Finite Group Databasebeta

← #5024503

Subgroups — #5024503

Order 512.

9346 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 9346 conjugacy classes of subgroups — too many to draw. Only its 538 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#15986C213yesmaximal, complemented
2#15480C213yesmaximal, complemented
3#15297C219yesmaximal, complemented
4#15282C219yesmaximal, complemented
5#14679C213yesmaximal
6#14673C213yesmaximal, complemented
7#13827C211yesmaximal, characteristic, complemented
8C27C2211yescharacteristic
9C22×#126C2211yescharacteristic, complemented
10C2×#397C2219yescomplemented
11C2×#396C2213yescomplemented
12C2×#388C2219yescomplemented
13C2×#379C2213yes—
14C2×#379C2213yes—
15C2×#379C2219yes—
16C2×#379C2219yes—
17C2×#379C2219yes—
18C2×#379C2219yes—
19#1440C2219yescomplemented
20#1440C2219yescomplemented
21#1439C2219yescomplemented
22#1439C2219yescomplemented
23#1439C2219yescomplemented
24#1439C2219yescomplemented
25#1439C2219yescomplemented
26#1439C2219yescomplemented
27#1437C2213yescomplemented
28#1437C2219yescomplemented
29#1437C2213yescomplemented
30#1437C2213yescomplemented
31C26—212no—
32C26—418no—
33C26—46no—
34C26C2313yes—
35C26C2313yes—
36C26C2311yescharacteristic
37C2×#126—46no—
38C22×C4⋊C4C2319yescomplemented
39C22×C4⋊C4C2313yescomplemented
40C22×C22⋊C4C2319yes—
41C22×C22⋊C4C2313yes—
42C22×C22⋊C4C2319yes—
43C22×C22⋊C4C2319yes—
44C22×C22⋊C4C2319yes—
45C22×C22⋊C4C2319yes—
46C22×C22⋊C4C2319yes—
47C22×C22⋊C4C2319yes—
48C22×C22⋊C4C2319yes—
49C22×C22⋊C4C2319yes—
50C22×C42C2311yescharacteristic, complemented
51C22×C42C2313yescomplemented
52#397—418no—
53#396—46no—
54#388—418no—
55#379—212no—
56#379—212no—
57#379—418no—
58#379—418no—
59#379—418no—
60#379—418no—
61C2×#95C2313yescomplemented
62C2×#95C2319yescomplemented
63C2×#95C2319yescomplemented
64C2×#95C2319yescomplemented
65C2×#95C2319yescomplemented
66C2×#95C2319yescomplemented
67C25—218no—
68C25—218no—
69C25—218no—
70C25—218no—
71C25—218no—
72C25—218no—
73C25—49no—
74C25—43no—
75C25—49no—
76C25—49no—
77C25—49no—
78C25—49no—
79C25—4144no—
80C25—4144no—
81C25—448no—
82C25—4144no—
83C25—448no—
84C25—416no—
85C25—416no—
86C25#41112yes—
87C25C2413yes—
88C25C2413yes—
89C25C2411yescharacteristic
90C23×C4C2413yes—
91C23×C4C2419yes—
92C23×C4C2413yes—
93C23×C4C2419yes—
94#126—416no—
95C2×C4⋊C4—49no—
96C2×C4⋊C4—49no—
97C2×C4⋊C4—49no—
98C2×C4⋊C4—49no—
99C2×C22⋊C4—218no—
100C2×C22⋊C4—218no—
101C2×C22⋊C4—218no—
102C2×C22⋊C4—218no—
103C2×C22⋊C4—218no—
104C2×C22⋊C4—218no—
105C2×C22⋊C4—218no—
106C2×C22⋊C4—218no—
107C2×C22⋊C4—218no—
108C2×C22⋊C4—218no—
109C2×C22⋊C4—218no—
110C2×C22⋊C4—218no—
111C2×C22⋊C4—49no—
112C2×C22⋊C4—49no—
113C2×C22⋊C4—49no—
114C2×C22⋊C4—49no—
115C2×C22⋊C4—49no—
116C2×C22⋊C4—49no—
117C2×C22⋊C4—49no—
118C2×C22⋊C4—49no—
119C2×C22⋊C4—49no—
120C2×C22⋊C4—49no—
121C2×C22⋊C4—49no—
122C2×C22⋊C4—49no—
123C2×C22⋊C4—49no—
124C2×C22⋊C4—49no—
125C2×C22⋊C4—49no—
126C2×C22⋊C4—49no—
127C2×C42—43no—
128C2×C42—49no—
129#95—49no—
130#95—49no—
131#95—43no—
132#95—49no—
133#95—49no—
134#95—49no—
135C24—29no—
136C24—23no—
137C24—29no—
138C24—29no—
139C24—29no—
140C24—29no—
141C24—29no—
142C24—29no—
143C24—29no—
144C24—29no—
145C24—272no—
146C24—272no—
147C24—436no—
148C24—436no—
149C24—436no—
150C24—436no—
151C24—436no—
152C24—436no—
153C24—436no—
154C24—436no—
155C24—436no—
156C24—436no—
157C24—436no—
158C24—436no—
159C24—436no—
160C24—436no—
161C24—436no—
162C24—436no—
163C24—436no—
164C24—436no—
165C24—436no—
166C24—436no—
167C24—4384no—
168C24—41152no—
169C24—4384no—
170C24—412no—
171C24—412no—
172C24—412no—
173C24—412no—
174C24—44no—
175C24—412no—
176C24—412no—
177C24—436no—
178C24—412no—
179C24—412no—
180C24—412no—
181C24—412no—
182C24—412no—
183C24—412no—
184C24#126116yescomplemented
185C24C2×#41118yes—
186C24H(5,𝔽2)118yes—
187C24H(5,𝔽2)16yes—
188C24C2511yescharacteristic
189C22×C4—29no—
190C22×C4—23no—
191C22×C4—29no—
192C22×C4—23no—
193C22×C4—29no—
194C22×C4—29no—
195C22×C4—29no—
196C22×C4—29no—
197C22×C4—29no—
198C22×C4—29no—
199C22×C4—29no—
200C22×C4—29no—
201C22×C4—29no—
202C22×C4—29no—
203C22×C4—29no—
204C22×C4—29no—
205C22×C4—29no—
206C22×C4—29no—
207C22×C4—29no—
208C22×C4—29no—
209C4⋊C4—412no—
210C4⋊C4—818no—
211C22⋊C4—436no—
212C22⋊C4—412no—
213C22⋊C4—818no—
214C22⋊C4—818no—
215C22⋊C4—818no—
216C22⋊C4—818no—
217C22⋊C4—818no—
218C22⋊C4—818no—
219C22⋊C4—818no—
220C22⋊C4—818no—
221C42—44no—
222C42—86no—
223C23—218no—
224C23—218no—
225C23—218no—
226C23—218no—
227C23—218no—
228C23—218no—
229C23—218no—
230C23—218no—
231C23—218no—
232C23—218no—
233C23—218no—
234C23—218no—
235C23—218no—
236C23—218no—
237C23—448no—
238C23—4144no—
239C23—4144no—
240C23—4144no—
241C23—4144no—
242C23—4144no—
243C23—4144no—
244C23—41024no—
245C23—4144no—
246C23—4144no—
247C23—4144no—
248C23—4144no—
249C23—448no—
250C23—4144no—
251C23—43no—
252C23—41no—
253C23—49no—
254C23—49no—
255C23—49no—
256C23—49no—
257C23—43no—
258C23—43no—
259C23—43no—
260C23—43no—
261C23—43no—
262C23—43no—
263C23—43no—
264C23—43no—
265C23—49no—
266C23—43no—
267C23—49no—
268C23—49no—
269C23—49no—
270C23—49no—
271C23C2×#126112yes—
272C23#560112yes—
273C23H(3,𝔽4)14yes—
274C23C22×#4113yes—
275C23C2×H(5,𝔽2)19yes—
276C23C2×H(5,𝔽2)13yes—
277C2×C4—49no—
278C2×C4—49no—
279C2×C4—49no—
280C2×C4—49no—
281C2×C4—49no—
282C2×C4—49no—
283C2×C4—49no—
284C2×C4—49no—
285C2×C4—49no—
286C2×C4—49no—
287C2×C4—43no—
288C2×C4—43no—
289C2×C4—49no—
290C2×C4—49no—
291C2×C4—49no—
292C2×C4—49no—
293C2×C4—49no—
294C2×C4—49no—
295C2×C4—49no—
296C2×C4—49no—
297C22—236no—
298C22—236no—
299C22—212no—
300C22—464no—
301C22—4192no—
302C22—4192no—
303C22—418no—
304C22—46no—
305C22—418no—
306C22—418no—
307C22—418no—
308C22—418no—
309C22—418no—
310C22—418no—
311C22—418no—
312C22—418no—
313C22C22×#12611yescharacteristic
314C22#349019yes—
315C22#349019yes—
316C22C2×#56013yes—
317C22C2×H(3,𝔽4)11yescharacteristic
318C22#353213yes—
319C22#353219yes—
320C4—89no—
321C4—89no—
322C4—83no—
323C4—83no—
324C2—412no—
325C2—412no—
326C2—44no—
327C2#3536913yes—
328C2#4545719yes—
329C2#4545813yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC27C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #15986C2semidirect
2. #15480C2semidirect
3. #15297C2semidirect
4. #15282C2semidirect
6. #14673C2semidirect
7. #13827C2semidirect
9. C22×#126C22semidirect
10. C2×#397C22semidirect
11. C2×#396C22semidirect
12. C2×#388C22semidirect
19. #1440C22semidirect
20. #1440C22semidirect
21. #1439C22semidirect
22. #1439C22semidirect
23. #1439C22semidirect
24. #1439C22semidirect
25. #1439C22semidirect
26. #1439C22semidirect
27. #1437C22semidirect
28. #1437C22semidirect
29. #1437C22semidirect
30. #1437C22semidirect
38. C22×C4⋊C4C23semidirect
39. C22×C4⋊C4C23semidirect
50. C22×C42C23semidirect
51. C22×C42C23semidirect
61. C2×#95C23semidirect
62. C2×#95C23semidirect
63. C2×#95C23semidirect
64. C2×#95C23semidirect
65. C2×#95C23semidirect
66. C2×#95C23semidirect
184. C24#126semidirect

Maximal subgroups

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

subgroupindex
#159862
#154802
#152972
#152822
#146792
#146732
#138272