Finite Group Databasebeta

← #6132318

Subgroups — #6132318

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17081C211yesmaximal, characteristic, complemented
2#17073C213yesmaximal, complemented
3#16908C216yesmaximal, complemented
4#16829C216yesmaximal, complemented
5#16302C216yesmaximal
6#16300C216yesmaximal
7#14685C213yesmaximal
8C23×C42C2211yescharacteristic
9C2×#399C2212yes—
10C2×#398C2216yes—
11C2×#398C2216yes—
12C2×#396C2212yes—
13C2×#381C2216yes—
14C2×#380C2213yes—
15C2×#380C2213yes—
16#1445C2216yes—
17#1444C2216yes—
18#1444C2216yes—
19#1444C2216yes—
20#1443C2216yes—
21#1442C2216yes—
22#1442C2216yes—
23#1442C2216yes—
24#1442C2216yes—
25#1441C2216yes—
26#1441C2216yes—
27#1441C2216yes—
28#1441C2216yes—
29#1441C2216yes—
30#1439C2216yes—
31#1439C2216yes—
32C4⋊#95C2213yes—
33C4⋊#95C2213yes—
34C4⋊#95C2213yes—
35C4⋊#95C2213yes—
36#1434C2216yes—
37#1434C2216yes—
38C22⋊#95C2216yes—
39C24×C4C2313yes—
40C22×C4⋊C4C2313yes—
41C22×C4⋊C4C2313yes—
42C22×C4⋊C4C2313yes—
43C22×C4⋊C4C2313yes—
44C22×C4⋊C4C2316yes—
45C22×C4⋊C4C2316yes—
46C22×C4⋊C4C2316yes—
47C22×C4⋊C4C2316yes—
48C22×C22⋊C4C2316yes—
49C22×C22⋊C4C2316yes—
50C22×C42—42no—
51C22×C42—42no—
52C22×C42Q818yes—
53C22×C42C2313yes—
54C22×C42C2311yescharacteristic
55#399—44no—
56#398—412no—
57#398—412no—
58#396—44no—
59#381—412no—
60#380—46no—
61#380—46no—
62C2×#95C2316yes—
63C2×#95C2316yes—
64C2×#95C2316yes—
65C2×#95C2316yes—
66C2×#95C2312yes—
67C2×#95C2316yes—
68C2×#95C2316yes—
69C2×#95C2312yes—
70C2×#95C2316yes—
71C2×#95C2316yes—
72C2×#95C2316yes—
73C2×#95C2316yes—
74C2×#95C2316yes—
75C2×#95C2316yes—
76C2×#95C2316yes—
77C2×#95C2316yes—
78C2×#95C2316yes—
79C2×#95C2316yes—
80C25C2411yescharacteristic
81C23×C4—26no—
82C23×C4—26no—
83C23×C4—26no—
84C23×C4—26no—
85C23×C4—43no—
86C23×C4—43no—
87C23×C4C2×Q8112yes—
88C23×C4C2416yes—
89C23×C4C2416yes—
90C23×C4C2416yes—
91C23×C4C2416yes—
92C23×C4C2413yes—
93C23×C4C2413yes—
94C2×C4⋊C4—26no—
95C2×C4⋊C4—26no—
96C2×C4⋊C4—26no—
97C2×C4⋊C4—26no—
98C2×C4⋊C4—43no—
99C2×C4⋊C4—43no—
100C2×C4⋊C4—43no—
101C2×C4⋊C4—43no—
102C2×C4⋊C4—46no—
103C2×C4⋊C4—46no—
104C2×C4⋊C4—46no—
105C2×C4⋊C4—46no—
106C2×C4⋊C4—43no—
107C2×C4⋊C4—43no—
108C2×C4⋊C4—43no—
109C2×C4⋊C4—43no—
110C2×C4⋊C4—46no—
111C2×C4⋊C4—46no—
112C2×C4⋊C4—46no—
113C2×C4⋊C4—46no—
114C2×C4⋊C4—46no—
115C2×C4⋊C4—46no—
116C2×C4⋊C4—46no—
117C2×C4⋊C4—46no—
118C2×C22⋊C4—212no—
119C2×C22⋊C4—212no—
120C2×C22⋊C4—46no—
121C2×C22⋊C4—46no—
122C2×C22⋊C4—46no—
123C2×C22⋊C4—46no—
124C2×C42—416no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—43no—
128C2×C42—43no—
129C2×C42C2×Q8112yes—
130C2×C42C2×Q814yes—
131#95—212no—
132#95—212no—
133#95—212no—
134#95—212no—
135#95—46no—
136#95—46no—
137#95—46no—
138#95—46no—
139#95—46no—
140#95—46no—
141#95—46no—
142#95—46no—
143#95—46no—
144#95—46no—
145#95—42no—
146#95—46no—
147#95—46no—
148#95—42no—
149C24—23no—
150C24—23no—
151C24—23no—
152C24—23no—
153C24C22×Q812yes—
154C24H(5,𝔽2)12yes—
155C24H(5,𝔽2)12yes—
156C24C2511yescharacteristic
157C22×C4—23no—
158C22×C4—23no—
159C22×C4—23no—
160C22×C4—23no—
161C22×C4—224no—
162C22×C4—23no—
163C22×C4—23no—
164C22×C4—23no—
165C22×C4—23no—
166C22×C4—23no—
167C22×C4—23no—
168C22×C4—224no—
169C22×C4—26no—
170C22×C4—26no—
171C22×C4—26no—
172C22×C4—26no—
173C22×C4—26no—
174C22×C4—26no—
175C22×C4—26no—
176C22×C4—26no—
177C22×C4—26no—
178C22×C4—26no—
179C22×C4—26no—
180C22×C4—26no—
181C22×C4—26no—
182C22×C4—26no—
183C22×C4—26no—
184C22×C4—26no—
185C22×C4—26no—
186C22×C4—26no—
187C22×C4—26no—
188C22×C4—26no—
189C22×C4—26no—
190C22×C4—26no—
191C22×C4—26no—
192C22×C4—26no—
193C22×C4—26no—
194C22×C4—26no—
195C22×C4—26no—
196C22×C4—26no—
197C22×C4—412no—
198C22×C4—412no—
199C22×C4—412no—
200C22×C4—412no—
201C22×C4—412no—
202C22×C4C22×Q816yes—
203C22×C4C22×Q816yes—
204C22×C4H(5,𝔽2)16yes—
205C22×C42−1+416yes—
206C4⋊C4—86no—
207C4⋊C4—86no—
208C4⋊C4—86no—
209C4⋊C4—86no—
210C4⋊C4—812no—
211C4⋊C4—812no—
212C4⋊C4—812no—
213C4⋊C4—812no—
214C22⋊C4—812no—
215C22⋊C4—812no—
216C42—44no—
217C42—412no—
218C23—26no—
219C23—26no—
220C23—26no—
221C23—26no—
222C23—26no—
223C23—26no—
224C23—43no—
225C23—43no—
226C23—43no—
227C23—43no—
228C23—41no—
229C23—43no—
230C23#54314yes—
231C23C23×Q811yescharacteristic
232C23C2×H(5,𝔽2)11yescharacteristic
233C23C2×H(5,𝔽2)11yescharacteristic
234C23C2×H(5,𝔽2)13yes—
235C23C2×2−1+413yes—
236C23#58513yes—
237C23#58513yes—
238C2×C4—26no—
239C2×C4—26no—
240C2×C4—26no—
241C2×C4—26no—
242C2×C4—26no—
243C2×C4—26no—
244C2×C4—26no—
245C2×C4—26no—
246C2×C4—26no—
247C2×C4—448no—
248C2×C4—43no—
249C2×C4—43no—
250C2×C4—43no—
251C2×C4—43no—
252C2×C4—46no—
253C2×C4—46no—
254C2×C4—46no—
255C2×C4—46no—
256C2×C4—46no—
257C2×C4—46no—
258C2×C4—46no—
259C2×C4—46no—
260C2×C4—46no—
261C2×C4—46no—
262C2×C4—46no—
263C2×C4—46no—
264C2×C4—46no—
265C2×C4—46no—
266C2×C4—46no—
267C2×C4—46no—
268C2×C4—46no—
269C2×C4—46no—
270C2×C4—46no—
271C2×C4—46no—
272C2×C4—46no—
273C2×C4—46no—
274C2×C4—46no—
275C2×C4—46no—
276C2×C4—46no—
277C2×C4—46no—
278C2×C4—46no—
279C2×C4—46no—
280C2×C4#544112yes—
281C22—212no—
282C22—42no—
283C22—46no—
284C22—42no—
285C22—46no—
286C22—42no—
287C22—46no—
288C22C2×#54311yescharacteristic
289C22C2×#54413yes—
290C22#345813yes—
291C22#353413yes—
292C22#353413yes—
293C22#353513yes—
294C22#353513yes—
295C22#354613yes—
296C22#354713yes—
297C22#355413yes—
298C22#355513yes—
299C22#356711yescharacteristic
300C22#356713yes—
301C4—212no—
302C4—46no—
303C4—86no—
304C4—86no—
305C4—86no—
306C4—86no—
307C2—44no—
308C2#3747813yes—
309C2#3756411yescharacteristic
310C2#3762413yes—
311C2#5329611yescharacteristic
312C2#5329713yes—
313C2#5330011yescharacteristic
314C2#5330113yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17081C2semidirect
2. #17073C2semidirect
3. #16908C2semidirect
4. #16829C2semidirect

Maximal subgroups

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

subgroupindex
#170812
#170732
#169082
#168292
#163022
#163002
#146852