Finite Group Databasebeta

← #134792

Subgroups — #134792

Order 512.

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

normal subgroups only

This group has 329 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#21714C211yesmaximal, characteristic
2#12802C212yesmaximal
3#8617C211yesmaximal, characteristic
4#8615C211yesmaximal, characteristic
5#7238C212yesmaximal
6C2×#446C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9C2×#356—22no—
10#1984C411yescharacteristic
11#1984C2211yescharacteristic
12#1940—21no—
13#1940—21no—
14#1842C411yescharacteristic
15#1801C2212yes—
16#1801C2212yes—
17#1771—21no—
18#1771—21no—
19C4×#97C411yescharacteristic
20#1567—21no—
21#1566—21no—
22#1394—22no—
23#1292—21no—
24#1292—21no—
25C2×C4⋊Q8D411yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C2×C4×Q8C2311yescharacteristic
28#446—41no—
29#446—41no—
30C8×Q8—41no—
31C8×Q8—41no—
32#423—22no—
33#423—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×#101—22no—
42C2×#101—22no—
43C2×#97—21no—
44C2×#97—21no—
45C2×#97—21no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#391D411yescharacteristic
51#391Q811yescharacteristic
52#389—21no—
53#389—21no—
54#384Q811yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C43C2×C411yescharacteristic
58#356—44no—
59#343—22no—
60#343—22no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#335—41no—
66#335—41no—
67#334—41no—
68#334—41no—
69C22×Q8C4⋊C411yescharacteristic
70C22×Q8C2×D411yescharacteristic
71C2×M16—42no—
72C2×M16—42no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81C4×Q8—41no—
82C4×Q8—41no—
83C4×Q8—41no—
84C4×Q8—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C4⋊C411yescharacteristic
91C2×C4⋊C4C2×Q811yescharacteristic
92C2×C4⋊C4#4111yescharacteristic
93C2×C4⋊C4#4111yescharacteristic
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C4⋊C411yescharacteristic
98C2×C42C4⋊C411yescharacteristic
99C2×C42C22×C411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113#101—42no—
114#101—42no—
115#101—42no—
116#101—42no—
117#97—24no—
118#97—42no—
119#97—41no—
120#97—41no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—41no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C2×Q8—22no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8AGL(1,ℤ/8)12yes—
138C2×Q8#13712yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×C4⋊C411yescharacteristic
145C22×C4#11711yescharacteristic
146C22×C4#12111yescharacteristic
147C22×C4#12111yescharacteristic
148C22×C4C22⋊Q811yescharacteristic
149C22×C4C22⋊Q811yescharacteristic
150M16—82no—
151M16—82no—
152M16—82no—
153M16—82no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—82no—
168C2×C8—82no—
169C2×C8#9911yescharacteristic
170C2×C8#9911yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C42—22no—
182C42—22no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—22no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—81no—
194C23#38011yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4C2×#9911yescharacteristic
225C2×C4#41011yescharacteristic
226C2×C4#42812yes—
227C2×C4#46812yes—
228C2×C4#46911yescharacteristic
229C2×C4#47011yescharacteristic
230C8—21no—
231C8—42no—
232C8—41no—
233C8—41no—
234C8—41no—
235C8—81no—
236C8—162no—
237C8—162no—
238C8#35612yescomplemented
239C22—21no—
240C22—21no—
241C22#179911yescharacteristic
242C22#180911yescharacteristic
243C22#181211yescharacteristic
244C4—21no—
245C4—21no—
246C4—22no—
247C4—22no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—81no—
256C4C2×#35611yescharacteristic
257C4#213211yescharacteristic
258C2—21no—
259C2—21no—
260C2#861011yescharacteristic
261C2#1280211yescharacteristic
262C2#1281011yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#380
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
238. C8#356semidirect

Maximal subgroups

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

subgroupindex
#217142
#128022
#86172
#86152
#72382