Finite Group Databasebeta

← #150304

Subgroups — #150304

Order 512.

333 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 66 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#29963C211yesmaximal, characteristic
2#12586C211yesmaximal, characteristic
3#11434C211yesmaximal, characteristic
4#7437C212yesmaximal, complemented
5#7429C212yesmaximal, complemented
6C22×#101—21no—
7#2629C411yescharacteristic
8#2310C2211yescharacteristic
9#2096C2211yescharacteristic
10#2095C411yescharacteristic
11#2068—21no—
12#2066—21no—
13#2026C411yescharacteristic
14#2026C2211yescharacteristic
15#1854—21no—
16#1798C411yescharacteristic
17#1787—21no—
18#1458—21no—
19#1394—21no—
20#1394C2212yes—
21#1319—21no—
22#1319C2212yes—
23C23×Q8D411yescharacteristic
24C22×M16—41no—
25C2×#125C2×C411yescharacteristic
26C2×#117C2×C411yescharacteristic
27C2×#117D411yescharacteristic
28#431—41no—
29C2×#107—21no—
30C2×#106—21no—
31C2×C4⋊C8—21no—
32C2×#103—21no—
33C2×#103—21no—
34C2×#103—21no—
35C2×#103—21no—
36C2×#101—24no—
37C2×#101—22no—
38C2×#101—41no—
39C2×#101—42no—
40C2×C22⋊C8—21no—
41#398—41no—
42#396—41no—
43#391—41no—
44#391C2×C411yescharacteristic
45#391D411yescharacteristic
46#391D411yescharacteristic
47#391C2311yescharacteristic
48#389—21no—
49#384C2×C411yescharacteristic
50#381—41no—
51#380—21no—
52#340—21no—
53#340—21no—
54#336C2×C411yescharacteristic
55#336C2×C411yescharacteristic
56#324—22no—
57#323—41no—
58C22×Q8—21no—
59C22×Q8—22no—
60C22×Q8—81no—
61C22×Q8—81no—
62C22×Q8C22⋊C411yescharacteristic
63C22×Q8SD1612yes—
64C22×Q8Q1612yes—
65C22×Q8C2×D411yescharacteristic
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C2×M16—44no—
69C2×M16—42no—
70C2×M16—82no—
71C2×M16—81no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—81no—
75#125—81no—
76#117—81no—
77#117—81no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4C22⋊C411yescharacteristic
85C2×C4⋊C4C2×D411yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—41no—
89C2×C42—21no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22×C411yescharacteristic
92#107—81no—
93#106—81no—
94C4⋊C8—81no—
95C4⋊C8—81no—
96#103—81no—
97#103—81no—
98#103—81no—
99#103—81no—
100#101—24no—
101#101—44no—
102#101—42no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C4×C8—81no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112C24#9911yescharacteristic
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—81no—
122C2×Q8—81no—
123C2×Q8—82no—
124C2×Q8—82no—
125C2×Q8—82no—
126C2×Q8—82no—
127C2×Q8#10312yes—
128C2×Q8C2×SD1611yescharacteristic
129C2×Q8C2×Q1611yescharacteristic
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4#9911yescharacteristic
146C22×C4#10312yes—
147C22×C4#10412yes—
148C22×C4C2×C22⋊C411yescharacteristic
149M16—44no—
150M16—84no—
151M16—82no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—84no—
159C2×C8—82no—
160C2×C8—82no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—82no—
169C4⋊C4—161no—
170C4⋊C4—161no—
171C22⋊C4—81no—
172C22⋊C4—81no—
173C22⋊C4—161no—
174C42—41no—
175C42—41no—
176C42—41no—
177C42—41no—
178C23—21no—
179C23—21no—
180C23—41no—
181C23—41no—
182C23#32812yes—
183C23C2×#9911yescharacteristic
184Q8—42no—
185Q8—41no—
186Q8—41no—
187Q8—81no—
188Q8—81no—
189Q8—82no—
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4#32812yes—
225C2×C4C2×#10311yescharacteristic
226C2×C4C2×#10411yescharacteristic
227C8—81no—
228C8—81no—
229C8—84no—
230C8—162no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22C2×#32811yescharacteristic
239C22#212111yescharacteristic
240C22#212711yescharacteristic
241C4—21no—
242C4—22no—
243C4—21no—
244C4—41no—
245C4—42no—
246C4—41no—
247C4—81no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4—161no—
252C4—161no—
253C2—21no—
254C2—21no—
255C2—42no—
256C2#1171611yescharacteristic
257C2#1360811yescharacteristic
258C2#1361111yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2310 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect
5. #7429C2semidirect

Maximal subgroups

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

subgroupindex
#299632
#125862
#114342
#74372
#74292