Finite Group Databasebeta

← #137885

Subgroups — #137885

Order 512.

375 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 100 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#33489C211yesmaximal, characteristic
2#22208C211yesmaximal, characteristic
3#11637C212yesmaximal, complemented
4#11487C211yesmaximal, characteristic
5#7142C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8C4×#124C411yescharacteristic
9#2169—24no—
10#2162C2212yes—
11#2162C2212yes—
12#2112—22no—
13#1984C411yescharacteristic
14#1954C411yescharacteristic
15#1923—22no—
16#1850—22no—
17#1842C411yescharacteristic
18#1744C2211yescharacteristic
19Q8⋊C16—24no—
20#1314—22no—
21C23×C8D412yes—
22C2×#124C811yescharacteristic
23C2×C4×Q8C2×C411yescharacteristic
24C2×C4×D4C2×C411yescharacteristic
25C2×M32—24no—
26C8×Q8—22no—
27C8×Q8—22no—
28C8×D4—22no—
29C8×D4—22no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C22⋊C8D412yes—
36C2×C4×C8—21no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2311yescharacteristic
39#391C811yescharacteristic
40#390C811yescharacteristic
41#388—22no—
42#382C811yescharacteristic
43C4×C22⋊C4—22no—
44C43C2×C411yescharacteristic
45C4⋊C16—22no—
46C4⋊C16—22no—
47C22⋊C16—24no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50C4×C16—22no—
51C4×C16D414yes—
52#336—22no—
53#336—21no—
54#336—21no—
55C22×Q8C2×C811yescharacteristic
56C22×D4C2×C811yescharacteristic
57C23×C4C22⋊C412yes—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—82no—
63C22×C8C22⋊C412yes—
64C22×C8C2×D411yescharacteristic
65C22×C8C2×D411yescharacteristic
66#124—22no—
67#124—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×D4—22no—
71C4×D4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C2×C811yescharacteristic
74C2×C4⋊C4C2×C811yescharacteristic
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4M1612yes—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42C2×C811yescharacteristic
82C2×C42C2×C811yescharacteristic
83C2×C42C22×C411yescharacteristic
84M32—84no—
85C2×C16—24no—
86C2×C16—42no—
87C4⋊C8—22no—
88C4⋊C8—22no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—82no—
92C22⋊C8—24no—
93C22⋊C8—82no—
94C4×C8—22no—
95C4×C8—42no—
96C4×C8C2×D412yes—
97C4×C8#4112yes—
98#95—22no—
99#95—22no—
100C24C22⋊C812yes—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8M3212yescomplemented
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C22⋊C812yes—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C22×C811yescharacteristic
120C22×C4C2×M1611yescharacteristic
121C2×C8—22no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—44no—
125C2×C8—42no—
126C2×C8—41no—
127C2×C8—42no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—82no—
132C2×C8#12011yescharacteristic
133C2×C8#12112yes—
134C2×C8#12311yescharacteristic
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C22⋊C4—28no—
139C22⋊C4—24no—
140C22⋊C4—82no—
141C42—21no—
142C42—22no—
143C42—41no—
144C42—41no—
145C42—41no—
146C42—41no—
147C42—81no—
148C42M3212yescomplemented
149C42C4×D412yes—
150C42#13112yes—
151C16—44no—
152C16—82no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—42no—
157C23—42no—
158C23C2×C22⋊C811yescharacteristic
159Q8—24no—
160Q8—42no—
161D4—42no—
162D4—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—82no—
182C2×C4#38611yescharacteristic
183C2×C4#40811yescharacteristic
184C2×C4C8×D412yes—
185C2×C4#43512yes—
186C2×C4C2×M3211yescharacteristic
187C2×C4C2×M3211yescharacteristic
188C2×C4#50412yes—
189C8—22no—
190C8—21no—
191C8—21no—
192C8—42no—
193C8—81no—
194C8—81no—
195C22—21no—
196C22—21no—
197C22—44no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—82no—
202C22#186911yescharacteristic
203C22#211211yescharacteristic
204C22#211311yescharacteristic
205C4—21no—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4#190911yescharacteristic
213C4#191111yescharacteristic
214C4#216812yes—
215C4#216912yes—
216C2—21no—
217C2—21no—
218C2—82no—
219C2#909911yescharacteristic
220C2#1297711yescharacteristic
221C2#1298811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C22×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
3. #11637C2semidirect
5. #7142C2semidirect
104. C2×Q8M32semidirect
148. C42M32semidirect

Maximal subgroups

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

subgroupindex
#334892
#222082
#116372
#114872
#71422