Finite Group Databasebeta

← #125326

Subgroups — #125326

Order 512.

426 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 94 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#12599C211yesmaximal, characteristic
3#7875C211yesmaximal, characteristic
4#7113C214yesmaximal, complemented
5C2×#494C411yescharacteristic
6C2×#467—21no—
7C2×#467C412yes—
8C2×#432—21no—
9C22×C4×C8C2211yescharacteristic
10#2857—21no—
11C2×C22⋊C16—21no—
12C2×#346—21no—
13#1969—21no—
14#1969C2212yes—
15#1956—21no—
16#1845C411yescharacteristic
17#1374C2212yes—
18#1277—21no—
19#1277C2212yes—
20C22×Q16—21no—
21C22×Q16C2×C412yes—
22C23×C8C2×C411yescharacteristic
23C23×C8D411yescharacteristic
24C2×C4⋊Q8—21no—
25C2×C22⋊Q8—22no—
26C22×C42D411yescharacteristic
27C22×C16—22no—
28#494—42no—
29#467—44no—
30#467—44no—
31#432—24no—
32#432—42no—
33C2×#107—21no—
34C2×#107C2×C412yes—
35C2×C4⋊C8—22no—
36C2×#103—22no—
37C2×#103—22no—
38C2×C22⋊C8—41no—
39C2×#97—41no—
40C2×C4×C8—42no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#391—22no—
46#389—41no—
47#389—41no—
48#384—21no—
49#380—41no—
50#380—41no—
51C22⋊C16—24no—
52#346—42no—
53C22×Q8—22no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4D811yescharacteristic
56C23×C4SD1611yescharacteristic
57C2×Q16—24no—
58C2×Q16—42no—
59C2×Q16—42no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—42no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76C22⋊Q8—44no—
77C22⋊Q8—44no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C22⋊C4—42no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42D811yescharacteristic
89C2×C42SD1611yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C16—24no—
92C2×C16—42no—
93C2×C16—42no—
94#107—42no—
95#107—42no—
96C4⋊C8—24no—
97C4⋊C8—42no—
98#103—44no—
99#103—44no—
100C22⋊C8—44no—
101#97—44no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—41no—
105C4×C8—41no—
106#95—42no—
107#95—42no—
108C24#10211yescharacteristic
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—42no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4#10211yescharacteristic
130C22×C4#10211yescharacteristic
131C22×C4#10211yescharacteristic
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C2×D811yescharacteristic
134C22×C4C2×SD1611yescharacteristic
135Q16—84no—
136Q16—84no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—44no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8#10012yes—
151C2×C8#10412yes—
152C2×C8#13112yes—
153C2×C8AGL(1,ℤ/8)12yes—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—84no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C22⋊C4—84no—
162C42—22no—
163C42—22no—
164C42—21no—
165C42—21no—
166C42#9912yes—
167C42#10412yes—
168C16—44no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23#36212yes—
176C23C2×#10211yescharacteristic
177Q8—82no—
178Q8—82no—
179Q8—82no—
180Q8—82no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#32712yes—
206C2×C4#32712yes—
207C2×C4#40811yescharacteristic
208C2×C4C2×#9911yescharacteristic
209C2×C4C2×#10011yescharacteristic
210C2×C4#41811yescharacteristic
211C2×C4C2×#10411yescharacteristic
212C2×C4C2×#10411yescharacteristic
213C2×C4#51012yes—
214C8—22no—
215C8—22no—
216C8—22no—
217C8—22no—
218C8—84no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22#149211yescharacteristic
227C22C2×#32711yescharacteristic
228C22#154511yescharacteristic
229C22#214111yescharacteristic
230C22#214311yescharacteristic
231C22#214511yescharacteristic
232C22C2×#36211yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—162no—
239C4—162no—
240C4#147111yescharacteristic
241C4#148411yescharacteristic
242C4#150711yescharacteristic
243C4#152511yescharacteristic
244C2—42no—
245C2#768411yescharacteristic
246C2#1167611yescharacteristic
247C2#1167911yescharacteristic
248C2#1170611yescharacteristic
249C2#1171211yescharacteristic
250C2#1196011yescharacteristic
251C2#1196511yescharacteristic
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
#125992
#78752
#71132