Finite Group Databasebeta

← #320731

Subgroups — #320731

Order 512.

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

normal subgroups only

This group has 780 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#33231C211yesmaximal, characteristic
2#29193C212yesmaximal, complemented
3#28749C212yesmaximal
4#21727C211yesmaximal, characteristic, complemented
5#10673C211yesmaximal, characteristic, complemented
6#10671C211yesmaximal, characteristic, complemented
7#10312C211yesmaximal, characteristic, complemented
8#10312C211yesmaximal, characteristic, complemented
9#9838C212yesmaximal, complemented
10#9830C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#490—22no—
14C2×#486C2212yes—
15C2×#486C2212yes—
16#2867C2211yescharacteristic
17#2859C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19#2068—24no—
20#2041—24no—
21#1990C2212yes—
22#1982—24no—
23#1961—22no—
24#1957C2211yescharacteristic
25#1957C2211yescharacteristic
26#1957C2212yes—
27#1957C2212yes—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1956C2212yes—
31#1956C2212yes—
32#1944—22no—
33#1863—24no—
34#1845C2212yes—
35#1845C2211yescharacteristic
36#1834C2212yes—
37#1776—22no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C2×C4⋊Q8C2312yes—
44C2×#127—21no—
45C2×#127—21no—
46C2×#127—22no—
47C2×#127—22no—
48C2×#127C2311yescharacteristic
49C2×#127C2311yescharacteristic
50C2×#125—21no—
51C2×C4×Q8—22no—
52#498—24no—
53#490—48no—
54#486—24no—
55#486—24no—
56#486—42no—
57#486—42no—
58C2×#107—22no—
59C2×#107—24no—
60C2×#106—22no—
61C2×C4⋊C8D414yes—
62C2×#103—24no—
63C2×#103—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#97D412yes—
73C2×#97D412yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#384C2312yes—
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#382—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C43C2311yescharacteristic
90#340—24no—
91#340—24no—
92#336D414yes—
93#322—42no—
94C22×Q8—22no—
95C22×D4—21no—
96C22×D4—21no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—41no—
100C22×D4—41no—
101C22×D4—42no—
102C22×C8C2×D414yes—
103C22×C8C2×D414yes—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8#4114yes—
110#127—22no—
111#127—22no—
112#127—21no—
113#127—21no—
114#127—21no—
115#127—21no—
116#127—48no—
117#127—44no—
118#127—44no—
119#127—84no—
120#125—44no—
121C4×Q8—84no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C42—42no—
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2411yescharacteristic
140#107—44no—
141#107—84no—
142#106—28no—
143C4⋊C8—44no—
144#103—48no—
145#103—84no—
146#102—48no—
147#102—48no—
148#102—44no—
149#102—44no—
150#102—44no—
151#102—44no—
152#102—84no—
153#102—84no—
154#97—24no—
155#97—24no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—21no—
163#95—42no—
164#95—42no—
165#95—42no—
166C24—41no—
167C24—41no—
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—84no—
189C2×D4—82no—
190C2×D4—82no—
191C2×D4—84no—
192C2×D4—84no—
193C2×D4—84no—
194C2×D4—84no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4#12711yescharacteristic
200C22×C4#12711yescharacteristic
201C22×C4#12712yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D412yes—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8AGL(1,ℤ/8)14yes—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—84no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C42—22no—
231C42—24no—
232C42—24no—
233C42—84no—
234C42#13714yes—
235C42C2×#4112yes—
236C42H(5,𝔽2)11yescharacteristic
237C422−1+411yescharacteristic
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23—81no—
247C23—81no—
248C23C2×#12711yescharacteristic
249Q8—48no—
250Q8—84no—
251D4—82no—
252D4—82no—
253D4—84no—
254D4—84no—
255D4—82no—
256D4—82no—
257D4—84no—
258D4—84no—
259D4—168no—
260D4—164no—
261D4—164no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—44no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#49714yes—
279C2×C4#52511yescharacteristic
280C2×C4#53711yescharacteristic
281C2×C4#54612yes—
282C2×C4#54812yes—
283C2×C4C2×AGL(1,ℤ/8)12yes—
284C2×C4C2×#13712yes—
285C2×C4#57514yes—
286C2×C4#57614yes—
287C8—24no—
288C8—42no—
289C8—44no—
290C8—44no—
291C22—162no—
292C22—162no—
293C22—161no—
294C22—161no—
295C22—161no—
296C22—161no—
297C22—162no—
298C22—162no—
299C22—161no—
300C22—161no—
301C22#254011yescharacteristic
302C22C2×#49711yescharacteristic
303C22#315111yescharacteristic
304C22#315312yes—
305C22#315512yes—
306C4—24no—
307C4—41no—
308C4—41no—
309C4—84no—
310C4—161no—
311C4—162no—
312C4—161no—
313C4#311212yes—
314C4#325412yes—
315C4#330914yes—
316C4#337014yes—
317C2—321no—
318C2—321no—
319C2#1968212yes—
320C2#1972711yescharacteristic
321C2#1974911yescharacteristic
322C2#2838611yescharacteristic
323C2#2842512yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2867C22
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. #29193C2semidirect
4. #21727C2semidirect
5. #10673C2semidirect
6. #10671C2semidirect
7. #10312C2semidirect
8. #10312C2semidirect
9. #9838C2semidirect
10. #9830C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332312
#291932
#287492
#217272
#106732
#106712
#103122
#98382
#98302
#82632