Finite Group Databasebeta

← #329600

Subgroups — #329600

Order 512.

684 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 154 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#32957C212yesmaximal
2#29878C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic
4#10624C212yesmaximal, complemented
5#10483C214yesmaximal, complemented
6#10279C212yesmaximal, complemented
7#9128C212yesmaximal
8#8599C211yesmaximal, characteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#477C2214yes—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2087—22no—
16#2076—22no—
17#1990C2212yes—
18#1984—22no—
19#1984C2212yes—
20#1984C2212yes—
21#1863—24no—
22#1852—22no—
23#1834C2211yescharacteristic
24#1817—22no—
25#1803C2214yes—
26#1797C2212yes—
27#1787—22no—
28C2×#335C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C23×Q8D411yescharacteristic
33C2×C4⋊Q8C2312yes—
34C2×C22⋊Q8—22no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—42no—
38C2×C4×Q8—42no—
39C2×C4×Q8D412yes—
40C2×C4×Q8C2312yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45#500—42no—
46#477—44no—
47#477—44no—
48#431—42no—
49#431—42no—
50C2×#107—22no—
51C2×#107—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2314yes—
54C2×C4⋊C8C2312yes—
55C2×#103—24no—
56C2×#103—24no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8D412yes—
60C2×C4×C8—21no—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#391—22no—
64#389—22no—
65#389—22no—
66#384C2312yes—
67#380—22no—
68#380—22no—
69#340—22no—
70#340—24no—
71#340—24no—
72#336—24no—
73#336—24no—
74#336—22no—
75#335—42no—
76#333—44no—
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—42no—
81C22×Q8—42no—
82C22×Q8Q1614yes—
83C22×Q8C2×D412yes—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C22×C8—22no—
90C22×C8—24no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C4⋊Q8—24no—
94C4⋊Q8—24no—
95C22⋊Q8—84no—
96C4×Q8—24no—
97C4×Q8—24no—
98C4×Q8—44no—
99C4×Q8—42no—
100C4×Q8—42no—
101C4×Q8—44no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×D412yes—
114C2×C22⋊C4—42no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42C2411yescharacteristic
125#107—48no—
126#107—44no—
127C4⋊C8—24no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—42no—
131#103—48no—
132#103—48no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—44no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—42no—
139#95—42no—
140#95—42no—
141C24#12111yescharacteristic
142C2×Q8—22no—
143C2×Q8—24no—
144C2×Q8—24no—
145C2×Q8—24no—
146C2×Q8—48no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—48no—
150C2×Q8—44no—
151C2×Q8—82no—
152C2×Q8—82no—
153C2×Q8C2×Q1614yes—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—82no—
170C22×C4#12111yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4C2×Q1612yes—
174C22×C4#13512yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4111yescharacteristic
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—24no—
181C2×C8—24no—
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—44no—
185C2×C8—44no—
186C2×C8—44no—
187C4⋊C4—24no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C22⋊C4—84no—
199C42—21no—
200C42—22no—
201C42—21no—
202C42—22no—
203C42—44no—
204C42—42no—
205C42—42no—
206C42H(5,𝔽2)12yes—
207C42H(5,𝔽2)12yes—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23#46712yes—
213C23C2×#12111yescharacteristic
214Q8—28no—
215Q8—48no—
216Q8—44no—
217Q8—164no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—48no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4#46712yes—
244C2×C4#53412yes—
245C2×C4#53411yescharacteristic
246C2×C4#53511yescharacteristic
247C2×C4#53812yes—
248C2×C4C22×Q1611yescharacteristic
249C2×C4C2×#13511yescharacteristic
250C2×C4#57512yes—
251C2×C4#57814yes—
252C8—42no—
253C8—42no—
254C8—84no—
255C8—82no—
256C22—22no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22#261711yescharacteristic
262C22#304412yes—
263C22C2×#46711yescharacteristic
264C22#305411yescharacteristic
265C22#306812yes—
266C4—22no—
267C4—22no—
268C4—41no—
269C4—44no—
270C4—41no—
271C4—82no—
272C4—162no—
273C4—162no—
274C4#318912yes—
275C4#319112yes—
276C4#320212yes—
277C4#322312yes—
278C2—42no—
279C2#1940512yes—
280C2#1944212yes—
281C2#2972411yescharacteristic
282C2#2973111yescharacteristic
283C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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. #10624C2semidirect
5. #10483C2semidirect
6. #10279C2semidirect

Maximal subgroups

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

subgroupindex
#329572
#298782
#218182
#106242
#104832
#102792
#91282
#85992