Finite Group Databasebeta

← #124285

Subgroups — #124285

Order 512.

358 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 102 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#21145C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C4×#349C211yesmaximal, characteristic
4#7124C214yesmaximal, complemented
5C2×#424—21no—
6C22×C4⋊C8C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8#2285C411yescharacteristic
9#2150—21no—
10C2×#349—21no—
11C2×#349—21no—
12C4×M32—21no—
13#1940C412yes—
14#1940C2212yes—
15#1939—21no—
16#1939—21no—
17#1729—21no—
18#1377C2212yes—
19#1377C2212yes—
20C22×M16D411yescharacteristic
21C22×M16D411yescharacteristic
22C23×C8—21no—
23C2×#117—21no—
24C22×C42C2×C411yescharacteristic
25C2×M32—22no—
26C2×M32—22no—
27#424—42no—
28#424—42no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—41no—
33C2×C4⋊C8—41no—
34C2×C4⋊C8C2×C412yes—
35C4×M16—41no—
36C4×M16—41no—
37C2×#97C2×C411yescharacteristic
38C2×#97C2311yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#384—21no—
42#382C812yes—
43#380—21no—
44#380C812yes—
45C4×C4⋊C4—21no—
46C4×C4⋊C4C2×C412yes—
47C4⋊C16—41no—
48C4⋊C16—41no—
49#349—24no—
50#349—42no—
51#346—41no—
52C4×C16—41no—
53#336—22no—
54#336—22no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C2×C811yescharacteristic
57C23×C4M1611yescharacteristic
58C2×M16—22no—
59C2×M16—22no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16D812yes—
64C2×M16SD1612yes—
65C22×C8—22no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8—42no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73#117—42no—
74#117—42no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—22no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C2×C811yescharacteristic
86C2×C42M1611yescharacteristic
87C2×C42C22×C411yescharacteristic
88M32—24no—
89M32—42no—
90M32—42no—
91M32—42no—
92C2×C16—42no—
93C2×C16—42no—
94C4⋊C8—42no—
95C4⋊C8—44no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100#97—41no—
101#97—41no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106#95—22no—
107#95C2×C814yes—
108C24C22⋊C811yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—24no—
118C22×C4—42no—
119C22×C4—81no—
120C22×C4C22⋊C811yescharacteristic
121C22×C4C22⋊C811yescharacteristic
122C22×C4C22⋊C811yescharacteristic
123C22×C4#10212yes—
124C22×C4#10412yes—
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4C22×C811yescharacteristic
127C22×C4C2×M1611yescharacteristic
128M16—42no—
129M16—42no—
130M16—44no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—84no—
144C2×C8#10212yes—
145C2×C8C2×D811yescharacteristic
146C2×C8C2×SD1611yescharacteristic
147C2×C8#13712yes—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—82no—
151C22⋊C4—44no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—42no—
159C42—42no—
160C42#9911yescharacteristic
161C42#9911yescharacteristic
162C42#10111yescharacteristic
163C42#10111yescharacteristic
164C16—44no—
165C16—82no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23D4⋊C812yes—
171C23C2×C22⋊C811yescharacteristic
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4#32411yescharacteristic
192C2×C4#32411yescharacteristic
193C2×C4#32411yescharacteristic
194C2×C4#32411yescharacteristic
195C2×C4D4⋊C812yes—
196C2×C4C2×#9911yescharacteristic
197C2×C4C2×#10111yescharacteristic
198C2×C4C2×#10211yescharacteristic
199C2×C4#41911yescharacteristic
200C2×C4C2×#10411yescharacteristic
201C2×C4#42111yescharacteristic
202C8—22no—
203C8—21no—
204C8—22no—
205C8—21no—
206C8—41no—
207C8—82no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—42no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22C2×#32411yescharacteristic
216C22C2×D4⋊C811yescharacteristic
217C22#147911yescharacteristic
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—84no—
225C4#132812yes—
226C4#150811yescharacteristic
227C4#151911yescharacteristic
228C4#152811yescharacteristic
229C4#154611yescharacteristic
230C2—21no—
231C2—21no—
232C2—42no—
233C2#757311yescharacteristic
234C2#1145611yescharacteristic
235C2C2×#132811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#324
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7124C2semidirect

Maximal subgroups

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

subgroupindex
#211452
#127852
C4×#3492
#71242