Finite Group Databasebeta

← #331208

Subgroups — #331208

Order 512.

722 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 178 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#33207C211yesmaximal, characteristic
2#33172C211yesmaximal, characteristic
3#29877C212yesmaximal
4#29845C212yesmaximal
5#10686C211yesmaximal, characteristic, complemented
6#10685C211yesmaximal, characteristic, complemented
7#10680C212yesmaximal, complemented
8#10318C212yesmaximal, complemented
9#10317C211yesmaximal, characteristic, complemented
10#10316C211yesmaximal, characteristic, complemented
11#7849C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#498C2212yes—
14C2×#494C2211yescharacteristic
15C2×#487C2212yes—
16C2×#467—22no—
17C2×#431C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2212yes—
22#2088—24no—
23#2077—24no—
24#2066—24no—
25#2035—24no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1969—22no—
31#1956C2211yescharacteristic
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1941—22no—
35#1845C2211yescharacteristic
36#1844C2212yes—
37#1844C2211yescharacteristic
38#1834C2212yes—
39#1807—22no—
40#1273C2211yescharacteristic
41#1273C2211yescharacteristic
42#1273C2212yes—
43C22×Q16—22no—
44C23×C8D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C22×C42C2311yescharacteristic
52#498—42no—
53#498—42no—
54#498—42no—
55#494—42no—
56#487—42no—
57#487—42no—
58#467—48no—
59#431—42no—
60#431—42no—
61C2×#107—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—24no—
65C2×#106—24no—
66C2×C4⋊C8D414yes—
67C2×#103—24no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×C22⋊C8D414yes—
72C2×C4×C8—42no—
73C2×C4×C8D412yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78#391—22no—
79#391—21no—
80#391—21no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#384C2311yescharacteristic
86#384C2312yes—
87#384C2311yescharacteristic
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—22no—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C2×Q16—44no—
103C2×Q16—44no—
104C2×Q16—44no—
105C22×C8—42no—
106C22×C8—42no—
107C22×C8SD1618yes—
108C22×C8C2×D414yes—
109C22×C8C2×D414yes—
110C22×C8C2×D412yes—
111C22×C8C2×D412yes—
112C4⋊Q8—21no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C22⋊Q8—84no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2411yescharacteristic
148#107—44no—
149#106—44no—
150#106—44no—
151#106—48no—
152#106—84no—
153C4⋊C8—44no—
154#103—48no—
155#103—44no—
156#103—44no—
157#103—84no—
158C22⋊C8—44no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—42no—
166#95—42no—
167#95—42no—
168#95—42no—
169#95—42no—
170C24#12711yescharacteristic
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—44no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4#12712yes—
197C22×C4#12711yescharacteristic
198C22×C4C2×SD1614yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D412yes—
201Q16—88no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—24no—
212C2×C8—48no—
213C2×C8C2×SD1614yes—
214C2×C8C2×SD1614yes—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—84no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C22⋊C4—84no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C42—21no—
237C42—22no—
238C42—22no—
239C42—21no—
240C42H(5,𝔽2)12yes—
241C42H(5,𝔽2)11yescharacteristic
242C422−1+411yescharacteristic
243C23—21no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23#49212yes—
248C23C2×#12711yescharacteristic
249Q8—82no—
250Q8—82no—
251Q8—84no—
252Q8—82no—
253Q8—82no—
254Q8—84no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4#49212yes—
281C2×C4#53411yescharacteristic
282C2×C4#53412yes—
283C2×C4#53711yescharacteristic
284C2×C4#53712yes—
285C2×C4C22×SD1612yes—
286C2×C4#57512yes—
287C2×C4#57714yes—
288C2×C4#57814yes—
289C8—24no—
290C8—24no—
291C8—44no—
292C8—44no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22#265611yescharacteristic
299C22C2×#49211yescharacteristic
300C22#315011yescharacteristic
301C22#315412yes—
302C22#315612yes—
303C4—22no—
304C4—42no—
305C4—161no—
306C4—161no—
307C4—162no—
308C4—161no—
309C4—161no—
310C4—162no—
311C4#318712yes—
312C4#320114yes—
313C4#324612yes—
314C4#325814yes—
315C2—42no—
316C2#1965311yescharacteristic
317C2#1965411yescharacteristic
318C2#1970712yes—
319C2#3024011yescharacteristic
320C2#3025212yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
5. #10686C2semidirect
6. #10685C2semidirect
7. #10680C2semidirect
8. #10318C2semidirect
9. #10317C2semidirect
10. #10316C2semidirect

Maximal subgroups

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

subgroupindex
#332072
#331722
#298772
#298452
#106862
#106852
#106802
#103182
#103172
#103162
#78492