Finite Group Databasebeta

← #320599

Subgroups — #320599

Order 512.

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

normal subgroups only

This group has 945 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#33230C211yesmaximal, characteristic, complemented
2#28615C212yesmaximal, complemented
3#28509C212yesmaximal, complemented
4#21690C211yesmaximal, characteristic, complemented
5#10685C211yesmaximal, characteristic, complemented
6#10683C211yesmaximal, characteristic, complemented
7#10307C211yesmaximal, characteristic, complemented
8#10304C211yesmaximal, characteristic
9#9837C212yesmaximal, complemented
10#9827C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#493C2212yes—
14C2×#489—22no—
15C2×#486C2212yes—
16C2×#476—24no—
17C2×#442—22no—
18#2868C2211yescharacteristic
19#2858C2211yescharacteristic
20C4×#127C2212yes—
21#2094—24no—
22#2054—24no—
23#1990C2212yes—
24#1972—22no—
25#1959—22no—
26#1957C2212yes—
27#1957C2211yescharacteristic
28#1957C2212yes—
29#1957C2212yes—
30#1957C2211yescharacteristic
31#1957C2212yes—
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1879—24no—
35#1844C2212yes—
36#1844C2212yes—
37#1844C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×D8—22no—
44C22×D8—24no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×#127C2312yes—
50C2×#124—21no—
51C2×#124—21no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55#493—24no—
56#493—24no—
57#493—42no—
58#489—48no—
59#486—24no—
60#486—42no—
61#476—416no—
62#442—48no—
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—22no—
66C2×C4⋊C8D414yes—
67C2×#103—22no—
68C2×#103—22no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#97D412yes—
77C2×#97D412yes—
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2312yes—
81C2×C4×C8C2311yescharacteristic
82#390—22no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390—22no—
87#384—22no—
88#384—21no—
89#384C2312yes—
90#384C2311yescharacteristic
91#384C2311yescharacteristic
92C43C2311yescharacteristic
93#340—24no—
94#336D414yes—
95#322—42no—
96C22×Q8—21no—
97C22×Q8—41no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—22no—
103C23×C4—42no—
104C2×D8—28no—
105C2×D8—44no—
106C2×D8—48no—
107C2×D8—44no—
108C2×D8—48no—
109C2×D8—84no—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D414yes—
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—48no—
118C4⋊Q8—44no—
119#127—22no—
120#127—22no—
121#127—22no—
122#127—21no—
123#127—22no—
124#127—21no—
125#127#4114yes—
126#124—44no—
127#124—44no—
128C4×D4—48no—
129C4×D4—48no—
130C4×D4—84no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2411yescharacteristic
152#106—44no—
153#106—48no—
154#106—48no—
155C4⋊C8—28no—
156#103—44no—
157#103—44no—
158#102—48no—
159#102—48no—
160#102—48no—
161#102—44no—
162#102—48no—
163#102—44no—
164#102—84no—
165#97—24no—
166#97—24no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—21no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—21no—
174C24—42no—
175C24—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—84no—
181C2×Q8—82no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—44no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—84no—
209C22×C4#12711yescharacteristic
210C22×C4#12711yescharacteristic
211C22×C4#12712yes—
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D412yes—
214D8—416no—
215D8—816no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—22no—
219C2×C8—24no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8AGL(1,ℤ/8)14yes—
227C2×C8AGL(1,ℤ/8)18yes—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—84no—
238C4⋊C4—82no—
239C22⋊C4—84no—
240C22⋊C4—84no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—84no—
244C22⋊C4—84no—
245C42—22no—
246C42—24no—
247C42—24no—
248C42AGL(1,ℤ/8)14yes—
249C42C2×#4112yes—
250C42H(5,𝔽2)11yescharacteristic
251C422−1+411yescharacteristic
252C23—44no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—81no—
260C23—81no—
261C23—82no—
262C23—82no—
263C23—81no—
264C23C2×#12711yescharacteristic
265Q8—82no—
266Q8—82no—
267Q8—164no—
268D4—48no—
269D4—48no—
270D4—48no—
271D4—82no—
272D4—84no—
273D4—84no—
274D4—84no—
275D4—82no—
276D4—84no—
277D4—84no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—168no—
295C2×C4#49614yes—
296C2×C4#52511yescharacteristic
297C2×C4#53711yescharacteristic
298C2×C4#54612yes—
299C2×C4#54812yes—
300C2×C4C2×AGL(1,ℤ/8)12yes—
301C2×C4C2×AGL(1,ℤ/8)12yes—
302C2×C4C2×AGL(1,ℤ/8)14yes—
303C2×C4#57714yes—
304C8—24no—
305C8—28no—
306C8—42no—
307C8—44no—
308C22—84no—
309C22—84no—
310C22—84no—
311C22—162no—
312C22—162no—
313C22—162no—
314C22—162no—
315C22—161no—
316C22—162no—
317C22—161no—
318C22—162no—
319C22—161no—
320C22#254011yescharacteristic
321C22C2×#49611yescharacteristic
322C22#315111yescharacteristic
323C22#315112yes—
324C22#315412yes—
325C4—24no—
326C4—41no—
327C4—41no—
328C4—161no—
329C4—161no—
330C4—162no—
331C4—161no—
332C4#310912yes—
333C4#325312yes—
334C4#330614yes—
335C4#337314yes—
336C2—164no—
337C2—321no—
338C2#1967112yes—
339C2#1972111yescharacteristic
340C2#1974511yescharacteristic
341C2#2838111yescharacteristic
342C2#2843012yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#10304C2
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
1. #33230C2semidirect
2. #28615C2semidirect
3. #28509C2semidirect
4. #21690C2semidirect
5. #10685C2semidirect
6. #10683C2semidirect
7. #10307C2semidirect
9. #9837C2semidirect
10. #9827C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332302
#286152
#285092
#216902
#106852
#106832
#103072
#103042
#98372
#98272
#82632