Finite Group Databasebeta

← #302876

Subgroups — #302876

Order 512.

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

normal subgroups only

This group has 753 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
1#59869C211yesmaximal, characteristic
2#29206C212yesmaximal
3#28511C211yesmaximal, characteristic
4#10576C214yesmaximal
5#9865C212yesmaximal
6#8928C214yesmaximal
7C4×#333C211yesmaximal, characteristic
8C2×#477—22no—
9#2868C2211yescharacteristic
10#2809—22no—
11#2571C2212yes—
12#2568—22no—
13#2546—22no—
14#2508—24no—
15#2369C2212yes—
16#2355—22no—
17C4×C4⋊Q8C2212yes—
18C4×C4⋊Q8C2211yescharacteristic
19#2095—22no—
20#2058—24no—
21#1990—22no—
22#1956C2212yes—
23#1948C2218yes—
24#1845C2212yes—
25#1844—22no—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#333C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8C2311yescharacteristic
36C2×C4×Q8—22no—
37C2×C4×Q8D412yes—
38#477—48no—
39C2×#107—24no—
40C2×#106—28no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2314yes—
43C2×#103—24no—
44C2×C4×C8—22no—
45C2×C4×C8C2312yes—
46#400—24no—
47#400—24no—
48#398—28no—
49#391—24no—
50#391—24no—
51#391—22no—
52#391D412yes—
53#389—24no—
54#389—24no—
55#389—22no—
56#389—22no—
57#389D414yes—
58#387—28no—
59#384—21no—
60#384—22no—
61#384D412yes—
62#384C2312yes—
63#384C2311yescharacteristic
64#382—24no—
65#382—24no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—28no—
73#336—22no—
74#336—21no—
75#333Q818yescomplemented
76C22×Q8—22no—
77C22×Q8—41no—
78C22×Q8C2×D412yes—
79C22×C8—24no—
80C22×C8—22no—
81C4⋊Q8—24no—
82C4⋊Q8—24no—
83C4⋊Q8—22no—
84C4⋊Q8—44no—
85C4⋊Q8—84no—
86C4⋊Q8—84no—
87C4⋊Q8C2×Q814yes—
88C4×Q8—44no—
89C4×Q8—44no—
90C4×Q8—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C2×D414yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42—24no—
109C2×C42—24no—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D412yes—
114C2×C42C2411yescharacteristic
115#107—48no—
116#106—88no—
117C4⋊C8—24no—
118C4⋊C8—24no—
119C4⋊C8C2×Q818yes—
120#103—48no—
121C4×C8—24no—
122C4×C8—44no—
123#95—24no—
124#95—28no—
125#95—28no—
126#95—24no—
127#95—24no—
128#95—24no—
129C2×Q8—22no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—82no—
136C2×Q8—84no—
137C2×Q8C4⋊Q814yes—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—42no—
143C22×C4#12011yescharacteristic
144C22×C4#12012yes—
145C22×C4#12011yescharacteristic
146C22×C4C22×D411yescharacteristic
147C22×C4C22×D412yes—
148C2×C8—24no—
149C2×C8—28no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—44no—
153C2×C8—44no—
154C4⋊C4—28no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—28no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—48no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C4⋊C4—84no—
170C4⋊C4—82no—
171C42—24no—
172C42—22no—
173C42—41no—
174C42—44no—
175C42—44no—
176C42—44no—
177C42—84no—
178C42#13514yes—
179C42#13714yes—
180C42C22×Q812yes—
181C422−1+412yes—
182C23C2×#12011yescharacteristic
183Q8—44no—
184Q8—82no—
185Q8—88no—
186Q8—164no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—84no—
205C2×C4—82no—
206C2×C4#45214yes—
207C2×C4#45214yes—
208C2×C4#45312yes—
209C2×C4C2×C4⋊Q811yescharacteristic
210C2×C4#53611yescharacteristic
211C2×C4D4×Q814yes—
212C2×C4C2×#13512yes—
213C2×C4C2×#13712yes—
214C2×C4#57714yes—
215C8—44no—
216C8—48no—
217C22#243411yescharacteristic
218C22C2×#45212yes—
219C22#301012yes—
220C22C2×#45311yescharacteristic
221C22#302211yescharacteristic
222C4—24no—
223C4—21no—
224C4—21no—
225C4—41no—
226C4—44no—
227C4—82no—
228C4—84no—
229C4—84no—
230C4—162no—
231C4#163414yes—
232C4#338114yes—
233C4#338714yes—
234C2C2×#163411yescharacteristic
235C2#1871811yescharacteristic
236C2#1890912yes—
237C2#2582411yescharacteristic
238C2#2583612yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
75. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#598692
#292062
#285112
#105762
#98652
#89282
C4×#3332