Finite Group Databasebeta

← #237896

Subgroups — #237896

Order 512.

868 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 186 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#36691C211yesmaximal, characteristic
2#21841C212yesmaximal
3#21202C211yesmaximal, characteristic
4#10361C214yesmaximal, complemented
5#10359C214yesmaximal, complemented
6#8035C212yesmaximal
7#8031C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2212yes—
10C2×#419—22no—
11C2×C4×M16C2211yescharacteristic
12#2845C2212yes—
13#2844C2212yes—
14#2843—22no—
15#2841—24no—
16#2402—22no—
17#2399—22no—
18#2396—22no—
19#2285C2211yescharacteristic
20#2283C2212yes—
21#1993C2218yes—
22#1988—22no—
23#1988—22no—
24#1960C2212yes—
25#1960C2212yes—
26#1956—22no—
27#1824—24no—
28#1295C2212yes—
29#1295C2212yes—
30#1295C2218yes—
31C23×Q8—21no—
32C22×M16—21no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8Q814yescomplemented
38C2×C4⋊Q8C2312yes—
39C2×C4×Q8—22no—
40C2×#117D412yes—
41C22×C4⋊C4—22no—
42C22×C42C2311yescharacteristic
43#432—24no—
44#432—24no—
45C2×C4⋊C8—22no—
46#419—48no—
47C2×#103—24no—
48C2×#103—24no—
49C2×C22⋊C8—22no—
50C4×M16—24no—
51C2×#97C2312yes—
52C2×#97C2314yes—
53C2×C4×C8—21no—
54#400—24no—
55#400—24no—
56#400—22no—
57#399—22no—
58#398—24no—
59#398—24no—
60#398—24no—
61#398—24no—
62#398—24no—
63#396—24no—
64#395—24no—
65#395—24no—
66#395—22no—
67#393—22no—
68#384D412yes—
69#383—22no—
70#382C2311yescharacteristic
71#382C2314yes—
72#382C2311yescharacteristic
73#381—22no—
74#380—22no—
75#380D414yes—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4D412yes—
80C4×C4⋊C4D412yes—
81C4×C22⋊C4—22no—
82#376—22no—
83#340—28no—
84#340—28no—
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—22no—
88C22×Q8—44no—
89C22×Q8—44no—
90C22×Q8—41no—
91C22×Q8—42no—
92C23×C4—41no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C2×M16—24no—
96C2×M16—42no—
97C2×M16—42no—
98C22×C8—22no—
99C22×C8—24no—
100C4⋊Q8—28no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—22no—
104C4⋊Q8—42no—
105C4⋊Q8—44no—
106C4⋊Q8C2×Q814yes—
107C4×Q8—84no—
108#117—44no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×D414yes—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42C2×D412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×Q812yes—
131C2×C42#4114yes—
132C2×C42#4112yes—
133C2×C42C2411yescharacteristic
134C4⋊C8—44no—
135#103—48no—
136#103—48no—
137C22⋊C8—44no—
138#97—24no—
139#97—24no—
140#97—24no—
141C4×C8—42no—
142#95—24no—
143#95—22no—
144#95—24no—
145#95—22no—
146#95—24no—
147#95—22no—
148#95—22no—
149#95—24no—
150#95—24no—
151#95—22no—
152#95—22no—
153C24#12011yescharacteristic
154C2×Q8—24no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—416no—
158C2×Q8—42no—
159C2×Q8—44no—
160C2×Q8—88no—
161C2×Q8—82no—
162C2×Q8—84no—
163C2×Q8—82no—
164C2×Q8—82no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4#12012yes—
177C22×C4#12011yescharacteristic
178C22×C4C22⋊Q812yes—
179C22×C4C22⋊Q814yes—
180C22×C4#12412yes—
181C22×C4#12414yes—
182C22×C4C22×D412yes—
183C22×C4C22×D411yescharacteristic
184M16—48no—
185C2×C8—24no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—48no—
189C2×C8—44no—
190C4⋊C4—28no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—48no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C4⋊C4—84no—
200C4⋊C4—84no—
201C4⋊C4—84no—
202C4⋊C4—82no—
203C22⋊C4—84no—
204C22⋊C4—82no—
205C42—44no—
206C42—44no—
207C42—44no—
208C42C22⋊Q814yes—
209C42C22⋊Q812yes—
210C42#12412yes—
211C42#12414yes—
212C42C22×Q811yescharacteristic
213C42C2×#4112yes—
214C42C2×#4111yescharacteristic
215C23—21no—
216C23—21no—
217C23—24no—
218C23#39312yes—
219C23C2×#12011yescharacteristic
220Q8—48no—
221Q8—84no—
222Q8—88no—
223Q8—88no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—88no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4#39314yes—
247C2×C4#39312yes—
248C2×C4#45312yes—
249C2×C4#45412yes—
250C2×C4C2×C22⋊Q812yes—
251C2×C4C2×C22⋊Q811yescharacteristic
252C2×C4C2×#12412yes—
253C2×C4C2×#12411yescharacteristic
254C8—44no—
255C8—84no—
256C22—22no—
257C22—24no—
258C22—42no—
259C22—42no—
260C22C2×#39311yescharacteristic
261C22C2×#45311yescharacteristic
262C22C2×#45411yescharacteristic
263C22#302214yes—
264C4—21no—
265C4—21no—
266C4—22no—
267C4—22no—
268C4—24no—
269C4—82no—
270C4—82no—
271C4—84no—
272C4—84no—
273C4—84no—
274C4#202714yes—
275C2—42no—
276C2#1899612yes—
277C2#1900512yes—
278C2C2×#202711yescharacteristic
279C2#2346412yes—
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
4. #10361C2semidirect
5. #10359C2semidirect
37. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#366912
#218412
#212022
#103612
#103592
#80352
#80312