Finite Group Databasebeta

← #194921

Subgroups — #194921

Order 512.

1156 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 258 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
1C2×#2634C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#21838C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9026C214yesmaximal
6#9022C214yesmaximal
7#8018C212yesmaximal
8C22×#125C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2634—24no—
12#2634C418yescomplemented
13C2×#400—21no—
14C2×#400C2212yes—
15C2×#398C2212yes—
16C2×#395—21no—
17#2356—24no—
18#2355—24no—
19#2313C2214yes—
20#2310—21no—
21C2×#387—22no—
22#2285C2212yes—
23#2283—22no—
24C2×#383C2211yescharacteristic
25#2274—21no—
26#1992—24no—
27#1989—24no—
28#1824C2214yes—
29#1823C2214yes—
30#1467—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C2×#125—22no—
35C2×#125—22no—
36C2×#125—22no—
37C2×#125—24no—
38C2×#125C2×C414yes—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8C2×C414yes—
43C2×C4×Q8D414yes—
44C2×#117—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C42D412yes—
53C22×C42C2311yescharacteristic
54#432—24no—
55#432—24no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8C2314yes—
58C2×#97—22no—
59#400—28no—
60#400—24no—
61#400C2×C418yes—
62#398—24no—
63#398C2×C418yes—
64#395—28no—
65#391—28no—
66#387—216no—
67#387—216no—
68#384D414yes—
69#383—24no—
70#383—22no—
71#383C2×C414yes—
72#382—24no—
73#382—22no—
74#381—24no—
75#380—22no—
76#380C2314yes—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—28no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4D414yes—
81C4×C22⋊C4—24no—
82#376—24no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95C2312yes—
89#340—28no—
90#340—28no—
91#324—28no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—44no—
95C22×Q8C22⋊C414yes—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2411yescharacteristic
104C22×C8—24no—
105#125—28no—
106#125—28no—
107#125—44no—
108#125—44no—
109C4×Q8—28no—
110C4×Q8—28no—
111C4×Q8—28no—
112C4×Q8—28no—
113#117—48no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—48no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C22⋊C414yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C412yes—
138C2×C4⋊C4C22×C414yes—
139C2×C4⋊C4C2×D414yes—
140C2×C4⋊C4C2×D418yes—
141C2×C22⋊C4—24no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—28no—
147C2×C42C22⋊C418yes—
148C2×C42C22×C412yes—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42#4114yes—
153C4⋊C8—44no—
154C22⋊C8—24no—
155C22⋊C8—24no—
156#97—44no—
157#95—28no—
158#95—28no—
159#95—28no—
160#95—24no—
161#95—24no—
162#95—28no—
163#95—28no—
164#95—28no—
165#95—24no—
166#95—22no—
167#95—24no—
168#95C22×C414yes—
169C24C22×D411yescharacteristic
170C2×Q8—28no—
171C2×Q8—48no—
172C2×Q8—48no—
173C2×Q8—44no—
174C2×Q8—44no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—28no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4C2×C22⋊C414yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C414yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4#12011yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12012yes—
205C22×C4#12312yes—
206C22×C4#12312yes—
207C22×C4C23×C411yescharacteristic
208C22×C4C22×D412yes—
209C22×C4C22×D411yescharacteristic
210C22×C4C2×#4112yes—
211C2×C8—48no—
212C2×C8—44no—
213C4⋊C4—28no—
214C4⋊C4—28no—
215C4⋊C4—28no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C4⋊C4—416no—
221C4⋊C4—48no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—48no—
227C22⋊C4—48no—
228C42—28no—
229C42—28no—
230C42—48no—
231C42—88no—
232C42C4×D414yes—
233C42C4×D414yes—
234C42#12114yes—
235C42#12114yes—
236C23—22no—
237C23—22no—
238C23C22×C22⋊C411yescharacteristic
239C23C2×#12011yescharacteristic
240C23C2×#12311yescharacteristic
241C23#57514yes—
242Q8—416no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—88no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4#38612yes—
280C2×C4#38612yes—
281C2×C4#38614yes—
282C2×C4C2×C4×D412yes—
283C2×C4C2×#12112yes—
284C8—84no—
285C22—28no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22C2×#38611yescharacteristic
291C22#289612yes—
292C22#300512yes—
293C22#302012yes—
294C22#302812yes—
295C22#309312yes—
296C4—24no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—22no—
301C4—44no—
302C4—44no—
303C4—84no—
304C4—82no—
305C4—88no—
306C4—82no—
307C2—22no—
308C2—22no—
309C2#1836514yes—
310C2#2091011yescharacteristic
311C2#2095411yescharacteristic
312C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2634C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26342
#341872
#218382
C2×#14672
#90262
#90222
#80182