Finite Group Databasebeta

← #237918

Subgroups — #237918

Order 512.

794 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 178 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#37563C211yesmaximal, characteristic
2#21841C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic
4#21811C211yesmaximal, characteristic
5#10359C218yesmaximal, complemented
6#8032C211yesmaximal, characteristic
7#8032C211yesmaximal, characteristic
8#8032C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2844C2214yes—
14#2665—22no—
15#2665—22no—
16#2665—22no—
17#2605—22no—
18#2605—22no—
19#2605—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#1993C2214yes—
24#1993C2214yes—
25#1988—22no—
26#1987—22no—
27#1981—22no—
28#1960C2214yes—
29#1824—22no—
30#1823—22no—
31#1787—22no—
32#1295C2214yes—
33#1295C2214yes—
34#1295C2214yes—
35C23×Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8Q814yescomplemented
43C2×C4⋊Q8C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432—22no—
51#432—22no—
52#432—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×#103—28no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×#97C2312yes—
61C2×#97C2312yes—
62C2×#97C2312yes—
63#400—24no—
64#400—24no—
65#400—24no—
66#396—24no—
67#395—24no—
68#395—24no—
69#395—24no—
70#393—24no—
71#391—24no—
72#391—24no—
73#391—24no—
74#387—24no—
75#387—24no—
76#387—24no—
77#383—21no—
78#383—21no—
79#383—21no—
80#382C2312yes—
81#382C2312yes—
82#382C2312yes—
83#381—21no—
84#381—21no—
85#381—21no—
86#380D412yes—
87#380D412yes—
88#380D412yes—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C4×C22⋊C4—21no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95#376—21no—
96#376—21no—
97#376—21no—
98#340—28no—
99#340—28no—
100C22×Q8—24no—
101C22×Q8—24no—
102C22×Q8—22no—
103C22×Q8—41no—
104C22×Q8—42no—
105C22×Q8—84no—
106C23×C4—41no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C4⋊Q8—24no—
114C4⋊Q8—24no—
115C4⋊Q8—24no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—42no—
122C4⋊Q8C2×Q814yes—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×D414yes—
136C2×C4⋊C4C2×D414yes—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×Q812yes—
147C2×C42#4112yes—
148C2×C42#4112yes—
149C2×C42#4112yes—
150C2×C42C2411yescharacteristic
151C4⋊C8—42no—
152C4⋊C8—42no—
153C4⋊C8—42no—
154#103—88no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158#97—22no—
159#97—22no—
160#97—22no—
161#97—22no—
162#97—22no—
163#97—22no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180C24#12011yescharacteristic
181C2×Q8—24no—
182C2×Q8—48no—
183C2×Q8—42no—
184C2×Q8—44no—
185C2×Q8—84no—
186C2×Q8—84no—
187C2×Q8—84no—
188C2×Q8—84no—
189C2×Q8—88no—
190C2×Q8—88no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—84no—
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4C22⋊Q812yes—
204C22×C4C22⋊Q812yes—
205C22×C4C22⋊Q812yes—
206C22×C4#12412yes—
207C22×C4#12412yes—
208C22×C4#12412yes—
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C42—82no—
234C42—82no—
235C42—82no—
236C42C22⋊Q812yes—
237C42C22⋊Q812yes—
238C42C22⋊Q812yes—
239C42#12412yes—
240C42#12412yes—
241C42#12412yes—
242C42C22×Q811yescharacteristic
243C42C2×#4111yescharacteristic
244C42C2×#4111yescharacteristic
245C42C2×#4111yescharacteristic
246C23—22no—
247C23—22no—
248C23—22no—
249C23#39312yes—
250C23C2×#12011yescharacteristic
251Q8—48no—
252Q8—84no—
253Q8—88no—
254Q8—168no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—88no—
278C2×C4—84no—
279C2×C4#39312yes—
280C2×C4#39312yes—
281C2×C4#39312yes—
282C2×C4C2×C22⋊Q811yescharacteristic
283C2×C4C2×C22⋊Q811yescharacteristic
284C2×C4C2×C22⋊Q811yescharacteristic
285C2×C4C2×#12411yescharacteristic
286C2×C4C2×#12411yescharacteristic
287C2×C4C2×#12411yescharacteristic
288C8—82no—
289C8—82no—
290C8—82no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—44no—
295C22C2×#39311yescharacteristic
296C22#302112yes—
297C22#302212yes—
298C22#302212yes—
299C4—21no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—22no—
307C4—22no—
308C4—84no—
309C4—84no—
310C4—84no—
311C4—84no—
312C2—42no—
313C2#1909514yes—
314C2#2346311yescharacteristic
315C2#2346411yescharacteristic
316C2#2346411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10359C2semidirect
42. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#375632
#218412
#218402
#218112
#103592
#80322