Finite Group Databasebeta

← #331211

Subgroups — #331211

Order 512.

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

normal subgroups only

This group has 773 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, complemented
2#33170C211yesmaximal, characteristic
3#29845C212yesmaximal, complemented
4#29843C212yesmaximal
5#10685C212yesmaximal, complemented
6#10685C211yesmaximal, characteristic, complemented
7#10683C211yesmaximal, characteristic, complemented
8#10317C211yesmaximal, characteristic, complemented
9#10312C211yesmaximal, characteristic, complemented
10#10310C212yesmaximal, complemented
11#7849C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#492C2211yescharacteristic
14C2×#487C2212yes—
15C2×#486C2212yes—
16C2×#465—22no—
17C2×#431C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2212yes—
21#2856C2211yescharacteristic
22#2066—24no—
23#2065—24no—
24#2035—24no—
25#2034—24no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1968—22no—
30#1957C2212yes—
31#1957C2211yescharacteristic
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1941—22no—
35#1844C2211yescharacteristic
36#1844C2212yes—
37#1844C2212yes—
38#1844C2211yescharacteristic
39#1807—22no—
40#1273C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—22no—
44C23×C8D412yes—
45C2×C4⋊Q8C2312yes—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×#121—21no—
51C2×#121—22no—
52C22×C42C2311yescharacteristic
53#498—42no—
54#492—42no—
55#487—42no—
56#487—42no—
57#486—42no—
58#486—42no—
59#465—48no—
60#431—42no—
61#431—42no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—24no—
66C2×#106—24no—
67C2×C4⋊C8D414yes—
68C2×#103—24no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#102—24no—
72C2×#102—22no—
73C2×#102—24no—
74C2×C22⋊C8D414yes—
75C2×C4×C8—42no—
76C2×C4×C8D412yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80C2×C4×C8C2312yes—
81#391—22no—
82#391—21no—
83#390—22no—
84#390—21no—
85#389—22no—
86#389—22no—
87#389—22no—
88#384C2311yescharacteristic
89#384C2311yescharacteristic
90#384C2312yes—
91#380—22no—
92#380—22no—
93#380—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×D4—21no—
100C22×D4—22no—
101C22×D4—41no—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C2×SD16—44no—
105C2×SD16—44no—
106C2×SD16—44no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8SD1618yes—
110C22×C8C2×D414yes—
111C22×C8C2×D414yes—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—21no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—21no—
121#127—21no—
122#127—22no—
123#127—21no—
124C22⋊Q8—44no—
125C22⋊Q8—84no—
126#121—44no—
127#121—84no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
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×C22⋊C4—42no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—41no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D412yes—
149C2×C42C2411yescharacteristic
150#106—44no—
151#106—44no—
152#106—44no—
153#106—84no—
154#106—84no—
155C4⋊C8—44no—
156#103—48no—
157#103—44no—
158#103—84no—
159#102—48no—
160#102—44no—
161#102—84no—
162C22⋊C8—44no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—42no—
170#95—42no—
171#95—42no—
172#95—42no—
173C24—41no—
174C24#12711yescharacteristic
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—84no—
192C2×D4—82no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4#12712yes—
206C22×C4#12711yescharacteristic
207C22×C4C2×SD1614yes—
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D412yes—
210SD16—88no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—48no—
222C2×C8C2×SD1614yes—
223C2×C8C2×SD1614yes—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
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—
243C42—21no—
244C42—22no—
245C42—22no—
246C42—21no—
247C42H(5,𝔽2)12yes—
248C42H(5,𝔽2)11yescharacteristic
249C422−1+411yescharacteristic
250C23—21no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—81no—
255C23—81no—
256C23—82no—
257C23—82no—
258C23—81no—
259C23#49212yes—
260C23C2×#12711yescharacteristic
261Q8—82no—
262Q8—84no—
263Q8—82no—
264Q8—84no—
265D4—82no—
266D4—84no—
267D4—82no—
268D4—84no—
269D4—164no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4#49212yes—
293C2×C4#53411yescharacteristic
294C2×C4#53412yes—
295C2×C4#53711yescharacteristic
296C2×C4#53712yes—
297C2×C4C22×SD1612yes—
298C2×C4#57512yes—
299C2×C4#57714yes—
300C2×C4#57714yes—
301C8—24no—
302C8—24no—
303C8—44no—
304C8—44no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—161no—
311C22—162no—
312C22—161no—
313C22—161no—
314C22—162no—
315C22#265611yescharacteristic
316C22C2×#49211yescharacteristic
317C22#315011yescharacteristic
318C22#315412yes—
319C22#315412yes—
320C4—22no—
321C4—42no—
322C4—161no—
323C4—162no—
324C4—161no—
325C4—161no—
326C4—162no—
327C4#318712yes—
328C4#320414yes—
329C4#324612yes—
330C4#326314yes—
331C2—42no—
332C2—321no—
333C2#1965111yescharacteristic
334C2#1965411yescharacteristic
335C2#1970912yes—
336C2#3024011yescharacteristic
337C2#3025612yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
1. #33207C2semidirect
3. #29845C2semidirect
5. #10685C2semidirect
6. #10685C2semidirect
7. #10683C2semidirect
8. #10317C2semidirect
9. #10312C2semidirect
10. #10310C2semidirect
11. #7849C2semidirect

Maximal subgroups

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

subgroupindex
#332072
#331702
#298452
#298432
#106852
#106832
#103172
#103122
#103102
#78492