Finite Group Databasebeta

← #117113

Subgroups — #117113

Order 512.

728 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 178 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#21833C212yesmaximal
2C2×#1960C211yesmaximal, characteristic
3#20762C211yesmaximal, characteristic
4#8026C212yesmaximal
5C2×#1281C211yesmaximal, characteristic
6C22×C4⋊Q8C2211yescharacteristic
7C2×#431C2212yes—
8C22×#103—22no—
9C22×#97C2211yescharacteristic
10#2285C2212yes—
11C2×#382C2211yescharacteristic
12#1990—24no—
13#1988—24no—
14#1960—22no—
15#1960C414yescomplemented
16#1836—24no—
17#1823—24no—
18#1735—22no—
19#1294—24no—
20#1281—22no—
21C23×Q8—21no—
22C23×C8D412yes—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—21no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8C2×C412yes—
30C2×C4⋊Q8D412yes—
31C2×C4⋊Q8Q812yes—
32C2×#117—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C42C2311yescharacteristic
36#431—22no—
37#431—22no—
38#431—22no—
39#431—22no—
40C2×C4⋊C8—22no—
41C2×#103—28no—
42C2×#103—48no—
43C2×C22⋊C8D414yes—
44C2×#97—22no—
45C2×#97—22no—
46C2×#97C2×C412yes—
47C2×#97D412yes—
48C2×C4×C8—22no—
49#384D414yes—
50#382—22no—
51#382—22no—
52#382—21no—
53#382C2×C412yes—
54#380—22no—
55#380—22no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C2×#95—22no—
59#340—216no—
60#340—28no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—44no—
68C22×Q8—44no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4C2×D412yes—
72C23×C4C2×D411yescharacteristic
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—24no—
76C22×C8—24no—
77C22×C8—24no—
78C22×C8C22⋊C414yes—
79C4⋊Q8—22no—
80C4⋊Q8—24no—
81C4⋊Q8—22no—
82C4⋊Q8—24no—
83C4⋊Q8—22no—
84C4⋊Q8—24no—
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8C4⋊C414yes—
89#117—44no—
90#117—44no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—41no—
103C2×C22⋊C4—42no—
104C2×C42—42no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D412yes—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×Q811yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4112yes—
113C2×C42#4112yes—
114C4⋊C8—42no—
115C4⋊C8—42no—
116#103—416no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119#97—24no—
120#97—24no—
121C4×C8—42no—
122C4×C8—42no—
123#95—24no—
124#95—42no—
125#95—42no—
126#95—42no—
127C24#12711yescharacteristic
128C2×Q8—24no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—48no—
135C2×Q8—48no—
136C2×Q8—84no—
137C2×Q8—84no—
138C2×Q8—82no—
139C2×Q8—82no—
140C22×C4—44no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4—84no—
149C22×C4—82no—
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4C4×D412yes—
152C22×C4#12012yes—
153C22×C4#12011yescharacteristic
154C22×C4#12112yes—
155C22×C4#12112yes—
156C22×C4#12111yescharacteristic
157C22×C4C22⋊Q812yes—
158C22×C4C22⋊Q811yescharacteristic
159C22×C4#12312yes—
160C22×C4#12311yescharacteristic
161C22×C4#12411yescharacteristic
162C22×C4#13514yes—
163C22×C4AGL(1,ℤ/8)14yes—
164C2×C8—24no—
165C2×C8—24no—
166C2×C8—24no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—48no—
170C2×C8—44no—
171C4⋊C4—28no—
172C4⋊C4—48no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—48no—
178C4⋊C4—84no—
179C4⋊C4—82no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C42—84no—
183C42C2×C4⋊C411yescharacteristic
184C42#11711yescharacteristic
185C42C4×D412yes—
186C42#12111yescharacteristic
187C42#12112yes—
188C42C22⋊Q811yescharacteristic
189C42C22⋊Q812yes—
190C42#12312yes—
191C42#12412yes—
192C42#12612yes—
193C23—24no—
194C23#39011yescharacteristic
195C23#39211yescharacteristic
196C23#39911yescharacteristic
197C23#49512yes—
198C23#49612yes—
199Q8—48no—
200Q8—84no—
201Q8—84no—
202Q8—84no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—84no—
214C2×C4—84no—
215C2×C4—82no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4—164no—
220C2×C4#38011yescharacteristic
221C2×C4#38512yes—
222C2×C4#38611yescharacteristic
223C2×C4#38812yes—
224C2×C4#39311yescharacteristic
225C2×C4#39412yes—
226C2×C4#39612yes—
227C2×C4#39711yescharacteristic
228C2×C4#41612yes—
229C2×C4#41812yes—
230C2×C4#44914yes—
231C2×C4#46314yes—
232C2×C4#47814yes—
233C2×C4#48214yes—
234C2×C4#48812yes—
235C2×C4#49012yes—
236C8—44no—
237C8—84no—
238C22—22no—
239C22—22no—
240C22—24no—
241C22—24no—
242C22#143911yescharacteristic
243C22#195811yescharacteristic
244C22#196111yescharacteristic
245C22#200914yes—
246C22#201212yes—
247C22#209212yes—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—22no—
252C4—84no—
253C4—162no—
254C4—164no—
255C4#180412yes—
256C4#187512yes—
257C4#202512yes—
258C4#207312yes—
259C2—21no—
260C2—22no—
261C2—21no—
262C2#1022614yes—
263C2#1105111yescharacteristic
264C2#1113012yes—
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1960C4semidirect

Maximal subgroups

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

subgroupindex
#218332
C2×#19602
#207622
#80262
C2×#12812