Finite Group Databasebeta

← #237841

Subgroups — #237841

Order 512.

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

normal subgroups only

This group has 992 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#35181C211yesmaximal, characteristic
2#21804C212yesmaximal
3#14161C211yesmaximal, characteristic
4#10345C218yesmaximal
5#8014C212yesmaximal
6#7908C211yesmaximal, characteristic
7C22×C4⋊Q8C2211yescharacteristic
8C2×#431C2212yes—
9#2851—22no—
10#2646C2214yes—
11#2571—22no—
12#2570—24no—
13#2552—22no—
14#2317—22no—
15#2285C2212yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1981—24no—
19#1956—22no—
20#1956—22no—
21#1787C2218yes—
22#1446C2214yes—
23C4⋊#95—22no—
24C4⋊#95—22no—
25#1294C2218yes—
26#1280C2214yes—
27C23×Q8C2312yes—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8Q814yes—
33C2×#125—22no—
34C2×#117—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D412yes—
39C22×C42—21no—
40C22×C42D412yes—
41C22×C42C2311yescharacteristic
42#431—24no—
43#431—24no—
44C2×C4⋊C8—22no—
45C2×#103—28no—
46C2×#103—28no—
47C2×C22⋊C8C2314yes—
48C2×C4×C8—22no—
49#400—28no—
50#398—28no—
51#389—24no—
52#389—24no—
53#389—24no—
54#387—28no—
55#385—28no—
56#384—24no—
57#384D412yes—
58#384D412yes—
59#382—24no—
60#382—22no—
61#382—22no—
62#380—22no—
63#380C2314yes—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4D414yes—
69C2×#95—28no—
70C2×#95C2312yes—
71C2×#95C2312yes—
72C22×Q8—22no—
73C22×Q8—24no—
74C22×Q8—24no—
75C22×Q8—84no—
76C22×Q8C2×Q814yes—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D414yes—
81C23×C4C2×D411yescharacteristic
82C23×C4C2411yescharacteristic
83C22×C8—24no—
84C4⋊Q8—28no—
85C4⋊Q8—28no—
86C4⋊Q8—44no—
87C4⋊Q8—44no—
88#125—84no—
89#117—44no—
90#117—44no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D414yes—
113C2×C22⋊C4—24no—
114C2×C42—24no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—44no—
118C2×C42—42no—
119C2×C42—44no—
120C2×C42C2×D412yes—
121C2×C42C2×D412yes—
122C2×C42C2×Q812yes—
123C2×C42#4114yes—
124C2×C42#4112yes—
125C4⋊C8—44no—
126#103—88no—
127#103—88no—
128C22⋊C8—24no—
129C22⋊C8—24no—
130C4×C8—44no—
131#95—28no—
132#95—28no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—88no—
138C24C22×D411yescharacteristic
139C2×Q8—28no—
140C2×Q8—48no—
141C2×Q8—48no—
142C2×Q8—44no—
143C2×Q8—84no—
144C2×Q8—84no—
145C2×Q8—88no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—48no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4#12011yescharacteristic
166C22×C4#12012yes—
167C22×C4#12011yescharacteristic
168C22×C4C22⋊Q812yes—
169C22×C4C22⋊Q812yes—
170C22×C4#12412yes—
171C22×C4#12412yes—
172C22×C4#13514yes—
173C22×C4#13714yes—
174C22×C4C22×D412yes—
175C22×C4C22×Q811yescharacteristic
176C22×C4C2×#4112yes—
177C22×C4C2×#4111yescharacteristic
178C2×C8—44no—
179C2×C8—44no—
180C2×C8—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—88no—
194C4⋊C4—88no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C42—48no—
198C42—44no—
199C42—82no—
200C42—84no—
201C42C22⋊Q814yes—
202C42C22⋊Q814yes—
203C42#12414yes—
204C42#12414yes—
205C23—21no—
206C23—21no—
207C23—22no—
208C23C2×#12011yescharacteristic
209C23C2×C22⋊Q811yescharacteristic
210C23C2×#12411yescharacteristic
211C23C2×#13512yes—
212C23C2×#13712yes—
213Q8—416no—
214Q8—88no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—48no—
226C2×C4—48no—
227C2×C4—48no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4—84no—
243C2×C4—84no—
244C2×C4#39312yes—
245C2×C4#39314yes—
246C2×C4#39312yes—
247C2×C4#45214yes—
248C2×C4#45214yes—
249C2×C4#45512yes—
250C2×C4#45612yes—
251C2×C4#47918yes—
252C2×C4#48814yes—
253C2×C4#49114yes—
254C2×C4C2×C22⋊Q812yes—
255C2×C4C2×#12412yes—
256C8—84no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22C2×#39311yescharacteristic
266C22C2×#45212yes—
267C22C2×#45511yescharacteristic
268C22C2×#45611yescharacteristic
269C22#302712yes—
270C22C2×#47912yes—
271C22C2×#48811yescharacteristic
272C22C2×#49111yescharacteristic
273C4—22no—
274C4—22no—
275C4—22no—
276C4—84no—
277C4—84no—
278C4—84no—
279C4—84no—
280C4—84no—
281C4#202618yes—
282C4#202814yes—
283C2—21no—
284C2—22no—
285C2—21no—
286C2#1839212yes—
287C2#1840612yes—
288C2C2×#202611yescharacteristic
289C2C2×#202811yescharacteristic
290C2#2346911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351812
#218042
#141612
#103452
#80142
#79082