Finite Group Databasebeta

← #237824

Subgroups — #237824

Order 512.

928 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 190 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#35120C211yesmaximal, characteristic
2#21811C212yesmaximal
3#14282C211yesmaximal, characteristic
4#10345C214yesmaximal
5#10341C214yesmaximal
6#8017C212yesmaximal
7#7910C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2212yes—
10#2855—22no—
11#2659C2212yes—
12#2646C2212yes—
13#2612—24no—
14#2606—22no—
15#2605—22no—
16#2329—22no—
17#2283C2212yes—
18C2×#382C2211yescharacteristic
19C2×#376C2211yescharacteristic
20#1981—22no—
21#1981—22no—
22#1960—24no—
23#1787C2218yes—
24#1446C2212yes—
25#1446C2212yes—
26#1445—22no—
27C4⋊#95—22no—
28#1294C2218yes—
29#1280C2212yes—
30#1280C2212yes—
31C23×Q8C2312yes—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8Q814yescomplemented
37C2×C22⋊Q8—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43#432—24no—
44#432—24no—
45C2×C4⋊C8—22no—
46C2×#103—28no—
47C2×#103—28no—
48C2×C22⋊C8C2314yes—
49C2×#97—22no—
50#400—24no—
51#398—24no—
52#398—24no—
53#395—24no—
54#391—24no—
55#391—22no—
56#391—22no—
57#389—24no—
58#389—22no—
59#389—22no—
60#387—24no—
61#387—24no—
62#387—24no—
63#387—24no—
64#385—24no—
65#383—22no—
66#382—24no—
67#382D414yes—
68#381—22no—
69#380—22no—
70#380C2314yes—
71C4×C4⋊C4D412yes—
72C4×C4⋊C4D412yes—
73C4×C22⋊C4—22no—
74#376—22no—
75#376—22no—
76#376—22no—
77C2×#95—21no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95D412yes—
81C2×#95C2312yes—
82C2×#95C2311yescharacteristic
83C2×#95C2311yescharacteristic
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—24no—
88C22×Q8—84no—
89C22×Q8C2×Q814yes—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2411yescharacteristic
97C22×C8—24no—
98C4⋊Q8—28no—
99C4⋊Q8—28no—
100C4⋊Q8—44no—
101C4⋊Q8—44no—
102C22⋊Q8—44no—
103C22⋊Q8—44no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C22⋊C4—24no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—24no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—42no—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×Q812yes—
131C2×C42#4114yes—
132C2×C42#4112yes—
133C4⋊C8—44no—
134#103—88no—
135#103—88no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138#97—44no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95—24no—
150#95—24no—
151#95—42no—
152#95—84no—
153#95—84no—
154C24C22×D411yescharacteristic
155C2×Q8—28no—
156C2×Q8—44no—
157C2×Q8—48no—
158C2×Q8—48no—
159C2×Q8—44no—
160C2×Q8—84no—
161C2×Q8—84no—
162C2×Q8—82no—
163C2×Q8—82no—
164C2×Q8—88no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4#12011yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12012yes—
194C22×C4C22⋊Q812yes—
195C22×C4C22⋊Q812yes—
196C22×C4#12412yes—
197C22×C4#12412yes—
198C22×C4#13714yes—
199C22×C4C22×D412yes—
200C22×C4C22×Q811yescharacteristic
201C22×C4C2×#4112yes—
202C22×C4C2×#4111yescharacteristic
203C2×C8—48no—
204C2×C8—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—48no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—84no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C42—44no—
220C42—44no—
221C42—82no—
222C42—82no—
223C42C22⋊Q814yes—
224C42C22⋊Q814yes—
225C42#12414yes—
226C42#12414yes—
227C23—21no—
228C23—21no—
229C23—22no—
230C23C2×#12011yescharacteristic
231C23C2×C22⋊Q811yescharacteristic
232C23C2×#12411yescharacteristic
233C23C2×#13712yes—
234C23#57512yes—
235Q8—416no—
236Q8—84no—
237Q8—84no—
238Q8—164no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—84no—
276C2×C4—84no—
277C2×C4—82no—
278C2×C4#39312yes—
279C2×C4#39312yes—
280C2×C4#39314yes—
281C2×C4C2×C22⋊Q812yes—
282C2×C4C2×#12412yes—
283C8—84no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22C2×#39311yescharacteristic
293C22#300712yes—
294C22#302211yescharacteristic
295C22#302211yescharacteristic
296C22#302712yes—
297C22#307812yes—
298C22#313911yescharacteristic
299C22#314011yescharacteristic
300C4—22no—
301C4—22no—
302C4—24no—
303C4—22no—
304C4—22no—
305C4—44no—
306C4—84no—
307C4—84no—
308C4—84no—
309C4—82no—
310C4—162no—
311C2—21no—
312C2—22no—
313C2—21no—
314C2#1835712yes—
315C2#1836712yes—
316C2#2345511yescharacteristic
317C2#2346411yescharacteristic
318C2#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

Complements

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

NH ≅ G/Ntype
36. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351202
#218112
#142822
#103452
#103412
#80172
#79102