Finite Group Databasebeta

← #124337

Subgroups — #124337

Order 512.

423 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 88 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#21414C211yesmaximal, characteristic
2#12850C211yesmaximal, characteristic, complemented
3#12547C211yesmaximal, characteristic, complemented
4#7130C212yesmaximal, complemented
5#7116C212yesmaximal, complemented
6C2×#540C411yescharacteristic
7C22×M32—21no—
8C2×#405C2211yescharacteristic
9#2314C411yescharacteristic
10C2×#349—21no—
11#1992C2211yescharacteristic
12#1991C411yescharacteristic
13#1961C2211yescharacteristic
14#1957C411yescharacteristic
15#1895—21no—
16#1893—21no—
17#1762—21no—
18C4×#102—21no—
19#1376—21no—
20#1375—21no—
21#1375C2212yes—
22#1375C2212yes—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25#540—22no—
26#540—42no—
27#540C814yes—
28C2×#127C2×C411yescharacteristic
29C2×#125C2×C411yescharacteristic
30C2×#123—41no—
31C2×#121—21no—
32C2×#121—41no—
33C2×C4×D4—21no—
34C2×#117C2×C411yescharacteristic
35C2×M32—22no—
36C2×M32—22no—
37C2×M32—22no—
38C2×M32—42no—
39C2×M32—42no—
40C22×C16—22no—
41C2×#102—21no—
42C2×#102—21no—
43C2×#102—21no—
44C2×#102—21no—
45#405—42no—
46C2×#97D411yescharacteristic
47C2×#97C2311yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#388—41no—
51#387—41no—
52#384C2×C411yescharacteristic
53#382C2×C411yescharacteristic
54#380—21no—
55C4×C4⋊C4—21no—
56#349—42no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61C22×D4—21no—
62C22×D4—41no—
63C22×D4—41no—
64C23×C4—41no—
65C23×C4C22⋊C411yescharacteristic
66C2×M16—22no—
67C2×M16—22no—
68C2×M16—41no—
69C22×C8—22no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78#127—21no—
79#127—41no—
80#127C2×C812yes—
81#125—21no—
82#125—41no—
83#125C2×C812yes—
84#123—44no—
85#123—44no—
86#121—24no—
87#121—44no—
88#121—44no—
89#121—42no—
90C4×D4—24no—
91C4×D4—42no—
92#117—22no—
93#117C2×C812yes—
94#117M1612yes—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C42—41no—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22×C411yescharacteristic
108M32—44no—
109M32—42no—
110M32—42no—
111M32—42no—
112M32—42no—
113C2×C16—24no—
114C2×C16—44no—
115C2×C16—42no—
116#102—42no—
117#102—41no—
118#102—41no—
119#102—42no—
120#102—41no—
121#102—41no—
122#97—41no—
123#97—41no—
124C4×C8—41no—
125C4×C8—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129C24—41no—
130C24#10011yescharacteristic
131C2×D4—22no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—44no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—161no—
141C2×D4—161no—
142C2×D4—161no—
143C2×D4—161no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—44no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—161no—
155C22×C4—161no—
156C22×C4C22⋊C812yes—
157C22×C4#9912yes—
158C22×C4#10011yescharacteristic
159C22×C4C2×C22⋊C411yescharacteristic
160M16—44no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8#10411yescharacteristic
175C2×C8#10411yescharacteristic
176C2×C8#13512yes—
177C2×C8AGL(1,ℤ/8)11yescharacteristic
178C2×C8#13711yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—161no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—161no—
195C42—21no—
196C42—21no—
197C42—161no—
198C42C22⋊C812yes—
199C42C22×C811yescharacteristic
200C42C2×M1611yescharacteristic
201C16—44no—
202C16—82no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—81no—
209C23—81no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23#32312yes—
215C23C2×#10011yescharacteristic
216D4—42no—
217D4—42no—
218D4—81no—
219D4—81no—
220D4—162no—
221D4—162no—
222D4—164no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—164no—
246C2×C4#32312yes—
247C2×C4C2×C22⋊C811yescharacteristic
248C2×C4C2×#9911yescharacteristic
249C2×C4#41611yescharacteristic
250C2×C4#41711yescharacteristic
251C2×C4C2×#10411yescharacteristic
252C2×C4#42111yescharacteristic
253C8—21no—
254C8—21no—
255C8—21no—
256C8—21no—
257C8—22no—
258C8—41no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—81no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—161no—
270C22—161no—
271C22—161no—
272C22—161no—
273C22C2×#32311yescharacteristic
274C22#153011yescharacteristic
275C22#155111yescharacteristic
276C4—22no—
277C4—42no—
278C4—41no—
279C4—41no—
280C4—82no—
281C4—161no—
282C4—161no—
283C4#147611yescharacteristic
284C4#147711yescharacteristic
285C4#150411yescharacteristic
286C4#151611yescharacteristic
287C2—21no—
288C2—21no—
289C2—42no—
290C2—162no—
291C2#757811yescharacteristic
292C2#1191211yescharacteristic
293C2#1191311yescharacteristic
subgroupG/H
center, socleC22#1530
commutatorC42C22×C8
Frattini, cosocleC2×#97C23
prime radicalC2×#540C4
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12850C2semidirect
3. #12547C2semidirect
4. #7130C2semidirect
5. #7116C2semidirect

Maximal subgroups

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

subgroupindex
#214142
#128502
#125472
#71302
#71162