Finite Group Databasebeta

← #417003

Subgroups — #417003

Order 512.

1324 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1324 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#35020C211yesmaximal, characteristic
2#13924C211yesmaximal, characteristic
3#13924C212yesmaximal
4#11287C218yesmaximal
5#10854C212yesmaximal
6C4×#342C211yesmaximal, characteristic
7C23×C4⋊C4C2211yescharacteristic
8#2851—22no—
9#2646—22no—
10#2569—24no—
11#2442—24no—
12#2315C2214yes—
13C2×#380C2211yescharacteristic
14C2×#380C2212yes—
15C2×C4×C22⋊C4C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17C2×#342C2218yes—
18C2×#342C2214yes—
19#1445—24no—
20#1445—22no—
21C4⋊#95—22no—
22C4⋊#95—24no—
23C22⋊#95C2214yes—
24C22⋊#95C2218yes—
25C24×C4C2312yes—
26C24×C4C2311yescharacteristic
27C2×#125—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—24no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4D414yes—
36C22×C4⋊C4Q814yes—
37C22×C22⋊C4C2312yes—
38C22×C22⋊C4C2312yes—
39C22×C22⋊C4C2314yes—
40C22×C22⋊C4C2314yes—
41C22×C42—22no—
42C22×C42D412yes—
43#400—28no—
44#398—28no—
45#389—24no—
46#387—28no—
47#385—28no—
48#385—28no—
49#382—22no—
50#380—22no—
51#380—28no—
52#380—22no—
53#380—22no—
54#380—22no—
55#380—24no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—24no—
59C4×C22⋊C4—24no—
60C2×#95—22no—
61C2×#95—21no—
62C2×#95—216no—
63C2×#95—28no—
64C2×#95D414yes—
65C2×#95D412yes—
66#342—28no—
67#342—216no—
68C25C2411yescharacteristic
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—24no—
72C23×C4—24no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—44no—
76C23×C4—84no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C23×C4C2×D418yes—
80C23×C4C2×D414yes—
81C23×C4C2×D412yes—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×Q814yes—
84C23×C4C2×Q812yes—
85C23×C4#4112yes—
86C23×C4#4114yes—
87#125—84no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4#4118yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—44no—
125C2×C42—48no—
126C2×C42—44no—
127#95—28no—
128#95—28no—
129#95—24no—
130#95—24no—
131#95—24no—
132#95—44no—
133#95—42no—
134#95—88no—
135#95—816no—
136C24—42no—
137C24—42no—
138C24C22×D411yescharacteristic
139C24C22×D412yes—
140C24C22×Q811yescharacteristic
141C24C2×#4112yes—
142C24C2×#4111yescharacteristic
143C24C2×#4114yes—
144C242−1+414yes—
145C22×C4—24no—
146C22×C4—28no—
147C22×C4—24no—
148C22×C4—28no—
149C22×C4—24no—
150C22×C4—28no—
151C22×C4—24no—
152C22×C4—28no—
153C22×C4—24no—
154C22×C4—216no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—48no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4—416no—
165C22×C4—48no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—84no—
178C22×C4—88no—
179C22×C4—84no—
180C22×C4#12011yescharacteristic
181C22×C4#12012yes—
182C22×C4#12011yescharacteristic
183C22×C4C22⋊Q812yes—
184C22×C4C22⋊Q814yes—
185C22×C4C22⋊Q812yes—
186C22×C4C22⋊Q814yes—
187C22×C4#12414yes—
188C22×C4#12412yes—
189C22×C4#12412yes—
190C22×C4#12414yes—
191C4⋊C4—44no—
192C4⋊C4—48no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C4⋊C4—88no—
198C4⋊C4—88no—
199C22⋊C4—28no—
200C22⋊C4—28no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—48no—
207C22⋊C4—44no—
208C22⋊C4—48no—
209C42—44no—
210C42—82no—
211C42—88no—
212C23—22no—
213C23—24no—
214C23—22no—
215C23—24no—
216C23—41no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—41no—
222C23—48no—
223C23—48no—
224C23C2×#12011yescharacteristic
225C23C2×C22⋊Q811yescharacteristic
226C23C2×C22⋊Q812yes—
227C23C2×#12411yescharacteristic
228C23C2×#12412yes—
229C23#52914yes—
230C23#53212yes—
231C23#53612yes—
232C23#54114yes—
233C23#54714yes—
234C23#54814yes—
235C23#55214yes—
236C23#55714yes—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—28no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—28no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—416no—
248C2×C4—48no—
249C2×C4—48no—
250C2×C4—416no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—48no—
254C2×C4—48no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—48no—
259C2×C4—84no—
260C2×C4—88no—
261C2×C4—84no—
262C2×C4—88no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4—816no—
267C2×C4—82no—
268C2×C4—88no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#39312yes—
272C2×C4#39314yes—
273C2×C4#39312yes—
274C2×C4#45814yes—
275C22—24no—
276C22—28no—
277C22—24no—
278C22—28no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—21no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—44no—
291C22C2×#39311yescharacteristic
292C22#243212yes—
293C22#245314yes—
294C22#248914yes—
295C22#256512yes—
296C22#264211yescharacteristic
297C22#286111yescharacteristic
298C22C2×#45812yes—
299C22#302712yes—
300C4—22no—
301C4—24no—
302C4—88no—
303C4—84no—
304C4—88no—
305C4—84no—
306C4—162no—
307C4#203014yes—
308C2—24no—
309C2—21no—
310C2—22no—
311C2—21no—
312C2—24no—
313C2#1523811yescharacteristic
314C2C2×#203011yescharacteristic
315C2#2346911yescharacteristic
316C2#2566414yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#350202
#139242
#112872
#108542
C4×#3422