Finite Group Databasebeta

← #317161

Subgroups — #317161

Order 512.

500 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 144 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#21735C211yesmaximal, characteristic
2#21724C211yesmaximal, characteristic
3#19207C212yesmaximal
4#10633C212yesmaximal
5#10224C212yesmaximal
6#9298C212yesmaximal
7#9297C212yesmaximal
8#8780C211yesmaximal, characteristic
9#8675C212yesmaximal
10#3230C2212yes—
11#3209C2212yes—
12#3162C2211yescharacteristic
13C2×#498C2211yescharacteristic
14#2960C2212yes—
15#2331C2212yes—
16#1995C2212yes—
17#1993C2212yes—
18#1946—21no—
19#1946—21no—
20#1944—21no—
21#1941—21no—
22#1849C2211yescharacteristic
23#1846C2211yescharacteristic
24#1776—21no—
25C4×#106—21no—
26#1772—21no—
27#1772—21no—
28#1726C2211yescharacteristic
29#1692—24no—
30#1672—24no—
31#1564C2212yes—
32#1549C2212yes—
33#1548C2212yes—
34#1518C2212yes—
35#1518C2212yes—
36#1286C2212yes—
37#1286C2212yes—
38#1281C2212yes—
39#1274C2211yescharacteristic
40#1274C2211yescharacteristic
41#533C2312yes—
42C2×C4⋊Q8C2312yes—
43#501—22no—
44#501—22no—
45#500—22no—
46#498—21no—
47#498—21no—
48#498D412yes—
49#498D412yes—
50#494—22no—
51#487—22no—
52#487—22no—
53#484—42no—
54#477—24no—
55#462—24no—
56#451—42no—
57#446—22no—
58#446—22no—
59C8×Q8—22no—
60C8×Q8—22no—
61#431C2312yes—
62#431C2312yes—
63C2×#108—21no—
64C2×#108—21no—
65#427—21no—
66#427—21no—
67C2×#107—21no—
68C2×#106—21no—
69C2×#106—21no—
70C2×#106—21no—
71#424—22no—
72#423C2312yes—
73#405—22no—
74C2×#97D411yescharacteristic
75C2×#97D411yescharacteristic
76C2×#97C2311yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391—42no—
81#389—22no—
82#387—42no—
83#382C2312yes—
84C4×C4⋊C4—22no—
85#376C2311yescharacteristic
86#340—22no—
87#340—22no—
88#339—22no—
89#339—22no—
90#335—22no—
91#334—22no—
92#333—24no—
93#332—24no—
94#328—42no—
95#328—42no—
96C22×Q8—22no—
97C2×Q16—44no—
98C2×M16—22no—
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8C2×D414yes—
107C4⋊Q8#4112yes—
108C4⋊Q8#4112yes—
109#125—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114#117—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C42—42no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2411yescharacteristic
123#108—42no—
124#108—42no—
125#107—22no—
126#107—22no—
127#107—44no—
128#107—42no—
129#106—21no—
130#106—21no—
131#106—21no—
132#106—21no—
133#106—22no—
134#106—22no—
135#106—22no—
136#106—22no—
137#106—42no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144#103—44no—
145#103—44no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148#97—22no—
149#97—22no—
150#97—41no—
151#97—42no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—42no—
156C4×C8Q1614yescomplemented
157C4×C8C2×D411yescharacteristic
158C4×C8C2×D411yescharacteristic
159C4×C8#4111yescharacteristic
160C4×C8#4111yescharacteristic
161#95—22no—
162#95—22no—
163#95—22no—
164#95—42no—
165#95—42no—
166C2×Q8—22no—
167C2×Q8—24no—
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—162no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C22×D411yescharacteristic
176C22×C4H(5,𝔽2)12yes—
177Q16—164no—
178M16—42no—
179M16—42no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8#12111yescharacteristic
190C2×C8#12111yescharacteristic
191C2×C8#12111yescharacteristic
192C2×C8#12111yescharacteristic
193C2×C8C2×Q1612yes—
194C2×C8C2×Q1612yes—
195C2×C8#13712yes—
196C2×C8#13712yes—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—162no—
210C22⋊C4—42no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—42no—
215C42—162no—
216C42C2×Q1612yes—
217C42C22×D411yescharacteristic
218C42C2×#4111yescharacteristic
219C42C2×#4112yes—
220C23#53511yescharacteristic
221Q8—44no—
222Q8—84no—
223Q8—82no—
224Q8—162no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4C2×#12111yescharacteristic
239C2×C4#52511yescharacteristic
240C2×C4#54612yes—
241C2×C4#55312yes—
242C2×C4C22×Q1611yescharacteristic
243C2×C4C2×#13711yescharacteristic
244C2×C4C2×#13711yescharacteristic
245C2×C4#57511yescharacteristic
246C8—22no—
247C8—22no—
248C8—21no—
249C8—21no—
250C8—82no—
251C8—82no—
252C8—82no—
253C8#46212yes—
254C8#46212yes—
255C22—22no—
256C22#253511yescharacteristic
257C22#320911yescharacteristic
258C22#321611yescharacteristic
259C4—22no—
260C4—22no—
261C4—42no—
262C4—81no—
263C4—84no—
264C4—82no—
265C4—82no—
266C4—162no—
267C4C2×#46211yescharacteristic
268C4#303711yescharacteristic
269C4#310411yescharacteristic
270C4#311311yescharacteristic
271C2—21no—
272C2—21no—
273C2#2816111yescharacteristic
274C2#2818212yes—
subgroupG/H
center, socleC22#2535
commutator, Frattini, cosocleC2×C42C24
prime radicalC23#535
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
156. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#217352
#217242
#192072
#106332
#102242
#92982
#92972
#87802
#86752