Finite Group Databasebeta

← #194792

Subgroups — #194792

Order 512.

1312 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1312 conjugacy classes of subgroups — too many to draw. Only its 266 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×#2399C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#20740C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9026C214yesmaximal
6#9014C214yesmaximal
7#8022C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2399—24no—
12#2399C418yescomplemented
13C2×#398—21no—
14C2×#398C2212yes—
15C2×#395—21no—
16C2×#395C2212yes—
17#2356—24no—
18#2355—24no—
19#2313C2214yes—
20#2310—21no—
21#2285C2212yes—
22#2283—22no—
23#2274—21no—
24C2×C4×C4⋊C4—22no—
25C2×#376C2211yescharacteristic
26#1823C2214yes—
27#1803—24no—
28#1787C2214yes—
29#1763—24no—
30#1467—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C23×Q8C2311yescharacteristic
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—24no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8C2×C414yes—
43C2×C4×Q8C2×C414yes—
44C2×C4×Q8D414yes—
45C2×#117—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#432—24no—
57#432—24no—
58C2×C4⋊C8—22no—
59C2×#103—28no—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62#398—28no—
63#398—24no—
64#398C2×C418yes—
65#395—28no—
66#395—24no—
67#395C2×C418yes—
68#391—28no—
69#387—216no—
70#384D414yes—
71#383—24no—
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—216no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4D414yes—
82C4×C22⋊C4—24no—
83#376—24no—
84#376—22no—
85#376C2×C414yes—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95C2312yes—
90#340—28no—
91#324—28no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—21no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—44no—
98C22×Q8—44no—
99C22×Q8C22⋊C414yes—
100C22×Q8C22×C412yes—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2411yescharacteristic
109C22×C8—24no—
110C4×Q8—28no—
111C4×Q8—28no—
112C4×Q8—28no—
113C4×Q8—28no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—44no—
117C4×Q8—44no—
118#117—48no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—48no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C22⋊C414yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2×D418yes—
140C2×C22⋊C4—24no—
141C2×C42—28no—
142C2×C42—24no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—24no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—28no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—42no—
152C2×C42C22⋊C418yes—
153C2×C42C22×C412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42#4114yes—
158C4⋊C8—44no—
159#103—416no—
160C22⋊C8—24no—
161C22⋊C8—24no—
162#97—44no—
163#95—28no—
164#95—28no—
165#95—28no—
166#95—24no—
167#95—28no—
168#95—28no—
169#95—24no—
170#95—22no—
171#95—24no—
172#95C22×C414yes—
173C24C22×D411yescharacteristic
174C2×Q8—28no—
175C2×Q8—28no—
176C2×Q8—24no—
177C2×Q8—48no—
178C2×Q8—44no—
179C2×Q8—48no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—48no—
185C2×Q8—42no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—28no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—28no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—44no—
210C22×C4C2×C22⋊C414yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C414yes—
214C22×C4#12011yescharacteristic
215C22×C4#12012yes—
216C22×C4#12011yescharacteristic
217C22×C4#12312yes—
218C22×C4#12312yes—
219C22×C4C23×C411yescharacteristic
220C22×C4C22×D412yes—
221C22×C4C22×D411yescharacteristic
222C22×C4C2×#4112yes—
223C2×C8—48no—
224C2×C8—44no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—28no—
228C4⋊C4—24no—
229C4⋊C4—48no—
230C4⋊C4—416no—
231C4⋊C4—416no—
232C4⋊C4—48no—
233C4⋊C4—42no—
234C4⋊C4—48no—
235C4⋊C4—48no—
236C22⋊C4—48no—
237C42—216no—
238C42—28no—
239C42—28no—
240C42—48no—
241C42—44no—
242C42—44no—
243C42—88no—
244C42C4×D414yes—
245C42C4×D414yes—
246C42#12114yes—
247C42#12114yes—
248C23—22no—
249C23—22no—
250C23C22×C22⋊C411yescharacteristic
251C23C2×#12011yescharacteristic
252C23C2×#12311yescharacteristic
253C23#57514yes—
254Q8—28no—
255Q8—416no—
256Q8—48no—
257Q8—44no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—416no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4#38612yes—
295C2×C4#38614yes—
296C2×C4#38612yes—
297C2×C4#45614yes—
298C2×C4C2×C4×D412yes—
299C2×C4C2×#12112yes—
300C8—84no—
301C22—28no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22C2×#38611yescharacteristic
307C22#289612yes—
308C22#300512yes—
309C22C2×#45612yes—
310C22#302512yes—
311C22#309312yes—
312C4—24no—
313C4—22no—
314C4—22no—
315C4—22no—
316C4—48no—
317C4—44no—
318C4—44no—
319C4—44no—
320C4—88no—
321C4—82no—
322C4#188614yes—
323C2—22no—
324C2—22no—
325C2#1818314yes—
326C2#2091011yescharacteristic
327C2C2×#188611yescharacteristic
328C2#2095911yescharacteristic
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. #2399C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
#341872
#207402
C2×#14672
#90262
#90142
#80222