Finite Group Databasebeta

← #323532

Subgroups — #323532

Order 512.

822 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 186 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#33297C211yesmaximal, characteristic, complemented
2#29192C212yesmaximal, complemented
3#28509C212yesmaximal, complemented
4#21690C211yesmaximal, characteristic
5#10677C212yesmaximal, complemented
6#10672C212yesmaximal, complemented
7#9832C212yesmaximal
8#8819C212yesmaximal, complemented
9#8263C211yesmaximal, characteristic, complemented
10C2×#493C2211yescharacteristic
11C2×#476—24no—
12C2×#442—22no—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×#127C2212yes—
16#2094—24no—
17#2067—24no—
18#1993—22no—
19#1992—22no—
20#1972—22no—
21#1957C2212yes—
22#1957C2212yes—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2212yes—
26#1862—24no—
27#1845C2212yes—
28#1844C2212yes—
29#1844C2212yes—
30#1844C2211yescharacteristic
31#1834C2212yes—
32#1834C2212yes—
33#1834C2212yes—
34C4×C4⋊C8C2212yes—
35C4×#97C2211yescharacteristic
36C2×#322C2212yes—
37#1273C2211yescharacteristic
38#1273C2211yescharacteristic
39C22×D8—22no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—22no—
42C2×#127C2312yes—
43C2×#125—21no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47#493—22no—
48#493Q814yescomplemented
49#476—416no—
50#442—48no—
51C2×#107—24no—
52C2×#106—24no—
53C2×#106—24no—
54C2×#106—22no—
55C2×C4⋊C8D414yes—
56C2×#102—24no—
57C2×#102—24no—
58C2×#102—24no—
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#390—22no—
70#389—22no—
71#389—22no—
72#384—22no—
73#384—21no—
74#384C2312yes—
75#384C2312yes—
76#384C2312yes—
77#382—21no—
78C43C2311yescharacteristic
79#340—24no—
80#340—24no—
81#340—24no—
82#336D414yes—
83#322—24no—
84C22×Q8—41no—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—22no—
88C23×C4—42no—
89C2×D8—28no—
90C2×D8—44no—
91C2×D8—44no—
92C22×C8C2×D414yes—
93C22×C8C2×D412yes—
94C22×C8C2×D412yes—
95C22×C8C2×D414yes—
96C4⋊Q8—44no—
97C4⋊Q8—48no—
98#127—22no—
99#127—22no—
100#127—22no—
101#127C2×Q814yes—
102#125—44no—
103C4×D4—48no—
104C4×D4—48no—
105C4×D4—84no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C42C2×D412yes—
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2411yescharacteristic
126#107—84no—
127#106—44no—
128#106—48no—
129#106—48no—
130C4⋊C8—28no—
131#102—48no—
132#102—48no—
133#102—48no—
134#102—48no—
135#102—84no—
136#97—24no—
137#97—42no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8C2×Q812yes—
144C4×C8#4112yes—
145#95—41no—
146#95—41no—
147C24—42no—
148C2×Q8—84no—
149C2×Q8—82no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—44no—
157C2×D4—44no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4#12711yescharacteristic
169C22×C4#12711yescharacteristic
170C22×C4#12712yes—
171C22×C4C22×D411yescharacteristic
172C22×C4C22×D412yes—
173D8—416no—
174C2×C8—24no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—24no—
178C2×C8—22no—
179C2×C8—24no—
180C2×C8—22no—
181C2×C8—24no—
182C2×C8—22no—
183C2×C8—24no—
184C2×C8C22⋊Q814yes—
185C2×C8AGL(1,ℤ/8)18yes—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—84no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C22⋊C4—84no—
199C22⋊C4—84no—
200C22⋊C4—84no—
201C42—24no—
202C42—22no—
203C42—24no—
204C42AGL(1,ℤ/8)14yes—
205C42C22×Q811yescharacteristic
206C42C2×#4111yescharacteristic
207C422−1+412yes—
208C23—44no—
209C23—82no—
210C23—82no—
211C23—82no—
212C23—82no—
213C23—82no—
214C23—82no—
215C23C2×#12711yescharacteristic
216Q8—164no—
217D4—48no—
218D4—48no—
219D4—48no—
220D4—84no—
221D4—84no—
222D4—84no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—168no—
239C2×C4#49614yes—
240C2×C4C2×C22⋊Q811yescharacteristic
241C2×C4#53611yescharacteristic
242C2×C4#54712yes—
243C2×C4D4×Q812yes—
244C2×C4C2×AGL(1,ℤ/8)12yes—
245C2×C4C2×AGL(1,ℤ/8)14yes—
246C2×C4#57512yes—
247C2×C4#57714yes—
248C8—24no—
249C8—28no—
250C8—44no—
251C8—42no—
252C22—84no—
253C22—84no—
254C22—84no—
255C22—162no—
256C22—162no—
257C22—162no—
258C22—162no—
259C22#254211yescharacteristic
260C22C2×#49611yescharacteristic
261C22#315112yes—
262C22#315311yescharacteristic
263C22#315412yes—
264C4—24no—
265C4—41no—
266C4—41no—
267C4—162no—
268C4—162no—
269C4—162no—
270C4#307712yes—
271C4#323512yes—
272C4#334514yes—
273C4#338414yes—
274C2—164no—
275C2#1967112yes—
276C2#1972411yescharacteristic
277C2#1974711yescharacteristic
278C2#2852111yescharacteristic
279C2#2854812yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #33297C2semidirect
2. #29192C2semidirect
3. #28509C2semidirect
5. #10677C2semidirect
6. #10672C2semidirect
8. #8819C2semidirect
9. #8263C2semidirect
48. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#332972
#291922
#285092
#216902
#106772
#106722
#98322
#88192
#82632