Finite Group Databasebeta

← #135377

Subgroups — #135377

Order 512.

432 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 110 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#20817C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#10820C211yesmaximal, characteristic
4#7258C214yesmaximal
5C22×C4×C8C2211yescharacteristic
6#2285—21no—
7C2×C22⋊C16C2212yes—
8#1845C411yescharacteristic, complemented
9#1845C411yescharacteristic, complemented
10#1834—21no—
11#1808—21no—
12#1808C2212yes—
13C4×#107—21no—
14C4×#107C412yescomplemented
15C2×#340—21no—
16C2×#340C2212yes—
17C4⋊#95—21no—
18#1374—21no—
19#1374—21no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#117—41no—
23C22×C4⋊C4—22no—
24C22×C42D411yescharacteristic
25C22×C16—22no—
26C2×#107—21no—
27C2×#107—21no—
28C2×#107C2×C412yes—
29C2×#107C2×C412yes—
30C2×C4×C8—24no—
31C2×C4×C8C2×C411yescharacteristic
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35#384—21no—
36#384—21no—
37#382—41no—
38#380—22no—
39#380—41no—
40C4×C4⋊C4—22no—
41C4×C4⋊C4—41no—
42C2×#95—41no—
43C2×#95—41no—
44C22⋊C16—24no—
45#340—24no—
46#340—22no—
47#340—22no—
48#340—24no—
49C23×C4—42no—
50C23×C4D811yescharacteristic
51C23×C4D811yescharacteristic
52C23×C4C2×D411yescharacteristic
53C22×C8—22no—
54C22×C8—22no—
55C22×C8—41no—
56C22×C8—41no—
57C22×C8—41no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22×C411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C22×C8#4114yes—
63#117—84no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C22⋊C4—42no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—42no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22⋊C411yescharacteristic
82C2×C42SD1611yescharacteristic
83C2×C42SD1611yescharacteristic
84C2×C16—42no—
85C2×C16—42no—
86C2×C16—42no—
87#107—42no—
88#107—42no—
89#107—42no—
90#107—42no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—22no—
95#95—42no—
96#95—42no—
97#95—82no—
98#95—82no—
99C24C2×D811yescharacteristic
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—41no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—82no—
113C22×C4—82no—
114C22×C4—82no—
115C22×C4—82no—
116C22×C4—82no—
117C22×C4—82no—
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4#10211yescharacteristic
122C22×C4SD3212yes—
123C22×C4Q3212yes—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4#12312yes—
126C22×C4C2×SD1611yescharacteristic
127C22×C4#13712yes—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—44no—
138C2×C8—44no—
139C2×C8#10412yes—
140C2×C8#11712yes—
141C2×C8#12312yes—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C4⋊C4—82no—
154C22⋊C4—84no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—22no—
160C42—22no—
161C42—84no—
162C16—82no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23C2×#10211yescharacteristic
168C23#48012yes—
169C23C2×SD3211yescharacteristic
170C23C2×Q3211yescharacteristic
171C23#50912yes—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—44no—
190C2×C4—84no—
191C2×C4—84no—
192C2×C4—84no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—162no—
200C2×C4—162no—
201C2×C4—162no—
202C2×C4—162no—
203C2×C4#35812yes—
204C2×C4#35812yes—
205C2×C4#36012yes—
206C2×C4#38111yescharacteristic
207C2×C4#41011yescharacteristic
208C2×C4#41011yescharacteristic
209C2×C4#41911yescharacteristic
210C2×C4C2×#10411yescharacteristic
211C2×C4#42111yescharacteristic
212C2×C4#48312yes—
213C8—22no—
214C8—22no—
215C8—22no—
216C8—41no—
217C22—22no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—22no—
222C22—21no—
223C22—21no—
224C22—22no—
225C22#150211yescharacteristic
226C22#151711yescharacteristic
227C22#182311yescharacteristic
228C22C2×#35811yescharacteristic
229C22#214011yescharacteristic
230C22C2×#36011yescharacteristic
231C22#214511yescharacteristic
232C22#223312yes—
233C22#223612yes—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—41no—
239C4—164no—
240C4—162no—
241C4—162no—
242C4#150611yescharacteristic
243C4#151511yescharacteristic
244C4#182011yescharacteristic
245C4#182611yescharacteristic
246C2—22no—
247C2—22no—
248C2#871611yescharacteristic
249C2#1174311yescharacteristic
250C2#1174411yescharacteristic
251C2#1185711yescharacteristic
252C2#1185811yescharacteristic
253C2#1286711yescharacteristic
254C2#1287011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #1845C4semidirect
9. #1845C4semidirect
14. C4×#107C4semidirect

Maximal subgroups

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

subgroupindex
#208172
#128442
#108202
#72582