Finite Group Databasebeta

← #122194

Subgroups — #122194

Order 512.

816 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 138 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#37480C211yesmaximal, characteristic
2#10737C212yesmaximal
3#10729C211yesmaximal, characteristic
4#8115C212yesmaximal
5#8112C211yesmaximal, characteristic
6C23×M16C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2212yes—
9#2842C412yes—
10#2840C412yes—
11#2665—24no—
12#2628—22no—
13#2607—22no—
14#2439—24no—
15#2280C2211yescharacteristic
16#2280C2212yes—
17C2×#344—22no—
18C2×#344—24no—
19#1306—22no—
20#1306—24no—
21C24×C4C2311yescharacteristic
22C22×M16—21no—
23C22×M16—22no—
24C22×M16—22no—
25C22×M16—21no—
26C22×M16—22no—
27C22×M16—44no—
28C22×M16Q812yescomplemented
29C23×C8—21no—
30C2×C4⋊Q8—22no—
31C2×#117D412yes—
32C2×#117D414yes—
33#407—22no—
34#407—22no—
35#407—22no—
36#407—22no—
37#407—22no—
38#407—21no—
39#407Q812yescomplemented
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8Q814yescomplemented
43#400—24no—
44#399—24no—
45#396—28no—
46#395—24no—
47#393—24no—
48#393—28no—
49#391—28no—
50#389—24no—
51#387—28no—
52#382—22no—
53#381D412yes—
54#381D414yes—
55#380—22no—
56#380—22no—
57#380—22no—
58#379C2×C411yescharacteristic
59#379C2×C414yes—
60#379C2×C411yescharacteristic
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64#344—44no—
65#344—48no—
66C25C22×C411yescharacteristic
67C22×Q8—24no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4C2×D412yes—
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C23×C4C2×Q811yescharacteristic
75C23×C4C2×Q812yes—
76C23×C4C2×Q811yescharacteristic
77C23×C4#4111yescharacteristic
78C23×C4#4112yes—
79C23×C4#4111yescharacteristic
80C2×M16—24no—
81C2×M16—28no—
82C2×M16—24no—
83C2×M16—22no—
84C2×M16—44no—
85C2×M16—48no—
86C2×M16—42no—
87C2×M16—44no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—44no—
91C2×M16—44no—
92C2×M16—42no—
93C2×M16—41no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—42no—
97C22×C8—22no—
98C22×C8—24no—
99C22×C8—44no—
100C22×C8—42no—
101C22×C8—41no—
102C4⋊Q8—82no—
103C4⋊Q8—82no—
104#117—42no—
105#117—42no—
106#117—44no—
107#117—44no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4C22⋊C412yes—
117C2×C22⋊C4C22⋊C418yes—
118C2×C22⋊C4C22⋊C412yes—
119C2×C42—22no—
120C2×C42—24no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123C22⋊C8—24no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127#95—24no—
128#95—24no—
129#95—28no—
130#95—24no—
131#95—24no—
132#95—24no—
133C24—42no—
134C24—82no—
135C24C2×C22⋊C412yes—
136C24C2×C22⋊C411yescharacteristic
137C24C4×Q811yescharacteristic
138C24C4×Q811yescharacteristic
139C24C4×Q812yes—
140C2×Q8—84no—
141C2×Q8—88no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—28no—
148C22×C4—22no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4#12012yes—
161C22×C4#12011yescharacteristic
162C22×C4#12011yescharacteristic
163C22×C4C22⋊Q812yes—
164C22×C4C22⋊Q811yescharacteristic
165C22×C4C22⋊Q811yescharacteristic
166C22×C4C22⋊Q812yes—
167C22×C4C22⋊Q812yes—
168C22×C4C22⋊Q812yes—
169C22×C4C22⋊Q812yes—
170C22×C4#12412yes—
171C22×C4#12411yescharacteristic
172C22×C4#12411yescharacteristic
173C22×C4#12412yes—
174C22×C4C4⋊Q812yes—
175C22×C4C4⋊Q811yescharacteristic
176C22×C4C4⋊Q811yescharacteristic
177C22×C4C4⋊Q812yes—
178M16—28no—
179M16—44no—
180M16—44no—
181M16—44no—
182M16—44no—
183M16—84no—
184M16—82no—
185M16—82no—
186M16—82no—
187C2×C8—24no—
188C2×C8—42no—
189C2×C8—48no—
190C2×C8—42no—
191C2×C8—48no—
192C2×C8—44no—
193C2×C8—42no—
194C2×C8—81no—
195C2×C8—81no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—82no—
199C4⋊C4—88no—
200C4⋊C4—84no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—48no—
204C22⋊C4—82no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C22⋊C4—82no—
208C42—42no—
209C42—42no—
210C42—84no—
211C23—22no—
212C23—24no—
213C23—41no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—88no—
224C23#37911yescharacteristic
225C23#39111yescharacteristic
226C23#39112yes—
227C23#39112yes—
228C23#39111yescharacteristic
229Q8—168no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—48no—
253C2×C4—48no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—48no—
258C2×C4#39312yes—
259C2×C4#39311yescharacteristic
260C2×C4#39311yescharacteristic
261C2×C4#39512yes—
262C2×C4#39511yescharacteristic
263C2×C4#39511yescharacteristic
264C2×C4B(4,𝔽2)12yes—
265C2×C4#45812yes—
266C8—44no—
267C8—84no—
268C8—82no—
269C22—22no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—44no—
281C22—82no—
282C22—84no—
283C22—82no—
284C22—84no—
285C22#144611yescharacteristic
286C22#179012yes—
287C4—22no—
288C4—21no—
289C4—22no—
290C4—21no—
291C4—82no—
292C4—88no—
293C4—82no—
294C4#202911yescharacteristic
295C4#203011yescharacteristic
296C4#206212yes—
297C2—22no—
298C2—41no—
299C2—42no—
300C2—42no—
301C2—41no—
302C2#1139011yescharacteristic
303C2#1139512yes—
subgroupG/H
center, socleC22#1446
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. C22×M16Q8semidirect
39. #407Q8semidirect
42. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#107372
#107292
#81152
#81122