Finite Group Databasebeta

← #122289

Subgroups — #122289

Order 512.

379 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 86 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#21302C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic
3C2×#1322C211yesmaximal, characteristic
4#7061C214yesmaximal, complemented
5C2×#533C411yescharacteristic
6C22×C4⋊C8C2211yescharacteristic
7C2×C4×M16C411yescharacteristic
8#2112—21no—
9C2×C22⋊C16—21no—
10#1985C412yes—
11#1985C2212yes—
12#1984—21no—
13#1984—21no—
14#1729—21no—
15#1729—21no—
16#1322—21no—
17#1322—21no—
18#1322C2212yes—
19#1322C2212yes—
20C22×M16—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23#533—41no—
24#533—41no—
25#533—41no—
26#533—41no—
27C2×C4⋊Q8C811yescharacteristic
28C2×C4⋊Q8C811yescharacteristic
29C2×#125—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8C812yes—
32C2×C4×Q8—21no—
33C2×C4×Q8C2×C412yes—
34C2×#117—21no—
35C22×C42C2×C411yescharacteristic
36C2×M32—41no—
37C22×C16—41no—
38C2×C4⋊C8—41no—
39C2×C4⋊C8—41no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8C2311yescharacteristic
44C4×M16—41no—
45C4×M16—41no—
46C2×#97C2×C412yes—
47C2×C4×C8—21no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50C22⋊C16—41no—
51C22⋊C16—41no—
52#336—22no—
53#336—22no—
54#336—22no—
55#336—22no—
56C22×Q8—21no—
57C22×Q8C2×C812yes—
58C23×C4C22⋊C411yescharacteristic
59C23×C4C2×C811yescharacteristic
60C23×C4M1611yescharacteristic
61C2×M16—24no—
62C2×M16—42no—
63C2×M16—41no—
64C2×M16—41no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—22no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C4⋊Q8—41no—
79C4⋊Q8—41no—
80#125—42no—
81#125—42no—
82C22⋊Q8—42no—
83C22⋊Q8—42no—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C4×Q8—42no—
87C4×Q8—42no—
88C4×Q8—42no—
89C4×Q8—42no—
90#117—41no—
91#117—41no—
92#117—41no—
93#117—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4C2×C812yes—
100C2×C22⋊C4—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—22no—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C2×C811yescharacteristic
108C2×C42M1611yescharacteristic
109C2×C42C22×C411yescharacteristic
110M32—82no—
111C2×C16—42no—
112C2×C16—42no—
113C2×C16—82no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118#97—42no—
119C4×C8—42no—
120C24C22⋊C811yescharacteristic
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—82no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4—81no—
137C22×C4C22⋊C811yescharacteristic
138C22×C4C22⋊C811yescharacteristic
139C22×C4C22⋊C811yescharacteristic
140C22×C4#9911yescharacteristic
141C22×C4#9911yescharacteristic
142C22×C4#10011yescharacteristic
143C22×C4#10011yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C22×C811yescharacteristic
146C22×C4C2×M1611yescharacteristic
147M16—44no—
148M16—44no—
149C2×C8—24no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—81no—
163C2×C8—81no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—42no—
183C42—42no—
184C16—82no—
185C16—82no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—41no—
190C23#32311yescharacteristic
191C23#32311yescharacteristic
192C23C2×C22⋊C811yescharacteristic
193Q8—84no—
194Q8—84no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4#32311yescharacteristic
221C2×C4#32311yescharacteristic
222C2×C4#35012yes—
223C2×C4#35412yes—
224C2×C4#35612yes—
225C2×C4C2×#9911yescharacteristic
226C2×C4C2×#10011yescharacteristic
227C8—44no—
228C8—41no—
229C8—41no—
230C8—41no—
231C8—41no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—42no—
240C22—42no—
241C22#131912yes—
242C22C2×#32311yescharacteristic
243C22C2×#35011yescharacteristic
244C22#211711yescharacteristic
245C4—22no—
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—82no—
252C4—82no—
253C4#131912yes—
254C4C2×#35411yescharacteristic
255C4C2×#35611yescharacteristic
256C2—22no—
257C2—21no—
258C2—21no—
259C2—41no—
260C2#1142211yescharacteristic
261C2C2×#131911yescharacteristic
262C2#1143611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7061C2semidirect

Maximal subgroups

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

subgroupindex
#213022
#114402
C2×#13222
#70612