Finite Group Databasebeta

← #214027

Subgroups — #214027

Order 512.

805 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 805 conjugacy classes of subgroups — too many to draw. Only its 190 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#35670C211yesmaximal, characteristic
2#17371C212yesmaximal
3#9452C214yesmaximal, complemented
4#9450C214yesmaximal, complemented
5#8745C212yesmaximal
6#8527C212yesmaximal
7C2×#541C2211yescharacteristic
8C2×#519C2211yescharacteristic
9#2983C2214yes—
10#2565C2212yes—
11#2546C2212yes—
12#2499C412yes—
13#2478—22no—
14#2478—22no—
15#2462—22no—
16#2462—22no—
17#2434C412yes—
18#2327C412yes—
19C4×C4⋊Q8C412yes—
20#2298C2211yescharacteristic
21#1558C2214yes—
22#1534C2214yes—
23#1467—22no—
24#1467C2214yes—
25#1464—22no—
26#1464C2214yes—
27#1461—22no—
28#1458—22no—
29#1285C2218yes—
30C23×Q8C2×C411yescharacteristic
31#541—44no—
32C2×C4⋊Q8C2×C412yes—
33C2×C4⋊Q8C2×C412yes—
34C2×C4⋊Q8C2311yescharacteristic
35C2×#126—22no—
36C2×#125—22no—
37C2×#125C2311yescharacteristic
38C2×C22⋊Q8C2×C412yes—
39#519—44no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8C2×C412yes—
42C2×C4×Q8D412yes—
43C2×C4×Q8D412yes—
44C2×C4×Q8C2312yes—
45C2×#117C2×C411yescharacteristic
46C2×#117D412yes—
47C2×#117D412yes—
48C2×#117C2311yescharacteristic
49#446—48no—
50#432—24no—
51#432C2318yes—
52#431—24no—
53#398—22no—
54#398—22no—
55#398—22no—
56#398—22no—
57#397—22no—
58#397—22no—
59#395—22no—
60#395—22no—
61#391C2×C412yes—
62#391C2×C412yes—
63#391C2×C412yes—
64#391C2×C412yes—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#388—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#385—22no—
80#385—22no—
81#384C2×C412yes—
82#382C2×C412yes—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2312yes—
86C4×C22⋊C4C2×C412yes—
87#376C2×C411yescharacteristic
88C43C2×C411yescharacteristic
89#333—44no—
90#330—44no—
91#324—24no—
92#324—24no—
93#324—24no—
94#323—24no—
95C22×Q8—42no—
96C22×Q8—44no—
97C22×Q8C22⋊C412yes—
98C22×Q8C22×C412yes—
99C22×Q8C22×C412yes—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C4⋊Q8—24no—
104C4⋊Q8—44no—
105C4⋊Q8—44no—
106#126—48no—
107#125—24no—
108#125—48no—
109C22⋊Q8—44no—
110C4×Q8—28no—
111C4×Q8—44no—
112C4×Q8—44no—
113C4×Q8—44no—
114C4×Q8—44no—
115#117—24no—
116#117—44no—
117#117—44no—
118#117—42no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22×C412yes—
133C2×C4⋊C4C22×C412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4C22⋊C412yes—
140C2×C22⋊C4C2×D412yes—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42C22⋊C411yescharacteristic
144C2×C42C22⋊C411yescharacteristic
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22×C411yescharacteristic
148C2×C42C22×C412yes—
149C2×C42C22×C412yes—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42C2411yescharacteristic
153C4⋊C8—28no—
154C4⋊C8—44no—
155C4⋊C8—44no—
156C22⋊C8—28no—
157#97—28no—
158#97—44no—
159C4×C8—44no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173C24C2×C22⋊C411yescharacteristic
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—24no—
177C2×Q8—24no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—416no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C22⋊C412yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C23×C411yescharacteristic
201C22×C4C22×D412yes—
202C2×C8—48no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C22⋊C4—24no—
222C22⋊C4—42no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C42—22no—
226C42—24no—
227C42—24no—
228C42—42no—
229C42—42no—
230C42—41no—
231C42—41no—
232C42—42no—
233C42—42no—
234C42—44no—
235C42—44no—
236C42H(5,𝔽2)14yes—
237C23—21no—
238C23—22no—
239C23—42no—
240C23C22×C22⋊C411yescharacteristic
241Q8—48no—
242Q8—48no—
243Q8—84no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4#41314yes—
275C2×C4#51812yes—
276C2×C4#53414yes—
277C8—88no—
278C22—22no—
279C22—42no—
280C22—41no—
281C22—82no—
282C22#230811yescharacteristic
283C22C2×#41312yes—
284C4—24no—
285C4—22no—
286C4—24no—
287C4—44no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4—82no—
296C2—22no—
297C2—81no—
298C2#2175411yescharacteristic
299C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9452C2semidirect
4. #9450C2semidirect

Maximal subgroups

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

subgroupindex
#356702
#173712
#94522
#94502
#87452
#85272