Finite Group Databasebeta

← #496399

Subgroups — #496399

Order 512.

908 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 198 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#35333C211yesmaximal, characteristic
2#20723C212yesmaximal
3#13300C218yesmaximal, complemented
4#12859C214yesmaximal
5C22×#125C411yescharacteristic, complemented
6C2×C4×#41C2211yescharacteristic
7#2851—22no—
8#2629—22no—
9#2579—22no—
10#2579—22no—
11#2566C2214yes—
12#2565C414yescomplemented
13#2317C412yescomplemented
14#2311C2212yes—
15#2285C411yescharacteristic, complemented
16#2283—22no—
17#1820—22no—
18#1799—22no—
19#1799C2214yes—
20#1782C2214yes—
21C4×#99—22no—
22C4×#99C2214yes—
23#1393C22116yes—
24C22×#41C2311yescharacteristic
25C2×#125—21no—
26C2×#125—21no—
27C2×#125—41no—
28C2×#125—42no—
29C2×#125—41no—
30C2×#125C2×C414yes—
31C4×#41—41no—
32C4×#41—42no—
33C4×#41—41no—
34C2×C4×Q8C2×C412yes—
35C2×C4×D4D412yes—
36C2×C4×D4D412yes—
37C2×C4×D4C2312yes—
38C2×#117C2×C411yescharacteristic
39C2×#117C2×C412yes—
40C2×#117D412yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C412yes—
43C22×C42C2×C411yescharacteristic
44C22×C42D412yes—
45C2×#99—24no—
46C2×#99C2318yes—
47#398—24no—
48#398—24no—
49#396—24no—
50#396—24no—
51#391C2×C414yes—
52#390C2314yes—
53#389—22no—
54#389—22no—
55#387—24no—
56#387—24no—
57#386—22no—
58#386—22no—
59#385—24no—
60#385—24no—
61#385—22no—
62#385—22no—
63#384—21no—
64#384C2×C414yes—
65#383—42no—
66#382—21no—
67#382—42no—
68#382C2×C414yes—
69#381—24no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380—24no—
74C4×C4⋊C4—42no—
75C4×C4⋊C4C2×C414yes—
76C4×C22⋊C4—24no—
77#376—42no—
78#342—24no—
79#342—24no—
80C2×#41—24no—
81C2×#41—41no—
82C2×#41—41no—
83C2×#41—42no—
84C22×Q8C22×C411yescharacteristic
85C22×D4C2×D412yes—
86C22×D4C2411yescharacteristic
87C23×C4—22no—
88C23×C4C22⋊C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93#125—44no—
94#125—44no—
95#125—42no—
96#125—42no—
97#125—41no—
98#125—42no—
99#125—41no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×D4—42no—
103C4×D4—42no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108#117—41no—
109#117—41no—
110#117—42no—
111#117—42no—
112#117—44no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4C22⋊C412yes—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C22×C414yes—
128C2×C4⋊C4C22×C412yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4C22⋊C411yescharacteristic
134C2×C22⋊C4C22⋊C411yescharacteristic
135C2×C22⋊C4C2×D412yes—
136C2×C22⋊C4C2×D412yes—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—44no—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C411yescharacteristic
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C412yes—
149C2×C42C22⋊C411yescharacteristic
150C2×C42C22×C412yes—
151C2×C42C22×C414yes—
152#99—216no—
153#99—48no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—44no—
160#95—44no—
161#95—44no—
162#95—44no—
163C24C2×C22⋊C411yescharacteristic
164C24C22×D412yes—
165#41—48no—
166#41—48no—
167C2×Q8—22no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—42no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D42−1+414yes—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—84no—
197C22×C4—82no—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4C23×C411yescharacteristic
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4—88no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—44no—
224C22⋊C4—48no—
225C22⋊C4—48no—
226C22⋊C4—84no—
227C22⋊C4—84no—
228C42—22no—
229C42—22no—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—44no—
239C42—88no—
240C42#9911yescharacteristic
241C42#9912yes—
242C42#9911yescharacteristic
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23#41012yes—
251C23C22×C22⋊C411yescharacteristic
252C23#53614yes—
253Q8—44no—
254D4—44no—
255D4—48no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—88no—
287C2×C4—88no—
288C2×C4—88no—
289C2×C4—88no—
290C2×C4C2×#9911yescharacteristic
291C2×C4C2×#9911yescharacteristic
292C2×C4C2×#9912yes—
293C2×C4C2×#9912yes—
294C2×C4#51912yes—
295C22—22no—
296C22—22no—
297C22—24no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22#231011yescharacteristic
306C22C22×#9911yescharacteristic
307C22C2×#41011yescharacteristic
308C4—21no—
309C4—21no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—84no—
318C4—84no—
319C4—168no—
320C2—21no—
321C2—21no—
322C2—44no—
323C2—42no—
324C2#2182511yescharacteristic
325C2#3335812yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13300C2semidirect
5. C22×#125C4semidirect
12. #2565C4semidirect
13. #2317C4semidirect
15. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#353332
#207232
#133002
#128592