54216no—
259C2×C4—5422no—
260C2×C4—5422no—
261C2×C4—5421no—
262C2×C4—5422no—
263C2×C4—5422no—
264C2×C4—5422no—
265C2×C4—5422no—
266C2×C4—10844no—
267C2×C4D4×D27112yes—
268C2×C4D4×D27114yes—
269C2×C4Q8×D27112yescomplemented
270C22—2714no—
271C22—5422no—
272C22—5424no—
273C22—5424no—
274C22—5422no—
275C22—5424no—
276C22—5422no—
277C22—10841no—
278C22—10842no—
279C22—10841no—
280C22C2×D4×D27111yescharacteristic
281C22C2×D4×D27112yes—
282C22C2×Q8×D27111yescharacteristic
283C22#41×D27111yescharacteristic
284C22#2257620812yes—
285C4—22no—
286C4—24no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—2714no—
291C4—5424no—
292C4—10841no—
293C4—10842no—
294C2—5424no—
295C2—10842no—
296C2#120×D27111yescharacteristic
297C2C22⋊Q8×D27111yescharacteristic
298C2#3177447211yescharacteristic
299C2#3177455712yes—
300C2#3177456212yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#395×C271C2
prime radicalC22×D1084C22
hypercenter#395D271
hypocenterC271#2409
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D271C2semidirect
2. C2×#22576138C2semidirect
4. #31773817C2semidirect
5. #31773810C2semidirect
6. #31773808C2semidirect
7. #31773801C2semidirect
9. #31773768C2semidirect
10. #31773756C2semidirect
11. #395×C271C2semidirect
45. C2×Dic542D4semidirect
49. C2×D1084Q8semidirect
86. C22×D271C2×Q8semidirect
118. C2×C4×C271C2×Q8semidirect
133. C2×D271C22⋊Q8semidirect
150. C271#2409semidirect
156. #395D271semidirect
269. C2×C4Q8×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×#225761382
C2×#225761222
#317738172
#317738102
#317738082
#317738012
#317737752
#317737682
#317737562
#395×C2712
#2409271