Finite Group Databasebeta

← #141464

Subgroups — #141464

Order 512.

399 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 82 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#27661C211yesmaximal, characteristic
2#13217C212yesmaximal
3#11874C212yesmaximal
4#8609C211yesmaximal, characteristic
5C8×#100C211yesmaximal, characteristic
6C2×C8×D4C2211yescharacteristic
7C22×#107C411yescharacteristic, complemented
8#2486C2212yes—
9C2×#352—24no—
10#2086—21no—
11#2075—21no—
12#1942—21no—
13#1823C412yescomplemented
14#1808—21no—
15#1808—21no—
16#1808C411yescharacteristic, complemented
17#1799C2212yes—
18C4×#100C2211yescharacteristic
19#1460C2211yescharacteristic
20C8×M16—21no—
21#1344—24no—
22#1289—21no—
23#1283—21no—
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#123—22no—
27C2×C4×D4C2311yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C8×D4—41no—
30C8×D4—41no—
31#431—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—41no—
36C2×#107—41no—
37C2×#107—41no—
38C2×C4⋊C8D411yescharacteristic
39C2×#100D412yes—
40C2×#99—22no—
41C2×C22⋊C8C2×C411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C4×M16—21no—
44C2×C4×C8D411yescharacteristic
45#398—22no—
46#397—22no—
47#389—22no—
48#381—22no—
49#380—22no—
50#380—22no—
51#380C2×C412yes—
52#352—48no—
53#343—21no—
54#342—22no—
55#340—22no—
56#340—21no—
57#340—22no—
58#340—41no—
59#323—21no—
60#323D412yes—
61#322—41no—
62C82—41no—
63C22×D4C2×D411yescharacteristic
64C23×C4—22no—
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C2×M16—22no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78#123—44no—
79#123—44no—
80C4×D4—41no—
81C4×D4—41no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4#4111yescharacteristic
90C2×C22⋊C4—22no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4C2×D411yescharacteristic
94C2×C22⋊C4#4111yescharacteristic
95C2×C42C2×D411yescharacteristic
96#107—42no—
97#107—42no—
98#107—41no—
99#107—41no—
100#107—82no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103#100—22no—
104#100D814yes—
105#99—44no—
106C22⋊C8—22no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109#97—41no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—41no—
113C4×C8—41no—
114#95—22no—
115#95—42no—
116#95—42no—
117#95—42no—
118C24C4×D411yescharacteristic
119C24#12711yescharacteristic
120C2×D4—41no—
121C2×D4—41no—
122C2×D4C2×D812yes—
123C2×D4#13512yes—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—44no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C4×D411yescharacteristic
140C22×C4#12111yescharacteristic
141C22×C4#12111yescharacteristic
142C22×C4#12411yescharacteristic
143M16—24no—
144M16—42no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—81no—
158C2×C8#9911yescharacteristic
159C2×C8#9911yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—84no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—44no—
172C22⋊C4—82no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C22⋊C4—82no—
176C42—21no—
177C42—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23#39011yescharacteristic
187C23C4×D812yes—
188C23#49311yescharacteristic
189C23#49511yescharacteristic
190D4—42no—
191D4—42no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—164no—
215C2×C4—162no—
216C2×C4—162no—
217C2×C4—162no—
218C2×C4C2×#9911yescharacteristic
219C2×C4#41011yescharacteristic
220C2×C4#46612yes—
221C8—21no—
222C8—44no—
223C8—41no—
224C8—41no—
225C8—41no—
226C8—82no—
227C8#35212yescomplemented
228C22—21no—
229C22—21no—
230C22—22no—
231C22—21no—
232C22—21no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22#196411yescharacteristic
243C22#197011yescharacteristic
244C22#197611yescharacteristic
245C4—22no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—162no—
251C4—162no—
252C4—322no—
253C4C2×#35211yescharacteristic
254C4#212311yescharacteristic
255C2—21no—
256C2—21no—
257C2—41no—
258C2—42no—
259C2—41no—
260C2#978311yescharacteristic
261C2#1321711yescharacteristic
262C2#1323011yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C22×#107C4semidirect
13. #1823C4semidirect
16. #1808C4semidirect
227. C8#352semidirect

Maximal subgroups

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

subgroupindex
#276612
#132172
#118742
#86092
C8×#1002