Finite Group Databasebeta

← #343686

Subgroups — #343686

Order 512.

546 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 134 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#26091C212yesmaximal, complemented
2#24966C212yesmaximal
3#10655C212yesmaximal, complemented
4#10619C212yesmaximal, complemented
5#10576C211yesmaximal, characteristic, complemented
6#10430C211yesmaximal, characteristic
7#9560C212yesmaximal, complemented
8#8889C212yesmaximal, complemented
9#8819C211yesmaximal, characteristic, complemented
10#2571C2211yescharacteristic
11#2537C2211yescharacteristic
12#2354C2212yes—
13#2092—22no—
14#2077—22no—
15#2065—22no—
16#2034—22no—
17#1954C2212yes—
18#1949—22no—
19#1948C2212yes—
20#1947C2212yes—
21#1939C2212yes—
22#1919—22no—
23#1875—22no—
24#1845C2211yescharacteristic
25#1844C2212yes—
26#1844C2211yescharacteristic
27#1834C2212yes—
28#1825—22no—
29#1802C2212yes—
30#1802C2212yes—
31C4×#102C2212yes—
32C2×D4⋊C8C2212yes—
33C2×#322C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1277C2212yes—
39C2×#121—22no—
40C2×C4×D4C2312yes—
41C2×#117—22no—
42C2×#107—22no—
43C2×#107—41no—
44C2×#106—22no—
45C2×#106—22no—
46C2×#106—22no—
47C2×#106—41no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#102—22no—
52C2×#102—22no—
53C2×#102D412yes—
54C2×#102D412yes—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C4×C8C2312yes—
58C2×C4×C8C2311yescharacteristic
59#400—41no—
60#394—41no—
61#390—22no—
62#389—22no—
63#388—42no—
64#384C2311yescharacteristic
65#384C2312yes—
66#384C2311yescharacteristic
67#382—22no—
68#380—22no—
69#380—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2312yes—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340D412yes—
79#340D412yes—
80#336—22no—
81D4⋊C8—24no—
82#322—22no—
83C22×D4—41no—
84C22×D4C2×D412yes—
85C23×C4—22no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8C2×D412yes—
90C22×C8C2×D412yes—
91#121—44no—
92#121—82no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97#117—44no—
98#117—82no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C42—22no—
115C2×C42—42no—
116C2×C42C2411yescharacteristic
117#107—82no—
118#107—82no—
119#106—44no—
120#106—82no—
121#106—82no—
122#106—82no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—44no—
128#102—42no—
129#102—42no—
130#102—82no—
131#102—82no—
132C22⋊C8—82no—
133C22⋊C8—82no—
134C4×C8—22no—
135C4×C8—21no—
136C4×C8—22no—
137C4×C8#4112yes—
138#95—22no—
139#95—42no—
140#95—41no—
141#95—41no—
142#95—42no—
143C24—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—84no—
151C2×D4—161no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4C22×D412yes—
165C22×C4H(5,𝔽2)11yescharacteristic
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—82no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—161no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—84no—
202C22⋊C4—84no—
203C22⋊C4—162no—
204C42—42no—
205C42—42no—
206C42—84no—
207C42C2×#4111yescharacteristic
208C422−1+411yescharacteristic
209C422−1+412yes—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23D4211yescharacteristic
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—164no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4#53612yes—
246C2×C4#53911yescharacteristic
247C2×C4#54712yes—
248C2×C4#56311yescharacteristic
249C2×C4#57512yes—
250C2×C4#57512yes—
251C2×C4#57612yes—
252C2×C4#57712yes—
253C8—24no—
254C8—42no—
255C8—42no—
256C8—82no—
257C8—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22#268011yescharacteristic
266C22#326712yes—
267C22#326812yes—
268C22#329111yescharacteristic
269C22#329111yescharacteristic
270C4—21no—
271C4—21no—
272C4—82no—
273C4—82no—
274C4—82no—
275C4—162no—
276C4—161no—
277C4—161no—
278C4#324012yes—
279C4#324112yes—
280C4#334712yes—
281C4#335112yes—
282C2—162no—
283C2#2013212yes—
284C2#2014311yescharacteristic
285C2#2015211yescharacteristic
286C2#3110612yes—
287C2#3111211yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#2537C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26091C2semidirect
3. #10655C2semidirect
4. #10619C2semidirect
5. #10576C2semidirect
7. #9560C2semidirect
8. #8889C2semidirect
9. #8819C2semidirect

Maximal subgroups

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

subgroupindex
#260912
#249662
#106552
#106192
#105762
#104302
#95602
#88892
#88192