Finite Group Databasebeta

← #46840031

Subgroups — #46840031

Order 34688.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#22576138C212yesmaximal, complemented
2C2×C22⋊C4×D271C212yesmaximal
3#31773820C212yesmaximal, complemented
4#31773809C211yesmaximal, characteristic, complemented
5#31773771C212yesmaximal, complemented
6#31773768C212yesmaximal, complemented
7#31773754C212yesmaximal, complemented
8#31773752C211yesmaximal, characteristic
9#399×C271C211yesmaximal, characteristic, complemented
10C24×D271C2211yescharacteristic
11C22×Dic542C2211yescharacteristic, complemented
12C22×C4×D271C2212yes—
13C22×C4×D271C2212yes—
14#22576138—28no—
15C2×C4⋊Dic271C2212yescomplemented
16C2×Dic271⋊C4C2212yescomplemented
17C2×Dic271⋊C4C2212yescomplemented
18C2×C22⋊Dic271C2212yes—
19C2×C22⋊Dic271C2211yescharacteristic
20C2×(C2×D271)⋊C4C2212yes—
21C2×(C2×D271)⋊C4C2212yes—
22C2×(C2×D271)⋊C4C2212yes—
23C2×(C2×D271)⋊C4C2212yes—
24C22⋊C4×D271—28no—
25C271⋊#95C2211yescharacteristic, complemented
26C271⋊#95C2212yescomplemented
27C271⋊#95C2212yescomplemented
28C2×C4⋊C4×C271C2212yescomplemented
29C2×C4⋊C4×C271C2211yescharacteristic, complemented
30C2×C22⋊C4×C271C2212yes—
31C2×C22⋊C4×C271C2211yescharacteristic
32#95×C271C2211yescharacteristic, complemented
33C23×D271—24no—
34C23×D271—24no—
35C23×D271—41no—
36C23×D271—41no—
37C23×D271Q814yescomplemented
38C23×D271C2312yes—
39C2×Dic542—24no—
40C2×Dic542—41no—
41C22×Dic271C2312yes—
42C22×Dic271C2312yes—
43C22×Dic271C2312yes—
44C2×C4×D271—24no—
45C2×C4×D271—24no—
46C2×C4×D271—24no—
47C2×C4×D271—24no—
48C2×C4×D271—42no—
49C2×C4×D271—42no—
50C4⋊Dic271—42no—
51Dic271⋊C4—24no—
52Dic271⋊C4—24no—
53C22⋊Dic271—24no—
54C22⋊Dic271—41no—
55(C2×D271)⋊C4—24no—
56(C2×D271)⋊C4—24no—
57(C2×D271)⋊C4—24no—
58(C2×D271)⋊C4D418yes—
59C24×C271C2311yescharacteristic
60C22×C4×C271C2312yes—
61C22×C4×C271C2311yescharacteristic
62C22×C4×C271C2311yescharacteristic
63C22×C4×C271C2312yes—
64C4⋊C4×C271—24no—
65C4⋊C4×C271—41no—
66C22⋊C4×C271—24no—
67C22⋊C4×C271—41no—
68C22×D271—22no—
69C22×D271—22no—
70C22×D271—22no—
71C22×D271—22no—
72C22×D271—216no—
73C22×D271—44no—
74C22×D271—44no—
75C22×D271—48no—
76C22×D271—44no—
77C22×D271C2×D414yes—
78C22×D271C2×Q814yes—
79C22×D271#4114yes—
80Dic542—44no—
81C2×Dic271—22no—
82C2×Dic271—22no—
83C2×Dic271—22no—
84C2×Dic271—22no—
85C2×Dic271—22no—
86C2×Dic271—22no—
87C2×Dic271—22no—
88C2×Dic271—22no—
89C2×Dic271C2×D414yes—
90C4×D271—48no—
91C4×D271—48no—
92C23×C271—22no—
93C23×C271—22no—
94C23×C271—21no—
95C23×C271—21no—
96C23×C271C2×Q812yes—
97C23×C271C2411yescharacteristic
98C2×C4×C271—22no—
99C2×C4×C271—22no—
100C2×C4×C271—22no—
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271—21no—
105C2×C4×C271—21no—
106C2×C4×C271—22no—
107C2×C4×C271C2×D414yes—
108C22×C271—24no—
109C22×C271—22no—
110C22×C271—41no—
111C22×C271—42no—
112C22×C271—41no—
113C22×C271C22×D412yes—
114C22×C271C22×Q811yescharacteristic
115C22×C271C2×#4112yes—
116C22×C271H(5,𝔽2)11yescharacteristic
117C22×C271H(5,𝔽2)11yescharacteristic
118C2×D271—24no—
119C2×D271—24no—
120C2×D271—24no—
121C2×D271—24no—
122C2×D271—24no—
123C2×D271—24no—
124C2×D271—416no—
125C2×D271C22⋊Q818yescomplemented
126C4×C271—24no—
127C4×C271—42no—
128C4×C271—41no—
129C4×C271—41no—
130Dic271—24no—
131Dic271—42no—
132Dic271—42no—
133C2×C271—42no—
134C2×C271C2×C22⋊Q812yes—
135C2×C271#54311yescharacteristic
136C2×C271D4211yescharacteristic
137C2×C271#54612yes—
138C2×C271#55111yescharacteristic
139D271—28no—
140C271#243711yescharacteristic, complemented, Sylow
141#2437—2711nomaximal, Sylow
142C2×C22⋊Q8—2712no—
143C22×C22⋊C4—2712no—
144#399D27111yescharacteristic, complemented
145#396—2712no—
146#393—2711no—
147#385—2712no—
148#385—2712no—
149#380—2712no—
150#379—2711no—
151C25—2711no—
152C22×Q8—2711no—
153C23×C4—2712no—
154C23×C4—2712no—
155C22⋊Q8—5428no—
156C2×C4⋊C4—2712no—
157C2×C4⋊C4—2712no—
158C2×C4⋊C4—2712no—
159C2×C4⋊C4C2×D27112yescomplemented
160C2×C4⋊C4C2×D27111yescharacteristic, complemented
161C2×C22⋊C4—2712no—
162C2×C22⋊C4—2712no—
163C2×C22⋊C4—2712no—
164C2×C22⋊C4—2712no—
165C2×C22⋊C4—2712no—
166C2×C22⋊C4—2711no—
167C2×C22⋊C4—5428no—
168C2×C22⋊C4C2×D27112yes—
169C2×C22⋊C4C2×D27111yescharacteristic
170#95—2711no—
171#95—2712no—
172#95—2712no—
173#95C2×D27111yescharacteristic, complemented
174C24—2712no—
175C24—2714no—
176C24—5424no—
177C24—5424no—
178C24—10841no—
179C24—10841no—
180C24C22×D27111yescharacteristic
181C2×Q8—5424no—
182C2×Q8—10841no—
183C22×C4—2712no—
184C22×C4—2712no—
185C22×C4—2712no—
186C22×C4—5424no—
187C22×C4—5424no—
188C22×C4—5424no—
189C22×C4—5424no—
190C22×C4—10842no—
191C22×C4—10842no—
192C22×C4C22×D27111yescharacteristic
193C22×C4C22×D27111yescharacteristic
194C22×C4C22×D27112yes—
195C22×C4C22×D27112yes—
196C4⋊C4—24no—
197C4⋊C4—41no—
198C4⋊C4—5424no—
199C4⋊C4—5424no—
200C4⋊C4—10842no—
201C22⋊C4—24no—
202C22⋊C4—41no—
203C22⋊C4—2718no—
204C22⋊C4—5424no—
205C22⋊C4—5424no—
206C22⋊C4—5424no—
207C22⋊C4—5424no—
208C22⋊C4—10841no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—2714no—
214C23—2714no—
215C23—2714no—
216C23—54216no—
217C23—5422no—
218C23—5422no—
219C23—5422no—
220C23—5422no—
221C23—10844no—
222C23—10844no—
223C23—10848no—
224C23—10844no—
225C23Q8×D27112yes—
226C23C23×D27111yescharacteristic
227Q8—10844no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—2714no—
238C2×C4—5422no—
239C2×C4—5422no—
240C2×C4—5422no—
241C2×C4—5422no—
242C2×C4—5422no—
243C2×C4—5422no—
244C2×C4—5422no—
245C2×C4—5422no—
246C2×C4—10848no—
247C2×C4—10848no—
248C2×C4D4×D27114yes—
249C22—22no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—2718no—
255C22—5424no—
256C22—5424no—
257C22—5424no—
258C22—5424no—
259C22—5424no—
260C22—5424no—
261C22—108416no—
262C22C2×D4×D27112yes—
263C22C2×Q8×D27111yescharacteristic
264C22#41×D27112yes—
265C22#2257620511yescharacteristic
266C22#2257620611yescharacteristic
267C4—24no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—5424no—
272C4—10842no—
273C4—10842no—
274C2—42no—
275C2—5428no—
276C2C22⋊Q8×D27112yes—
277C2#3177452211yescharacteristic
278C2#3177453211yescharacteristic
279C2#3177453912yes—
280C2#3177457511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#399×C271C2
prime radicalC24×D271C22
hypercenter#399D271
hypocenterC271#2437
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#22576138C2semidirect
3. #31773820C2semidirect
4. #31773809C2semidirect
5. #31773771C2semidirect
6. #31773768C2semidirect
7. #31773754C2semidirect
9. #399×C271C2semidirect
11. C22×Dic542C22semidirect
15. C2×C4⋊Dic271C22semidirect
16. C2×Dic271⋊C4C22semidirect
17. C2×Dic271⋊C4C22semidirect
25. C271⋊#95C22semidirect
26. C271⋊#95C22semidirect
27. C271⋊#95C22semidirect
28. C2×C4⋊C4×C271C22semidirect
29. C2×C4⋊C4×C271C22semidirect
32. #95×C271C22semidirect
37. C23×D271Q8semidirect
125. C2×D271C22⋊Q8semidirect
140. C271#2437semidirect
144. #399D271semidirect
159. C2×C4⋊C4C2×D271semidirect
160. C2×C4⋊C4C2×D271semidirect
173. #95C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761382
C2×C22⋊C4×D2712
#317738202
#317738092
#317737712
#317737682
#317737542
#317737522
#399×C2712
#2437271