Finite Group Databasebeta

← #320718

Subgroups — #320718

Order 512.

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

normal subgroups only

This group has 868 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#33231C211yesmaximal, characteristic
2#29179C212yesmaximal, complemented
3#28441C212yesmaximal
4#21727C211yesmaximal, characteristic, complemented
5#10673C211yesmaximal, characteristic, complemented
6#10671C211yesmaximal, characteristic, complemented
7#10315C211yesmaximal, characteristic, complemented
8#10315C211yesmaximal, characteristic, complemented
9#9842C212yesmaximal, complemented
10#9829C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#492C2212yes—
14C2×#492C2212yes—
15C2×#490—22no—
16C2×#460—24no—
17#2867C2211yescharacteristic
18#2859C2211yescharacteristic
19C4×C4⋊Q8C2212yes—
20#2087—24no—
21#2014—24no—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1961—22no—
26#1957C2212yes—
27#1957C2212yes—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1944—22no—
33#1921—24no—
34#1845C2212yes—
35#1845C2211yescharacteristic
36#1834C2212yes—
37#1776—22no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—24no—
44C22×SD16—24no—
45C2×C4⋊Q8C2312yes—
46C2×#127—21no—
47C2×#127—21no—
48C2×#127—22no—
49C2×#127—22no—
50C2×#127C2311yescharacteristic
51C2×#127C2311yescharacteristic
52C2×#125—21no—
53C2×C4×Q8—22no—
54#498—24no—
55#492—24no—
56#492—24no—
57#492—42no—
58#492—42no—
59#490—48no—
60#460—416no—
61C2×#107—24no—
62C2×#107—22no—
63C2×#106—22no—
64C2×C4⋊C8D414yes—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#97D412yes—
72C2×#97D412yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384C2312yes—
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85#382—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C43C2311yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#336D414yes—
94#322—42no—
95C22×Q8—22no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—21no—
99C22×D4—21no—
100C22×D4—41no—
101C22×D4—41no—
102C22×D4—42no—
103C2×SD16—48no—
104C2×SD16—48no—
105C2×SD16—48no—
106C2×SD16—48no—
107C2×SD16—84no—
108C2×SD16—84no—
109C22×C8C2×D414yes—
110C22×C8C2×D412yes—
111C22×C8C2×D414yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8#4114yes—
117#127—22no—
118#127—22no—
119#127—21no—
120#127—21no—
121#127—21no—
122#127—21no—
123#127—48no—
124#127—44no—
125#127—44no—
126#127—84no—
127#125—44no—
128C4×Q8—48no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C42—42no—
142C2×C42C2×D412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42C2411yescharacteristic
147#107—44no—
148#107—48no—
149#106—28no—
150C4⋊C8—28no—
151#102—48no—
152#102—44no—
153#102—44no—
154#102—48no—
155#102—44no—
156#102—44no—
157#97—24no—
158#97—24no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—21no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—21no—
166#95—42no—
167#95—42no—
168#95—42no—
169C24—41no—
170C24—41no—
171C2×Q8—24no—
172C2×Q8—42no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—84no—
192C2×D4—84no—
193C2×D4—82no—
194C2×D4—82no—
195C2×D4—84no—
196C2×D4—84no—
197C2×D4—84no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4#12711yescharacteristic
203C22×C4#12711yescharacteristic
204C22×C4#12712yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D412yes—
207SD16—816no—
208SD16—816no—
209C2×C8—22no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—24no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—24no—
219C2×C8AGL(1,ℤ/8)14yes—
220C2×C8AGL(1,ℤ/8)18yes—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—84no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C42—22no—
236C42—24no—
237C42—24no—
238C42—84no—
239C42#13714yes—
240C42C2×#4112yes—
241C42H(5,𝔽2)11yescharacteristic
242C422−1+411yescharacteristic
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23—81no—
252C23—81no—
253C23C2×#12711yescharacteristic
254Q8—48no—
255Q8—84no—
256D4—82no—
257D4—82no—
258D4—84no—
259D4—84no—
260D4—82no—
261D4—82no—
262D4—84no—
263D4—84no—
264D4—168no—
265D4—164no—
266D4—164no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—44no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4#49714yes—
284C2×C4#52511yescharacteristic
285C2×C4#53711yescharacteristic
286C2×C4#54612yes—
287C2×C4#54812yes—
288C2×C4C2×AGL(1,ℤ/8)12yes—
289C2×C4C2×AGL(1,ℤ/8)14yes—
290C2×C4C2×#13712yes—
291C2×C4#57714yes—
292C8—24no—
293C8—28no—
294C8—42no—
295C8—44no—
296C22—161no—
297C22—161no—
298C22—162no—
299C22—162no—
300C22—161no—
301C22—161no—
302C22—162no—
303C22—162no—
304C22—161no—
305C22—161no—
306C22#254011yescharacteristic
307C22C2×#49711yescharacteristic
308C22#315111yescharacteristic
309C22#315112yes—
310C22#315412yes—
311C4—24no—
312C4—41no—
313C4—41no—
314C4—84no—
315C4—161no—
316C4—162no—
317C4—161no—
318C4#311212yes—
319C4#325412yes—
320C4#331014yes—
321C4#336814yes—
322C2—321no—
323C2—321no—
324C2#1967912yes—
325C2#1971311yescharacteristic
326C2#1974511yescharacteristic
327C2#2838611yescharacteristic
328C2#2843512yes—
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. #29179C2semidirect
4. #21727C2semidirect
5. #10673C2semidirect
6. #10671C2semidirect
7. #10315C2semidirect
8. #10315C2semidirect
9. #9842C2semidirect
10. #9829C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332312
#291792
#284412
#217272
#106732
#106712
#103152
#98422
#98292
#82632