Finite Group Databasebeta

← #134265

Subgroups — #134265

Order 512.

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

normal subgroups only

This group has 400 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#20429C211yesmaximal, characteristic
2#12785C212yesmaximal
3C2×#1327C212yesmaximal
4#7193C212yesmaximal
5C2×#423C411yescharacteristic
6C2×#423C2211yescharacteristic
7C2×C4×M16C411yescharacteristic
8C2×C4×M16C2211yescharacteristic
9C2×#376C411yescharacteristic
10#2150—22no—
11C2×#349C2214yes—
12#1768—22no—
13#1768—22no—
14#1728C411yescharacteristic
15#1728C2211yescharacteristic
16#1377—22no—
17#1327—24no—
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C22×M16C2311yescharacteristic
22C22×C42C2×C411yescharacteristic
23C22×C42C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C2×M32—24no—
26#423—22no—
27#423—22no—
28#423—22no—
29#423D412yes—
30#423Q812yes—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C4×M16—42no—
34C4×M16—42no—
35C2×#97—21no—
36C2×#97D412yes—
37C2×C4×C8—21no—
38C2×C4×C8D412yes—
39#376—21no—
40#376—21no—
41#376C814yes—
42C2×#95—22no—
43C4⋊C16—44no—
44#349—24no—
45#349—24no—
46#336—22no—
47#336—22no—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4M1612yes—
51C23×C4C22×C411yescharacteristic
52C2×M16—21no—
53C2×M16—21no—
54C2×M16—21no—
55C2×M16—21no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16C4⋊C412yes—
65C2×M16M1612yes—
66C2×M16C2×D411yescharacteristic
67C2×M16C2×Q811yescharacteristic
68C2×M16#4112yes—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8C22⋊C412yes—
72C22×C8C2×D412yes—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—21no—
79C2×C42—41no—
80C2×C42—41no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C4⋊C412yes—
84C2×C42C2×C812yes—
85C2×C42C2×C812yes—
86C2×C42C2×C812yes—
87C2×C42M1612yes—
88M32—44no—
89M32—44no—
90C2×C16—44no—
91C4⋊C8—28no—
92C4⋊C8—24no—
93C4⋊C8—24no—
94#97—42no—
95#97—42no—
96C4×C8—24no—
97C4×C8—42no—
98#95—24no—
99#95—24no—
100C24C2×M1611yescharacteristic
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—82no—
112C22×C4—82no—
113C22×C4—82no—
114C22×C4C22⋊C812yes—
115C22×C4C22⋊C812yes—
116C22×C4C2×C22⋊C411yescharacteristic
117C22×C4C2×C4⋊C411yescharacteristic
118C22×C4#11711yescharacteristic
119C22×C4C22×C811yescharacteristic
120C22×C4C2×M1611yescharacteristic
121C22×C4C2×M1611yescharacteristic
122M16—41no—
123M16—41no—
124M16—42no—
125M16—42no—
126M16—42no—
127M16—42no—
128M16—42no—
129M16—42no—
130M16—42no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8#12112yes—
143C2×C8C22⋊Q812yes—
144C42—24no—
145C42—22no—
146C42—22no—
147C42—21no—
148C42—21no—
149C42—22no—
150C42—42no—
151C42—41no—
152C42—41no—
153C42#9912yes—
154C42#10112yes—
155C42C4⋊C814yes—
156C16—84no—
157C23—21no—
158C23—21no—
159C23—42no—
160C23C2×C22⋊C811yescharacteristic
161C23#42311yescharacteristic
162C23#43211yescharacteristic
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—21no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—84no—
184C2×C4#32414yes—
185C2×C4#38011yescharacteristic
186C2×C4#40711yescharacteristic
187C2×C4C2×#9911yescharacteristic
188C2×C4#41011yescharacteristic
189C2×C4C2×#10111yescharacteristic
190C2×C4#41311yescharacteristic
191C2×C4C2×C4⋊C811yescharacteristic
192C2×C4#43111yescharacteristic
193C8—22no—
194C8—22no—
195C8—42no—
196C8—42no—
197C8—42no—
198C22—22no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22—42no—
205C22C2×#32411yescharacteristic
206C22#145811yescharacteristic
207C22#179711yescharacteristic
208C4—21no—
209C4—21no—
210C4—24no—
211C4—44no—
212C4—42no—
213C4—82no—
214C4#146511yescharacteristic
215C4#146711yescharacteristic
216C4#179911yescharacteristic
217C4#180111yescharacteristic
218C2—21no—
219C2—21no—
220C2—41no—
221C2—41no—
222C2#855811yescharacteristic
223C2#1145612yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22C2×#324
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ 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
#204292
#127852
C2×#13272
#71932