Finite Group Databasebeta

← #237828

Subgroups — #237828

Order 512.

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

normal subgroups only

This group has 896 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#21841C212yesmaximal
3#14282C211yesmaximal, characteristic
4#10347C214yesmaximal
5#10343C214yesmaximal
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#1993—24no—
21#1988—22no—
22#1988—22no—
23#1824C2218yes—
24#1446—22no—
25#1445C2212yes—
26#1445C2212yes—
27C4⋊#95—22no—
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#389—24no—
56#389—22no—
57#389—22no—
58#387—24no—
59#387—24no—
60#387—24no—
61#387—24no—
62#385—24no—
63#383—22no—
64#382—24no—
65#382D414yes—
66#381—22no—
67#380—22no—
68#380C2314yes—
69C4×C4⋊C4D412yes—
70C4×C4⋊C4D412yes—
71C4×C22⋊C4—22no—
72#376—22no—
73#376—22no—
74#376—22no—
75C2×#95—21no—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95D412yes—
79C2×#95C2312yes—
80C2×#95C2311yescharacteristic
81C2×#95C2311yescharacteristic
82#340—28no—
83#340—28no—
84C22×Q8—22no—
85C22×Q8—24no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—82no—
89C22×Q8—82no—
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—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×Q814yes—
122C2×C22⋊C4—24no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—24no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—42no—
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C2×C42C2×Q812yes—
132C2×C42#4114yes—
133C2×C42#4112yes—
134C4⋊C8—44no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137#97—44no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—24no—
149#95—24no—
150#95—42no—
151#95—84no—
152#95—84no—
153C24C22×D411yescharacteristic
154C2×Q8—44no—
155C2×Q8—48no—
156C2×Q8—84no—
157C2×Q8—84no—
158C2×Q8—88no—
159C2×Q8—82no—
160C2×Q8—82no—
161C2×Q8—88no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4#12011yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12012yes—
191C22×C4C22⋊Q812yes—
192C22×C4C22⋊Q812yes—
193C22×C4#12412yes—
194C22×C4#12412yes—
195C22×C4#13714yes—
196C22×C4C22×D412yes—
197C22×C4C22×Q811yescharacteristic
198C22×C4C2×#4112yes—
199C22×C4C2×#4111yescharacteristic
200C2×C8—48no—
201C2×C8—44no—
202C4⋊C4—28no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—84no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C42—44no—
218C42—44no—
219C42—82no—
220C42—82no—
221C42C22⋊Q814yes—
222C42C22⋊Q814yes—
223C42#12414yes—
224C42#12414yes—
225C23—21no—
226C23—21no—
227C23—22no—
228C23C2×#12011yescharacteristic
229C23C2×C22⋊Q811yescharacteristic
230C23C2×#12411yescharacteristic
231C23C2×#13712yes—
232C23#57512yes—
233Q8—816no—
234Q8—164no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—48no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—82no—
274C2×C4#39312yes—
275C2×C4#39312yes—
276C2×C4#39314yes—
277C2×C4C2×C22⋊Q812yes—
278C2×C4C2×#12412yes—
279C8—84no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22C2×#39311yescharacteristic
289C22#300712yes—
290C22#302111yescharacteristic
291C22#302111yescharacteristic
292C22#302812yes—
293C22#307812yes—
294C22#313911yescharacteristic
295C22#314011yescharacteristic
296C4—22no—
297C4—22no—
298C4—24no—
299C4—22no—
300C4—22no—
301C4—44no—
302C4—84no—
303C4—84no—
304C4—84no—
305C4—82no—
306C4—162no—
307C2—21no—
308C2—22no—
309C2—21no—
310C2#1836212yes—
311C2#1837212yes—
312C2#2345511yescharacteristic
313C2#2346311yescharacteristic
314C2#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
#218412
#142822
#103472
#103432
#80172
#79102