Finite Group Databasebeta

← #116917

Subgroups — #116917

Order 512.

650 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 134 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#21820C211yesmaximal, characteristic
2#20740C212yesmaximal
3#17818C211yesmaximal, characteristic
4#7981C211yesmaximal, characteristic
5#7952C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10#2285C2212yes—
11#1985C414yescomplemented
12#1984—22no—
13#1823—22no—
14#1817—22no—
15#1803—24no—
16#1787—22no—
17#1763—24no—
18#1573—24no—
19#1568—22no—
20#1289—22no—
21#1285—24no—
22C23×Q8D411yescharacteristic
23C2×C4×Q8—21no—
24C2×C4×Q8—21no—
25C2×C4×Q8—22no—
26C2×C4×Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8C2×C412yes—
30C2×C4×Q8Q812yes—
31C2×#117—22no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4D411yescharacteristic
34C22×C42—21no—
35C22×C42C2311yescharacteristic
36#432—22no—
37#432—22no—
38#432—22no—
39#432—22no—
40#432—22no—
41#432—22no—
42#431—22no—
43#431—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2×C412yes—
46C2×C4⋊C8D414yes—
47C2×#103—28no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8D412yes—
51C2×#97—22no—
52C2×#97C2×C412yes—
53C2×C4×C8D412yes—
54#384—22no—
55#382—22no—
56#380—22no—
57#380—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4D414yes—
60#340—28no—
61#336—28no—
62#335—24no—
63#334—24no—
64C22×Q8—21no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—44no—
68C22×Q8C4⋊C412yes—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4#4111yescharacteristic
74C22×C8—22no—
75C22×C8—24no—
76C22×C8C22⋊C414yes—
77C4×Q8—24no—
78C4×Q8—24no—
79C4×Q8—24no—
80C4×Q8—24no—
81C4×Q8—41no—
82C4×Q8—42no—
83C4×Q8—41no—
84C4×Q8—41no—
85C4×Q8—42no—
86C4×Q8—41no—
87#117—82no—
88#117—82no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4C4⋊C412yes—
100C2×C22⋊C4—42no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—24no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D412yes—
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×Q811yescharacteristic
111C2×C42#4112yes—
112C2×C42#4112yes—
113C2×C42#4111yescharacteristic
114C4⋊C8—24no—
115C4⋊C8—24no—
116C4⋊C8—24no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#103—416no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#97—24no—
125#97—42no—
126#97—42no—
127C4×C8—22no—
128C4×C8—22no—
129#95—24no—
130#95—42no—
131C24#12111yescharacteristic
132C2×Q8—24no—
133C2×Q8—28no—
134C2×Q8—24no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—42no—
138C2×Q8—48no—
139C2×Q8—41no—
140C2×Q8—41no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—44no—
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×C4⋊C411yescharacteristic
157C22×C4#11711yescharacteristic
158C22×C4#12012yes—
159C22×C4#12112yes—
160C22×C4#12111yescharacteristic
161C22×C4#12112yes—
162C22×C4C22⋊Q811yescharacteristic
163C22×C4C22⋊Q811yescharacteristic
164C22×C4#12312yes—
165C22×C4#12312yes—
166C22×C4#12412yes—
167C2×C8—24no—
168C2×C8—44no—
169C2×C8—48no—
170C2×C8—44no—
171C2×C8—42no—
172C2×C8—44no—
173C4⋊C4—24no—
174C4⋊C4—48no—
175C4⋊C4—44no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C42—42no—
185C42—42no—
186C42—44no—
187C42—84no—
188C42C4×D412yes—
189C42C4×D412yes—
190C42#12011yescharacteristic
191C42#12112yes—
192C42C22⋊Q812yes—
193C42C22⋊Q812yes—
194C42#12311yescharacteristic
195C42#12411yescharacteristic
196C42#12612yes—
197C42#12711yescharacteristic
198C23—41no—
199C23—41no—
200C23#38011yescharacteristic
201C23#39412yes—
202C23#47112yes—
203C23#47211yescharacteristic
204C23#47311yescharacteristic
205Q8—28no—
206Q8—44no—
207Q8—44no—
208Q8—44no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—48no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4#38512yes—
238C2×C4#38611yescharacteristic
239C2×C4#38812yes—
240C2×C4#39011yescharacteristic
241C2×C4#39211yescharacteristic
242C2×C4#39311yescharacteristic
243C2×C4#39612yes—
244C2×C4#39711yescharacteristic
245C2×C4#39911yescharacteristic
246C8—44no—
247C8—82no—
248C8—84no—
249C22—22no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—42no—
254C22—41no—
255C22—41no—
256C22#143911yescharacteristic
257C22#181511yescharacteristic
258C22#181611yescharacteristic
259C22#205112yes—
260C22#205212yes—
261C4—21no—
262C4—22no—
263C4—22no—
264C4—21no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—44no—
269C4—82no—
270C4—164no—
271C4#190912yes—
272C4#201012yes—
273C2—22no—
274C2—22no—
275C2—21no—
276C2—21no—
277C2#1107411yescharacteristic
278C2#1110011yescharacteristic
279C2#1110511yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1985C4semidirect

Maximal subgroups

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

subgroupindex
#218202
#207402
#178182
#79812
#79522