Finite Group Databasebeta

← #122182

Subgroups — #122182

Order 512.

848 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 170 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#35275C211yesmaximal, characteristic
2#10857C212yesmaximal
3#8041C211yesmaximal, characteristic
4#8028C211yesmaximal, characteristic
5#7903C212yesmaximal
6C2×#544C412yes—
7C2×#423C2212yes—
8C2×C4×M16C2212yes—
9#2868—21no—
10#2859—21no—
11#2660—22no—
12#2552—22no—
13#2551—22no—
14#2440—24no—
15#2317C412yes—
16#2285C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C4×#117C2211yescharacteristic
19C2×#343—24no—
20#1296—21no—
21#1296—21no—
22#1295—22no—
23#1279—22no—
24#1279—22no—
25C22×M16—22no—
26C22×M16—22no—
27#544—44no—
28#544—44no—
29C2×C4⋊Q8—24no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×#125—22no—
33C2×#125—22no—
34C2×C22⋊Q8—24no—
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C412yes—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4D414yes—
40C22×C42C2311yescharacteristic
41#423—22no—
42#423—22no—
43#423—22no—
44#423Q814yescomplemented
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C4×M16—22no—
48C4×M16—22no—
49C4×M16—22no—
50C4×M16Q814yescomplemented
51C2×#97—22no—
52C2×C4×C8—22no—
53#398—28no—
54#395—28no—
55#389—24no—
56#389—24no—
57#389—24no—
58#385—24no—
59#384—24no—
60#384—22no—
61#384—21no—
62#384—21no—
63#384—22no—
64#384D411yescharacteristic
65#384D411yescharacteristic
66#382—22no—
67#382D412yes—
68#380—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D412yes—
71C4×C4⋊C4D412yes—
72C4×C22⋊C4—22no—
73#376—21no—
74C43—21no—
75#343—48no—
76C22×Q8—24no—
77C23×C4C22⋊C414yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C2×M16—24no—
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—44no—
89C2×M16—44no—
90C22×C8—42no—
91C22×C8—42no—
92C4⋊Q8—48no—
93C4⋊Q8—44no—
94C4⋊Q8—44no—
95C4⋊Q8—84no—
96C4⋊Q8—84no—
97#125—48no—
98#125—44no—
99#125—44no—
100C22⋊Q8—416no—
101#117—22no—
102#117—22no—
103#117—24no—
104#117—42no—
105#117—42no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C22⋊C412yes—
124C2×C4⋊C4C22⋊C412yes—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—22no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×Q812yes—
137C2×C42C2×Q812yes—
138C2×C42#4112yes—
139C2×C42#4112yes—
140C4⋊C8—24no—
141C4⋊C8—24no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144#97—24no—
145#97—42no—
146C4×C8—24no—
147C4×C8—42no—
148#95—28no—
149#95—22no—
150#95—22no—
151#95—24no—
152C24C2×C22⋊C411yescharacteristic
153C2×Q8—48no—
154C2×Q8—88no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—44no—
169C22×C4—48no—
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×Q812yes—
173C22×C4C4×Q812yes—
174C22×C4#12012yes—
175C22×C4#12012yes—
176C22×C4C22⋊Q812yes—
177C22×C4C22⋊Q812yes—
178C22×C4#12412yes—
179C22×C4C4⋊Q812yes—
180M16—28no—
181M16—44no—
182M16—44no—
183M16—44no—
184M16—48no—
185M16—84no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—82no—
191C2×C8—82no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—48no—
199C4⋊C4—48no—
200C4⋊C4—48no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—84no—
210C4⋊C4—88no—
211C4⋊C4#10012yes—
212C4⋊C4#10012yes—
213C4⋊C4#10114yes—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—44no—
217C42—24no—
218C42—24no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42C22⋊Q812yes—
226C42C22⋊Q812yes—
227C42C22⋊Q812yes—
228C42C22⋊Q812yes—
229C42#12412yes—
230C42#12411yescharacteristic
231C42#12411yescharacteristic
232C42C4⋊Q811yescharacteristic
233C42C4⋊Q811yescharacteristic
234C42C4⋊Q812yes—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23#37911yescharacteristic
240C23#39112yes—
241Q8—168no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—48no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—164no—
272C2×C4#39111yescharacteristic
273C2×C4#39111yescharacteristic
274C2×C4#39112yes—
275C2×C4#39312yes—
276C2×C4#39312yes—
277C2×C4#39512yes—
278C2×C4#39512yes—
279C2×C4C2×#10011yescharacteristic
280C2×C4C2×#10011yescharacteristic
281C2×C4C2×#10112yes—
282C2×C4#45312yes—
283C2×C4#45412yes—
284C2×C4#45512yes—
285C2×C4#45612yes—
286C8—48no—
287C8—84no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—42no—
294C22#144611yescharacteristic
295C22#178311yescharacteristic
296C22#178411yescharacteristic
297C22#179212yes—
298C22#179312yes—
299C4—21no—
300C4—21no—
301C4—22no—
302C4—22no—
303C4—42no—
304C4—42no—
305C4—44no—
306C4—84no—
307C4—82no—
308C4—82no—
309C4#199712yes—
310C4#199712yes—
311C4#202712yes—
312C4#202812yes—
313C4#206012yes—
314C4#206112yes—
315C2—42no—
316C2#846911yescharacteristic
317C2#847111yescharacteristic
318C2#847312yes—
319C2#1138211yescharacteristic
320C2#1139212yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
44. #423Q8semidirect
50. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#108572
#80412
#80282
#79032