Finite Group Databasebeta

← #237827

Subgroups — #237827

Order 512.

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

normal subgroups only

This group has 924 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#21840C212yesmaximal
3#14274C211yesmaximal, characteristic
4#10346C214yesmaximal
5#10342C214yesmaximal
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—
17C2×#391—22no—
18#2283C2212yes—
19C2×#382C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1993—24no—
22#1987—22no—
23#1987—22no—
24#1823C2218yes—
25#1445—22no—
26#1445C2212yes—
27#1445C2212yes—
28#1294C2218yes—
29#1280C2212yes—
30#1280C2212yes—
31C23×Q8—21no—
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⋊C4C2312yes—
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43#432—24no—
44#432—24no—
45C2×C4⋊C8—22no—
46C2×C22⋊C8C2314yes—
47C2×#97—22no—
48#400—24no—
49#398—24no—
50#398—24no—
51#395—24no—
52#391—24no—
53#391—22no—
54#391—22no—
55#391—48no—
56#389—24no—
57#389—22no—
58#389—22no—
59#387—24no—
60#387—24no—
61#387—24no—
62#387—24no—
63#385—24no—
64#383—22no—
65#382—24no—
66#382D414yes—
67#381—22no—
68#380—22no—
69#380C2314yes—
70C4×C4⋊C4D412yes—
71C4×C4⋊C4D412yes—
72C4×C22⋊C4—22no—
73#376—22no—
74#376—22no—
75#376—22no—
76C2×#95—21no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95D412yes—
80C2×#95C2312yes—
81C2×#95C2311yescharacteristic
82C2×#95C2311yescharacteristic
83#340—28no—
84#340—28no—
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—44no—
88C22×Q8—44no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8—42no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2411yescharacteristic
99C22×C8—24no—
100C4⋊Q8—28no—
101C4⋊Q8—28no—
102C4⋊Q8—44no—
103C4⋊Q8—44no—
104C22⋊Q8—44no—
105C22⋊Q8—44no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
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—22no—
126C2×C42—24no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—42no—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×Q812yes—
133C2×C42#4114yes—
134C2×C42#4112yes—
135C4⋊C8—44no—
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—48no—
152#95—42no—
153#95—48no—
154C24C22×D411yescharacteristic
155C2×Q8—416no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—84no—
160C2×Q8—84no—
161C2×Q8—84no—
162C2×Q8—84no—
163C2×Q8—82no—
164C2×Q8—82no—
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—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4#12012yes—
192C22×C4#12011yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4C22⋊Q812yes—
195C22×C4C22⋊Q812yes—
196C22×C4#12412yes—
197C22×C4#12412yes—
198C22×C4AGL(1,ℤ/8)14yes—
199C22×C4C22×D412yes—
200C22×C4C22×Q811yescharacteristic
201C22×C4C2×#4112yes—
202C22×C4C2×#4111yescharacteristic
203C2×C8—48no—
204C2×C8—44no—
205C4⋊C4—28no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
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×AGL(1,ℤ/8)12yes—
234C23#57512yes—
235Q8—88no—
236Q8—88no—
237Q8—164no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—82no—
277C2×C4#39314yes—
278C2×C4#39312yes—
279C2×C4#39312yes—
280C2×C4C2×C22⋊Q812yes—
281C2×C4C2×#12412yes—
282C8—84no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22C2×#39311yescharacteristic
292C22#300612yes—
293C22#302211yescharacteristic
294C22#302211yescharacteristic
295C22#302812yes—
296C22#307712yes—
297C22#313811yescharacteristic
298C22#314011yescharacteristic
299C4—22no—
300C4—22no—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—44no—
305C4—84no—
306C4—84no—
307C4—84no—
308C4—82no—
309C4—162no—
310C2—21no—
311C2—22no—
312C2—21no—
313C2#1836112yes—
314C2#1837112yes—
315C2#2345411yescharacteristic
316C2#2346411yescharacteristic
317C2#2347011yescharacteristic
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
#218402
#142742
#103462
#103422
#80172
#79102