Finite Group Databasebeta

← #137901

Subgroups — #137901

Order 512.

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

normal subgroups only

This group has 423 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, complemented
2#22260C211yesmaximal, characteristic
3#11629C212yesmaximal, complemented
4#11487C211yesmaximal, characteristic, complemented
5#7142C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8C4×#127C411yescharacteristic
9#2169—24no—
10#2162C2212yes—
11#2162C2212yes—
12#2112—22no—
13#1954C411yescharacteristic
14#1954C411yescharacteristic
15#1869—22no—
16#1869—22no—
17#1831C411yescharacteristic
18#1744C2211yescharacteristic
19D4⋊C16—24no—
20#1314—22no—
21C23×C8—21no—
22C23×C8D412yes—
23C2×#127C811yescharacteristic
24C2×C4×D4—22no—
25C2×C4×D4C2×C411yescharacteristic
26C2×C4×D4C2×C411yescharacteristic
27C2×M32—24no—
28C8×D4—22no—
29C8×D4—22no—
30C8×D4—22no—
31C8×D4—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C22⋊C8—21no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8D412yes—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8C2311yescharacteristic
40#390—22no—
41#390C811yescharacteristic
42#390C811yescharacteristic
43#384C811yescharacteristic
44C43C2×C411yescharacteristic
45C4⋊C16—22no—
46C4⋊C16—22no—
47C22⋊C16—24no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50C4×C16—22no—
51C4×C16D414yescomplemented
52#336—22no—
53#336—21no—
54#336—21no—
55C22×D4—22no—
56C22×D4C2×C811yescharacteristic
57C22×D4C2×C811yescharacteristic
58C23×C4—21no—
59C23×C4C22⋊C412yes—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—42no—
65C22×C8—82no—
66C22×C8—82no—
67C22×C8C22⋊C412yes—
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70#127—22no—
71#127—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—82no—
77C4×D4—82no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4C2×C811yescharacteristic
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4M1612yes—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42C2×C811yescharacteristic
88C2×C42C2×C811yescharacteristic
89C2×C42C22×C411yescharacteristic
90M32—84no—
91C2×C16—24no—
92C2×C16—42no—
93C4⋊C8—22no—
94C4⋊C8—22no—
95C22⋊C8—24no—
96C22⋊C8—41no—
97C22⋊C8—41no—
98C22⋊C8—82no—
99C22⋊C8—82no—
100C4×C8—22no—
101C4×C8C2×D412yes—
102C4×C8#4112yes—
103C24—21no—
104C24C22⋊C812yes—
105C2×D4—28no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—82no—
114C2×D4—82no—
115C2×D4M3212yescomplemented
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4C22⋊C812yes—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C22×C811yescharacteristic
130C22×C4C2×M1611yescharacteristic
131C2×C8—22no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—44no—
135C2×C8—42no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—42no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—82no—
142C2×C8—84no—
143C2×C8#12011yescharacteristic
144C2×C8#12112yes—
145C2×C8#12311yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—82no—
149C22⋊C4—24no—
150C22⋊C4—42no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C42—21no—
154C42—22no—
155C42—41no—
156C42—41no—
157C42—81no—
158C42M3212yescomplemented
159C42C4×D412yes—
160C42#13112yes—
161C16—44no—
162C16—82no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—41no—
172C23—41no—
173C23C2×C22⋊C811yescharacteristic
174D4—24no—
175D4—42no—
176D4—42no—
177D4—42no—
178D4—88no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4#38611yescharacteristic
199C2×C4#40811yescharacteristic
200C2×C4C8×D412yes—
201C2×C4#43512yes—
202C2×C4C2×M3211yescharacteristic
203C2×C4C2×M3211yescharacteristic
204C2×C4#50412yes—
205C8—22no—
206C8—21no—
207C8—21no—
208C8—42no—
209C8—81no—
210C8—81no—
211C22—21no—
212C22—21no—
213C22—44no—
214C22—44no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—82no—
220C22—82no—
221C22—81no—
222C22—81no—
223C22#186911yescharacteristic
224C22#211211yescharacteristic
225C22#211311yescharacteristic
226C4—21no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—81no—
232C4#191011yescharacteristic
233C4#191111yescharacteristic
234C4#216812yes—
235C4#216912yes—
236C2—21no—
237C2—21no—
238C2—82no—
239C2—82no—
240C2#910111yescharacteristic
241C2#1297711yescharacteristic
242C2#1298811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC2×#127C8
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
1. #33489C2semidirect
3. #11629C2semidirect
4. #11487C2semidirect
5. #7142C2semidirect
51. C4×C16D4semidirect
115. C2×D4M32semidirect
158. C42M32semidirect

Maximal subgroups

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

subgroupindex
#334892
#222602
#116292
#114872
#71422