Finite Group Databasebeta

← #128406

Subgroups — #128406

Order 512.

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

normal subgroups only

This group has 396 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#20436C211yesmaximal, characteristic
2#12608C212yesmaximal
3C2×#1328C212yesmaximal
4#7193C212yesmaximal
5C2×#424C2211yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×#97C411yescharacteristic
9#2296C411yescharacteristic
10C2×#349C2214yes—
11C2×#347—22no—
12#1771—22no—
13#1770—22no—
14#1729C411yescharacteristic
15#1728C2211yescharacteristic
16#1376—22no—
17#1328C418yescomplemented
18C22×M16C2×C411yescharacteristic
19C22×M16C2311yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C2×M32—24no—
26#424—22no—
27#424C2×C414yes—
28C2×C4⋊C8—22no—
29#405—22no—
30#405—22no—
31#405—42no—
32C2×#97—21no—
33C2×#97—21no—
34C2×#97—21no—
35C2×#97—42no—
36C2×#97D412yes—
37C2×C4×C8—21no—
38C2×C4×C8D412yes—
39#389—22no—
40#385—22no—
41C4×C4⋊C4—22no—
42#349—24no—
43#349C2×C418yes—
44#347—44no—
45#336—22no—
46#336—22no—
47C23×C4—21no—
48C23×C4C22×C411yescharacteristic
49C2×M16—21no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16M1612yes—
55C2×M16C22×C412yes—
56C2×M16#4112yes—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8C4212yes—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C2×D412yes—
70#117—21no—
71#117—21no—
72#117—41no—
73#117—41no—
74#117C4212yes—
75#117M1612yes—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4M1612yes—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C42C22⋊C412yes—
86C2×C42M1612yes—
87M32—28no—
88M32—44no—
89C2×C16—44no—
90C4⋊C8—28no—
91#97—24no—
92#97—24no—
93#97—42no—
94#97—42no—
95C4×C8—24no—
96C4×C8—42no—
97#95—24no—
98C24C2×M1611yescharacteristic
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—82no—
108C22×C4C2×C4211yescharacteristic
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4#11711yescharacteristic
111C22×C4C2×M1611yescharacteristic
112C22×C4C2×M1611yescharacteristic
113C22×C4C2×M1611yescharacteristic
114M16—42no—
115M16—42no—
116M16—42no—
117M16—42no—
118C2×C8—24no—
119C2×C8—24no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—42no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—42no—
134C2×C8#9912yes—
135C2×C8#10112yes—
136C2×C8C4×D414yes—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4—84no—
141C22⋊C4—22no—
142C22⋊C4—42no—
143C22⋊C4#9714yes—
144C42—22no—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—42no—
149C42—41no—
150C42—41no—
151C16—48no—
152C23—21no—
153C23—21no—
154C23—42no—
155C23C2×#9711yescharacteristic
156C23#40711yescharacteristic
157C23#43211yescharacteristic
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—84no—
176C2×C4C4×C22⋊C411yescharacteristic
177C2×C4C4×M1611yescharacteristic
178C2×C4#40711yescharacteristic
179C2×C4C2×#9911yescharacteristic
180C2×C4#41011yescharacteristic
181C2×C4C2×#10111yescharacteristic
182C2×C4#41311yescharacteristic
183C2×C4#43211yescharacteristic
184C8—22no—
185C8—22no—
186C8—24no—
187C8—42no—
188C8—42no—
189C22—22no—
190C22—21no—
191C22—21no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22#146511yescharacteristic
197C22#146711yescharacteristic
198C22#174911yescharacteristic
199C4—21no—
200C4—21no—
201C4—44no—
202C4—82no—
203C4—82no—
204C4#146311yescharacteristic
205C4#146411yescharacteristic
206C4C4×#9911yescharacteristic
207C4C4×#10111yescharacteristic
208C2—21no—
209C2—21no—
210C2—41no—
211C2—41no—
212C2#815311yescharacteristic
213C2#1154012yes—
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. #1328C4semidirect

Maximal subgroups

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

subgroupindex
#204362
#126082
C2×#13282
#71932