Finite Group Databasebeta

← #451124

Subgroups — #451124

Order 512.

2462 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 2462 conjugacy classes of subgroups — too many to draw. Only its 206 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#35369C211yesmaximal, characteristic
2#14681C213yesmaximal
3#11196C213yesmaximal
4#11070C216yesmaximal, complemented
5#10936C212yesmaximal, complemented
6C25×C4C2211yescharacteristic
7#2480—212no—
8#2315C2213yes—
9C2×#381C2216yes—
10C2×#381C2216yes—
11C2×#379C2213yes—
12C2×#379C2213yes—
13#2273C2211yescharacteristic
14C2×#342C2216yes—
15C2×#342C2216yes—
16C22⋊#95—212no—
17C26C2311yescharacteristic
18C24×C4—26no—
19C24×C4—212no—
20C24×C4—46no—
21C24×C4C2312yes—
22C2×#126—24no—
23C22×C22⋊C4—26no—
24C22×C22⋊C4D4112yes—
25C22×C22⋊C4C2316yes—
26C22×C22⋊C4C2313yes—
27C22×C22⋊C4C2313yes—
28#397—212no—
29#396—24no—
30#388—212no—
31#385—26no—
32#385—26no—
33#381—212no—
34#381—212no—
35#381—46no—
36#381—46no—
37#380—26no—
38#379—26no—
39#379—26no—
40#379—46no—
41C4×C22⋊C4—26no—
42C2×#95—212no—
43C2×#95—212no—
44#342—46no—
45#342—46no—
46C25—26no—
47C25—26no—
48C25—48no—
49C25C2×D416yes—
50C25C2411yescharacteristic
51C23×C4—26no—
52C23×C4—26no—
53C23×C4—26no—
54C23×C4—26no—
55C23×C4—224no—
56C23×C4—26no—
57C23×C4—424no—
58C23×C4—424no—
59C23×C4—424no—
60C23×C4—424no—
61C23×C4—412no—
62C23×C4—48no—
63C23×C4—44no—
64C23×C4—48no—
65C23×C4C2×D416yes—
66C23×C4C2×D416yes—
67C23×C4#41112yes—
68#126—88no—
69C2×C4⋊C4—212no—
70C2×C22⋊C4—212no—
71C2×C22⋊C4—23no—
72C2×C22⋊C4—23no—
73C2×C22⋊C4—23no—
74C2×C22⋊C4—23no—
75C2×C22⋊C4—26no—
76C2×C22⋊C4—26no—
77C2×C22⋊C4—26no—
78C2×C22⋊C4—26no—
79C2×C22⋊C4—26no—
80C2×C22⋊C4—26no—
81C2×C22⋊C4—26no—
82C2×C22⋊C4—26no—
83C2×C22⋊C4—26no—
84C2×C22⋊C4—26no—
85C2×C22⋊C4—212no—
86C2×C22⋊C4—424no—
87C2×C22⋊C4—424no—
88C2×C22⋊C4—412no—
89C2×C42—24no—
90#95—24no—
91#95—212no—
92#95—424no—
93#95—424no—
94C24—26no—
95C24—26no—
96C24—224no—
97C24—212no—
98C24—212no—
99C24—412no—
100C24—44no—
101C24—448no—
102C24—448no—
103C24—42no—
104C24—46no—
105C24—46no—
106C24—42no—
107C24#12014yes—
108C24C22×D413yes—
109C24C2×#4116yes—
110C24H(5,𝔽2)16yes—
111C22×C4—212no—
112C22×C4—212no—
113C22×C4—212no—
114C22×C4—212no—
115C22×C4—212no—
116C22×C4—212no—
117C22×C4—26no—
118C22×C4—26no—
119C22×C4—212no—
120C22×C4—212no—
121C22×C4—496no—
122C22×C4—448no—
123C22×C4—496no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—46no—
127C22×C4—46no—
128C22×C4—46no—
129C22×C4—46no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—412no—
133C22×C4—412no—
134C22×C4—424no—
135C22×C4—46no—
136C22×C4—46no—
137C22×C4—46no—
138C22×C4#12618yescomplemented
139C4⋊C4—812no—
140C22⋊C4—43no—
141C22⋊C4—43no—
142C22⋊C4—43no—
143C22⋊C4—43no—
144C22⋊C4—46no—
145C22⋊C4—46no—
146C22⋊C4—424no—
147C22⋊C4—46no—
148C22⋊C4—46no—
149C22⋊C4—46no—
150C22⋊C4—46no—
151C22⋊C4—812no—
152C22⋊C4—824no—
153C42—84no—
154C23—212no—
155C23—224no—
156C23—224no—
157C23—212no—
158C23—212no—
159C23—412no—
160C23—412no—
161C23—448no—
162C23—448no—
163C23—412no—
164C23—412no—
165C23—416no—
166C23—448no—
167C23—448no—
168C23—416no—
169C23—412no—
170C23—412no—
171C23#45818yes—
172C23C2×#12011yescharacteristic
173C23#52513yes—
174C23C2×#12612yes—
175C23#53513yes—
176C23#53816yes—
177C23#54616yes—
178C23#54616yes—
179C23#55116yes—
180C23#56012yes—
181C2×C4—224no—
182C2×C4—464no—
183C2×C4—412no—
184C2×C4—412no—
185C2×C4—412no—
186C2×C4—412no—
187C2×C4—46no—
188C2×C4—46no—
189C2×C4—412no—
190C2×C4—412no—
191C2×C4—812no—
192C2×C4—812no—
193C2×C4—812no—
194C2×C4—812no—
195C2×C4—812no—
196C2×C4—86no—
197C22—26no—
198C22—212no—
199C22—212no—
200C22—26no—
201C22—224no—
202C22—42no—
203C22—46no—
204C22—46no—
205C22—42no—
206C22—448no—
207C22—448no—
208C22—44no—
209C22—412no—
210C22#240413yes—
211C22#248016yes—
212C22#268516yes—
213C22C2×#45814yes—
214C4—48no—
215C4—86no—
216C4—86no—
217C4—166no—
218C2—26no—
219C2—26no—
220C2—48no—
221C2#1598611yescharacteristic
222C2#2398516yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C26 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11070C2semidirect
5. #10936C2semidirect
138. C22×C4#126semidirect

Maximal subgroups

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

subgroupindex
#353692
#146812
#111962
#110702
#109362