Finite Group Databasebeta

← #135375

Subgroups — #135375

Order 512.

410 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 102 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#20818C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#10820C211yesmaximal, characteristic
4#7258C212yesmaximal
5#7258C212yesmaximal
6C22×C4×C8C2211yescharacteristic
7#2283—21no—
8C2×C22⋊C16C2212yes—
9#1846C411yescharacteristic, complemented
10#1846C411yescharacteristic, complemented
11#1835—21no—
12#1808C2211yescharacteristic
13#1808C2211yescharacteristic
14#1807—21no—
15C4×#107C411yescharacteristic, complemented
16C4×#107C411yescharacteristic, complemented
17C4×#106—21no—
18C2×#340—21no—
19C2×#340C2212yes—
20C4⋊#95—21no—
21#1374—21no—
22#1374—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C22×C4⋊C4—22no—
26C22×C42D411yescharacteristic
27C22×C16—22no—
28C2×#107C2×C411yescharacteristic
29C2×#107C2×C411yescharacteristic
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#106—21no—
33C2×#106—21no—
34C2×C4×C8—24no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#383—41no—
40#382—21no—
41#382—21no—
42#381—41no—
43#380—21no—
44#380—21no—
45C4×C4⋊C4—21no—
46C4×C4⋊C4—21no—
47C4×C22⋊C4—41no—
48#376—41no—
49C2×#95—41no—
50C2×#95—41no—
51C22⋊C16—24no—
52#340—24no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—24no—
58C23×C4—42no—
59C23×C4D811yescharacteristic
60C23×C4D811yescharacteristic
61C23×C4C2×D411yescharacteristic
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22×C411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C22×C8#4114yes—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—41no—
87C2×C42—41no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42SD1611yescharacteristic
91C2×C42SD1611yescharacteristic
92C2×C16—42no—
93C2×C16—42no—
94C2×C16—42no—
95#107—41no—
96#107—41no—
97#107—41no—
98#107—41no—
99#106—42no—
100#106—42no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—22no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—82no—
114#95—82no—
115C24C2×D811yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—44no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—82no—
133C22×C4—82no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4#12312yes—
144C22×C4C2×SD1611yescharacteristic
145C22×C4#13712yes—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8#10412yes—
158C2×C8#11712yes—
159C2×C8#12312yes—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C22⋊C4—82no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—22no—
180C42—22no—
181C42—82no—
182C42—82no—
183C16—82no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23C2×#10211yescharacteristic
189C23#48012yes—
190C23#50812yes—
191C23#50911yescharacteristic
192C23#51011yescharacteristic
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
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—42no—
212C2×C4—42no—
213C2×C4—84no—
214C2×C4—84no—
215C2×C4—84no—
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—162no—
229C2×C4—162no—
230C2×C4—162no—
231C2×C4—162no—
232C2×C4#36012yes—
233C2×C4#38111yescharacteristic
234C2×C4#41011yescharacteristic
235C2×C4#41011yescharacteristic
236C2×C4#41911yescharacteristic
237C2×C4C2×#10411yescharacteristic
238C2×C4#42111yescharacteristic
239C2×C4#48312yes—
240C8—22no—
241C8—22no—
242C8—22no—
243C8—41no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—21no—
251C22—21no—
252C22#150211yescharacteristic
253C22#151711yescharacteristic
254C22#182311yescharacteristic
255C22#213811yescharacteristic
256C22#213911yescharacteristic
257C22C2×#36011yescharacteristic
258C22#214511yescharacteristic
259C22#223412yes—
260C22#223712yes—
261C4—22no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—164no—
267C4—161no—
268C4—161no—
269C4—161no—
270C4—161no—
271C4#150611yescharacteristic
272C4#151511yescharacteristic
273C4#182011yescharacteristic
274C4#182611yescharacteristic
275C2—22no—
276C2—22no—
277C2#871611yescharacteristic
278C2#1174111yescharacteristic
279C2#1174611yescharacteristic
280C2#1185511yescharacteristic
281C2#1186011yescharacteristic
282C2#1286811yescharacteristic
283C2#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
9. #1846C4semidirect
10. #1846C4semidirect
15. C4×#107C4semidirect
16. C4×#107C4semidirect

Maximal subgroups

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

subgroupindex
#208182
#128442
#108202
#72582