Finite Group Databasebeta

← #237753

Subgroups — #237753

Order 512.

1046 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 194 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#34642C211yesmaximal, characteristic
2#20740C212yesmaximal
3#13961C211yesmaximal, characteristic
4#10342C214yesmaximal
5#10341C214yesmaximal
6#8018C212yesmaximal
7#7910C211yesmaximal, characteristic
8C2×#533—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2660—24no—
12#2659—22no—
13#2659C2214yes—
14#2399—24no—
15#2285—21no—
16#2285C2212yes—
17C2×#383C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1823C2214yes—
20#1803—24no—
21#1787C2214yes—
22#1763—24no—
23#1446C2212yes—
24#1445C2212yes—
25#1436—21no—
26#1436—21no—
27#1434—22no—
28#1294C2218yes—
29#1280C2214yes—
30C23×Q8C2311yescharacteristic
31#533—48no—
32C2×C4⋊Q8—22no—
33C2×#125—22no—
34C2×C22⋊Q8—24no—
35C2×C4×Q8—24no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8Q814yescomplemented
41C2×#117—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2311yescharacteristic
44C22×C42—21no—
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42C2311yescharacteristic
48#432—24no—
49#432—24no—
50C2×C4⋊C8—22no—
51C2×#103—28no—
52C2×C22⋊C8C2314yes—
53C2×#97—22no—
54#398—28no—
55#398—28no—
56#395—28no—
57#395—28no—
58#384—22no—
59#384D414yes—
60#383—24no—
61#382—22no—
62#382—22no—
63#380—22no—
64#380—24no—
65#380C2314yes—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4D414yes—
69#376—22no—
70#376—22no—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95D412yes—
78C2×#95C2312yes—
79C2×#95C2312yes—
80#340—28no—
81C22×Q8—24no—
82C22×Q8—21no—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×Q8—44no—
86C22×Q8C2×Q812yes—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2411yescharacteristic
95C22×C8—24no—
96C4⋊Q8—48no—
97#125—48no—
98C22⋊Q8—48no—
99C22⋊Q8—48no—
100C4×Q8—28no—
101C4×Q8—28no—
102C4×Q8—416no—
103C4×Q8—44no—
104C4×Q8—44no—
105#117—48no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D418yes—
120C2×C4⋊C4C2×Q812yes—
121C2×C22⋊C4—24no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—24no—
127C2×C42—24no—
128C2×C42—24no—
129C2×C42—24no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×Q812yes—
135C2×C42#4114yes—
136C2×C42#4112yes—
137C4⋊C8—44no—
138#103—416no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141#97—44no—
142#95—28no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—44no—
149#95—44no—
150#95—42no—
151#95—42no—
152#95—44no—
153#95—44no—
154C24C22×D411yescharacteristic
155C2×Q8—28no—
156C2×Q8—24no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—48no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—84no—
164C2×Q8—84no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—28no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4#12012yes—
196C22×C4#12011yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4C22⋊Q812yes—
199C22×C4C22⋊Q812yes—
200C22×C4#12412yes—
201C22×C4#12412yes—
202C22×C4AGL(1,ℤ/8)14yes—
203C22×C4#13714yes—
204C22×C4C22×D412yes—
205C22×C4C22×Q811yescharacteristic
206C22×C4C2×#4112yes—
207C22×C4C2×#4111yescharacteristic
208C2×C8—48no—
209C2×C8—44no—
210C4⋊C4—24no—
211C4⋊C4—48no—
212C4⋊C4—48no—
213C4⋊C4—416no—
214C4⋊C4—48no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C22⋊C4—48no—
222C42—24no—
223C42—24no—
224C42—24no—
225C42—24no—
226C42—48no—
227C42—48no—
228C42C22⋊Q814yes—
229C42C22⋊Q814yes—
230C42#12414yes—
231C42#12414yes—
232C23—22no—
233C23—22no—
234C23C2×#12011yescharacteristic
235C23C2×C22⋊Q811yescharacteristic
236C23C2×#12411yescharacteristic
237C23C2×AGL(1,ℤ/8)12yes—
238C23C2×#13712yes—
239Q8—28no—
240Q8—48no—
241Q8—44no—
242Q8—816no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#39314yes—
287C2×C4#39312yes—
288C2×C4#39312yes—
289C2×C4C2×C22⋊Q812yes—
290C2×C4C2×#12412yes—
291C8—84no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22C2×#39311yescharacteristic
301C22#300311yescharacteristic
302C22#300411yescharacteristic
303C22#302212yes—
304C22#302512yes—
305C22#307912yes—
306C22#314112yes—
307C4—22no—
308C4—22no—
309C4—24no—
310C4—22no—
311C4—22no—
312C4—44no—
313C4—44no—
314C4—44no—
315C4—82no—
316C4—88no—
317C4—162no—
318C2—22no—
319C2—22no—
320C2#1815312yes—
321C2#1815512yes—
322C2#2345211yescharacteristic
323C2#2346411yescharacteristic
324C2#2346711yescharacteristic
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
40. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#346422
#207402
#139612
#103422
#103412
#80182
#79102