Finite Group Databasebeta

← #122432

Subgroups — #122432

Order 512.

302 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 74 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#21638C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic
3#11465C211yesmaximal, characteristic
4#7065C212yesmaximal, complemented
5#7061C212yesmaximal, complemented
6C2×#446C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2398C411yescharacteristic
9#2112—21no—
10C2×C22⋊C16—21no—
11#1985C2211yescharacteristic
12#1955C411yescharacteristic
13#1923—21no—
14#1923—21no—
15#1832C2211yescharacteristic
16#1752—21no—
17#1752—21no—
18#1726C411yescharacteristic
19#1322C2212yes—
20#1322C2212yes—
21#1314—21no—
22#1314—21no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4×Q8C2×C411yescharacteristic
26C2×C4×D4C2×C411yescharacteristic
27C2×M32—41no—
28C22×C16—41no—
29#446—41no—
30#446—41no—
31C8×D4—41no—
32C8×D4—41no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2311yescharacteristic
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8D411yescharacteristic
39C2×C22⋊C8D411yescharacteristic
40C2×#97—21no—
41C2×#97—21no—
42C2×#97C2×C411yescharacteristic
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45#400—21no—
46#397C811yescharacteristic
47#397C811yescharacteristic
48#394—21no—
49#393C811yescharacteristic
50#393C811yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53#376C2×C411yescharacteristic
54C22⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16—41no—
57C22⋊C16—41no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×Q8C2×C811yescharacteristic
63C22×D4M1611yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C4×Q8—41no—
79C4×Q8—41no—
80C4×D4—41no—
81C4×D4—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2×C811yescharacteristic
84C2×C4⋊C4M1611yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4C2×C811yescharacteristic
90C2×C22⋊C4C2×C811yescharacteristic
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C22×C411yescharacteristic
94M32—82no—
95C2×C16—42no—
96C2×C16—42no—
97C2×C16—82no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—42no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—41no—
106#97—42no—
107#97—41no—
108#97—41no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—42no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95C2×C811yescharacteristic
118#95C2×C811yescharacteristic
119C24C22⋊C811yescharacteristic
120C24C22⋊C811yescharacteristic
121C2×Q8—41no—
122C2×Q8—81no—
123C2×D4—22no—
124C2×D4—41no—
125C2×D4—41no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C22⋊C811yescharacteristic
139C22×C4C22⋊C811yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C22×C811yescharacteristic
142C22×C4C2×M1611yescharacteristic
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8#9911yescharacteristic
161C2×C8#9911yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C22⋊C4—41no—
166C22⋊C4—41no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C22⋊C4—81no—
170C22⋊C4—81no—
171C42—21no—
172C42—21no—
173C42—41no—
174C42—41no—
175C42—41no—
176C42—41no—
177C16—82no—
178C16—82no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23#35012yes—
189C23C2×C22⋊C811yescharacteristic
190Q8—82no—
191D4—44no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4C2×#9911yescharacteristic
222C2×C4#41311yescharacteristic
223C8—21no—
224C8—42no—
225C8—41no—
226C8—41no—
227C8—41no—
228C8—81no—
229C8#35412yes—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—42no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—81no—
242C22#145911yescharacteristic
243C22C2×#35011yescharacteristic
244C22#211711yescharacteristic
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4—81no—
254C4C2×#35411yescharacteristic
255C4#213211yescharacteristic
256C2—21no—
257C2—21no—
258C2—42no—
259C2—81no—
260C2#1147011yescharacteristic
261C2#1147711yescharacteristic
262C2#1148411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22#1459
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7065C2semidirect
5. #7061C2semidirect

Maximal subgroups

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

subgroupindex
#216382
#114662
#114652
#70652
#70612