Finite Group Databasebeta

← #327975

Subgroups — #327975

Order 512.

1182 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 192 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
1C2×#2551C211yesmaximal, characteristic, complemented
2#33106C212yesmaximal
3#27909C211yesmaximal, characteristic
4#10573C214yesmaximal, complemented
5#10348C214yesmaximal, complemented
6#8723C211yesmaximal, characteristic, complemented
7#8706C212yesmaximal, complemented
8C22×C4⋊Q8—22no—
9C2×#480—22no—
10C2×#480—22no—
11#2856C2211yescharacteristic
12#2551—22no—
13#2551—24no—
14#2551C2214yes—
15C2×#395—22no—
16C2×#389—22no—
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2212yes—
19#2093—22no—
20#1982C2214yes—
21#1957C2212yes—
22#1947C2214yes—
23#1925—24no—
24#1874—22no—
25#1845C2212yes—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28#1296C2214yes—
29#1279C2218yes—
30C23×Q8—22no—
31C2×C4⋊Q8—28no—
32C2×C4⋊Q8—28no—
33C2×C4⋊Q8—48no—
34C2×C4⋊Q8—48no—
35C2×#127C2311yescharacteristic
36C2×#121—21no—
37C2×#121—22no—
38C22×C4⋊C4—24no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D412yes—
43C22×C42—22no—
44C22×C42C2311yescharacteristic
45#480—48no—
46#480—48no—
47C2×#107—24no—
48C2×#107—24no—
49C2×C4⋊C8C2314yes—
50C2×#102—24no—
51C2×#102—24no—
52C2×#102—24no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—22no—
55C2×C4×C8C2312yes—
56#395—28no—
57#395—28no—
58#390—21no—
59#390—22no—
60#389—28no—
61#389—28no—
62#384—22no—
63#384D412yes—
64#384C2311yescharacteristic
65#384C2312yes—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4D414yes—
69C4×C4⋊C4C2314yes—
70C2×#95—22no—
71C2×#95—22no—
72#340—24no—
73#336—22no—
74#336—21no—
75C22×Q8—28no—
76C22×Q8—416no—
77C22×Q8—84no—
78C22×D4—21no—
79C22×D4—22no—
80C22×D4—41no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D412yes—
86C22×C8—22no—
87C22×C8—24no—
88C4⋊Q8—416no—
89C4⋊Q8—432no—
90C4⋊Q8—416no—
91#127—24no—
92#121—44no—
93#121—44no—
94#121—44no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—28no—
99C2×C4⋊C4—216no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—416no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4D818yes—
110C2×C4⋊C4C2×D414yes—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D414yes—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—42no—
115C2×C42—24no—
116C2×C42—24no—
117C2×C42—24no—
118C2×C42—44no—
119C2×C42—44no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D412yes—
122C2×C42C2411yescharacteristic
123#107—48no—
124#107—48no—
125C4⋊C8—28no—
126#102—48no—
127#102—48no—
128#102—48no—
129C22⋊C8—24no—
130C22⋊C8—44no—
131C4×C8—24no—
132#95—28no—
133#95—24no—
134#95—24no—
135C24—41no—
136C24#12011yescharacteristic
137C2×Q8—432no—
138C2×Q8—88no—
139C2×Q8—832no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—44no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—44no—
147C2×D4—84no—
148C2×D4—82no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—28no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—41no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—44no—
165C22×C4—41no—
166C22×C4#12011yescharacteristic
167C22×C4#12012yes—
168C22×C4C2×D814yes—
169C22×C4AGL(1,ℤ/8)14yes—
170C22×C4C22×D411yescharacteristic
171C22×C4C22×D412yes—
172C2×C8—24no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—44no—
177C2×C8—44no—
178C4⋊C4—28no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—28no—
182C4⋊C4—44no—
183C4⋊C4—432no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—48no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—82no—
191C4⋊C4C2×D818yes—
192C4⋊C4#13718yes—
193C22⋊C4—82no—
194C22⋊C4—84no—
195C42—22no—
196C42—24no—
197C42—44no—
198C42—44no—
199C42—48no—
200C422−1+414yes—
201C23—21no—
202C23—22no—
203C23—21no—
204C23—81no—
205C23—82no—
206C23—82no—
207C23—81no—
208C23—81no—
209C23#44714yes—
210C23#45312yes—
211C23C2×#12011yescharacteristic
212Q8—832no—
213D4—84no—
214D4—84no—
215D4—84no—
216D4—164no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—28no—
222C2×C4—28no—
223C2×C4—21no—
224C2×C4—28no—
225C2×C4—28no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—44no—
229C2×C4—416no—
230C2×C4—416no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—48no—
236C2×C4—44no—
237C2×C4—88no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4#44714yes—
241C2×C4#53612yes—
242C2×C4#53614yes—
243C2×C4C22×D812yes—
244C2×C4C2×AGL(1,ℤ/8)12yes—
245C2×C4C2×#13714yes—
246C8—44no—
247C8—84no—
248C22—24no—
249C22—22no—
250C22—24no—
251C22—22no—
252C22—161no—
253C22—162no—
254C22—161no—
255C22—162no—
256C22—161no—
257C22#161914yes—
258C22#261411yescharacteristic
259C22C2×#44712yes—
260C22#300412yes—
261C22C2×#45311yescharacteristic
262C22#302011yescharacteristic
263C4—24no—
264C4—22no—
265C4—48no—
266C4—48no—
267C4—41no—
268C4—84no—
269C4—161no—
270C4#322614yes—
271C4#323814yes—
272C2—24no—
273C2—321no—
274C2C2×#161911yescharacteristic
275C2#1854611yescharacteristic
276C2#1877612yes—
277C2#2963812yes—
278C2#2965011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
1. C2×#2551C2semidirect
4. #10573C2semidirect
5. #10348C2semidirect
6. #8723C2semidirect
7. #8706C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#331062
#279092
#105732
#103482
#87232
#87062