Finite Group Databasebeta

← #161138

Subgroups — #161138

Order 512.

1274 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 242 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
1C2×#2296C211yesmaximal, characteristic
2#20715C212yesmaximal
3#7613C218yesmaximal, complemented
4#7593C214yesmaximal, complemented
5C22×#117C411yescharacteristic
6C22×#117C411yescharacteristic
7C23×C42C2211yescharacteristic
8C2×#432—22no—
9#2296—22no—
10#2296—22no—
11#2296C414yes—
12#2296C2214yes—
13C2×#389C412yes—
14C2×#389C2212yes—
15C2×#385—22no—
16C2×C4×C4⋊C4—22no—
17#1817C2214yes—
18#1797—22no—
19#1797C2214yes—
20#1779—22no—
21#1749—22no—
22#1749C2214yes—
23#1271C22116yescomplemented
24C24×C4C2×C411yescharacteristic
25C24×C4D412yes—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—28no—
29C2×#117—44no—
30C2×#117C2×C414yes—
31C2×#117C2×C414yes—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42—21no—
39C22×C42—21no—
40C22×C42—44no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C412yes—
43C22×C42D412yes—
44C22×C42D412yes—
45C22×C42D412yes—
46C22×C42C2312yes—
47C22×C42C2311yescharacteristic
48#432—48no—
49C2×C4⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8—24no—
53C2×#97—24no—
54#389—22no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389C2×C414yes—
59#389C2×C414yes—
60#389C2×C414yes—
61#389C2314yes—
62#385—24no—
63#385—24no—
64#385—24no—
65#385—24no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—24no—
70C2×#95—22no—
71C2×#95—22no—
72#336—24no—
73#336C2318yes—
74C25C22⋊C411yescharacteristic
75C23×C4—24no—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—44no—
81C23×C4—44no—
82C23×C4—44no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22×C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C22×C8—28no—
94#117—216no—
95#117—216no—
96#117—48no—
97#117—48no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—48no—
107C2×C4⋊C4—48no—
108C2×C4⋊C4M1618yes—
109C2×C4⋊C4C22×C414yes—
110C2×C4⋊C4C22×C414yes—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—44no—
115C2×C42—28no—
116C2×C42—28no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—44no—
120C2×C42—44no—
121C2×C42—44no—
122C2×C42—44no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—44no—
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C412yes—
128C2×C42C22⋊C411yescharacteristic
129C2×C42C22⋊C411yescharacteristic
130C2×C42C22×C411yescharacteristic
131C2×C42C22×C412yes—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D412yes—
135C2×C42C2411yescharacteristic
136C4⋊C8—48no—
137C22⋊C8—48no—
138C22⋊C8—48no—
139C22⋊C8—48no—
140#97—48no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145C24—22no—
146C24—22no—
147C24—42no—
148C24—42no—
149C24#10114yes—
150C24C2×C22⋊C412yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—232no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—28no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—432no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4#9914yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4C2×C22⋊C412yes—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C412yes—
195C22×C4C2×C22⋊C412yes—
196C22×C4C2×M1614yes—
197C22×C4C23×C411yescharacteristic
198C22×C4C22×D412yes—
199C2×C8—48no—
200C2×C8—48no—
201C2×C8—48no—
202C4⋊C4—24no—
203C4⋊C4—28no—
204C4⋊C4—44no—
205C4⋊C4—48no—
206C4⋊C4—48no—
207C4⋊C4—44no—
208C4⋊C4—48no—
209C4⋊C4—48no—
210C4⋊C4C2×M1618yes—
211C22⋊C4—216no—
212C22⋊C4—48no—
213C42—216no—
214C42—24no—
215C42—24no—
216C42—24no—
217C42—24no—
218C42—41no—
219C42—41no—
220C42—42no—
221C42—41no—
222C42—42no—
223C42—41no—
224C42—48no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—28no—
234C23—44no—
235C23—44no—
236C23—48no—
237C23—48no—
238C23#40712yes—
239C23C2×#9912yes—
240C23C2×#10114yes—
241C23C2×#10112yes—
242C23C22×C22⋊C411yescharacteristic
243C2×C4—216no—
244C2×C4—216no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—28no—
256C2×C4—28no—
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—48no—
265C2×C4—48no—
266C2×C4—48no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—416no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—88no—
277C2×C4#40712yes—
278C2×C4C2×#9914yes—
279C2×C4#42114yes—
280C2×C4C22×M1612yes—
281C2×C4#57514yes—
282C8—88no—
283C22—22no—
284C22—24no—
285C22—22no—
286C22—24no—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—416no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22#146314yes—
296C22C2×#40711yescharacteristic
297C22C22×#9911yescharacteristic
298C22C22×#10111yescharacteristic
299C22#289612yes—
300C22C2×#42112yes—
301C4—24no—
302C4—24no—
303C4—22no—
304C4—22no—
305C4—24no—
306C4—48no—
307C4—42no—
308C4—42no—
309C4—84no—
310C2—24no—
311C2—44no—
312C2C2×#146311yescharacteristic
313C2#1745012yes—
314C2#1755912yes—
315C2#1767611yescharacteristic
316C2#1775511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7613C2semidirect
4. #7593C2semidirect
23. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#207152
#76132
#75932