Finite Group Databasebeta

← #128402

Subgroups — #128402

Order 512.

410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 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#12608C212yesmaximal
3C2×#1327C212yesmaximal
4#7192C212yesmaximal
5C2×#423C2211yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2285C411yescharacteristic
10C2×#349C2214yes—
11C2×#347—22no—
12#1770—22no—
13C8×C4⋊C4—22no—
14#1729C411yescharacteristic
15#1729C2211yescharacteristic
16#1376—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—42no—
30#405—42no—
31C2×#97—21no—
32C2×#97D412yes—
33C2×C4×C8—21no—
34C2×C4×C8—21no—
35C2×C4×C8—22no—
36C2×C4×C8—22no—
37C2×C4×C8—21no—
38C2×C4×C8D412yes—
39#384—21no—
40#382—21no—
41#380—22no—
42C4×C4⋊C4—22no—
43#349—24no—
44#349C2×C418yes—
45#347—44no—
46#336—22no—
47#336—22no—
48C23×C4—21no—
49C23×C4C22×C411yescharacteristic
50C2×M16—21no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16C4212yes—
56C2×M16C22×C412yes—
57C2×M16#4112yes—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8M1612yes—
70C22×C8C2×D412yes—
71#117—41no—
72#117—41no—
73#117—41no—
74#117—41no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4M1612yes—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42C4212yes—
85C2×C42C22⋊C412yes—
86C2×C42M1611yescharacteristic
87C2×C42M1611yescharacteristic
88C2×C42M1612yes—
89M32—28no—
90M32—44no—
91C2×C16—44no—
92C4⋊C8—28no—
93#97—24no—
94#97—42no—
95C4×C8—24no—
96C4×C8—24no—
97C4×C8—42no—
98C4×C8—42no—
99#95—24no—
100C24C2×M1611yescharacteristic
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—44no—
112C22×C4C2×C4211yescharacteristic
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4#11711yescharacteristic
115C22×C4C2×M1611yescharacteristic
116C22×C4C2×M1611yescharacteristic
117C22×C4C2×M1611yescharacteristic
118M16—42no—
119M16—42no—
120M16—42no—
121M16—42no—
122C2×C8—24no—
123C2×C8—24no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—42no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—42no—
138C2×C8#9912yes—
139C2×C8#10112yes—
140C2×C8C4×D414yes—
141C4⋊C4—48no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C42—22no—
147C42—22no—
148C42—21no—
149C42—21no—
150C42—22no—
151C42—22no—
152C42—41no—
153C42—41no—
154C42#9712yes—
155C42#9712yes—
156C16—48no—
157C23—21no—
158C23—21no—
159C23—42no—
160C23C4×M1611yescharacteristic
161C23#40711yescharacteristic
162C23#43211yescharacteristic
163C2×C4—24no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4C4×C22⋊C411yescharacteristic
184C2×C4C2×#9711yescharacteristic
185C2×C4#40711yescharacteristic
186C2×C4C2×#9911yescharacteristic
187C2×C4#41011yescharacteristic
188C2×C4C2×#10111yescharacteristic
189C2×C4#41311yescharacteristic
190C2×C4#43211yescharacteristic
191C8—22no—
192C8—22no—
193C8—24no—
194C8—42no—
195C8—42no—
196C22—22no—
197C22—21no—
198C22—21no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22#146511yescharacteristic
204C22#146711yescharacteristic
205C22#174911yescharacteristic
206C4—24no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—82no—
211C4—82no—
212C4#146311yescharacteristic
213C4#146411yescharacteristic
214C4C4×#9911yescharacteristic
215C4C4×#10111yescharacteristic
216C2—21no—
217C2—21no—
218C2—41no—
219C2—41no—
220C2#815311yescharacteristic
221C2#1153912yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C2×C42
Frattini, cosocleC22×M16C23
socleC22#1467
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#126082
C2×#13272
#71922