Finite Group Databasebeta

← #107283

Subgroups — #107283

Order 512.

862 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 190 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#34154C211yesmaximal, characteristic
2#21329C211yesmaximal, characteristic
3#17739C211yesmaximal, characteristic
4#8097C212yesmaximal
5#8048C212yesmaximal
6C2×#533C2211yescharacteristic
7#2920C2212yes—
8C2×#423C2211yescharacteristic
9#2371C414yescomplemented
10#2365—22no—
11#2362—24no—
12#2298—22no—
13#2295—22no—
14C42×Q8—22no—
15C4×#117C2211yescharacteristic
16C4×#117C2212yes—
17#1995C414yescomplemented
18#1994—22no—
19#1733—22no—
20#1561C414yescomplemented
21#1556—22no—
22#1544—22no—
23#1303—22no—
24#1303—22no—
25#1297—24no—
26C22×M16—21no—
27#533—22no—
28#533—22no—
29#533C2×C412yes—
30#533C2×C412yes—
31#533D412yes—
32#533Q812yes—
33C2×C4⋊Q8D412yes—
34C2×#125C2×C412yes—
35C2×C22⋊Q8—22no—
36C2×C4×Q8—24no—
37C2×C4×Q8—22no—
38C2×#117—22no—
39C2×#117—21no—
40C2×#117D412yes—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#431—22no—
45#431—22no—
46#423—22no—
47#423—21no—
48#423C2×C414yes—
49#423C2×C412yes—
50C2×C4⋊C8—22no—
51#407—22no—
52C4×M16—22no—
53#391—24no—
54#389—24no—
55#388—28no—
56#387—28no—
57#387—28no—
58C4×C4⋊C4—24no—
59C4×C4⋊C4—24no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—28no—
63C4×C4⋊C4D412yes—
64C4×C4⋊C4D412yes—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4D414yes—
68#376—22no—
69#376C2×C412yes—
70#376C2×C414yes—
71C43—22no—
72C43—22no—
73C43—21no—
74#339—28no—
75#332—28no—
76C22×Q8—22no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C2×M16—22no—
80C2×M16—24no—
81C2×M16—41no—
82C2×M16—44no—
83C22×C8—41no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8C22⋊C414yes—
87C4⋊Q8C4⋊C414yes—
88#125—24no—
89#125C4214yes—
90#125C4⋊C414yes—
91C22⋊Q8—24no—
92C22⋊Q8—24no—
93C22⋊Q8—24no—
94C22⋊Q8—24no—
95C4×Q8—24no—
96C4×Q8—24no—
97C4×Q8—24no—
98C4×Q8—24no—
99C4×Q8—416no—
100#117—22no—
101#117—22no—
102#117—24no—
103#117—48no—
104#117—42no—
105#117C22⋊C414yes—
106#117C4⋊C414yes—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C22⋊C414yes—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—41no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—28no—
120C2×C42—24no—
121C2×C42—24no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—42no—
125C2×C42C22×C411yescharacteristic
126C2×C42C22×C411yescharacteristic
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2×Q811yescharacteristic
132C2×C42#4112yes—
133C2×C42#4112yes—
134C4⋊C8—24no—
135C4⋊C8—24no—
136C4⋊C8—24no—
137C4⋊C8—42no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140#97—42no—
141C4×C8—42no—
142#95—28no—
143#95—28no—
144#95—24no—
145#95—24no—
146#95C22⋊C414yes—
147#95C22⋊C414yes—
148C24#12011yescharacteristic
149C2×Q8—24no—
150C2×Q8—24no—
151C2×Q8—48no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4C2×C22⋊C412yes—
162C22×C4C2×C4⋊C411yescharacteristic
163C22×C4C4×D412yes—
164C22×C4C4×D412yes—
165C22×C4#12012yes—
166C22×C4#12011yescharacteristic
167C22×C4#12112yes—
168C22×C4C22⋊Q812yes—
169C22×C4#12311yescharacteristic
170C22×C4#12311yescharacteristic
171M16—44no—
172M16—82no—
173M16—84no—
174C2×C8—44no—
175C2×C8—42no—
176C2×C8—82no—
177C4⋊C4—28no—
178C4⋊C4—28no—
179C4⋊C4—28no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—44no—
184C4⋊C4—416no—
185C4⋊C4—44no—
186C4⋊C4#9518yes—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—48no—
190C22⋊C4—44no—
191C42—24no—
192C42—24no—
193C42—24no—
194C42—42no—
195C42—48no—
196C42—48no—
197C42—416no—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—84no—
203C42C2×C4211yescharacteristic
204C42C2×C22⋊C411yescharacteristic
205C42C2×C4⋊C411yescharacteristic
206C42C2×C4⋊C411yescharacteristic
207C42#11712yes—
208C42C4×D412yes—
209C42C4×D412yes—
210C42#12112yes—
211C42C22⋊Q812yes—
212C42#12312yes—
213C23—22no—
214C23—42no—
215C23#37911yescharacteristic
216C23#38511yescharacteristic
217C23#38611yescharacteristic
218Q8—48no—
219Q8—48no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—28no—
231C2×C4—28no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—84no—
243C2×C4C2×#9511yescharacteristic
244C2×C4C4×C22⋊C412yes—
245C2×C4#38012yes—
246C2×C4#38111yescharacteristic
247C2×C4#38512yes—
248C2×C4#38511yescharacteristic
249C2×C4#38612yes—
250C2×C4#38611yescharacteristic
251C2×C4#45412yes—
252C2×C4#45612yes—
253C8—82no—
254C8—84no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—21no—
259C22—21no—
260C22—82no—
261C22C22⋊#9511yescharacteristic
262C22#179312yes—
263C4—22no—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—48no—
269C4—48no—
270C4—84no—
271C4—84no—
272C4#185712yes—
273C4#188611yescharacteristic
274C4#188811yescharacteristic
275C4#190614yes—
276C2—22no—
277C2—41no—
278C2—41no—
279C2#1075011yescharacteristic
280C2#1077212yes—
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2371C4semidirect
17. #1995C4semidirect
20. #1561C4semidirect

Maximal subgroups

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

subgroupindex
#341542
#213292
#177392
#80972
#80482