Finite Group Databasebeta

← #327994

Subgroups — #327994

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#33107C212yesmaximal
3#27906C211yesmaximal, characteristic
4#10574C214yesmaximal, complemented
5#10348C214yesmaximal, complemented
6#8725C211yesmaximal, characteristic, complemented
7#8707C212yesmaximal, complemented
8C22×C4⋊Q8—22no—
9C2×#481—22no—
10C2×#481—22no—
11#2856C2211yescharacteristic
12#2551—22no—
13#2551—24no—
14#2551C2214yes—
15C2×#395—22no—
16C2×#389—22no—
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2212yes—
19#2094—22no—
20#1982C2214yes—
21#1957C2212yes—
22#1948C2214yes—
23#1926—24no—
24#1874—22no—
25#1844C2212yes—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28#1296C2214yes—
29#1279C2218yes—
30C23×Q8—22no—
31C2×C4⋊Q8—28no—
32C2×C4⋊Q8—28no—
33C2×C4⋊Q8—416no—
34C2×#127C2311yescharacteristic
35C2×#121—21no—
36C2×#121—22no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D412yes—
42C22×C42—22no—
43C22×C42C2311yescharacteristic
44#481—48no—
45#481—48no—
46C2×#106—24no—
47C2×#106—24no—
48C2×C4⋊C8C2314yes—
49C2×#102—24no—
50C2×#102—24no—
51C2×#102—24no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—22no—
54C2×C4×C8C2312yes—
55#395—28no—
56#395—28no—
57#390—21no—
58#390—22no—
59#389—28no—
60#389—28no—
61#384—22no—
62#384D412yes—
63#384C2311yescharacteristic
64#384C2312yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4D414yes—
68C4×C4⋊C4C2314yes—
69C2×#95—22no—
70C2×#95—22no—
71#340—24no—
72#336—22no—
73#336—21no—
74C22×Q8—28no—
75C22×Q8—416no—
76C22×Q8—84no—
77C22×D4—21no—
78C22×D4—22no—
79C22×D4—41no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C22×C8—22no—
86C22×C8—24no—
87C4⋊Q8—432no—
88C4⋊Q8—416no—
89C4⋊Q8—416no—
90#127—24no—
91#121—44no—
92#121—44no—
93#121—44no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—216no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—416no—
105C2×C4⋊C4—48no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4SD1618yes—
108C2×C4⋊C4C2×D414yes—
109C2×C4⋊C4C2×D414yes—
110C2×C4⋊C4C2×D414yes—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—42no—
113C2×C42—24no—
114C2×C42—24no—
115C2×C42—24no—
116C2×C42—48no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C2×C42C2411yescharacteristic
120#106—48no—
121#106—48no—
122C4⋊C8—28no—
123#102—48no—
124#102—48no—
125#102—48no—
126C22⋊C8—24no—
127C22⋊C8—44no—
128C4×C8—24no—
129#95—28no—
130#95—24no—
131#95—24no—
132C24—41no—
133C24#12011yescharacteristic
134C2×Q8—432no—
135C2×Q8—88no—
136C2×Q8—832no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—44no—
141C2×D4—42no—
142C2×D4—44no—
143C2×D4—42no—
144C2×D4—84no—
145C2×D4—82no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—28no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—41no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—48no—
161C22×C4—44no—
162C22×C4—41no—
163C22×C4#12011yescharacteristic
164C22×C4#12012yes—
165C22×C4C2×SD1614yes—
166C22×C4#13714yes—
167C22×C4C22×D411yescharacteristic
168C22×C4C22×D412yes—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—44no—
173C2×C8—44no—
174C2×C8—44no—
175C4⋊C4—28no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—28no—
179C4⋊C4—44no—
180C4⋊C4—48no—
181C4⋊C4—432no—
182C4⋊C4—42no—
183C4⋊C4—48no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—82no—
187C4⋊C4C2×SD1618yes—
188C4⋊C4AGL(1,ℤ/8)18yes—
189C22⋊C4—82no—
190C22⋊C4—84no—
191C42—22no—
192C42—24no—
193C42—48no—
194C42—48no—
195C422−1+414yes—
196C23—21no—
197C23—22no—
198C23—21no—
199C23—81no—
200C23—82no—
201C23—82no—
202C23—81no—
203C23—81no—
204C23#44814yes—
205C23#45312yes—
206C23C2×#12011yescharacteristic
207Q8—832no—
208D4—84no—
209D4—84no—
210D4—84no—
211D4—164no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—28no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—28no—
220C2×C4—28no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—44no—
224C2×C4—416no—
225C2×C4—416no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—48no—
231C2×C4—44no—
232C2×C4—88no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4#44814yes—
236C2×C4#53612yes—
237C2×C4#53614yes—
238C2×C4C22×SD1612yes—
239C2×C4C2×AGL(1,ℤ/8)14yes—
240C2×C4C2×#13712yes—
241C8—44no—
242C8—84no—
243C22—24no—
244C22—22no—
245C22—24no—
246C22—22no—
247C22—162no—
248C22—161no—
249C22—161no—
250C22—162no—
251C22—161no—
252C22#162314yes—
253C22#261411yescharacteristic
254C22C2×#44812yes—
255C22#300312yes—
256C22C2×#45311yescharacteristic
257C22#302011yescharacteristic
258C4—24no—
259C4—22no—
260C4—48no—
261C4—48no—
262C4—41no—
263C4—84no—
264C4—161no—
265C4#322714yes—
266C4#323914yes—
267C2—24no—
268C2—321no—
269C2C2×#162311yescharacteristic
270C2#1852411yescharacteristic
271C2#1875812yes—
272C2#2963912yes—
273C2#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. #10574C2semidirect
5. #10348C2semidirect
6. #8725C2semidirect
7. #8707C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#331072
#279062
#105742
#103482
#87252
#87072