Finite Group Databasebeta

← #125514

Subgroups — #125514

Order 512.

344 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 86 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#33130C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3#8595C211yesmaximal, characteristic
4#7120C212yesmaximal, complemented
5#7113C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2855—21no—
9C2×C4⋊C16—21no—
10C2×C22⋊C16—21no—
11#1969C411yescharacteristic
12#1969C2211yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1933—21no—
16#1933—21no—
17#1892—21no—
18#1808C411yescharacteristic
19#1797—21no—
20C2×#334—21no—
21#1374C2212yes—
22#1277C2212yes—
23C22×Q16C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C22⋊Q8—21no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#467—42no—
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×C4⋊C8—22no—
34C2×#103—21no—
35C2×#103—21no—
36C2×#103—21no—
37C2×C22⋊C8—41no—
38C2×C4×C8—42no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8C2311yescharacteristic
43#391—21no—
44#391—21no—
45#389—21no—
46#389—21no—
47#387—41no—
48#387—41no—
49#385—41no—
50#381—41no—
51#380—21no—
52C4⋊C16—42no—
53C22⋊C16—42no—
54#340—21no—
55#340—21no—
56#340—21no—
57#336—41no—
58#334—42no—
59C22×Q8—21no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4D811yescharacteristic
62C23×C4SD1611yescharacteristic
63C2×Q16—22no—
64C2×Q16—41no—
65C2×Q16—41no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22⋊Q8—42no—
79C22⋊Q8—42no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C42—41no—
89C2×C42—41no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42D811yescharacteristic
92C2×C42SD1611yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C16—42no—
95C2×C16—42no—
96C2×C16—42no—
97#107—22no—
98#107—41no—
99#107—41no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102#103—42no—
103#103—42no—
104#103—41no—
105#103—41no—
106C22⋊C8—82no—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—41no—
110C4×C8—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118C24#10211yescharacteristic
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4#10211yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C2×D811yescharacteristic
146C22×C4C2×SD1611yescharacteristic
147Q16—82no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8#9912yes—
164C2×C8#13112yes—
165C2×C8AGL(1,ℤ/8)11yescharacteristic
166C2×C8#13711yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—41no—
183C42—41no—
184C16—82no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23#36212yes—
192C23C2×#10211yescharacteristic
193Q8—81no—
194Q8—81no—
195Q8—81no—
196Q8—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4#36012yes—
229C2×C4#40811yescharacteristic
230C2×C4C2×#9911yescharacteristic
231C2×C4#41311yescharacteristic
232C2×C4#41711yescharacteristic
233C2×C4#42111yescharacteristic
234C2×C4#42111yescharacteristic
235C2×C4#50812yes—
236C8—22no—
237C8—22no—
238C8—21no—
239C8—21no—
240C8—41no—
241C8—162no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22#148011yescharacteristic
250C22#151911yescharacteristic
251C22#154711yescharacteristic
252C22#213811yescharacteristic
253C22#214311yescharacteristic
254C22C2×#36011yescharacteristic
255C22C2×#36211yescharacteristic
256C4—22no—
257C4—22no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—161no—
262C4—161no—
263C4—161no—
264C4—161no—
265C4#146811yescharacteristic
266C4#149511yescharacteristic
267C4#151311yescharacteristic
268C4#153511yescharacteristic
269C2—42no—
270C2#771411yescharacteristic
271C2#1164711yescharacteristic
272C2#1165511yescharacteristic
273C2#1188511yescharacteristic
274C2#1189611yescharacteristic
275C2#1198011yescharacteristic
276C2#1198411yescharacteristic
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. #7120C2semidirect
5. #7113C2semidirect

Maximal subgroups

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

subgroupindex
#331302
#127832
#85952
#71202
#71132