Finite Group Databasebeta

← #134267

Subgroups — #134267

Order 512.

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

normal subgroups only

This group has 392 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#20427C211yesmaximal, characteristic
2#12785C212yesmaximal
3C2×#1328C212yesmaximal
4#7192C212yesmaximal
5C2×#424C411yescharacteristic
6C2×#424C2211yescharacteristic
7C2×C4×M16C2211yescharacteristic
8C22×#97C411yescharacteristic
9C4×#117C411yescharacteristic
10#2150—22no—
11C2×#349C2214yes—
12#1768—22no—
13#1767—22no—
14#1729C411yescharacteristic
15#1729C2211yescharacteristic
16#1377—22no—
17#1328—24no—
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#424—22no—
28#424—22no—
29#424D412yes—
30#424Q812yes—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C4×M16—42no—
34C2×#97—21no—
35C2×#97—21no—
36C2×#97—42no—
37C2×#97D412yes—
38C2×C4×C8D412yes—
39C4×C4⋊C4—22no—
40C4×C22⋊C4C814yes—
41#376—21no—
42C43—21no—
43C4⋊C16—44no—
44#349—24no—
45#349—24no—
46#336—22no—
47#336—22no—
48C23×C4C2×C812yes—
49C23×C4C22×C411yescharacteristic
50C2×M16—21no—
51C2×M16—21no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16C2×D411yescharacteristic
57C2×M16C2×Q811yescharacteristic
58C2×M16#4112yes—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C22⋊C412yes—
68C22×C8C4⋊C412yes—
69C22×C8M1612yes—
70C22×C8C2×D412yes—
71#117—21no—
72#117—21no—
73#117—41no—
74#117—41no—
75#117C4⋊C412yes—
76#117M1612yes—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C22⋊C4C2×C812yes—
80C2×C22⋊C4C2×C812yes—
81C2×C42—21no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—21no—
85C2×C42—24no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42M1612yes—
89M32—44no—
90M32—44no—
91C2×C16—44no—
92C4⋊C8—28no—
93C4⋊C8—24no—
94C4⋊C8—24no—
95#97—42no—
96#97—42no—
97#97—42no—
98C4×C8—24no—
99#95—24no—
100C24C22×C811yescharacteristic
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—82no—
109C22×C4C22⋊C812yes—
110C22×C4C22⋊C812yes—
111C22×C4C2×C22⋊C411yescharacteristic
112C22×C4C2×C4⋊C411yescharacteristic
113C22×C4#11711yescharacteristic
114C22×C4C2×M1611yescharacteristic
115C22×C4C2×M1611yescharacteristic
116C22×C4C2×M1611yescharacteristic
117M16—41no—
118M16—41no—
119M16—42no—
120M16—42no—
121M16—42no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8#12112yes—
138C2×C8C22⋊Q812yes—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4C4⋊C814yes—
145C42—24no—
146C42—22no—
147C42—22no—
148C42—21no—
149C42—21no—
150C42—41no—
151C42—41no—
152C42—84no—
153C42#9912yes—
154C42#10112yes—
155C16—84no—
156C23—21no—
157C23—21no—
158C23—42no—
159C23C2×C22⋊C811yescharacteristic
160C23C2×C4⋊C811yescharacteristic
161C23#43111yescharacteristic
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—84no—
180C2×C4#32414yes—
181C2×C4#38011yescharacteristic
182C2×C4#40711yescharacteristic
183C2×C4C2×#9911yescharacteristic
184C2×C4#41011yescharacteristic
185C2×C4C2×#10111yescharacteristic
186C2×C4#41311yescharacteristic
187C2×C4#42311yescharacteristic
188C2×C4#43211yescharacteristic
189C8—22no—
190C8—22no—
191C8—42no—
192C8—42no—
193C8—42no—
194C22—22no—
195C22—21no—
196C22—21no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22—42no—
201C22C2×#32411yescharacteristic
202C22#145811yescharacteristic
203C22#179711yescharacteristic
204C4—21no—
205C4—21no—
206C4—24no—
207C4—44no—
208C4—42no—
209C4—82no—
210C4#146511yescharacteristic
211C4#146711yescharacteristic
212C4#179911yescharacteristic
213C4#180111yescharacteristic
214C2—21no—
215C2—21no—
216C2—41no—
217C2—41no—
218C2#855811yescharacteristic
219C2#1145812yes—
subgroupG/H
centerC2×C4C2×#99
commutator, prime radicalC24C22×C8
Frattini, cosocleC22×M16C23
socleC22C2×#324
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#204272
#127852
C2×#13282
#71922