Finite Group Databasebeta

← #238160

Subgroups — #238160

Order 512.

924 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 202 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#37642C211yesmaximal, characteristic
2#21834C212yesmaximal
3#21202C211yesmaximal, characteristic
4#10352C214yesmaximal, complemented
5#10349C214yesmaximal, complemented
6#8038C211yesmaximal, characteristic
7#7896C212yesmaximal
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2212yes—
10C2×#419—22no—
11C2×C4×M16C2211yescharacteristic
12#2844—22no—
13#2628—24no—
14#2628—22no—
15#2571—22no—
16#2571C2212yes—
17#2551—22no—
18#2551C2212yes—
19#2285C2212yes—
20#2285C2211yescharacteristic
21#1990—22no—
22#1990—22no—
23#1987C2218yes—
24#1960—22no—
25#1956C2212yes—
26#1956C2212yes—
27#1824—24no—
28#1296C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C23×Q8—21no—
32C22×M16—21no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8Q814yescomplemented
40C2×C4⋊Q8C2312yes—
41C2×#117—22no—
42C2×#117D412yes—
43C22×C4⋊C4—22no—
44C22×C42C2311yescharacteristic
45#431—24no—
46#431—24no—
47C2×C4⋊C8C2314yes—
48#419—48no—
49C2×#103—24no—
50C2×#103—24no—
51C2×C22⋊C8—22no—
52C4×M16—24no—
53C2×#97—21no—
54C2×C4×C8—22no—
55C2×C4×C8C2312yes—
56#400—28no—
57#400—24no—
58#399—24no—
59#395—28no—
60#395—24no—
61#393—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#384D412yes—
66#384D412yes—
67#384C2311yescharacteristic
68#384C2311yescharacteristic
69#382—22no—
70#382—24no—
71#382—22no—
72#382D412yes—
73#380—22no—
74#380—22no—
75#380D414yes—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2314yes—
79#340—28no—
80#340—28no—
81C22×Q8—24no—
82C22×Q8—24no—
83C22×Q8—22no—
84C22×Q8—44no—
85C22×Q8—44no—
86C22×Q8—41no—
87C22×Q8—42no—
88C23×C4—41no—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C2×M16—24no—
92C2×M16—42no—
93C2×M16—42no—
94C22×C8—22no—
95C22×C8—24no—
96C4⋊Q8—28no—
97C4⋊Q8—24no—
98C4⋊Q8—24no—
99C4⋊Q8—22no—
100C4⋊Q8—44no—
101C4⋊Q8—42no—
102C4⋊Q8—88no—
103C4⋊Q8—84no—
104C4⋊Q8C2×Q814yes—
105#117—44no—
106#117—84no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D412yes—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—22no—
125C2×C42—22no—
126C2×C42—24no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×Q812yes—
130C2×C42#4114yes—
131C2×C42#4112yes—
132C2×C42C2411yescharacteristic
133C4⋊C8—24no—
134C4⋊C8—24no—
135#103—48no—
136#103—48no—
137C22⋊C8—44no—
138#97—42no—
139C4×C8—24no—
140C4×C8—44no—
141#95—24no—
142#95—24no—
143#95—28no—
144#95—24no—
145#95—24no—
146#95—24no—
147C24#12011yescharacteristic
148C2×Q8—24no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—416no—
152C2×Q8—42no—
153C2×Q8—44no—
154C2×Q8—88no—
155C2×Q8—88no—
156C2×Q8—84no—
157C2×Q8—82no—
158C2×Q8—82no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4#12012yes—
172C22×C4#12011yescharacteristic
173C22×C4C22⋊Q814yes—
174C22×C4C22⋊Q812yes—
175C22×C4#12414yes—
176C22×C4#12412yes—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D412yes—
179M16—48no—
180C2×C8—24no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—44no—
184C2×C8—44no—
185C2×C8—44no—
186C4⋊C4—28no—
187C4⋊C4—28no—
188C4⋊C4—28no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4—88no—
200C4⋊C4—84no—
201C22⋊C4—82no—
202C22⋊C4—84no—
203C42—48no—
204C42—44no—
205C42C22⋊Q812yes—
206C42C22⋊Q814yes—
207C42#12414yes—
208C42#12412yes—
209C42C22×Q811yescharacteristic
210C42C2×#4112yes—
211C42C2×#4111yescharacteristic
212C23—22no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23#39312yes—
217C23C2×#12011yescharacteristic
218Q8—48no—
219Q8—84no—
220Q8—88no—
221Q8—168no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—28no—
227C2×C4—28no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—88no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#39314yes—
245C2×C4#39312yes—
246C2×C4#45312yes—
247C2×C4#45412yes—
248C2×C4C2×C22⋊Q811yescharacteristic
249C2×C4C2×C22⋊Q812yes—
250C2×C4C2×#12411yescharacteristic
251C2×C4C2×#12412yes—
252C2×C4#57714yes—
253C2×C4#57814yes—
254C8—44no—
255C8—84no—
256C22—24no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22C2×#39311yescharacteristic
262C22#300912yes—
263C22#301112yes—
264C22C2×#45311yescharacteristic
265C22C2×#45411yescharacteristic
266C4—24no—
267C4—21no—
268C4—21no—
269C4—84no—
270C4—82no—
271C4—82no—
272C4—84no—
273C4—84no—
274C4#202714yes—
275C4#308214yes—
276C4#313714yes—
277C2—42no—
278C2#1871511yescharacteristic
279C2#1871911yescharacteristic
280C2#1873012yes—
281C2#2345712yes—
282C2C2×#202711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10352C2semidirect
5. #10349C2semidirect
39. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#218342
#212022
#103522
#103492
#80382
#78962