Finite Group Databasebeta

← #125156

Subgroups — #125156

Order 512.

430 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 106 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#33172C211yesmaximal, characteristic
2#12544C211yesmaximal, characteristic
3#8595C211yesmaximal, characteristic
4#7113C214yesmaximal, complemented
5C22×M32—21no—
6C2×#494C411yescharacteristic
7C2×#467—21no—
8C2×#467C412yes—
9C22×C4×C8C2211yescharacteristic
10#2857—21no—
11#1969—21no—
12#1969C2212yes—
13#1956—21no—
14#1845C411yescharacteristic
15#1797—21no—
16C2×#334—21no—
17#1374—21no—
18#1374C2212yes—
19#1277C2212yes—
20C22×Q16—21no—
21C22×Q16C2×C412yes—
22C23×C8C2×C411yescharacteristic
23C23×C8D411yescharacteristic
24C2×C4⋊Q8—21no—
25C2×C22⋊Q8—22no—
26C22×C42D411yescharacteristic
27C2×M32—24no—
28C2×M32—42no—
29C2×M32—42no—
30C22×C16—22no—
31#494—42no—
32#467—42no—
33#467—42no—
34#467—44no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37C2×C4⋊C8—22no—
38C2×#103—22no—
39C2×#103—22no—
40C2×C22⋊C8—41no—
41C2×C4×C8—42no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#391—22no—
47#389—41no—
48#389—41no—
49#384—21no—
50#380—41no—
51#380—41no—
52#336—41no—
53#334—42no—
54C22×Q8—22no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4D811yescharacteristic
57C23×C4SD1611yescharacteristic
58C2×Q16—24no—
59C2×Q16—42no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8D812yes—
73C22×C8SD1612yes—
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C22⋊Q8—44no—
79C22⋊Q8—44no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C42—41no—
88C2×C42—41no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42D811yescharacteristic
91C2×C42SD1611yescharacteristic
92C2×C42C2×D411yescharacteristic
93M32—48no—
94C2×C16—24no—
95C2×C16—42no—
96C2×C16—42no—
97#107—42no—
98#107—42no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101#103—44no—
102#103—42no—
103#103—42no—
104C22⋊C8—82no—
105C4×C8—22no—
106C4×C8—22no—
107C4×C8—41no—
108C4×C8—41no—
109#95—42no—
110#95—42no—
111C24#10211yescharacteristic
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#10012yes—
131C22×C4#10211yescharacteristic
132C22×C4#10212yes—
133C22×C4#10211yescharacteristic
134C22×C4#10211yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C2×D811yescharacteristic
137C22×C4C2×SD1611yescharacteristic
138Q16—84no—
139Q16—84no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—44no—
150C2×C8—82no—
151C2×C8—82no—
152C2×C8—82no—
153C2×C8#9912yes—
154C2×C8#10212yes—
155C2×C8#10412yes—
156C2×C8#13112yes—
157C2×C8C2×D811yescharacteristic
158C2×C8C2×SD1611yescharacteristic
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C4⋊C4—84no—
165C4⋊C4—82no—
166C22⋊C4—84no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—41no—
172C42—41no—
173C16—44no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23#33112yes—
181C23C2×#10211yescharacteristic
182Q8—82no—
183Q8—82no—
184Q8—82no—
185Q8—82no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#33112yes—
210C2×C4#36012yes—
211C2×C4#36212yes—
212C2×C4#40811yescharacteristic
213C2×C4C2×#9911yescharacteristic
214C2×C4C2×#10011yescharacteristic
215C2×C4C2×#10211yescharacteristic
216C2×C4C2×#10411yescharacteristic
217C2×C4#42111yescharacteristic
218C2×C4#51012yes—
219C8—22no—
220C8—21no—
221C8—21no—
222C8—41no—
223C8—162no—
224C8#32714yes—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22#149211yescharacteristic
233C22#151911yescharacteristic
234C22C2×#33111yescharacteristic
235C22#214111yescharacteristic
236C22#214311yescharacteristic
237C22C2×#36011yescharacteristic
238C22C2×#36211yescharacteristic
239C4—22no—
240C4—22no—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—162no—
245C4—162no—
246C4#147111yescharacteristic
247C4#148011yescharacteristic
248C4C2×#32711yescharacteristic
249C4#153011yescharacteristic
250C2—42no—
251C2#766411yescharacteristic
252C2#1167611yescharacteristic
253C2#1168111yescharacteristic
254C2#1188911yescharacteristic
255C2#1189611yescharacteristic
256C2#1192511yescharacteristic
257C2#1193211yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7113C2semidirect

Maximal subgroups

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

subgroupindex
#331722
#125442
#85952
#71132