Finite Group Databasebeta

← #213801

Subgroups — #213801

Order 512.

817 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 194 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#35670C211yesmaximal, characteristic
2#17371C212yesmaximal
3#9104C214yesmaximal, complemented
4#8962C214yesmaximal, complemented
5#8733C212yesmaximal
6#8327C212yesmaximal
7C2×#541C2211yescharacteristic
8C2×#519C2211yescharacteristic
9#2983C2214yes—
10#2565C412yes—
11#2546C412yes—
12#2499C412yes—
13#2478—24no—
14#2462—24no—
15#2434C412yes—
16#2327C2212yes—
17C4×C4⋊Q8C2212yes—
18#2298C2211yescharacteristic
19#1479C2218yes—
20#1467—24no—
21#1464—24no—
22#1461C2214yes—
23#1458C2214yes—
24#1276C2214yes—
25#1274C2214yes—
26C23×Q8C2×C411yescharacteristic
27#541—44no—
28C2×C4⋊Q8C2×C412yes—
29C2×C4⋊Q8C2×C412yes—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×#126—22no—
32C2×#125—22no—
33C2×#125C2×C411yescharacteristic
34C2×C22⋊Q8C2×C412yes—
35#519—44no—
36C2×C4×Q8C2×C412yes—
37C2×C4×Q8D414yes—
38C2×C4×Q8C2312yes—
39C2×C4×Q8C2312yes—
40C2×#117C2×C411yescharacteristic
41C2×#117D414yes—
42C2×#117C2311yescharacteristic
43#446—48no—
44#432—28no—
45#431C2318yes—
46#398—24no—
47#398—24no—
48#397—24no—
49#395—24no—
50#391C2×C412yes—
51#391C2×C412yes—
52#391C2×C412yes—
53#391C2×C412yes—
54#389—24no—
55#389—24no—
56#388—22no—
57#387—24no—
58#387—22no—
59#387—24no—
60#387—24no—
61#387—24no—
62#385—24no—
63#384C2×C412yes—
64#382C2×C412yes—
65C4×C4⋊C4—24no—
66C4×C4⋊C4C2×C412yes—
67C4×C22⋊C4C2×C412yes—
68#376C2311yescharacteristic
69C43C2311yescharacteristic
70Q8⋊C8—48no—
71#324—24no—
72#324—28no—
73#323—24no—
74C22×Q8—42no—
75C22×Q8—44no—
76C22×Q8C22⋊C412yes—
77C22×Q8C22×C412yes—
78C22×Q8C22×C412yes—
79C23×C4C22⋊C412yes—
80C23×C4C22×C411yescharacteristic
81C4⋊Q8—44no—
82C4⋊Q8—44no—
83C4⋊Q8—42no—
84#126—48no—
85#125—42no—
86#125—48no—
87C22⋊Q8—44no—
88C4×Q8—28no—
89C4×Q8—28no—
90C4×Q8—48no—
91C4×Q8—44no—
92#117—24no—
93#117—48no—
94#117—42no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4C22⋊C414yes—
104C2×C4⋊C4C22×C412yes—
105C2×C4⋊C4C22×C412yes—
106C2×C4⋊C4C2×D414yes—
107C2×C4⋊C4C2×D412yes—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4C22⋊C412yes—
111C2×C22⋊C4C2×D412yes—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42C22⋊C412yes—
115C2×C42C22⋊C414yes—
116C2×C42C22×C412yes—
117C2×C42C22×C412yes—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D414yes—
120C2×C42C2411yescharacteristic
121C4⋊C8—28no—
122C4⋊C8—48no—
123C22⋊C8—28no—
124#97—48no—
125C4×C8—28no—
126#95—22no—
127#95—24no—
128#95—24no—
129#95—22no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—22no—
134C24C2×C22⋊C411yescharacteristic
135C2×Q8—24no—
136C2×Q8—24no—
137C2×Q8—24no—
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—44no—
143C2×Q8—416no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C22⋊C414yes—
154C22×C4C2×C22⋊C414yes—
155C22×C4C2×C22⋊C412yes—
156C22×C4C23×C411yescharacteristic
157C22×C4C22×D412yes—
158C2×C8—48no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—48no—
170C4⋊C4—48no—
171C4⋊C4—44no—
172C22⋊C4—24no—
173C22⋊C4—42no—
174C22⋊C4—44no—
175C22⋊C4—44no—
176C42—22no—
177C42—28no—
178C42—24no—
179C42—24no—
180C42—44no—
181C42—42no—
182C42—44no—
183C42—44no—
184C42H(5,𝔽2)14yes—
185C23—21no—
186C23—22no—
187C23—42no—
188C23C22×C22⋊C411yescharacteristic
189Q8—48no—
190Q8—48no—
191Q8—48no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—48no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4#51812yes—
216C2×C4#53414yes—
217C2×C4#56814yes—
218C8—88no—
219C22—22no—
220C22—42no—
221C22—41no—
222C22—82no—
223C22#230811yescharacteristic
224C22#287912yes—
225C4—24no—
226C4—44no—
227C4—42no—
228C4—44no—
229C4—44no—
230C4—41no—
231C4—82no—
232C4—82no—
233C4—82no—
234C4#297814yes—
235C2—22no—
236C2—81no—
237C2#2174311yescharacteristic
238C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9104C2semidirect
4. #8962C2semidirect

Maximal subgroups

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

subgroupindex
#356702
#173712
#91042
#89622
#87332
#83272