Finite Group Databasebeta

← #122339

Subgroups — #122339

Order 512.

365 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 86 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#20793C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic
3C2×#1322C211yesmaximal, characteristic
4#7066C214yesmaximal, complemented
5C2×#424—21no—
6C2×#424—21no—
7C2×#423C411yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9C4×#117C411yescharacteristic
10#2112—21no—
11C2×C22⋊C16—21no—
12#1843C412yes—
13#1843C2212yes—
14#1842—21no—
15#1842—21no—
16#1322—21no—
17#1322—21no—
18#1322C2212yes—
19#1322C2212yes—
20C22×M16—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23C2×#117—21no—
24C22×C42C2×C411yescharacteristic
25C2×M32—41no—
26C22×C16—41no—
27#424—44no—
28#424—44no—
29#423—41no—
30#423—41no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38C2×C4⋊C8C2×C412yes—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×C4⋊C8C2311yescharacteristic
43C4×C4⋊C4—21no—
44C4×C4⋊C4C811yescharacteristic
45C4×C4⋊C4C811yescharacteristic
46C4×C22⋊C4—21no—
47C4×C22⋊C4C812yes—
48#376C2×C412yes—
49C43—21no—
50C22⋊C16—22no—
51C22⋊C16—22no—
52C22⋊C16—41no—
53C22⋊C16—41no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C2×C811yescharacteristic
56C23×C4M1611yescharacteristic
57C2×M16—42no—
58C2×M16—42no—
59C2×M16—41no—
60C2×M16—41no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—22no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72#117—24no—
73#117—24no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—22no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—22no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C2×C811yescharacteristic
87C2×C42M1611yescharacteristic
88C2×C42C22×C411yescharacteristic
89M32—82no—
90C2×C16—42no—
91C2×C16—42no—
92C2×C16—82no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—44no—
100C4⋊C8—44no—
101C4⋊C8—44no—
102C4⋊C8—44no—
103#95—22no—
104#95C2×C812yes—
105#95C2×C812yes—
106C24C22⋊C811yescharacteristic
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—81no—
118C22×C4C22⋊C811yescharacteristic
119C22×C4C22⋊C811yescharacteristic
120C22×C4C22⋊C811yescharacteristic
121C22×C4#9911yescharacteristic
122C22×C4#9911yescharacteristic
123C22×C4#10111yescharacteristic
124C22×C4#10111yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4C22×C811yescharacteristic
127C22×C4C2×M1611yescharacteristic
128M16—82no—
129M16—82no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—81no—
144C2×C8—81no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C22⋊C4—28no—
148C42—24no—
149C42—24no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—42no—
159C42—42no—
160C42—82no—
161C16—82no—
162C16—82no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23—41no—
167C23#32411yescharacteristic
168C23#32411yescharacteristic
169C23C2×C22⋊C811yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4#32411yescharacteristic
197C2×C4#32411yescharacteristic
198C2×C4#35012yes—
199C2×C4#35312yes—
200C2×C4#35612yes—
201C2×C4C2×#9911yescharacteristic
202C2×C4C2×#10111yescharacteristic
203C8—41no—
204C8—41no—
205C8—41no—
206C8—41no—
207C8—82no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—42no—
217C22#132512yes—
218C22C2×#32411yescharacteristic
219C22C2×#35011yescharacteristic
220C22#211711yescharacteristic
221C4—22no—
222C4—21no—
223C4—21no—
224C4—44no—
225C4—44no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4#132512yes—
230C4C2×#35311yescharacteristic
231C4C2×#35611yescharacteristic
232C2—21no—
233C2—21no—
234C2—22no—
235C2—41no—
236C2#1144411yescharacteristic
237C2C2×#132511yescharacteristic
238C2#1145811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#324
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7066C2semidirect

Maximal subgroups

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

subgroupindex
#207932
#114402
C2×#13222
#70662