Finite Group Databasebeta

← #417007

Subgroups — #417007

Order 512.

1084 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1084 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#14155C211yesmaximal, 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—
15C2×#391—22no—
16#2285C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1993—24no—
20#1990—22no—
21#1990—22no—
22C2×#343C2214yes—
23#1736C2218yes—
24C2×#339C2218yes—
25#1445—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#391—48no—
50#389—24no—
51#389—24no—
52#389—24no—
53#387—28no—
54#385—28no—
55#384—24no—
56#384D412yes—
57#384D412yes—
58#382—22no—
59#382—24no—
60#382D414yes—
61#380—22no—
62#380—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C2×#95—28no—
69C2×#95—21no—
70C2×#95D412yes—
71#343—28no—
72#340—28no—
73#340—28no—
74#339—216no—
75C22×Q8—24no—
76C22×Q8—44no—
77C22×Q8—44no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8—42no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D414yes—
85C23×C4C2×D411yescharacteristic
86C23×C4C2411yescharacteristic
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16#4118yes—
93C22×C8—24no—
94C4⋊Q8—28no—
95C4⋊Q8—28no—
96C4⋊Q8—44no—
97C4⋊Q8—44no—
98#125—84no—
99#117—28no—
100#117—28no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×Q814yes—
123C2×C22⋊C4—24no—
124C2×C42—22no—
125C2×C42—24no—
126C2×C42—28no—
127C2×C42—24no—
128C2×C42—22no—
129C2×C42—44no—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×Q812yes—
133C2×C42#4114yes—
134C2×C42#4112yes—
135#97—44no—
136C4×C8—44no—
137#95—28no—
138#95—28no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—416no—
143#95—42no—
144C24C22×D411yescharacteristic
145C2×Q8—416no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8—88no—
151C2×Q8—82no—
152C2×Q8—82no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4#12011yescharacteristic
174C22×C4#12012yes—
175C22×C4#12011yescharacteristic
176C22×C4C22⋊Q812yes—
177C22×C4C22⋊Q812yes—
178C22×C4#12412yes—
179C22×C4#12412yes—
180C22×C4C22×D412yes—
181C22×C4C22×Q811yescharacteristic
182C22×C4C2×#4112yes—
183C22×C4C2×#4111yescharacteristic
184C22×C4C2×#4114yes—
185C22×C42−1+414yes—
186M16—28no—
187M16—28no—
188M16—44no—
189M16—44no—
190C2×C8—28no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—28no—
194C4⋊C4—28no—
195C4⋊C4—28no—
196C4⋊C4—48no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—48no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—88no—
206C4⋊C4—88no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C42—48no—
210C42—44no—
211C42—48no—
212C42—82no—
213C42C22⋊Q814yes—
214C42C22⋊Q814yes—
215C42#12414yes—
216C42#12414yes—
217C23—21no—
218C23—21no—
219C23—22no—
220C23C2×#12011yescharacteristic
221C23C2×C22⋊Q811yescharacteristic
222C23C2×#12411yescharacteristic
223C23#53212yes—
224C23#53612yes—
225Q8—88no—
226Q8—88no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—84no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4#39314yes—
257C2×C4#39312yes—
258C2×C4#39312yes—
259C2×C4#45312yes—
260C2×C4#45412yes—
261C2×C4C2×C22⋊Q812yes—
262C2×C4C2×#12412yes—
263C2×C4#52914yes—
264C2×C4#54114yes—
265C2×C4#54714yes—
266C2×C4#54814yes—
267C2×C4#55214yes—
268C2×C4#55714yes—
269C8—48no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22C2×#39311yescharacteristic
279C22#243212yes—
280C22#256512yes—
281C22#264211yescharacteristic
282C22#286111yescharacteristic
283C22C2×#45311yescharacteristic
284C22C2×#45411yescharacteristic
285C22#302212yes—
286C4—22no—
287C4—24no—
288C4—84no—
289C4—84no—
290C4—84no—
291C4—84no—
292C4—84no—
293C4#202714yes—
294C4#245314yes—
295C4#248914yes—
296C2—21no—
297C2—22no—
298C2—21no—
299C2#1523811yescharacteristic
300C2C2×#202711yescharacteristic
301C2#2346411yescharacteristic
302C2#2569412yes—
303C2#2569612yes—
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
#141552
#112762
#108512
C4×#3432