Finite Group Databasebeta

← #122108

Subgroups — #122108

Order 512.

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

normal subgroups only

This group has 836 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#10729C211yesmaximal, characteristic
3#10717C212yesmaximal
4#8100C211yesmaximal, characteristic
5#7946C212yesmaximal
6C23×M16C2211yescharacteristic
7C2×#407C2211yescharacteristic
8#2842—21no—
9#2840—21no—
10#2665—24no—
11#2628—22no—
12#2607C414yescomplemented
13#2439—24no—
14#2280C2212yes—
15#2280C2211yescharacteristic
16#2280C2212yes—
17C2×#344—22no—
18C2×H(3,ℤ/4)—24no—
19#1304—21no—
20#1304—21no—
21#1284—24no—
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C22×M16—21no—
25C22×M16—21no—
26C22×M16—22no—
27C22×M16—21no—
28C22×M16—44no—
29C23×C8Q812yescomplemented
30C2×C4⋊Q8—22no—
31C2×#117—22no—
32C2×#117D412yes—
33C2×#117D414yes—
34#407—22no—
35#407—21no—
36#407Q812yescomplemented
37C2×C22⋊C8—22no—
38#400—24no—
39#399—24no—
40#396—28no—
41#395—24no—
42#393—28no—
43#393—24no—
44#391—28no—
45#389—24no—
46#387—28no—
47#382—22no—
48#381—22no—
49#381—22no—
50#381C2×C412yes—
51#380—22no—
52#380—22no—
53#380—22no—
54#380—22no—
55#380—22no—
56#379—22no—
57#379—22no—
58#379D411yescharacteristic
59#379D411yescharacteristic
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4C2×C414yes—
63C4×C22⋊C4D414yes—
64C4×C22⋊C4Q814yescomplemented
65#344—44no—
66H(3,ℤ/4)—48no—
67C25C2×D411yescharacteristic
68C22×Q8—24no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D412yes—
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D412yes—
75C23×C4C2×Q811yescharacteristic
76C23×C4C2×Q811yescharacteristic
77C23×C4C2×Q812yes—
78C23×C4#4111yescharacteristic
79C23×C4#4111yescharacteristic
80C23×C4#4112yes—
81C2×M16—28no—
82C2×M16—24no—
83C2×M16—22no—
84C2×M16—48no—
85C2×M16—44no—
86C2×M16—44no—
87C2×M16—42no—
88C2×M16—41no—
89C2×M16—42no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—42no—
93C2×M16—42no—
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—44no—
97C22×C8—41no—
98C4⋊Q8—82no—
99C4⋊Q8—82no—
100#117—44no—
101#117—44no—
102#117—44no—
103#117—44no—
104#117—42no—
105#117—42no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4C22⋊C412yes—
120C2×C22⋊C4C22⋊C412yes—
121C2×C42—22no—
122C2×C42—24no—
123C2×C42C22⋊C418yes—
124C22⋊C8—24no—
125C22⋊C8—42no—
126#95—24no—
127#95—24no—
128#95—28no—
129#95—24no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—42no—
134C24—42no—
135C24—82no—
136C24C2×C22⋊C411yescharacteristic
137C24#12012yes—
138C24C22⋊Q812yes—
139C24#12411yescharacteristic
140C24C4⋊Q811yescharacteristic
141C2×Q8—84no—
142C2×Q8—88no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—28no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4C2×C22⋊C412yes—
163C22×C4C4×Q811yescharacteristic
164C22×C4C4×Q811yescharacteristic
165C22×C4C4×Q812yes—
166C22×C4#12011yescharacteristic
167C22×C4#12011yescharacteristic
168C22×C4C22⋊Q812yes—
169C22×C4C22⋊Q812yes—
170C22×C4C22⋊Q811yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4C22⋊Q812yes—
173C22×C4C22⋊Q812yes—
174C22×C4#12412yes—
175C22×C4#12411yescharacteristic
176C22×C4#12412yes—
177C22×C4C4⋊Q812yes—
178C22×C4C4⋊Q811yescharacteristic
179C22×C4C4⋊Q812yes—
180M16—48no—
181M16—44no—
182M16—44no—
183M16—44no—
184M16—82no—
185C2×C8—28no—
186C2×C8—24no—
187C2×C8—44no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—81no—
192C2×C8—81no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—88no—
199C4⋊C4—84no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—84no—
203C22⋊C4—82no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C22⋊C4—82no—
207C22⋊C4—84no—
208C22⋊C4—84no—
209C22⋊C4—84no—
210C42—48no—
211C42—48no—
212C42—42no—
213C42—42no—
214C23—22no—
215C23—24no—
216C23—41no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—41no—
225C23—42no—
226C23—88no—
227C23#37911yescharacteristic
228C23#39111yescharacteristic
229C23#39111yescharacteristic
230C23#39312yes—
231C23#39512yes—
232Q8—168no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4#39112yes—
265C2×C4#39112yes—
266C2×C4#39311yescharacteristic
267C2×C4#39311yescharacteristic
268C2×C4#39511yescharacteristic
269C2×C4#39511yescharacteristic
270C2×C4B(4,𝔽2)12yes—
271C2×C4#45812yes—
272C8—44no—
273C8—82no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—44no—
286C22—84no—
287C22—82no—
288C22—82no—
289C22—84no—
290C22#144611yescharacteristic
291C22#179012yes—
292C4—21no—
293C4—22no—
294C4—22no—
295C4—21no—
296C4—82no—
297C4—88no—
298C4—82no—
299C4—164no—
300C4#202911yescharacteristic
301C4#203011yescharacteristic
302C4#206212yes—
303C2—22no—
304C2—42no—
305C2—41no—
306C2—42no—
307C2—41no—
308C2#1138912yes—
309C2#1139011yescharacteristic
subgroupG/H
center, socleC22#1446
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ 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
12. #2607C4semidirect
29. C23×C8Q8semidirect
36. #407Q8semidirect
64. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#107292
#107172
#81002
#79462