Finite Group Databasebeta

← #320729

Subgroups — #320729

Order 512.

737 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 170 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#33230C211yesmaximal, characteristic
2#29218C212yesmaximal, complemented
3#28614C212yesmaximal
4#21727C211yesmaximal, characteristic, complemented
5#10676C211yesmaximal, characteristic, complemented
6#10671C211yesmaximal, characteristic
7#10317C211yesmaximal, characteristic, complemented
8#10315C211yesmaximal, characteristic
9#9840C212yesmaximal, complemented
10#9829C212yesmaximal
11#8263C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#492C2212yes—
14C2×#489—22no—
15C2×#487C2212yes—
16#2868C2211yescharacteristic
17#2858C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19#2100—24no—
20#2055—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1987—24no—
26#1959—22no—
27#1957C2211yescharacteristic
28#1957C2211yescharacteristic
29#1957C2212yes—
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1944—22no—
33#1880—24no—
34#1845C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37#1776—22no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—24no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2312yes—
48C2×#127C2311yescharacteristic
49C2×#124—21no—
50C2×#124—21no—
51C2×C4×Q8—22no—
52#498—24no—
53#492—24no—
54#492—42no—
55#489—48no—
56#487—24no—
57#487—42no—
58C2×#107—22no—
59C2×#106—22no—
60C2×C4⋊C8D414yes—
61C2×#103—24no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#102—22no—
65C2×#102—22no—
66C2×#102—24no—
67C2×#97D412yes—
68C2×#97D412yes—
69C2×C4×C8C2312yes—
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—22no—
74#390—22no—
75#390—22no—
76#389—22no—
77#389—22no—
78#384—22no—
79#384—21no—
80#384C2312yes—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C43C2311yescharacteristic
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#336D414yes—
93#322—42no—
94C22×Q8—22no—
95C22×Q8—21no—
96C22×Q8—41no—
97C22×D4—21no—
98C22×D4—22no—
99C2×SD16—48no—
100C2×SD16—48no—
101C2×SD16—84no—
102C22×C8C2×D414yes—
103C22×C8C2×D414yes—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—21no—
110C4⋊Q8—22no—
111C4⋊Q8—21no—
112C4⋊Q8—48no—
113C4⋊Q8—44no—
114C4⋊Q8#4114yes—
115#127—22no—
116#127—21no—
117#127—21no—
118#124—44no—
119#124—44no—
120C4×Q8—84no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
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—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C42—42no—
138C2×C42C2×D412yes—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2411yescharacteristic
143#107—44no—
144#106—28no—
145C4⋊C8—44no—
146#103—48no—
147#103—44no—
148#103—44no—
149#102—44no—
150#102—44no—
151#102—84no—
152#97—24no—
153#97—24no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—21no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—21no—
161#95—42no—
162#95—42no—
163C24—41no—
164C2×Q8—24no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—82no—
173C2×Q8—84no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—44no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4#12711yescharacteristic
188C22×C4#12711yescharacteristic
189C22×C4#12712yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D412yes—
192SD16—816no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8AGL(1,ℤ/8)14yes—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—82no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—84no—
224C42—22no—
225C42—24no—
226C42—24no—
227C42—84no—
228C42#13714yes—
229C42C2×#4112yes—
230C42H(5,𝔽2)11yescharacteristic
231C422−1+411yescharacteristic
232C23—81no—
233C23—81no—
234C23—82no—
235C23—82no—
236C23—81no—
237C23C2×#12711yescharacteristic
238Q8—48no—
239Q8—84no—
240Q8—82no—
241Q8—82no—
242Q8—164no—
243D4—82no—
244D4—84no—
245D4—82no—
246D4—84no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4#49714yes—
267C2×C4#52511yescharacteristic
268C2×C4#53711yescharacteristic
269C2×C4#54612yes—
270C2×C4#54812yes—
271C2×C4C2×AGL(1,ℤ/8)12yes—
272C2×C4C2×#13712yes—
273C2×C4#57514yes—
274C2×C4#57714yes—
275C8—24no—
276C8—42no—
277C8—44no—
278C8—44no—
279C22—161no—
280C22—162no—
281C22—161no—
282C22—161no—
283C22—162no—
284C22#254011yescharacteristic
285C22C2×#49711yescharacteristic
286C22#315111yescharacteristic
287C22#315312yes—
288C22#315412yes—
289C4—24no—
290C4—41no—
291C4—41no—
292C4—84no—
293C4—162no—
294C4—161no—
295C4—161no—
296C4—161no—
297C4#311212yes—
298C4#325412yes—
299C4#331114yes—
300C4#337114yes—
301C2—321no—
302C2#1968512yes—
303C2#1972111yescharacteristic
304C2#1974911yescharacteristic
305C2#2838611yescharacteristic
306C2#2843612yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #29218C2semidirect
4. #21727C2semidirect
5. #10676C2semidirect
7. #10317C2semidirect
9. #9840C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332302
#292182
#286142
#217272
#106762
#106712
#103172
#103152
#98402
#98292
#82632