Finite Group Databasebeta

← #331631

Subgroups — #331631

Order 512.

627 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 158 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#33263C211yesmaximal, characteristic, complemented
2#33230C211yesmaximal, characteristic
3#30305C212yesmaximal, complemented
4#30204C212yesmaximal
5#10685C211yesmaximal, characteristic, complemented
6#10683C211yesmaximal, characteristic
7#10681C212yesmaximal, complemented
8#10677C212yesmaximal
9#10676C211yesmaximal, characteristic, complemented
10#10671C211yesmaximal, characteristic
11#8134C211yesmaximal, characteristic, complemented
12C2×#489—22no—
13#2868C2211yescharacteristic
14#2859C2212yes—
15#2858C2211yescharacteristic
16#2084—24no—
17#2073—24no—
18#2058—24no—
19#2025—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1959—22no—
23#1957C2212yes—
24#1957C2211yescharacteristic
25#1957C2211yescharacteristic
26#1957C2212yes—
27#1956C2211yescharacteristic
28#1956C2211yescharacteristic
29#1944—22no—
30#1943—22no—
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1844C2212yes—
34#1844C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38#1744C2212yes—
39C4×#97C2211yescharacteristic
40#1273C2211yescharacteristic
41#1273C2211yescharacteristic
42#1273C2212yes—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2311yescharacteristic
47C2×#125—22no—
48C2×#124—21no—
49C2×#124—21no—
50#489—48no—
51C2×#107—22no—
52C2×#106—22no—
53C2×#106—24no—
54C2×#106—24no—
55C2×#103—22no—
56C2×#103—22no—
57C2×#102—22no—
58C2×#102—22no—
59C2×#102—24no—
60C2×#102—24no—
61C2×#97D412yes—
62C2×#97D412yes—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2312yes—
65C2×C4×C8C2312yes—
66C2×C4×C8C2311yescharacteristic
67#390—22no—
68#390—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#384—22no—
74#384—21no—
75#384C2312yes—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#384C2312yes—
79#382—22no—
80C43C2311yescharacteristic
81#340—24no—
82#340—24no—
83#340—24no—
84#340—24no—
85#340—24no—
86#336D414yes—
87#336D414yes—
88C22×Q8—21no—
89C22×Q8—41no—
90C22×D4—21no—
91C22×D4—22no—
92C22×C8C2×D414yes—
93C22×C8C2×D414yes—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C4⋊Q8—22no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—44no—
100C4⋊Q8—84no—
101#127—22no—
102#127—21no—
103#127—21no—
104#125—84no—
105#124—44no—
106#124—44no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2411yescharacteristic
130#107—44no—
131#106—44no—
132#106—84no—
133#106—84no—
134#103—44no—
135#103—44no—
136#102—44no—
137#102—44no—
138#102—84no—
139#102—84no—
140#97—24no—
141#97—42no—
142C4×C8—22no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149#95—42no—
150#95—42no—
151C24—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—82no—
157C2×Q8—84no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—44no—
165C2×D4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4#12711yescharacteristic
172C22×C4#12712yes—
173C22×C4#12711yescharacteristic
174C22×C4C22×D411yescharacteristic
175C22×C4C22×D412yes—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—24no—
180C2×C8—24no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—24no—
184C2×C8—24no—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8AGL(1,ℤ/8)14yes—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—84no—
201C4⋊C4—84no—
202C4⋊C4—82no—
203C4⋊C4—84no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4—84no—
207C42—22no—
208C42—22no—
209C42—24no—
210C42—24no—
211C422−1+412yes—
212C422−1+411yescharacteristic
213C422−1+411yescharacteristic
214C23—81no—
215C23—81no—
216C23—82no—
217C23—82no—
218C23—81no—
219C23C2×#12711yescharacteristic
220Q8—82no—
221Q8—82no—
222Q8—164no—
223D4—82no—
224D4—82no—
225D4—84no—
226D4—84no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—82no—
238C2×C4—81no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4#53611yescharacteristic
248C2×C4#53611yescharacteristic
249C2×C4#53612yes—
250C2×C4#53612yes—
251C2×C4C2×AGL(1,ℤ/8)12yes—
252C2×C4#57512yes—
253C2×C4#57714yes—
254C2×C4#57714yes—
255C8—24no—
256C8—42no—
257C8—44no—
258C8—44no—
259C22—161no—
260C22—162no—
261C22—161no—
262C22—161no—
263C22—162no—
264C22#265811yescharacteristic
265C22#315111yescharacteristic
266C22#315311yescharacteristic
267C22#315412yes—
268C22#315412yes—
269C4—41no—
270C4—42no—
271C4—41no—
272C4—161no—
273C4—161no—
274C4—162no—
275C4—162no—
276C4—161no—
277C4#323412yes—
278C4#323512yes—
279C4#324414yes—
280C4#324414yes—
281C2—321no—
282C2#1971311yescharacteristic
283C2#1971812yes—
284C2#1972111yescharacteristic
285C2#3028311yescharacteristic
286C2#3029012yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33263C2semidirect
3. #30305C2semidirect
5. #10685C2semidirect
7. #10681C2semidirect
9. #10676C2semidirect
11. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#332632
#332302
#303052
#302042
#106852
#106832
#106812
#106772
#106762
#106712
#81342