Finite Group Databasebeta

← #325360

Subgroups — #325360

Order 512.

616 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 166 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#33289C211yesmaximal, characteristic, complemented
2#28726C212yesmaximal
3C8×#124C211yesmaximal, characteristic
4#10623C212yesmaximal, complemented
5#10621C212yesmaximal
6#9905C212yesmaximal, complemented
7#8649C212yesmaximal, complemented
8#8648C212yesmaximal
9#8274C211yesmaximal, characteristic, complemented
10C2×#489—22no—
11C2×C8×Q8C2211yescharacteristic
12C2×C8×D4C2211yescharacteristic
13#2868C2212yes—
14C4×#124C2211yescharacteristic
15#2072—22no—
16#2071—22no—
17#1993—22no—
18#1990—22no—
19#1990—22no—
20#1984C2211yescharacteristic
21#1954C2211yescharacteristic
22#1923—22no—
23#1834C2212yes—
24#1834C2212yes—
25#1831C2211yescharacteristic
26#1803C2214yes—
27#1802C2214yes—
28#1767C2212yes—
29C8×C22⋊C4—22no—
30C42×C8C2211yescharacteristic
31C2×#335C2212yes—
32#1296C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C23×C8—21no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×#124D411yescharacteristic
40C2×C4×Q8C2311yescharacteristic
41C2×C4×D4C2311yescharacteristic
42#489—48no—
43C8×Q8—24no—
44C8×D4—24no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8C2314yes—
47C2×C4⋊C8C2311yescharacteristic
48C2×C4⋊C8C2311yescharacteristic
49C2×#103—24no—
50C2×#102—24no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—22no—
53C2×#97—22no—
54C2×C4×C8—21no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59#391D411yescharacteristic
60#390D411yescharacteristic
61#388—22no—
62#384—22no—
63#384—22no—
64#384—22no—
65#384C2314yes—
66#382D411yescharacteristic
67C4×C22⋊C4—22no—
68C43C2311yescharacteristic
69#340—24no—
70#340—22no—
71#340—22no—
72#340—24no—
73#340—24no—
74#336—22no—
75#336—22no—
76#336—22no—
77#335—24no—
78C22×Q8—42no—
79C22×Q8C2×D411yescharacteristic
80C22×D4C2×D411yescharacteristic
81C23×C4—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—24no—
85C22×C8—44no—
86C22×C8—42no—
87C22×C8C2×D412yes—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C4⋊Q8—44no—
91C4⋊Q8—44no—
92C4⋊Q8—44no—
93C4⋊Q8—44no—
94C4⋊Q8—84no—
95#124—22no—
96#124D814yes—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×D4—22no—
100C4×D4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4C2×D411yescharacteristic
109C2×C4⋊C4C2×D411yescharacteristic
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—21no—
113C2×C42—21no—
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42#4112yes—
118C2×C42C2411yescharacteristic
119C4⋊C8—24no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—24no—
123C4⋊C8—42no—
124#103—48no—
125#102—48no—
126C22⋊C8—44no—
127C22⋊C8—42no—
128#97—44no—
129C4×C8—22no—
130C4×C8—42no—
131C4×C8—42no—
132C4×C8—42no—
133C4×C8#4114yes—
134#95—22no—
135#95—22no—
136C24—21no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—84no—
141C2×Q8—84no—
142C2×Q8—84no—
143C2×Q8C2×D812yes—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4C2×D812yes—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4#12112yes—
157C22×C4#12112yes—
158C22×C4C22×D411yescharacteristic
159C22×C4C22×D411yescharacteristic
160C22×C4C2×#4111yescharacteristic
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—44no—
169C2×C8—44no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—44no—
173C2×C8—84no—
174C2×C8#12412yes—
175C2×C8#12412yes—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C4⋊C4—84no—
186C4⋊C4—84no—
187C22⋊C4—28no—
188C22⋊C4—42no—
189C22⋊C4—44no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—24no—
194C42—41no—
195C42—41no—
196C42C2×D812yes—
197C42#13512yes—
198C42C2×#4112yes—
199C422−1+412yes—
200C23—22no—
201C23—42no—
202C23—41no—
203C23—41no—
204C23—42no—
205C23C2×#12111yescharacteristic
206Q8—24no—
207Q8—42no—
208Q8—168no—
209D4—24no—
210D4—42no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4#46614yes—
230C2×C4#47114yes—
231C2×C4C2×#12411yescharacteristic
232C2×C4#53611yescharacteristic
233C2×C4#54712yes—
234C2×C4#55612yes—
235C2×C4C22×D811yescharacteristic
236C2×C4C2×#13511yescharacteristic
237C2×C4#57512yes—
238C2×C4#57512yes—
239C2×C4#57812yes—
240C8—22no—
241C8—22no—
242C8—42no—
243C8—84no—
244C8—81no—
245C8—81no—
246C22—42no—
247C22—44no—
248C22—81no—
249C22—81no—
250C22—82no—
251C22#254711yescharacteristic
252C22#304012yes—
253C22C2×#46611yescharacteristic
254C22#305911yescharacteristic
255C22C2×#47112yes—
256C4—22no—
257C4—24no—
258C4—24no—
259C4—42no—
260C4—81no—
261C4—81no—
262C4—164no—
263C4#313511yescharacteristic
264C4#314011yescharacteristic
265C4#324012yes—
266C4#333412yes—
267C4#334212yes—
268C2—82no—
269C2#1927612yes—
270C2#1936512yes—
271C2#2870411yescharacteristic
272C2#2872711yescharacteristic
273C2#2873211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #33289C2semidirect
4. #10623C2semidirect
6. #9905C2semidirect
7. #8649C2semidirect
9. #8274C2semidirect

Maximal subgroups

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

subgroupindex
#332892
#287262
C8×#1242
#106232
#106212
#99052
#86492
#86482
#82742