Finite Group Databasebeta

← #304716

Subgroups — #304716

Order 512.

774 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 166 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#36192C211yesmaximal, characteristic
2#33197C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#29226C211yesmaximal, characteristic, complemented
5#10355C214yesmaximal, complemented
6#9902C211yesmaximal, characteristic, complemented
7#9872C211yesmaximal, characteristic, complemented
8#8976C214yesmaximal
9#8174C211yesmaximal, characteristic
10C2×#476C2212yes—
11#2868C2211yescharacteristic
12#2859C2211yescharacteristic
13#2851—21no—
14#2552C2212yes—
15#2522—24no—
16#2454—22no—
17#2441—22no—
18#2370—22no—
19#2354C2212yes—
20C4×#125C2211yescharacteristic
21#2285—21no—
22C42×D4C2211yescharacteristic
23#2099—22no—
24#2043—24no—
25#1992—21no—
26#1987C2214yes—
27#1987C2212yes—
28#1956C2214yes—
29#1948—22no—
30#1842C2211yescharacteristic
31#1836—21no—
32#1802—22no—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×D4⋊C8C2212yes—
36#1296C2214yes—
37#1279C2214yes—
38#1279C2212yes—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2312yes—
41C2×#125—22no—
42C2×#125—21no—
43C2×#125D411yescharacteristic
44C2×C4×D4D412yes—
45C2×C4×D4C2311yescharacteristic
46C2×#117—22no—
47C22×C42—21no—
48#476—28no—
49C2×#106—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—28no—
54C2×#102—22no—
55C2×#97—21no—
56C2×C4×C8C2312yes—
57#399—24no—
58#398—24no—
59#396—24no—
60#390—22no—
61#389—24no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389D412yes—
66#389D414yes—
67#388—24no—
68#387—24no—
69#387—24no—
70#387—22no—
71#385—24no—
72#384—22no—
73#384—22no—
74#384—21no—
75#384—21no—
76#384C2312yes—
77#382—21no—
78#382D411yescharacteristic
79#380—24no—
80#380—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4C2312yes—
86C4×C4⋊C4C2311yescharacteristic
87C4×C22⋊C4—21no—
88C4×C22⋊C4—22no—
89C43C2311yescharacteristic
90#340—24no—
91#340—24no—
92#340—24no—
93#336—21no—
94#336—22no—
95D4⋊C8—24no—
96C22×Q8—22no—
97C22×D4C2×D411yescharacteristic
98C23×C4—21no—
99C23×C4—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—84no—
106#125—22no—
107#125—22no—
108#125—44no—
109#125—44no—
110#125—82no—
111C4×D4—24no—
112C4×D4—22no—
113C4×D4—42no—
114C4×D4#4114yes—
115#117—84no—
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—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D414yes—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—21no—
140C2×C42—24no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42C2×D412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2411yescharacteristic
148#106—44no—
149C4⋊C8—22no—
150C4⋊C8—24no—
151C4⋊C8—22no—
152C4⋊C8—44no—
153#103—88no—
154#102—44no—
155#97—42no—
156C4×C8—22no—
157C4×C8—22no—
158#95—24no—
159#95—24no—
160#95—22no—
161#95—22no—
162#95—24no—
163#95—22no—
164#95—21no—
165#95—22no—
166C24—21no—
167C2×Q8—48no—
168C2×Q8—44no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4#12412yes—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—44no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—82no—
186C22×C4#12012yes—
187C22×C4#12012yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C2×C8—44no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—44no—
197C2×C8—42no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—24no—
203C4⋊C4—44no—
204C4⋊C4—48no—
205C4⋊C4—48no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4#12412yes—
219C4⋊C4#13714yes—
220C22⋊C4—24no—
221C22⋊C4—44no—
222C22⋊C4—84no—
223C22⋊C4—84no—
224C42—22no—
225C42—22no—
226C42—22no—
227C42—42no—
228C42—41no—
229C42—41no—
230C42—42no—
231C42—42no—
232C42—44no—
233C42—84no—
234C42#13512yes—
235C42#13712yes—
236C42C2×#4112yes—
237C422−1+412yes—
238C23—22no—
239C23—41no—
240C23—42no—
241C23—41no—
242C23—42no—
243C23C2×#12011yescharacteristic
244Q8—88no—
245D4—22no—
246D4—22no—
247D4—42no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—88no—
272C2×C4—82no—
273C2×C4—84no—
274C2×C4#45214yes—
275C2×C4C2×#12411yescharacteristic
276C2×C4#53611yescharacteristic
277C2×C4#54812yes—
278C2×C4#54812yes—
279C2×C4C2×#13511yescharacteristic
280C2×C4C2×#13711yescharacteristic
281C2×C4C2×#13712yes—
282C2×C4#57512yes—
283C2×C4#57712yes—
284C8—82no—
285C8—82no—
286C8—82no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—82no—
291C22—82no—
292C22#243611yescharacteristic
293C22C2×#45211yescharacteristic
294C22#300712yes—
295C22#300911yescharacteristic
296C22#302112yes—
297C4—22no—
298C4—24no—
299C4—42no—
300C4—44no—
301C4—42no—
302C4—82no—
303C4—82no—
304C4—84no—
305C4—162no—
306C4#313511yescharacteristic
307C4#313911yescharacteristic
308C4#323712yes—
309C4#336012yes—
310C4#337212yes—
311C2—82no—
312C2#1879312yes—
313C2#1885112yes—
314C2#2591011yescharacteristic
315C2#2592811yescharacteristic
316C2#2593411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30220C2semidirect
4. #29226C2semidirect
5. #10355C2semidirect
6. #9902C2semidirect
7. #9872C2semidirect

Maximal subgroups

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

subgroupindex
#361922
#331972
#302202
#292262
#103552
#99022
#98722
#89762
#81742