Finite Group Databasebeta

← #129690

Subgroups — #129690

Order 512.

673 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 148 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×#2982C211yesmaximal, characteristic
2#11637C214yesmaximal, complemented
3#11459C212yesmaximal
4C22×C4×Q8C411yescharacteristic
5#2982—28no—
6C2×C8×Q8—22no—
7C2×C8×Q8C412yes—
8C2×C8×Q8C2212yes—
9C2×#431—22no—
10C2×#431C411yescharacteristic
11C2×#431C2211yescharacteristic
12#2162C2214yes—
13Q8⋊C16—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
27C8×Q8—216no—
28C8×Q8—24no—
29C8×Q8—24no—
30C8×Q8—24no—
31C8×Q8D418yes—
32#431—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×C4×C8—22no—
42C2×C4×C8—22no—
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46C4⋊C16—24no—
47C22⋊C16—24no—
48C4×C16—24no—
49C22×Q8—24no—
50C22×Q8—24no—
51C22×Q8—21no—
52C22×Q8—22no—
53C22×Q8C1614yescomplemented
54C22×Q8C2×C812yes—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C2×C811yescharacteristic
59C23×C4M1611yescharacteristic
60C22×C8—22no—
61C22×C8C22⋊C412yes—
62C22×C8C2×D412yes—
63C4×Q8—28no—
64C4×Q8—24no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8C22⋊C414yes—
71C4×Q8M1614yes—
72C2×C4⋊C4—28no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4C2×C812yes—
78C2×C42—24no—
79C2×C42—24no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C2×C811yescharacteristic
86C2×C42M1611yescharacteristic
87C2×C42C22×C411yescharacteristic
88C2×C16—44no—
89C4⋊C8—28no—
90C4⋊C8—28no—
91C4⋊C8—24no—
92C4⋊C8—22no—
93C4⋊C8C22⋊C414yes—
94C22⋊C8—24no—
95C22⋊C8—24no—
96C4×C8—28no—
97C4×C8—24no—
98C4×C8—28no—
99C4×C8—22no—
100C4×C8C2×D414yes—
101C24C22⋊C811yescharacteristic
102C2×Q8—24no—
103C2×Q8—216no—
104C2×Q8—24no—
105C2×Q8—216no—
106C2×Q8—24no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8C22⋊C814yes—
111C2×Q8C2×C1614yes—
112C2×Q8M3214yes—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—82no—
126C22×C4—81no—
127C22×C4C22⋊C811yescharacteristic
128C22×C4C22⋊C811yescharacteristic
129C22×C4C22⋊C811yescharacteristic
130C22×C4C2×C1612yes—
131C22×C4M3212yes—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C22×C811yescharacteristic
134C22×C4C2×M1611yescharacteristic
135C2×C8—22no—
136C2×C8—24no—
137C2×C8—42no—
138C2×C8—44no—
139C2×C8—42no—
140C2×C8#12014yes—
141C4⋊C4—28no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4C22⋊C814yes—
147C42—28no—
148C42—24no—
149C42—21no—
150C42—22no—
151C42—24no—
152C42—21no—
153C42—22no—
154C42C2×C22⋊C412yes—
155C42C2×M1612yes—
156C16—84no—
157C23—21no—
158C23—21no—
159C23—41no—
160C23—41no—
161C23C22⋊C1612yes—
162C23C2×C22⋊C811yescharacteristic
163Q8—216no—
164Q8—24no—
165Q8—24no—
166Q8—24no—
167Q8C22⋊C1618yes—
168C2×C4—216no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—24no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—82no—
187C2×C4—84no—
188C2×C4C22⋊C1612yes—
189C2×C4#37911yescharacteristic
190C2×C4C2×C22⋊C812yes—
191C2×C4#40711yescharacteristic
192C2×C4#40712yes—
193C2×C4C22×C1611yescharacteristic
194C2×C4C2×M3212yes—
195C2×C4C2×M3211yescharacteristic
196C8—24no—
197C8—42no—
198C8—44no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22#177911yescharacteristic
207C22C2×C22⋊C1611yescharacteristic
208C22#211211yescharacteristic
209C4—21no—
210C4—28no—
211C4—22no—
212C4—22no—
213C4—41no—
214C4—44no—
215C4—41no—
216C4#179612yes—
217C4C2×C22⋊C1612yes—
218C4#211212yes—
219C2—21no—
220C2—21no—
221C2—42no—
222C2#832311yescharacteristic
223C2#1267911yescharacteristic
224C2#1268611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC24C22⋊C8
LCSG ▷ C2×C4 ▷ C2 ▷ 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. #11637C2semidirect
53. C22×Q8C16semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
#116372
#114592