Finite Group Databasebeta

← #213563

Subgroups — #213563

Order 512.

927 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 927 conjugacy classes of subgroups — too many to draw. Only its 202 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
1C2×#2637C211yesmaximal, characteristic
2#21822C212yesmaximal, complemented
3#9086C214yesmaximal, complemented
4#9084C214yesmaximal
5#8703C212yesmaximal, complemented
6#8309C212yesmaximal
7C22×#126C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2214yes—
10#2637—28no—
11C2×#398C411yescharacteristic
12C2×#398C412yes—
13C2×#397C411yescharacteristic
14C2×#396C412yes—
15C2×#395C411yescharacteristic
16C2×#393C411yescharacteristic
17C2×#388C2212yes—
18C2×#387C2212yes—
19C2×#382C2211yescharacteristic
20#1985—28no—
21#1817C2214yes—
22#1817C2214yes—
23C2×#324C2214yes—
24C2×#323C2214yes—
25#1280C2218yes—
26C23×Q8C2×C411yescharacteristic
27C2×#126—24no—
28C2×#126—44no—
29C2×C4×Q8—44no—
30C2×C4×Q8D418yes—
31C22×C4⋊C4C2×C412yes—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4C2×C412yes—
40C22×C22⋊C4C2×C411yescharacteristic
41C22×C22⋊C4C2×C411yescharacteristic
42C22×C22⋊C4C2×C412yes—
43C22×C22⋊C4C2311yescharacteristic
44C22×C42C2311yescharacteristic
45C22×C42C2312yes—
46#432—48no—
47C2×C4⋊C8—24no—
48C2×C22⋊C8C2318yes—
49C2×#97—24no—
50#398—24no—
51#398—28no—
52#397—24no—
53#396—28no—
54#395—24no—
55#393—24no—
56#388—24no—
57#388—24no—
58#387—24no—
59#387—24no—
60#382—24no—
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C412yes—
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C412yes—
65C2×#95C2×C412yes—
66C2×#95C2×C412yes—
67C2×#95C2312yes—
68#336—28no—
69#336—28no—
70#324—44no—
71#323—44no—
72C25C22×C411yescharacteristic
73C22×Q8—42no—
74C22×Q8—44no—
75C22×Q8C22⋊C414yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C412yes—
79C23×C4C22×C412yes—
80C23×C4C22×C412yes—
81C23×C4C22×C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C22×C412yes—
85C23×C4C2411yescharacteristic
86C22×C8—28no—
87#126—416no—
88C4×Q8—48no—
89C4×Q8—48no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—82no—
103C2×C4⋊C4—82no—
104C2×C4⋊C4C22⋊C418yes—
105C2×C4⋊C4C22⋊C414yes—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—82no—
116C2×C42—24no—
117C2×C42—28no—
118C2×C42C2×D414yes—
119C2×C42C2×D414yes—
120C2×C42C2×D414yes—
121C4⋊C8—48no—
122C22⋊C8—28no—
123C22⋊C8—28no—
124#97—48no—
125#95—22no—
126#95—22no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—24no—
131#95—22no—
132#95—22no—
133C24—21no—
134C24—22no—
135C24—24no—
136C24—41no—
137C24—42no—
138C24—41no—
139C24C23×C411yescharacteristic
140C2×Q8—44no—
141C2×Q8—48no—
142C2×Q8—416no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4C2×C22⋊C414yes—
174C22×C4C2×C22⋊C412yes—
175C22×C4C2×C22⋊C414yes—
176C22×C4C2×C22⋊C412yes—
177C22×C4C22×D412yes—
178C22×C4H(5,𝔽2)14yes—
179C2×C8—416no—
180C2×C8—48no—
181C4⋊C4—48no—
182C4⋊C4—44no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—84no—
188C4⋊C4—84no—
189C22⋊C4—48no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—84no—
194C42—24no—
195C42—24no—
196C42—28no—
197C42—28no—
198C23—22no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—41no—
205C23—41no—
206C23—44no—
207C23—84no—
208C23—84no—
209C23—82no—
210C23—82no—
211C23C22×C22⋊C411yescharacteristic
212C23#51812yes—
213C23#56814yes—
214Q8—416no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—48no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4#53414yes—
251C8—88no—
252C22—24no—
253C22—24no—
254C22—22no—
255C22—22no—
256C22—82no—
257C22—82no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—164no—
262C22#230811yescharacteristic
263C22#287912yes—
264C22#288412yes—
265C22#288811yescharacteristic
266C22#288911yescharacteristic
267C22#297814yes—
268C4—24no—
269C4—42no—
270C4—44no—
271C4—42no—
272C4—44no—
273C4—82no—
274C4—82no—
275C4—81no—
276C4—81no—
277C4—81no—
278C2—24no—
279C2—161no—
280C2#1740912yes—
281C2#1741012yes—
282C2#2174311yescharacteristic
283C2#2175111yescharacteristic
284C2#2175711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21822C2semidirect
3. #9086C2semidirect
5. #8703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26372
#218222
#90862
#90842
#87032
#83092