Finite Group Databasebeta

← #135503

Subgroups — #135503

Order 512.

259 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 58 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#21434C211yesmaximal, characteristic
2#8745C211yesmaximal, characteristic
3#8741C211yesmaximal, characteristic
4#7216C211yesmaximal, characteristic
5#7216C211yesmaximal, characteristic
6#7214C211yesmaximal, characteristic
7#7214C211yesmaximal, characteristic
8#2298C2211yescharacteristic
9#1924—21no—
10#1923—21no—
11#1917—21no—
12#1916—21no—
13#1822C2211yescharacteristic
14#1820C2211yescharacteristic
15#1768C411yescharacteristic, complemented
16#1768C411yescharacteristic, complemented
17#1750C2211yescharacteristic
18#1750C2211yescharacteristic
19#1744C411yescharacteristic, complemented
20C4×#97C411yescharacteristic, complemented
21#1464C2211yescharacteristic
22#1464C2211yescharacteristic
23#1285—21no—
24#1285—21no—
25#1284—21no—
26#1284—21no—
27C2×#117C2311yescharacteristic
28#432—21no—
29#432—21no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32#407—21no—
33C2×C22⋊C8—21no—
34C2×C22⋊C8—21no—
35C2×#97—21no—
36C2×#97C2×C411yescharacteristic
37C2×#97C2×C411yescharacteristic
38C2×C4×C8—21no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41#388—21no—
42#388—21no—
43#387—21no—
44#387—21no—
45#381—21no—
46C4×C4⋊C4D411yescharacteristic
47C4×C4⋊C4D411yescharacteristic
48C4×C22⋊C4D411yescharacteristic
49C4×C22⋊C4D411yescharacteristic
50#376C2×C411yescharacteristic
51C43C2×C411yescharacteristic
52#343—21no—
53#343—21no—
54#342—21no—
55#342—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#324—21no—
63#324—21no—
64#323—21no—
65#323—21no—
66C23×C4C2×D411yescharacteristic
67C2×M16—41no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72#117—41no—
73#117—41no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4#4111yescharacteristic
77C2×C4⋊C4#4111yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4#4111yescharacteristic
82C2×C22⋊C4#4111yescharacteristic
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22×C411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C4⋊C8—41no—
92C4⋊C8—41no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C22⋊C8—42no—
96C22⋊C8—42no—
97C22⋊C8—41no—
98C22⋊C8—41no—
99C22⋊C8—41no—
100C22⋊C8—41no—
101#97—41no—
102#97—41no—
103#97—41no—
104#97—41no—
105#97—42no—
106C4×C8—41no—
107C4×C8—41no—
108C4×C8—42no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—41no—
116#95—41no—
117C24#12311yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—81no—
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4#11711yescharacteristic
129C22×C4#11711yescharacteristic
130C22×C4#12311yescharacteristic
131C22×C4#12311yescharacteristic
132C22×C4#12311yescharacteristic
133M16—81no—
134M16—81no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—81no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—42no—
153C4⋊C4—81no—
154C22⋊C4—41no—
155C22⋊C4—41no—
156C22⋊C4—42no—
157C22⋊C4—81no—
158C22⋊C4—81no—
159C22⋊C4—81no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—41no—
165C42—41no—
166C42—41no—
167C42—41no—
168C42—42no—
169C42—81no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23#38111yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—161no—
204C2×C4—161no—
205C2×C4—161no—
206C2×C4#41011yescharacteristic
207C2×C4#41011yescharacteristic
208C8—42no—
209C8—42no—
210C8—81no—
211C8—81no—
212C8—161no—
213C8—161no—
214C8—161no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#182011yescharacteristic
223C22#182711yescharacteristic
224C22#182711yescharacteristic
225C4—21no—
226C4—21no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—81no—
234C4—81no—
235C4—81no—
236C4—321no—
237C2—21no—
238C2—21no—
239C2—81no—
240C2#874011yescharacteristic
241C2#1286411yescharacteristic
242C2#1286511yescharacteristic
subgroupG/H
center, socleC22#1820
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2298 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1768C4semidirect
16. #1768C4semidirect
19. #1744C4semidirect
20. C4×#97C4semidirect

Maximal subgroups

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

subgroupindex
#214342
#87452
#87412
#72162
#72142