Finite Group Databasebeta

← #128348

Subgroups — #128348

Order 512.

428 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 142 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#20437C211yesmaximal, characteristic
2#12602C212yesmaximal
3C2×#1327C212yesmaximal
4#7194C212yesmaximal
5C2×#423C2211yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2285C411yescharacteristic
10C2×#349C2214yes—
11C4×M32—22no—
12#1770—22no—
13C8×C4⋊C4—22no—
14#1729C411yescharacteristic
15#1729C2211yescharacteristic
16#1377—22no—
17#1327C418yescomplemented
18C22×M16C2×C411yescharacteristic
19C22×M16C2311yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C2×M32—24no—
26#423—22no—
27#423C2×C414yes—
28C2×C4⋊C8—22no—
29#405—41no—
30#405—41no—
31#405—41no—
32#405—41no—
33C2×#97—21no—
34C2×#97D412yes—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—22no—
38C2×C4×C8—21no—
39C2×C4×C8C814yes—
40C2×C4×C8D412yes—
41#384C812yes—
42#382C812yes—
43#380—22no—
44C4×C4⋊C4—22no—
45#349—24no—
46#349C2×C418yes—
47#346—42no—
48C4×C16—42no—
49#336—22no—
50#336—22no—
51C23×C4—21no—
52C23×C4C22×C411yescharacteristic
53C2×M16—21no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16C4212yes—
59C2×M16C22×C412yes—
60C2×M16#4112yes—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×C812yes—
72C22×C8C2×C812yes—
73C22×C8M1612yes—
74C22×C8C2×D412yes—
75#117—41no—
76#117—41no—
77#117—41no—
78#117—41no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4C2×C812yes—
81C2×C4⋊C4C2×C812yes—
82C2×C22⋊C4—21no—
83C2×C22⋊C4M1612yes—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C4212yes—
89C2×C42C22⋊C412yes—
90C2×C42C2×C811yescharacteristic
91C2×C42C2×C811yescharacteristic
92C2×C42C2×C812yes—
93M32—28no—
94M32—44no—
95C2×C16—44no—
96C4⋊C8—28no—
97#97—42no—
98#97—42no—
99C4×C8—24no—
100C4×C8—24no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—41no—
104C4×C8—41no—
105#95—24no—
106C24C2×M1611yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—44no—
117C22×C4C22⋊C814yes—
118C22×C4C2×C4211yescharacteristic
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4#11711yescharacteristic
121C22×C4C22×C811yescharacteristic
122C22×C4C22×C811yescharacteristic
123C22×C4C2×M1611yescharacteristic
124M16—42no—
125M16—42no—
126M16—42no—
127M16—42no—
128C2×C8—22no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—21no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—42no—
144C2×C8C22⋊C812yes—
145C2×C8C22⋊C812yes—
146C2×C8#9911yescharacteristic
147C2×C8#9911yescharacteristic
148C2×C8#10112yes—
149C2×C8C4×D414yes—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—48no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—21no—
160C42—21no—
161C42—41no—
162C42—41no—
163C42C4×C812yes—
164C42C4×C812yes—
165C16—48no—
166C23—21no—
167C23—21no—
168C23—42no—
169C23C4×M1611yescharacteristic
170C23C2×C22⋊C811yescharacteristic
171C23#43111yescharacteristic
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4C4×C22⋊C411yescharacteristic
193C2×C4C2×C4×C811yescharacteristic
194C2×C4C2×C22⋊C811yescharacteristic
195C2×C4C2×#9911yescharacteristic
196C2×C4#41011yescharacteristic
197C2×C4C2×#10111yescharacteristic
198C2×C4#41311yescharacteristic
199C2×C4#43111yescharacteristic
200C8—22no—
201C8—24no—
202C8—42no—
203C8—42no—
204C8#32412yes—
205C8#32412yes—
206C22—22no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—42no—
212C22—42no—
213C22#145911yescharacteristic
214C22#146111yescharacteristic
215C22C4×C22⋊C811yescharacteristic
216C4—24no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—44no—
221C4—82no—
222C4C2×#32411yescharacteristic
223C4#145811yescharacteristic
224C4C4×#9911yescharacteristic
225C4C4×#10111yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2C4×#32411yescharacteristic
231C2#1148512yes—
subgroupG/H
centerC2×C8#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC22×M16C23
socleC22#1459
prime radicalC24C2×M16
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1327C4semidirect

Maximal subgroups

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

subgroupindex
#204372
#126022
C2×#13272
#71942