Finite Group Databasebeta

← #480766

Subgroups — #480766

Order 512.

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

normal subgroups only

This group has 838 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#17366C212yesmaximal
3#11577C218yesmaximal, complemented
4#11520C214yesmaximal, complemented
5C22×#125C411yescharacteristic
6C2×C4×#41C2211yescharacteristic
7#2920—22no—
8#2851—22no—
9#2629—22no—
10#2579C414yes—
11#2579C2214yes—
12#2566—22no—
13#2565—22no—
14#2317C2212yes—
15#2311C412yes—
16#2285C411yescharacteristic
17#1465—22no—
18#1465C2214yes—
19#1464—22no—
20#1464C2214yes—
21#1461—22no—
22#1460C2214yes—
23#1326C22116yes—
24C22×#41C2×C411yescharacteristic
25C2×#125—22no—
26C2×#125—21no—
27C2×#125—21no—
28C2×#125—42no—
29C2×#125—41no—
30C2×#125—41no—
31C4×#41—42no—
32C4×#41—41no—
33C4×#41—41no—
34C2×C4×Q8D412yes—
35C2×C4×D4C2×C412yes—
36C2×C4×D4D412yes—
37C2×C4×D4C2312yes—
38C2×#117C2×C412yes—
39C2×#117D412yes—
40C2×#117C2311yescharacteristic
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C414yes—
43C22×C42C2×C411yescharacteristic
44C22×C42D412yes—
45#432—24no—
46#432—42no—
47#431—24no—
48#431—42no—
49#423—42no—
50#407—24no—
51C4×M16—42no—
52#398—24no—
53#398—24no—
54#396—24no—
55#396—24no—
56#391—22no—
57#390—22no—
58#389—22no—
59#389—22no—
60#387—24no—
61#387—24no—
62#386C2×C414yes—
63#386C2×C414yes—
64#385—24no—
65#385—24no—
66#385C2×C414yes—
67#385C2314yes—
68#384—22no—
69#384—21no—
70#382—22no—
71#382—21no—
72#380—22no—
73#380—22no—
74#380C2×C414yes—
75C4×C4⋊C4—22no—
76#324—24no—
77#324—24no—
78#323—24no—
79#323C2318yes—
80C2×#41—24no—
81C2×#41—41no—
82C2×#41—41no—
83C2×#41—42no—
84C22×Q8C22⋊C411yescharacteristic
85C22×D4C22⋊C411yescharacteristic
86C22×D4C22×C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C412yes—
91C23×C4C22×C414yes—
92C23×C4C2×D412yes—
93C2×M16—44no—
94#125—42no—
95#125—42no—
96#125—44no—
97#125—44no—
98#125—42no—
99#125—41no—
100#125—41no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×D4—42no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108C4×D4—42no—
109#117—42no—
110#117—42no—
111#117—41no—
112#117—41no—
113#117—44no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4C22⋊C412yes—
132C2×C22⋊C4C22×C414yes—
133C2×C22⋊C4C22×C411yescharacteristic
134C2×C22⋊C4C2×D412yes—
135C2×C22⋊C4C2411yescharacteristic
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42C22⋊C411yescharacteristic
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C411yescharacteristic
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C4⋊C8—44no—
151C4⋊C8—44no—
152C22⋊C8—28no—
153C22⋊C8—44no—
154#97—44no—
155C4×C8—44no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161C24C2×C22⋊C412yes—
162C24C23×C411yescharacteristic
163#41—48no—
164#41—48no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8#10114yes—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—24no—
171C2×D4—41no—
172C2×D4—42no—
173C2×D4—41no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—84no—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C22×D412yes—
202M16—88no—
203C2×C8—48no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—48no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C22⋊C4—28no—
218C22⋊C4—42no—
219C22⋊C4—41no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—84no—
223C22⋊C4—84no—
224C42—22no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—21no—
229C42—21no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—44no—
235C42#9912yes—
236C42#9911yescharacteristic
237C42#9911yescharacteristic
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23C22×C22⋊C411yescharacteristic
247C23#56812yes—
248Q8—44no—
249D4—48no—
250D4—44no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—88no—
281C2×C4C2×#9912yes—
282C2×C4C2×#9912yes—
283C2×C4C2×#9911yescharacteristic
284C2×C4C2×#9911yescharacteristic
285C2×C4C2×#10112yes—
286C2×C4C2×#10114yes—
287C8—88no—
288C22—22no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—44no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—82no—
298C22#287911yescharacteristic
299C22C22×#9911yescharacteristic
300C22C22×#10111yescharacteristic
301C4—22no—
302C4—22no—
303C4—21no—
304C4—21no—
305C4—22no—
306C4—24no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—84no—
311C4—84no—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—82no—
316C2#1739711yescharacteristic
317C2#3334712yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11577C2semidirect
4. #11520C2semidirect

Maximal subgroups

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

subgroupindex
#353332
#173662
#115772
#115202