Finite Group Databasebeta

← #46840030

Subgroups — #46840030

Order 34688.

780 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 181 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×#22576137C212yesmaximal, complemented
2C22×C22⋊Dic271C212yesmaximal
3#31773833C211yesmaximal, characteristic, complemented
4#31773820C212yesmaximal, complemented
5#31773809C211yesmaximal, characteristic, complemented
6#31773772C212yesmaximal, complemented
7#31773769C212yesmaximal, complemented
8#31773755C212yesmaximal, complemented
9#379×C271C211yesmaximal, characteristic
10C22×Dic542C2211yescharacteristic, complemented
11C23×Dic271C2212yes—
12C23×Dic271C2212yes—
13#22576137—28no—
14C2×C4⋊Dic271C2212yescomplemented
15C2×C4⋊Dic271C2211yescharacteristic, complemented
16C2×Dic271⋊C4C2212yescomplemented
17C2×Dic271⋊C4C2212yescomplemented
18C2×Dic271⋊C4C2212yescomplemented
19C2×C22⋊Dic271—28no—
20C2×C22⋊Dic271C2212yes—
21C2×C22⋊Dic271C2212yes—
22C2×C22⋊Dic271C2212yes—
23C2×C22⋊Dic271C2212yes—
24C271⋊#95C2211yescharacteristic, complemented
25C271⋊#95C2212yescomplemented
26C271⋊#95C2211yescharacteristic, complemented
27C271⋊#95C2212yescomplemented
28C25×C271C2211yescharacteristic
29C2×C22⋊C4×C271C2212yes—
30C2×C22⋊C4×C271C2212yes—
31C2×C22⋊C4×C271C2211yescharacteristic
32C2×C22⋊C4×C271C2211yescharacteristic
33C2×Dic542—24no—
34C2×Dic542—41no—
35C22×Dic271—24no—
36C22×Dic271—24no—
37C22×Dic271—24no—
38C22×Dic271—24no—
39C22×Dic271—42no—
40C22×Dic271—42no—
41C22×Dic271C2312yes—
42C22×Dic271C2312yes—
43C22×Dic271C2312yes—
44C22×Dic271C2312yes—
45C4⋊Dic271—24no—
46C4⋊Dic271—41no—
47Dic271⋊C4—24no—
48Dic271⋊C4—24no—
49Dic271⋊C4—42no—
50C22⋊Dic271—24no—
51C22⋊Dic271—24no—
52C22⋊Dic271—24no—
53C22⋊Dic271D418yes—
54C24×C271—24no—
55C24×C271—24no—
56C24×C271—41no—
57C24×C271—41no—
58C24×C271Q814yescomplemented
59C24×C271C2311yescharacteristic
60C24×C271C2312yes—
61C22×C4×C271C2311yescharacteristic
62C22×C4×C271C2311yescharacteristic
63C22×C4×C271C2312yes—
64C22⋊C4×C271—24no—
65C22⋊C4×C271—24no—
66C22⋊C4×C271—41no—
67C22⋊C4×C271—41no—
68Dic542—44no—
69C2×Dic271—22no—
70C2×Dic271—22no—
71C2×Dic271—22no—
72C2×Dic271—22no—
73C2×Dic271—22no—
74C2×Dic271—22no—
75C2×Dic271—22no—
76C2×Dic271—22no—
77C2×Dic271—22no—
78C2×Dic271—22no—
79C2×Dic271—48no—
80C2×Dic271—48no—
81C2×Dic271C2×D414yes—
82C2×Dic271C2×D414yes—
83C23×C271—22no—
84C23×C271—22no—
85C23×C271—22no—
86C23×C271—22no—
87C23×C271—216no—
88C23×C271—22no—
89C23×C271—22no—
90C23×C271—21no—
91C23×C271—21no—
92C23×C271—44no—
93C23×C271—44no—
94C23×C271—48no—
95C23×C271—44no—
96C23×C271C2×D414yes—
97C23×C271C2×Q812yes—
98C23×C271C2×Q814yes—
99C23×C271#4114yes—
100C23×C271C2411yescharacteristic
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271—22no—
105C2×C4×C271—22no—
106C2×C4×C271—21no—
107C2×C4×C271—21no—
108C22×C271—24no—
109C22×C271—24no—
110C22×C271—24no—
111C22×C271—24no—
112C22×C271—24no—
113C22×C271—24no—
114C22×C271—24no—
115C22×C271—22no—
116C22×C271—416no—
117C22×C271—41no—
118C22×C271—42no—
119C22×C271—41no—
120C22×C271C22⋊Q818yescomplemented
121C22×C271C22×D412yes—
122C22×C271C22×Q811yescharacteristic
123C22×C271C2×#4112yes—
124C22×C271H(5,𝔽2)11yescharacteristic
125C22×C271H(5,𝔽2)11yescharacteristic
126C4×C271—41no—
127C4×C271—42no—
128C4×C271—41no—
129Dic271—24no—
130Dic271—24no—
131Dic271—42no—
132Dic271—42no—
133C2×C271—28no—
134C2×C271—42no—
135C2×C271C2×C22⋊Q812yes—
136C2×C271#54311yescharacteristic
137C2×C271D4211yescharacteristic
138C2×C271#54612yes—
139C2×C271#55111yescharacteristic
140C271#243711yescharacteristic, complemented, Sylow
141#2437—2711nomaximal, Sylow
142C2×C22⋊Q8—2712no—
143C22×C22⋊C4—2712no—
144#399—2711no—
145#396—2712no—
146#393—2711no—
147#385—2712no—
148#385—2712no—
149#380—2712no—
150#379D27111yescharacteristic
151C25C2×D27111yescharacteristic
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⋊C4—2712no—
160C2×C4⋊C4—2711no—
161C2×C22⋊C4—2712no—
162C2×C22⋊C4—2712no—
163C2×C22⋊C4—2712no—
164C2×C22⋊C4—2712no—
165C2×C22⋊C4—5428no—
166C2×C22⋊C4C2×D27112yes—
167C2×C22⋊C4C2×D27112yes—
168C2×C22⋊C4C2×D27111yescharacteristic
169C2×C22⋊C4C2×D27111yescharacteristic
170#95—2711no—
171#95—2712no—
172#95—2712no—
173#95—2711no—
174C24—24no—
175C24—24no—
176C24—41no—
177C24—41no—
178C24Dic54214yescomplemented
179C24C22×D27111yescharacteristic
180C24C22×D27112yes—
181C2×Q8—5424no—
182C2×Q8—10841no—
183C22×C4—2712no—
184C22×C4—2712no—
185C22×C4—2712no—
186C22×C4—2712no—
187C22×C4—5424no—
188C22×C4—5424no—
189C22×C4—5424no—
190C22×C4—5424no—
191C22×C4—10842no—
192C22×C4—10842no—
193C22×C4C22×D27111yescharacteristic
194C22×C4C22×D27112yes—
195C22×C4C22×D27111yescharacteristic
196C4⋊C4—5424no—
197C4⋊C4—5424no—
198C4⋊C4—5424no—
199C4⋊C4—10842no—
200C4⋊C4—10841no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—2718no—
206C22⋊C4—5424no—
207C22⋊C4—5424no—
208C22⋊C4—5424no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—216no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—44no—
222C23D4×D27114yes—
223C23C2×Dic54212yes—
224C23C2×Dic54214yes—
225C23#1686465314yes—
226C23C23×D27111yescharacteristic
227Q8—10844no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—2714no—
236C2×C4—2714no—
237C2×C4—5422no—
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—5422no—
247C2×C4—10848no—
248C2×C4—10848no—
249C22—24no—
250C22—24no—
251C22—24no—
252C22—24no—
253C22—24no—
254C22—24no—
255C22—22no—
256C22—24no—
257C22—416no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22#2257613718yescomplemented
262C22C2×D4×D27112yes—
263C22C22×Dic54211yescharacteristic
264C22C2×#1686465312yes—
265C22#2257620511yescharacteristic
266C22#2257620511yescharacteristic
267C4—41no—
268C4—42no—
269C4—41no—
270C4—5424no—
271C4—5424no—
272C4—10842no—
273C4—10842no—
274C2—28no—
275C2—42no—
276C2C2×#2257613712yes—
277C2#3177452311yescharacteristic
278C2#3177452911yescharacteristic
279C2#3177453412yes—
280C2#3177457811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#379×C271C2
prime radicalC25×C271C22
hypercenter#379D271
hypocenterC271#2437
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#22576137C2semidirect
3. #31773833C2semidirect
4. #31773820C2semidirect
5. #31773809C2semidirect
6. #31773772C2semidirect
7. #31773769C2semidirect
8. #31773755C2semidirect
10. C22×Dic542C22semidirect
14. C2×C4⋊Dic271C22semidirect
15. C2×C4⋊Dic271C22semidirect
16. C2×Dic271⋊C4C22semidirect
17. C2×Dic271⋊C4C22semidirect
18. C2×Dic271⋊C4C22semidirect
24. C271⋊#95C22semidirect
25. C271⋊#95C22semidirect
26. C271⋊#95C22semidirect
27. C271⋊#95C22semidirect
58. C24×C271Q8semidirect
120. C22×C271C22⋊Q8semidirect
140. C271#2437semidirect
178. C24Dic542semidirect
261. C22#22576137semidirect

Maximal subgroups

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

subgroupindex
C2×#225761372
C22×C22⋊Dic2712
#317738332
#317738202
#317738092
#317737722
#317737692
#317737552
#379×C2712
#2437271