Finite Group Databasebeta

← #213932

Subgroups — #213932

Order 512.

886 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 186 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#36711C211yesmaximal, characteristic
2#21832C212yesmaximal, complemented
3#9512C214yesmaximal, complemented
4#9512C214yesmaximal, complemented
5#8755C212yesmaximal, complemented
6#8553C212yesmaximal
7C2×#544C2211yescharacteristic
8C22×#123C2211yescharacteristic
9#2921C2214yes—
10#2848C412yes—
11#2694—24no—
12#2674—24no—
13#2626C412yes—
14#2583C412yes—
15#2482C2212yes—
16#2482C2212yes—
17#2435C412yes—
18#2313C2211yescharacteristic
19#1998—24no—
20#1997—24no—
21#1822C2214yes—
22#1821C2214yes—
23#1462C2214yes—
24#1462C2214yes—
25#1306C2218yes—
26C23×D4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28#544—44no—
29C2×C4⋊Q8D414yes—
30C2×#125D414yes—
31C2×#124—22no—
32C2×#123—24no—
33C2×#123—44no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8C2×C412yes—
36C2×C22⋊Q8C2×C412yes—
37C2×#117C2312yes—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2311yescharacteristic
40C22×C22⋊C4C2×C411yescharacteristic
41C22×C22⋊C4C2311yescharacteristic
42C22×C22⋊C4C2311yescharacteristic
43#433—48no—
44#423—24no—
45#407C2318yes—
46C4×M16—24no—
47#399—22no—
48#399—22no—
49#399—22no—
50#398—22no—
51#397—22no—
52#397—22no—
53#396—24no—
54#395—22no—
55#394—24no—
56#393—22no—
57#390—22no—
58#389—24no—
59#388—24no—
60#388—24no—
61#387—24no—
62#385—24no—
63#385C2×C414yes—
64#385C2×C414yes—
65#381C2312yes—
66#380—22no—
67#380C2×C412yes—
68#379C2×C414yes—
69#379C2×C414yes—
70C4×C22⋊C4C2×C412yes—
71#343—28no—
72#343—24no—
73#343—24no—
74#323—44no—
75#323—44no—
76C25C22×C411yescharacteristic
77C25C22×C411yescharacteristic
78C22×Q8C22⋊C412yes—
79C22×Q8C22⋊C412yes—
80C22×D4—22no—
81C22×D4—42no—
82C22×D4—82no—
83C23×C4—22no—
84C23×C4—42no—
85C23×C4—82no—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C2411yescharacteristic
91C2×M16—28no—
92C4⋊Q8—44no—
93C4⋊Q8—44no—
94#125—44no—
95#125—44no—
96#124—84no—
97#123—48no—
98#123—48no—
99C22⋊Q8—44no—
100C22⋊Q8—44no—
101C22⋊Q8—84no—
102#117—28no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C4⋊C4C22⋊C414yes—
112C2×C4⋊C4C22⋊C414yes—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4C22×C414yes—
127C2×C22⋊C4C22×C414yes—
128C2×C42—22no—
129C2×C42C2×D414yes—
130C4⋊C8—48no—
131C22⋊C8—28no—
132C22⋊C8—28no—
133#97—44no—
134C4×C8—44no—
135#95—22no—
136#95—24no—
137#95—22no—
138#95—24no—
139#95—22no—
140#95—22no—
141C24—21no—
142C24—22no—
143C24—21no—
144C24—22no—
145C24—42no—
146C24—42no—
147C24—42no—
148C24—81no—
149C24—81no—
150C24—81no—
151C24C23×C411yescharacteristic
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—82no—
155C2×Q8—82no—
156C2×D4—44no—
157C2×D4—44no—
158C2×D4—82no—
159C2×D4—84no—
160C2×D4—84no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—82no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4C2×C22⋊C412yes—
179C22×C4C2×C22⋊C414yes—
180C22×C4C2×C22⋊C414yes—
181C22×C4C2×C22⋊C412yes—
182C22×C4C22×D412yes—
183C22×C4H(5,𝔽2)14yes—
184M16—416no—
185C2×C8—48no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C4⋊C4—82no—
198C22⋊C4—28no—
199C22⋊C4—44no—
200C22⋊C4—48no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C42—24no—
208C42—24no—
209C42—82no—
210C23—22no—
211C23—22no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—41no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—44no—
221C23—41no—
222C23—42no—
223C23—41no—
224C23—42no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23—88no—
230C23—164no—
231C23C22×C22⋊C411yescharacteristic
232C23#51812yes—
233Q8—162no—
234Q8—162no—
235D4—88no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—88no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#53414yes—
259C8—88no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—82no—
269C22—84no—
270C22—84no—
271C22—84no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—164no—
278C22#230811yescharacteristic
279C22#288412yes—
280C4—24no—
281C4—42no—
282C4—42no—
283C4—44no—
284C4—44no—
285C4—84no—
286C4—81no—
287C4—81no—
288C2—22no—
289C2—42no—
290C2—82no—
291C2—161no—
292C2#2175111yescharacteristic
293C2#2177811yescharacteristic
294C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21832C2semidirect
3. #9512C2semidirect
4. #9512C2semidirect
5. #8755C2semidirect

Maximal subgroups

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

subgroupindex
#367112
#218322
#95122
#87552
#85532