Finite Group Databasebeta

← #217276

Subgroups — #217276

Order 512.

888 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 238 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#20714C211yesmaximal, characteristic
2#20714C212yesmaximal
3#8831C218yesmaximal, complemented
4#8600C214yesmaximal, complemented
5C23×C42C2211yescharacteristic
6C2×#431C2214yes—
7C2×#431C2212yes—
8#1817—22no—
9#1817C412yes—
10#1797C414yes—
11#1797C2214yes—
12#1797C2214yes—
13#1779—22no—
14#1779C412yes—
15C4×C22⋊C8—22no—
16C4×C22⋊C8—22no—
17C4×C22⋊C8C2214yes—
18C2×#335C2214yescomplemented
19C2×#334C2214yescomplemented
20#1277C2218yescomplemented
21C24×C4C2×C411yescharacteristic
22C24×C4D411yescharacteristic
23C24×C4Q811yescharacteristic
24C22×C42—42no—
25C22×C42—42no—
26C22×C42C2×C412yes—
27C22×C42C2×C411yescharacteristic
28C22×C42D412yes—
29C22×C42D411yescharacteristic
30C22×C42Q811yescharacteristic
31C22×C42Q812yes—
32C22×C42C2311yescharacteristic
33C22×C42C2312yes—
34#431—42no—
35#431—42no—
36#431—44no—
37#431—44no—
38C2×C4⋊C8C2×C414yes—
39C2×C4⋊C8C2318yes—
40C2×C22⋊C8—24no—
41C2×C22⋊C8—24no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8C2×C418yes—
44C2×C22⋊C8C2×C414yes—
45C2×C4×C8—24no—
46C2×C4×C8C2314yes—
47#336—24no—
48#336—24no—
49#336C2×C418yes—
50#335—44no—
51#334—44no—
52C25C4⋊C411yescharacteristic
53C23×C4—24no—
54C23×C4—28no—
55C23×C4—42no—
56C23×C4—44no—
57C23×C4—42no—
58C23×C4—42no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4C4⋊C412yes—
62C23×C4C4⋊C411yescharacteristic
63C23×C4C4⋊C412yes—
64C23×C4C4⋊C412yes—
65C23×C4C4⋊C411yescharacteristic
66C23×C4C4⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C22×C412yes—
69C23×C4C2×D412yes—
70C23×C4C2×D411yescharacteristic
71C23×C4C2×Q811yescharacteristic
72C23×C4C2×Q812yes—
73C22×C8—28no—
74C22×C8C22×C418yes—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—24no—
78C2×C42—24no—
79C2×C42—44no—
80C2×C42—42no—
81C2×C42—42no—
82C2×C42—44no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C42—42no—
86C2×C42—42no—
87C2×C42C4⋊C412yes—
88C2×C42C4⋊C412yes—
89C2×C42C4⋊C412yes—
90C2×C42C22×C412yes—
91C2×C42C22×C411yescharacteristic
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D412yes—
94C2×C42C2×Q811yescharacteristic
95C2×C42C2×Q812yes—
96C2×C42C2411yescharacteristic
97C4⋊C8—28no—
98C4⋊C8—28no—
99C4⋊C8—44no—
100C22⋊C8—216no—
101C22⋊C8—44no—
102C22⋊C8—48no—
103C22⋊C8—48no—
104C22⋊C8—48no—
105C4×C8—24no—
106C4×C8—24no—
107C4×C8—48no—
108C24—24no—
109C24—22no—
110C24—41no—
111C24—42no—
112C24—41no—
113C24C2×C4⋊C412yes—
114C24C2×C4⋊C411yescharacteristic
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—416no—
148C22×C4—48no—
149C22×C4—48no—
150C22×C4—416no—
151C22×C4#10814yes—
152C22×C4C2×C4⋊C412yes—
153C22×C4C2×C4⋊C411yescharacteristic
154C22×C4C2×C4⋊C411yescharacteristic
155C22×C4C2×C4⋊C412yes—
156C22×C4C2×C4⋊C412yes—
157C22×C4C2×C4⋊C411yescharacteristic
158C22×C4#13118yes—
159C22×C4C23×C411yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C22×Q811yescharacteristic
162C2×C8—28no—
163C2×C8—216no—
164C2×C8—48no—
165C2×C8—48no—
166C2×C8—48no—
167C42—22no—
168C42—24no—
169C42—24no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—21no—
174C42—21no—
175C42—42no—
176C42—42no—
177C42—42no—
178C42H(5,𝔽2)12yes—
179C42H(5,𝔽2)12yes—
180C23—22no—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—44no—
191C23—48no—
192C23—44no—
193C23—48no—
194C23—44no—
195C23#42414yes—
196C23C2×#10812yes—
197C23C22×C4⋊C411yescharacteristic
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—216no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—48no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4—44no—
224C2×C4—48no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4#42414yes—
228C2×C4C2×#10814yes—
229C2×C4#43012yes—
230C2×C4#51812yes—
231C2×C4#53412yes—
232C2×C4#54311yescharacteristic
233C2×C4#54311yescharacteristic
234C2×C4C2×#13114yes—
235C2×C4#57514yes—
236C8—48no—
237C8—88no—
238C22—24no—
239C22—24no—
240C22—24no—
241C22—24no—
242C22—22no—
243C22—416no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—42no—
249C22—41no—
250C22—42no—
251C22#231511yescharacteristic
252C22C2×#42412yes—
253C22#291112yes—
254C22C22×#10811yescharacteristic
255C22C2×#43011yescharacteristic
256C4—24no—
257C4—21no—
258C4—21no—
259C4—22no—
260C4—24no—
261C4—41no—
262C4—41no—
263C4—42no—
264C4#297714yes—
265C4#319714yes—
266C2—44no—
267C2—42no—
268C2#1780612yes—
269C2#1780812yes—
270C2#2193111yescharacteristic
271C2#2194311yescharacteristic
272C2#2194811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C4⋊C4
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8831C2semidirect
4. #8600C2semidirect
18. C2×#335C22semidirect
19. C2×#334C22semidirect
20. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#207142
#88312
#86002