Finite Group Databasebeta

← #4127372

Subgroups — #4127372

Order 512.

2126 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17307C211yesmaximal, characteristic, complemented
2#15682C216yesmaximal, complemented
3#15363C213yesmaximal, complemented
4#15305C216yesmaximal
5#14204C216yesmaximal
6#14045C216yesmaximal, complemented
7#13965C213yesmaximal
8C2×#398C2216yes—
9C2×#396C2212yes—
10C2×#396C2212yes—
11C2×#389C2216yes—
12C2×#389C2216yes—
13C2×#389C2216yes—
14C2×#385C2216yes—
15C2×#385C2216yes—
16C2×#380C2213yes—
17C2×C4×C4⋊C4C2216yes—
18C2×C4×C22⋊C4C2213yes—
19C22×#95C2211yescharacteristic
20#1444C2213yes—
21#1444C2216yes—
22#1444C2213yes—
23#1444C2213yes—
24#1444C2216yes—
25#1444C2213yes—
26#1442C2216yes—
27#1441C2216yes—
28#1441C2216yes—
29#1441C2216yes—
30#1441C2216yes—
31#1439C2216yes—
32#1437C2216yes—
33#1436C2216yes—
34#1436C2216yes—
35#1436C2216yes—
36C4⋊#95C2216yes—
37C4⋊#95C2216yes—
38C4⋊#95C2216yes—
39C24×C4C2313yes—
40C22×C4⋊C4C2316yes—
41C22×C4⋊C4C2316yes—
42C22×C4⋊C4C2316yes—
43C22×C4⋊C4C2316yes—
44C22×C4⋊C4C2316yes—
45C22×C4⋊C4C2316yes—
46C22×C4⋊C4C2316yes—
47C22×C4⋊C4C2313yes—
48C22×C4⋊C4C2313yes—
49C22×C22⋊C4C2316yes—
50C22×C22⋊C4C2316yes—
51C22×C42C2316yes—
52C22×C42C2316yes—
53#398—412no—
54#396—44no—
55#396Q8116yes—
56#389—224no—
57#389—412no—
58#389—412no—
59#385—412no—
60#385—412no—
61#380—46no—
62C4×C4⋊C4—412no—
63C4×C22⋊C4—212no—
64C2×#95—42no—
65C2×#95C2316yes—
66C2×#95C2316yes—
67C2×#95C2316yes—
68C2×#95C2316yes—
69C2×#95C2316yes—
70C2×#95C2313yes—
71C2×#95C2312yes—
72C2×#95C2316yes—
73C2×#95C2313yes—
74C2×#95C2316yes—
75C2×#95C2316yes—
76C2×#95C2316yes—
77C2×#95C2316yes—
78C2×#95C2316yes—
79C2×#95C2311yescharacteristic
80C2×#95C2312yes—
81C2×#95C2313yes—
82C25C2411yescharacteristic
83C23×C4—26no—
84C23×C4—26no—
85C23×C4—26no—
86C23×C4—46no—
87C23×C4—46no—
88C23×C4C2416yes—
89C23×C4C2416yes—
90C23×C4C2416yes—
91C23×C4C2413yes—
92C23×C4C2413yes—
93C23×C4C2416yes—
94C2×C4⋊C4—212no—
95C2×C4⋊C4—212no—
96C2×C4⋊C4—212no—
97C2×C4⋊C4—212no—
98C2×C4⋊C4—212no—
99C2×C4⋊C4—212no—
100C2×C4⋊C4—212no—
101C2×C4⋊C4—212no—
102C2×C4⋊C4—212no—
103C2×C4⋊C4—46no—
104C2×C4⋊C4—46no—
105C2×C4⋊C4—43no—
106C2×C4⋊C4—43no—
107C2×C4⋊C4—46no—
108C2×C4⋊C4—46no—
109C2×C4⋊C4—46no—
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×C4⋊C4C2×Q8124yes—
119C2×C22⋊C4—212no—
120C2×C22⋊C4—212no—
121C2×C22⋊C4—212no—
122C2×C22⋊C4—46no—
123C2×C22⋊C4—46no—
124C2×C22⋊C4C2×Q8124yes—
125C2×C42—212no—
126C2×C42—212no—
127C2×C42—212no—
128C2×C42—46no—
129C2×C42—46no—
130C2×C42—46no—
131#95—26no—
132#95—212no—
133#95—212no—
134#95—212no—
135#95—212no—
136#95—212no—
137#95—43no—
138#95—41no—
139#95—46no—
140#95—46no—
141#95—46no—
142#95—46no—
143#95—46no—
144#95—42no—
145#95—46no—
146#95C2×Q818yes—
147#95#41112yes—
148C24—23no—
149C24—26no—
150C24—23no—
151C24C22×Q814yes—
152C24H(5,𝔽2)12yes—
153C24C2511yescharacteristic
154C22×C4—26no—
155C22×C4—26no—
156C22×C4—26no—
157C22×C4—26no—
158C22×C4—26no—
159C22×C4—26no—
160C22×C4—23no—
161C22×C4—26no—
162C22×C4—26no—
163C22×C4—26no—
164C22×C4—23no—
165C22×C4—26no—
166C22×C4—26no—
167C22×C4—23no—
168C22×C4—26no—
169C22×C4—26no—
170C22×C4—26no—
171C22×C4—23no—
172C22×C4—224no—
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—812no—
188C22×C4—812no—
189C22×C4—86no—
190C22×C4—86no—
191C22×C4C22×Q8112yes—
192C22×C4C22×Q8112yes—
193C22×C4C2×#4116yes—
194C22×C4C2×#41112yes—
195C22×C4H(5,𝔽2)112yes—
196C22×C42−1+416yes—
197C4⋊C4—248no—
198C4⋊C4—424no—
199C4⋊C4—424no—
200C4⋊C4—812no—
201C4⋊C4—812no—
202C4⋊C4—812no—
203C4⋊C4—86no—
204C4⋊C4—86no—
205C4⋊C4—812no—
206C22⋊C4—248no—
207C22⋊C4—812no—
208C42—424no—
209C42—812no—
210C23—212no—
211C23—26no—
212C23—212no—
213C23—26no—
214C23—43no—
215C23—46no—
216C23—46no—
217C23—41no—
218C23Q8214yes—
219C23C23×Q812yes—
220C23C22×#4113yes—
221C23C2×H(5,𝔽2)16yes—
222C23C2×H(5,𝔽2)11yescharacteristic
223C23C2×2−1+413yes—
224C2×C4—212no—
225C2×C4—212no—
226C2×C4—212no—
227C2×C4—212no—
228C2×C4—212no—
229C2×C4—212no—
230C2×C4—212no—
231C2×C4—212no—
232C2×C4—26no—
233C2×C4—26no—
234C2×C4—212no—
235C2×C4—212no—
236C2×C4—212no—
237C2×C4—46no—
238C2×C4—46no—
239C2×C4—46no—
240C2×C4—46no—
241C2×C4—46no—
242C2×C4—46no—
243C2×C4—43no—
244C2×C4—43no—
245C2×C4—43no—
246C2×C4—43no—
247C2×C4—46no—
248C2×C4—46no—
249C2×C4—46no—
250C2×C4—46no—
251C2×C4—46no—
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—824no—
262C2×C4#554124yes—
263C22—212no—
264C22—412no—
265C22—44no—
266C22—42no—
267C22—46no—
268C22C2×#55416yes—
269C22C2×Q8211yescharacteristic
270C22#348213yes—
271C22#349213yes—
272C22#350316yes—
273C22#351916yes—
274C22#353616yes—
275C22#354313yes—
276C22#355611yescharacteristic
277C4—224no—
278C4—412no—
279C4—46no—
280C4—46no—
281C4—86no—
282C4—86no—
283C2—44no—
284C2#4650713yes—
285C2#4773312yes—
286C2#5932213yes—
287C2#5945516yes—
288C2#6004911yescharacteristic
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. #17307C2semidirect
2. #15682C2semidirect
3. #15363C2semidirect
6. #14045C2semidirect

Maximal subgroups

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

subgroupindex
#173072
#156822
#153632
#153052
#142042
#140452
#139652