Finite Group Databasebeta

← #46814051

Subgroups — #46814051

Order 34432.

464 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 213 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#31770512C212yesmaximal, complemented
2#31770504C212yesmaximal, complemented
3#31770498C212yesmaximal
4#31770498C212yesmaximal
5#31770491C212yesmaximal, complemented
6#31770491C212yesmaximal, complemented
7#31770468C211yesmaximal, characteristic
8#31770465C211yesmaximal, characteristic
9C2×C22⋊Q8×C269C211yesmaximal, characteristic
10C2×C4⋊Dic269C2212yes—
11C2×C4⋊Dic269C2212yes—
12C2×C4⋊Dic269C2212yes—
13C2×C22⋊Dic269C2211yescharacteristic
14C2×C22⋊Dic269C2211yescharacteristic
15C2×C22⋊Dic269C2212yes—
16C2×C4×Dic269C2212yes—
17C2×C4×Dic269C2212yes—
18#22575360C2212yes—
19#22575360C2211yescharacteristic
20#22575360C2212yes—
21#22575360C2211yescharacteristic
22#22575360C2212yes—
23#22575360C2212yes—
24#22575360C2212yes—
25#22575360C2212yes—
26C22×Q8×C269C2211yescharacteristic
27C23×C4×C269C2211yescharacteristic
28C22⋊Q8×C269C418yescomplemented
29C2×C4⋊C4×C269C2211yescharacteristic
30C2×C4⋊C4×C269C2212yes—
31C2×C22⋊C4×C269C2212yes—
32C22×Dic269C2312yes—
33C22×Dic269C2312yes—
34C22×Dic269C2312yes—
35C22×Dic269C2312yes—
36C4⋊Dic269—42no—
37C4⋊Dic269—42no—
38C4⋊Dic269—42no—
39C22⋊Dic269—42no—
40C22⋊Dic269—41no—
41C22⋊Dic269—41no—
42C4×Dic269—42no—
43C4×Dic269—42no—
44C24×C269C2311yescharacteristic
45C2×Q8×C269—42no—
46C2×Q8×C269C2×C414yes—
47C22×C4×C269—42no—
48C22×C4×C269C2×C414yes—
49C22×C4×C269C2311yescharacteristic
50C22×C4×C269C2312yes—
51C22×C4×C269C2312yes—
52C22×C4×C269C2311yescharacteristic
53C4⋊C4×C269C2×C414yes—
54C4⋊C4×C269C2×C418yes—
55C22⋊C4×C269C2×C418yes—
56C2×Dic269—22no—
57C2×Dic269—22no—
58C2×Dic269—22no—
59C2×Dic269—22no—
60C2×Dic269—22no—
61C2×Dic269—22no—
62C2×Dic269—22no—
63C2×Dic269—22no—
64C2×Dic269—22no—
65C2×Dic269—22no—
66C2×Dic269—22no—
67C2×Dic269—22no—
68C23×C269—22no—
69C23×C269—21no—
70C23×C269—21no—
71C23×C269—22no—
72C23×C269C22×C412yes—
73C23×C269C2411yescharacteristic
74Q8×C269—44no—
75C2×C4×C269—22no—
76C2×C4×C269—22no—
77C2×C4×C269—22no—
78C2×C4×C269—22no—
79C2×C4×C269—22no—
80C2×C4×C269—22no—
81C2×C4×C269—44no—
82C2×C4×C269C22×C414yes—
83C2×C4×C269C22×C414yes—
84C2×C4×C269C22×C412yes—
85C2×C4×C269C22×C412yes—
86C22×C269—22no—
87C22×C269—22no—
88C22×C269—22no—
89C22×C269—42no—
90C22×C269—42no—
91C22×C269C23×C411yescharacteristic
92C22×C269H(5,𝔽2)12yes—
93C22×C269H(5,𝔽2)11yescharacteristic
94C22×C2692−1+411yescharacteristic
95C22×C2692−1+412yes—
96C4×C269—22no—
97C4×C269—22no—
98C4×C269—24no—
99C4×C269—24no—
100Dic269—42no—
101Dic269—42no—
102Dic269—42no—
103Dic269—42no—
104C2×C269—42no—
105C2×C269#51811yescharacteristic
106C2×C269#51911yescharacteristic
107C2×C269#52011yescharacteristic
108C2×C269#56212yes—
109C2×C269#56312yes—
110C269#237811yescharacteristic, complemented, Sylow
111#2378—2691nomaximal, Sylow
112C2×C22⋊Q8D26911yescharacteristic
113#391—2692no—
114#389—2692no—
115#388—2692no—
116#388—2692no—
117#387—2692no—
118#387—2692no—
119#381—2691no—
120#380—2691no—
121C22×Q8C2×D26911yescharacteristic
122C23×C4C2×D26911yescharacteristic
123C22⋊Q8Dic26918yescomplemented
124C2×C4⋊C4—2692no—
125C2×C4⋊C4—2692no—
126C2×C4⋊C4—2692no—
127C2×C4⋊C4C2×D26911yescharacteristic
128C2×C4⋊C4C2×D26912yes—
129C2×C22⋊C4—2691no—
130C2×C22⋊C4—2691no—
131C2×C22⋊C4—2692no—
132C2×C22⋊C4C2×D26912yes—
133C2×C42—2692no—
134C2×C42—2692no—
135#95—2692no—
136#95—2691no—
137#95—2692no—
138#95—2691no—
139#95—2692no—
140#95—2692no—
141#95—2692no—
142#95—2692no—
143C24C22×D26911yescharacteristic
144C2×Q8—42no—
145C2×Q8C2×Dic26914yes—
146C22×C4—42no—
147C22×C4—2692no—
148C22×C4—2692no—
149C22×C4—2692no—
150C22×C4—2692no—
151C22×C4C2×Dic26914yes—
152C22×C4C22×D26911yescharacteristic
153C22×C4C22×D26912yes—
154C22×C4C22×D26912yes—
155C22×C4C22×D26911yescharacteristic
156C4⋊C4—10762no—
157C4⋊C4—10762no—
158C4⋊C4—10762no—
159C4⋊C4C2×Dic26914yes—
160C4⋊C4C2×Dic26918yes—
161C22⋊C4—10762no—
162C22⋊C4—10761no—
163C22⋊C4—10761no—
164C22⋊C4C2×Dic26918yes—
165C42—10762no—
166C42—10762no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23C22×Dic26912yes—
172C23C23×D26911yescharacteristic
173Q8—44no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—5382no—
182C2×C4—5382no—
183C2×C4—5382no—
184C2×C4—5382no—
185C2×C4—5382no—
186C2×C4—5382no—
187C2×C4—5382no—
188C2×C4—5382no—
189C2×C4—5382no—
190C2×C4—5382no—
191C2×C4—5382no—
192C2×C4—5382no—
193C2×C4C22×Dic26914yes—
194C2×C4C22×Dic26914yes—
195C2×C4C22×Dic26912yes—
196C2×C4C22×Dic26912yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C23×Dic26911yescharacteristic
203C22#2257549812yes—
204C22#2257549911yescharacteristic
205C22#2257550011yescharacteristic
206C22#2257550112yes—
207C4—22no—
208C4—22no—
209C4—24no—
210C4—24no—
211C4—10762no—
212C4—10762no—
213C4—10762no—
214C4—10762no—
215C2—42no—
216C2#3177108011yescharacteristic
217C2#3177108311yescharacteristic
218C2#3177108811yescharacteristic
219C2#3177135112yes—
220C2#3177135612yes—
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
FittingC2×C22⋊Q8×C269C2
socle, cosocleC23×C269C24
prime radicalC24×C269C23
hypercenterC2×C22⋊Q8D269
hypocenterC269#2378
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #31770512C2semidirect
2. #31770504C2semidirect
5. #31770491C2semidirect
6. #31770491C2semidirect
28. C22⋊Q8×C269C4semidirect
110. C269#2378semidirect
123. C22⋊Q8Dic269semidirect

Maximal subgroups

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

subgroupindex
#317705122
#317705042
#317704982
#317704912
#317704682
#317704652
C2×C22⋊Q8×C2692
#2378269