Finite Group Databasebeta

← #237808

Subgroups — #237808

Order 512.

1152 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 234 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#35020C211yesmaximal, characteristic
2C2×#1823C212yesmaximal
3#13904C211yesmaximal, characteristic
4#10346C218yesmaximal, complemented
5#8023C212yesmaximal
6#7907C211yesmaximal, characteristic
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9#2851—22no—
10#2646C2214yes—
11#2569—24no—
12#2442—24no—
13#2315—22no—
14C2×#381—22no—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1823—24no—
19#1823C2218yes—
20C2×#340—24no—
21#1445C2214yes—
22C4⋊#95—22no—
23#1294C2218yes—
24#1280C2214yes—
25C24×C4—21no—
26C24×C4C2311yescharacteristic
27C23×C8—22no—
28C2×#125—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—24no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4D414yes—
36C22×C4⋊C4Q814yes—
37C22×C4⋊C4C2312yes—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—21no—
40C22×C22⋊C4D412yes—
41C22×C42D412yes—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8C2314yes—
46#400—28no—
47#398—28no—
48#389—24no—
49#387—28no—
50#385—28no—
51#385—28no—
52#382—22no—
53#381—48no—
54#380—28no—
55#380—22no—
56#380—22no—
57#380—24no—
58#380D414yes—
59#380C2314yes—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C2×#95—22no—
63C2×#95—28no—
64C2×#95C2312yes—
65C2×#95C2312yes—
66#340—216no—
67C25C2×D411yescharacteristic
68C23×C4—24no—
69C23×C4—22no—
70C23×C4—24no—
71C23×C4—22no—
72C23×C4—44no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—44no—
76C23×C4—44no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C23×C4C2×D414yes—
80C23×C4C2×D412yes—
81C23×C4C2×Q812yes—
82C23×C4#4114yes—
83C23×C4#4112yes—
84C23×C4C2411yescharacteristic
85C22×C8—24no—
86C22×C8—48no—
87C22×C8—44no—
88#125—84no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C2×D418yes—
107C2×C4⋊C4C2×Q814yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—44no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—44no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123C22⋊C8—44no—
124#95—28no—
125#95—28no—
126#95—24no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—416no—
131C24—42no—
132C24—42no—
133C24#12012yes—
134C24C22⋊Q812yes—
135C24#12412yes—
136C24#13514yes—
137C24AGL(1,ℤ/8)14yes—
138C24C22×D411yescharacteristic
139C22×C4—24no—
140C22×C4—28no—
141C22×C4—24no—
142C22×C4—28no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—44no—
147C22×C4—48no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—48no—
158C22×C4—416no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—84no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4—84no—
170C22×C4#12011yescharacteristic
171C22×C4#12011yescharacteristic
172C22×C4C22⋊Q814yes—
173C22×C4C22⋊Q812yes—
174C22×C4C22⋊Q814yes—
175C22×C4#12414yes—
176C22×C4#12412yes—
177C22×C4#12414yes—
178C22×C4C22×D412yes—
179C22×C4C22×Q811yescharacteristic
180C22×C4C2×#4112yes—
181C22×C4C2×#4111yescharacteristic
182C2×C8—48no—
183C2×C8—44no—
184C2×C8—88no—
185C4⋊C4—28no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—48no—
191C4⋊C4—84no—
192C4⋊C4—88no—
193C4⋊C4—88no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—48no—
199C22⋊C4—84no—
200C42—44no—
201C42—82no—
202C23—22no—
203C23—24no—
204C23—22no—
205C23—24no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—48no—
211C23—48no—
212C23#39314yes—
213C23#44914yes—
214C23#44914yes—
215C23#47818yes—
216C23#48814yes—
217C23#49014yes—
218C23C2×#12011yescharacteristic
219C23C2×C22⋊Q811yescharacteristic
220C23C2×#12411yescharacteristic
221C23C2×#13512yes—
222C23C2×AGL(1,ℤ/8)12yes—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—28no—
227C2×C4—28no—
228C2×C4—24no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—48no—
235C2×C4—48no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—48no—
239C2×C4—44no—
240C2×C4—48no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—84no—
246C2×C4—816no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—88no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—168no—
253C2×C4#39312yes—
254C2×C4#39312yes—
255C2×C4#45814yes—
256C2×C4C2×C22⋊Q812yes—
257C2×C4C2×#12412yes—
258C8—84no—
259C22—24no—
260C22—24no—
261C22—28no—
262C22—28no—
263C22—21no—
264C22—21no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—24no—
269C22—48no—
270C22—44no—
271C22—44no—
272C22#202518yes—
273C22C2×#39311yescharacteristic
274C22C2×#44912yes—
275C22#302212yes—
276C22C2×#45812yes—
277C22C2×#47812yes—
278C22C2×#48811yescharacteristic
279C22C2×#49011yescharacteristic
280C4—22no—
281C4—22no—
282C4—22no—
283C4—84no—
284C4—88no—
285C4—84no—
286C4—162no—
287C4#203014yes—
288C2—21no—
289C2—22no—
290C2—21no—
291C2—28no—
292C2#1820614yes—
293C2C2×#202511yescharacteristic
294C2#2346411yescharacteristic
295C2C2×#203011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10346C2semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#18232
#139042
#103462
#80232
#79072