Finite Group Databasebeta

← #129692

Subgroups — #129692

Order 512.

593 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 116 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×#2983C211yesmaximal, characteristic
2#11644C214yesmaximal, complemented
3#11460C212yesmaximal
4C22×C4×Q8C411yescharacteristic
5#2983—28no—
6C2×#446—22no—
7C2×#446C412yes—
8C2×#446C2212yes—
9C2×#432—22no—
10C2×#431C411yescharacteristic
11C2×#431C2211yescharacteristic
12#2163C2214yes—
13#1338—28no—
14#1322—22no—
15#1314—22no—
16C23×Q8C811yescharacteristic
17C2×C4×Q8—24no—
18C2×C4×Q8—24no—
19C2×C4×Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C812yes—
22C2×C4×Q8C2×C412yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C811yescharacteristic
25C22×C42—21no—
26C22×C42C2×C411yescharacteristic
27#446—216no—
28#446—24no—
29#446—24no—
30#446—24no—
31#446D418yes—
32#432—216no—
33#431—24no—
34#431—24no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8C2×C412yes—
38C2×C4⋊C8D412yes—
39C2×C22⋊C8—21no—
40C2×C22⋊C8D412yes—
41C2×#97—22no—
42C2×#97—22no—
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46C4⋊C16—24no—
47C22⋊C16—24no—
48#346—24no—
49C22×Q8—24no—
50C22×Q8—21no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8C2×C812yes—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C2×C811yescharacteristic
58C23×C4M1611yescharacteristic
59C22×C8—22no—
60C22×C8C22⋊C412yes—
61C22×C8C2×D412yes—
62C4×Q8—28no—
63C4×Q8—24no—
64C4×Q8—24no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8C22⋊C414yes—
70C4×Q8M1614yes—
71C2×C4⋊C4—28no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4C2×C812yes—
77C2×C42—24no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C2×C811yescharacteristic
85C2×C42M1611yescharacteristic
86C2×C42C22×C411yescharacteristic
87C2×C16—44no—
88C4⋊C8—28no—
89C4⋊C8—28no—
90C4⋊C8—24no—
91C4⋊C8—22no—
92C4⋊C8C22⋊C414yes—
93C22⋊C8—24no—
94C22⋊C8—24no—
95#97—28no—
96#97—28no—
97C4×C8—24no—
98C4×C8—22no—
99C4×C8C2×D414yes—
100C24C22⋊C811yescharacteristic
101C2×Q8—24no—
102C2×Q8—22no—
103C2×Q8—416no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8C22⋊C814yes—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—82no—
123C22×C4—81no—
124C22×C4C22⋊C811yescharacteristic
125C22×C4C22⋊C811yescharacteristic
126C22×C4C22⋊C811yescharacteristic
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4C22×C811yescharacteristic
129C22×C4C2×M1611yescharacteristic
130C2×C8—22no—
131C2×C8—24no—
132C2×C8—42no—
133C2×C8—44no—
134C2×C8—42no—
135C2×C8#12014yes—
136C4⋊C4—24no—
137C4⋊C4—28no—
138C4⋊C4—22no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4C22⋊C814yes—
142C42—28no—
143C42—21no—
144C42—24no—
145C42—22no—
146C42—24no—
147C42—21no—
148C42—22no—
149C42C2×C22⋊C412yes—
150C42C2×M1612yes—
151C16—84no—
152C23—21no—
153C23—21no—
154C23—41no—
155C23—41no—
156C23#34912yes—
157C23C2×C22⋊C811yescharacteristic
158Q8—44no—
159Q8—48no—
160Q8—44no—
161C2×C4—28no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—48no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—82no—
181C2×C4—84no—
182C2×C4#34912yes—
183C2×C4#37911yescharacteristic
184C2×C4C2×C22⋊C812yes—
185C2×C4#40711yescharacteristic
186C2×C4#40712yes—
187C8—24no—
188C8—42no—
189C8—44no—
190C22—21no—
191C22—21no—
192C22—22no—
193C22—42no—
194C22—42no—
195C22—41no—
196C22—41no—
197C22#177911yescharacteristic
198C22C2×#34911yescharacteristic
199C22C2×#34911yescharacteristic
200C4—21no—
201C4—22no—
202C4—22no—
203C4—22no—
204C4—41no—
205C4—44no—
206C4—44no—
207C4—41no—
208C4#179612yes—
209C2—21no—
210C2—21no—
211C2—42no—
212C2#832311yescharacteristic
213C2#1268111yescharacteristic
214C2#1268611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11644C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#116442
#114602