Finite Group Databasebeta

← #128352

Subgroups — #128352

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#20435C211yesmaximal, characteristic
2#12602C212yesmaximal
3C2×#1328C212yesmaximal
4#7191C212yesmaximal
5C2×#424C2211yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2296C411yescharacteristic
10C2×#349C2214yes—
11C4×M32—22no—
12#1771—22no—
13C8×C4⋊C4—22no—
14#1729C411yescharacteristic
15#1728C2211yescharacteristic
16#1377—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—41no—
31#405—41no—
32#405C814yes—
33C2×#97—21no—
34C2×#97D412yes—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—21no—
38C2×C4×C8—41no—
39C2×C4×C8—41no—
40C2×C4×C8D412yes—
41#389—22no—
42#385C814yes—
43C4×C4⋊C4—22no—
44#349—24no—
45#349C2×C418yes—
46#346—42no—
47C4×C16—42no—
48#336—22no—
49#336—22no—
50C23×C4C2×C812yes—
51C23×C4C22×C411yescharacteristic
52C2×M16—21no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16C2×C812yes—
58C2×M16C22×C412yes—
59C2×M16#4112yes—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—22no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C4212yes—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×C812yes—
72C22×C8M1612yes—
73C22×C8C2×D412yes—
74#117—21no—
75#117—21no—
76#117—41no—
77#117—41no—
78#117C4212yes—
79#117C2×C812yes—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4M1612yes—
83C2×C22⋊C4C2×C812yes—
84C2×C22⋊C4C2×C812yes—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—41no—
89C2×C42—41no—
90C2×C42C22⋊C412yes—
91M32—28no—
92M32—44no—
93C2×C16—44no—
94C4⋊C8—28no—
95#97—24no—
96#97—42no—
97C4×C8—24no—
98C4×C8—22no—
99C4×C8—22no—
100C4×C8—42no—
101C4×C8—41no—
102C4×C8—41no—
103#95—24no—
104C24C22×C811yescharacteristic
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—44no—
112C22×C4—42no—
113C22×C4C22⋊C814yes—
114C22×C4C2×C4211yescharacteristic
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4#11711yescharacteristic
117C22×C4C22×C811yescharacteristic
118C22×C4C2×M1611yescharacteristic
119C22×C4C2×M1611yescharacteristic
120M16—42no—
121M16—42no—
122M16—42no—
123M16—42no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8C22⋊C812yes—
141C2×C8C22⋊C812yes—
142C2×C8#9911yescharacteristic
143C2×C8#9911yescharacteristic
144C2×C8#10112yes—
145C2×C8C4×D414yes—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—48no—
150C22⋊C4—22no—
151C22⋊C4—42no—
152C22⋊C4C4×C814yes—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—22no—
157C42—42no—
158C42—41no—
159C42—41no—
160C16—48no—
161C23—21no—
162C23—21no—
163C23—42no—
164C23C2×C4×C811yescharacteristic
165C23C2×C22⋊C811yescharacteristic
166C23#43111yescharacteristic
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—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4C4×C22⋊C411yescharacteristic
187C2×C4C4×M1611yescharacteristic
188C2×C4C2×C22⋊C811yescharacteristic
189C2×C4C2×#9911yescharacteristic
190C2×C4#41011yescharacteristic
191C2×C4C2×#10111yescharacteristic
192C2×C4#41311yescharacteristic
193C2×C4#43111yescharacteristic
194C8—22no—
195C8—24no—
196C8—42no—
197C8—42no—
198C8#32412yes—
199C8#32412yes—
200C22—22no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—42no—
206C22—42no—
207C22#145911yescharacteristic
208C22#146111yescharacteristic
209C22C4×C22⋊C811yescharacteristic
210C4—21no—
211C4—21no—
212C4—44no—
213C4—44no—
214C4—44no—
215C4C2×#32411yescharacteristic
216C4#145811yescharacteristic
217C4C4×#9911yescharacteristic
218C4C4×#10111yescharacteristic
219C2—21no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2C4×#32411yescharacteristic
224C2#1148312yes—
subgroupG/H
centerC2×C8#99
commutatorC23C2×C4×C8
Frattini, cosocleC22×M16C23
socleC22#1459
prime radicalC24C22×C8
LCSG ▷ C23 ▷ 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. #1328C4semidirect

Maximal subgroups

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

subgroupindex
#204352
#126022
C2×#13282
#71912