Finite Group Databasebeta

← #124443

Subgroups — #124443

Order 512.

370 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 94 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#21537C211yesmaximal, characteristic
2#12551C211yesmaximal, characteristic
3#8602C211yesmaximal, characteristic
4#7127C212yesmaximal, complemented
5#7111C212yesmaximal, complemented
6C22×M32—21no—
7C2×#470C411yescharacteristic
8C2×C4×Q16C411yescharacteristic
9C2×#405C2211yescharacteristic
10#2323—21no—
11#1971C2211yescharacteristic
12#1944C2211yescharacteristic
13#1933—21no—
14#1892—21no—
15#1809C411yescharacteristic
16C4×#106C411yescharacteristic
17#1763—21no—
18#1564—21no—
19#1458—21no—
20#1377—21no—
21#1377C2212yes—
22#1275C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×M16C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C22⋊Q8—21no—
27C2×C4×Q8—21no—
28C2×#117D411yescharacteristic
29C2×M32—24no—
30C2×M32—22no—
31C2×M32—41no—
32C2×M32—42no—
33C22×C16—41no—
34#470—22no—
35#470C814yescomplemented
36C4×Q16—41no—
37C4×Q16—41no—
38#431—22no—
39C2×#107C2×C411yescharacteristic
40C2×#106C2×C411yescharacteristic
41C2×#106C2×C411yescharacteristic
42C2×#103—21no—
43C2×#103—21no—
44C2×#103—21no—
45#405—41no—
46#405—41no—
47C2×#97D411yescharacteristic
48C2×#97C2311yescharacteristic
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#391—21no—
52#389—21no—
53#389—41no—
54#388—41no—
55#388—41no—
56#387—41no—
57#380—21no—
58C4×C4⋊C4—21no—
59#340—21no—
60#340—21no—
61#340—21no—
62#335—41no—
63#334—41no—
64#324—41no—
65#323—41no—
66C22×Q8—21no—
67C23×C4C22⋊C411yescharacteristic
68C2×Q16—21no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16C2×C812yes—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16C2×C812yes—
75C2×M16M1612yes—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22⋊Q8—24no—
86C22⋊Q8—42no—
87C4×Q8—42no—
88C4×Q8—42no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4D811yescharacteristic
98C2×C4⋊C4SD1611yescharacteristic
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4D811yescharacteristic
102C2×C22⋊C4SD1611yescharacteristic
103C2×C42—41no—
104C2×C42—41no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107M32—24no—
108M32—44no—
109M32—42no—
110C2×C16—44no—
111C2×C16—42no—
112#107—41no—
113#106—21no—
114#106—41no—
115#106C2×C812yes—
116C4⋊C8—42no—
117#103—24no—
118#103—41no—
119#103—41no—
120#103—41no—
121#103—41no—
122C22⋊C8—42no—
123#97—41no—
124#97—41no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—42no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135C24#10211yescharacteristic
136C2×Q8—22no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4C22⋊C812yes—
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10412yes—
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4C2×SD1611yescharacteristic
158Q16—42no—
159Q16—42no—
160M16—42no—
161M16—42no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
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—82no—
175C2×C8C22⋊C812yes—
176C2×C8#9911yescharacteristic
177C2×C8#9911yescharacteristic
178C2×C8C22×C811yescharacteristic
179C2×C8C2×M1611yescharacteristic
180C2×C8AGL(1,ℤ/8)11yescharacteristic
181C2×C8#13711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—44no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—161no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—44no—
196C22⋊C4—161no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—82no—
202C42—161no—
203C16—44no—
204C23—21no—
205C23—21no—
206C23—41no—
207C23—41no—
208C23D4⋊C812yes—
209C23C2×#10211yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—81no—
213Q8—81no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4D4⋊C812yes—
241C2×C4C2×C22⋊C811yescharacteristic
242C2×C4C2×#9911yescharacteristic
243C2×C4#41311yescharacteristic
244C2×C4#41711yescharacteristic
245C2×C4C2×#10411yescharacteristic
246C2×C4#42111yescharacteristic
247C8—22no—
248C8—21no—
249C8—21no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—21no—
254C8—162no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22C2×D4⋊C811yescharacteristic
263C22#151911yescharacteristic
264C22#154711yescharacteristic
265C4—22no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—82no—
271C4—82no—
272C4—161no—
273C4—161no—
274C4#145911yescharacteristic
275C4#147711yescharacteristic
276C4#150411yescharacteristic
277C4#153211yescharacteristic
278C2—21no—
279C2—21no—
280C2—42no—
281C2#758611yescharacteristic
282C2#1186711yescharacteristic
283C2#1187111yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC2×C8C22×C8
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C2×C8 ▷ 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
4. #7127C2semidirect
5. #7111C2semidirect
35. #470C8semidirect

Maximal subgroups

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

subgroupindex
#215372
#125512
#86022
#71272
#71112