Finite Group Databasebeta

← #417006

Subgroups — #417006

Order 512.

1056 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1056 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
1#35181C211yesmaximal, characteristic
2#21324C212yesmaximal
3#14161C211yesmaximal, characteristic
4#11276C218yesmaximal
5#10851C212yesmaximal
6C4×#343C211yesmaximal, characteristic
7C22×C4⋊Q8C2211yescharacteristic
8C2×C4×M16C2212yes—
9#2851—22no—
10#2646—22no—
11#2571—22no—
12#2570—24no—
13#2552—22no—
14#2317C2214yes—
15#2285C2212yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1993—24no—
19#1990—22no—
20#1990—22no—
21C2×#343C2214yes—
22#1736C2218yes—
23C2×#339C2218yes—
24#1446—22no—
25C4⋊#95—22no—
26C4⋊#95C2214yes—
27C23×Q8—21no—
28C22×M16C2314yes—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8Q814yes—
34C2×#125—22no—
35C2×#117C2314yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2312yes—
39C22×C4⋊C4C2312yes—
40C22×C42D412yes—
41C22×C42C2312yes—
42C22×C42C2311yescharacteristic
43C4×M16—24no—
44C4×M16—24no—
45C2×#97—22no—
46C2×C4×C8—22no—
47#400—28no—
48#398—28no—
49#389—24no—
50#389—24no—
51#389—24no—
52#387—28no—
53#385—28no—
54#384—24no—
55#384D412yes—
56#384D412yes—
57#382—22no—
58#382—24no—
59#382D414yes—
60#380—22no—
61#380—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—24no—
67C2×#95—21no—
68C2×#95—28no—
69C2×#95D412yes—
70#343—28no—
71#340—28no—
72#340—28no—
73#339—216no—
74C22×Q8—24no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8—84no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D414yes—
82C23×C4C2×D411yescharacteristic
83C23×C4C2411yescharacteristic
84C2×M16—24no—
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—24no—
89C2×M16#4118yes—
90C22×C8—24no—
91C4⋊Q8—28no—
92C4⋊Q8—28no—
93C4⋊Q8—44no—
94C4⋊Q8—44no—
95#125—84no—
96#117—28no—
97#117—28no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×Q814yes—
121C2×C22⋊C4—24no—
122C2×C42—22no—
123C2×C42—24no—
124C2×C42—28no—
125C2×C42—24no—
126C2×C42—22no—
127C2×C42—44no—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×Q812yes—
131C2×C42#4114yes—
132C2×C42#4112yes—
133#97—44no—
134C4×C8—44no—
135#95—28no—
136#95—28no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—42no—
141#95—88no—
142C24C22×D411yescharacteristic
143C2×Q8—48no—
144C2×Q8—84no—
145C2×Q8—84no—
146C2×Q8—88no—
147C2×Q8—88no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4#12011yescharacteristic
168C22×C4#12012yes—
169C22×C4#12011yescharacteristic
170C22×C4C22⋊Q812yes—
171C22×C4C22⋊Q812yes—
172C22×C4#12412yes—
173C22×C4#12412yes—
174C22×C4C22×D412yes—
175C22×C4C22×Q811yescharacteristic
176C22×C4C2×#4112yes—
177C22×C4C2×#4111yescharacteristic
178C22×C4C2×#4114yes—
179C22×C42−1+414yes—
180M16—28no—
181M16—28no—
182M16—44no—
183M16—44no—
184C2×C8—28no—
185C2×C8—44no—
186C2×C8—44no—
187C4⋊C4—28no—
188C4⋊C4—28no—
189C4⋊C4—28no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—88no—
201C4⋊C4—88no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C42—48no—
205C42—44no—
206C42—48no—
207C42—82no—
208C42C22⋊Q814yes—
209C42C22⋊Q814yes—
210C42#12414yes—
211C42#12414yes—
212C23—21no—
213C23—21no—
214C23—22no—
215C23C2×#12011yescharacteristic
216C23C2×C22⋊Q811yescharacteristic
217C23C2×#12411yescharacteristic
218C23#53212yes—
219C23#53612yes—
220Q8—816no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—48no—
231C2×C4—48no—
232C2×C4—48no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—84no—
244C2×C4—82no—
245C2×C4—84no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—84no—
249C2×C4#39312yes—
250C2×C4#39314yes—
251C2×C4#39312yes—
252C2×C4#45512yes—
253C2×C4#45612yes—
254C2×C4C2×C22⋊Q812yes—
255C2×C4C2×#12412yes—
256C2×C4#52914yes—
257C2×C4#54114yes—
258C2×C4#54714yes—
259C2×C4#54814yes—
260C2×C4#55214yes—
261C2×C4#55714yes—
262C8—48no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22C2×#39311yescharacteristic
272C22#243212yes—
273C22#256512yes—
274C22#264211yescharacteristic
275C22#286111yescharacteristic
276C22C2×#45511yescharacteristic
277C22C2×#45611yescharacteristic
278C22#302112yes—
279C4—22no—
280C4—24no—
281C4—84no—
282C4—84no—
283C4—84no—
284C4—84no—
285C4—84no—
286C4#202814yes—
287C4#245314yes—
288C4#248914yes—
289C2—21no—
290C2—22no—
291C2—21no—
292C2#1523811yescharacteristic
293C2C2×#202811yescharacteristic
294C2#2346311yescharacteristic
295C2#2569312yes—
296C2#2569512yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#351812
#213242
#141612
#112762
#108512
C4×#3432