Finite Group Databasebeta

← #124288

Subgroups — #124288

Order 512.

353 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 100 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#21156C211yesmaximal, characteristic
2#12602C211yesmaximal, characteristic
3#8601C211yesmaximal, characteristic
4#7126C214yesmaximal, complemented
5C2×#427C411yescharacteristic
6C22×#107C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2317—21no—
9C2×#349—21no—
10C4×M32—21no—
11#1944C412yes—
12#1944C2212yes—
13#1942—21no—
14#1942—21no—
15#1736—21no—
16#1564—21no—
17#1458—21no—
18#1377C2212yes—
19#1275C2212yes—
20C22×M16D411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C2×#117—21no—
23C2×#117D411yescharacteristic
24C22×C4⋊C4—21no—
25C2×M32—22no—
26#431—22no—
27#427—22no—
28#427C814yescomplemented
29C2×#107—21no—
30C2×#107—21no—
31C2×#107—41no—
32C2×#107—41no—
33C2×#107C2×C412yes—
34C2×#106C2×C412yes—
35#405—41no—
36#405—41no—
37C2×#97C2×C411yescharacteristic
38C2×#97C2311yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#389—22no—
42#389—22no—
43#385—41no—
44#385—41no—
45#384—41no—
46#382—41no—
47#349—24no—
48#346—41no—
49C4×C16—41no—
50#340—22no—
51#340—22no—
52#335—41no—
53#334—41no—
54#324—41no—
55#323—41no—
56C23×C4—41no—
57C23×C4C22⋊C411yescharacteristic
58C2×M16—22no—
59C2×M16—22no—
60C2×M16—41no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—22no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C2×C812yes—
67C22×C8M1612yes—
68C22×C8C22×C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70#117—24no—
71#117—41no—
72#117—41no—
73#117—42no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4D811yescharacteristic
82C2×C4⋊C4SD1611yescharacteristic
83C2×C22⋊C4—41no—
84C2×C22⋊C4D811yescharacteristic
85C2×C22⋊C4SD1611yescharacteristic
86C2×C42—41no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C2×D411yescharacteristic
89M32—24no—
90M32—42no—
91C2×C16—42no—
92#107—24no—
93#107—41no—
94#107—41no—
95#107—42no—
96#106—22no—
97#106C2×C814yes—
98C4⋊C8—42no—
99C22⋊C8—42no—
100#97—41no—
101#97—41no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—42no—
107#95—42no—
108#95—42no—
109C24#10211yescharacteristic
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—82no—
117C22×C4—162no—
118C22×C4C22⋊C812yes—
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4#10211yescharacteristic
122C22×C4#10412yes—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C2×D811yescharacteristic
125C22×C4C2×SD1611yescharacteristic
126M16—44no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—82no—
139C2×C8C22⋊C812yes—
140C2×C8#9911yescharacteristic
141C2×C8#9911yescharacteristic
142C2×C8#10112yes—
143C2×C8C22×C811yescharacteristic
144C2×C8C2×M1611yescharacteristic
145C2×C8#13712yes—
146C4⋊C4—24no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—82no—
153C4⋊C4—162no—
154C4⋊C4—162no—
155C22⋊C4—41no—
156C22⋊C4—41no—
157C22⋊C4—44no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—44no—
163C16—44no—
164C23—21no—
165C23—21no—
166C23—41no—
167C23—41no—
168C23D4⋊C812yes—
169C23C2×#10211yescharacteristic
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—44no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—164no—
191C2×C4D4⋊C812yes—
192C2×C4C2×C22⋊C811yescharacteristic
193C2×C4C2×#9911yescharacteristic
194C2×C4C2×#10111yescharacteristic
195C2×C4#41911yescharacteristic
196C2×C4C2×#10411yescharacteristic
197C2×C4#42111yescharacteristic
198C8—21no—
199C8—21no—
200C8—22no—
201C8—41no—
202C8—162no—
203C8#32412yes—
204C8#32412yes—
205C22—21no—
206C22—21no—
207C22—22no—
208C22—42no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22C2×D4⋊C811yescharacteristic
213C22#151911yescharacteristic
214C22#154611yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—84no—
221C4—162no—
222C4C2×#32411yescharacteristic
223C4#147911yescharacteristic
224C4#150811yescharacteristic
225C4#152811yescharacteristic
226C2—21no—
227C2—21no—
228C2—42no—
229C2#757311yescharacteristic
230C2#1186911yescharacteristic
231C2#1187011yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC2×C8C22×C8
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7126C2semidirect
28. #427C8semidirect

Maximal subgroups

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

subgroupindex
#211562
#126022
#86012
#71262