Finite Group Databasebeta

← #331312

Subgroups — #331312

Order 512.

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

normal subgroups only

This group has 755 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#33210C211yesmaximal, characteristic
2#33210C212yesmaximal
3#33176C211yesmaximal, characteristic
4#33049C212yesmaximal
5#10684C212yesmaximal, complemented
6#10676C211yesmaximal, characteristic, complemented
7#10674C211yesmaximal, characteristic, complemented
8#10318C212yesmaximal, complemented
9#10317C211yesmaximal, characteristic, complemented
10#10316C211yesmaximal, characteristic, complemented
11#7852C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#498C2212yes—
14C2×#494C2211yescharacteristic
15C2×#487C2212yes—
16C2×#470—22no—
17C2×C4×M16C2212yes—
18C2×C4×M16C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2212yes—
21#2857C2211yescharacteristic
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1971—22no—
27#1956C2212yes—
28#1956C2211yescharacteristic
29#1956C2211yescharacteristic
30#1943—24no—
31#1943—22no—
32#1935—24no—
33#1894—24no—
34#1845C2211yescharacteristic
35#1845C2212yes—
36#1844C2211yescharacteristic
37#1834C2212yes—
38#1809—22no—
39#1809—24no—
40#1273C2211yescharacteristic
41#1273C2211yescharacteristic
42#1273C2212yes—
43C22×Q16—22no—
44C22×M16D412yes—
45C22×M16D414yes—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2312yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C22×C42C2311yescharacteristic
53#498—42no—
54#498—42no—
55#498—42no—
56#494—42no—
57#487—42no—
58#487—42no—
59#470—48no—
60C2×#107—24no—
61C2×#107—22no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#106—22no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—22no—
68C2×#103—22no—
69C4×M16—42no—
70C4×M16—42no—
71C4×M16—42no—
72C2×#97D414yes—
73C2×#97D412yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2312yes—
78#391—22no—
79#391—21no—
80#391—21no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#384C2311yescharacteristic
86#384C2312yes—
87#384C2311yescharacteristic
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C2×Q16—44no—
103C2×Q16—44no—
104C2×Q16—44no—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—28no—
108C2×M16—44no—
109C2×M16—42no—
110C2×M16—44no—
111C22×C8C2×D414yes—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C22×C8C2×D414yes—
115C4⋊Q8—21no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125C22⋊Q8—48no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
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—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42C2×D412yes—
149C2×C42C2×D411yescharacteristic
150C2×C42C2411yescharacteristic
151#107—48no—
152#107—44no—
153#106—44no—
154#106—44no—
155#106—48no—
156#103—48no—
157#103—44no—
158#103—44no—
159#103—48no—
160#97—42no—
161#97—44no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—42no—
172C24#12711yescharacteristic
173C2×Q8—42no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—44no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4#12712yes—
200C22×C4#12711yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D412yes—
203Q16—88no—
204M16—48no—
205M16—416no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8AGL(1,ℤ/8)14yes—
216C2×C8#13718yes—
217C2×C8#13714yes—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—84no—
239C42—21no—
240C42—22no—
241C42—22no—
242C42—21no—
243C42H(5,𝔽2)12yes—
244C42H(5,𝔽2)11yescharacteristic
245C422−1+411yescharacteristic
246C23—21no—
247C23—22no—
248C23—21no—
249C23—22no—
250C23—21no—
251C23C2×#12711yescharacteristic
252Q8—82no—
253Q8—82no—
254Q8—84no—
255Q8—84no—
256Q8—82no—
257Q8—82no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4#53411yescharacteristic
285C2×C4#53412yes—
286C2×C4#53712yes—
287C2×C4#53711yescharacteristic
288C2×C4C2×AGL(1,ℤ/8)12yes—
289C2×C4C2×#13712yes—
290C2×C4C2×#13714yes—
291C2×C4#57514yes—
292C8—24no—
293C8—24no—
294C8—28no—
295C8—44no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22#265611yescharacteristic
302C22#315111yescharacteristic
303C22#315212yes—
304C22#315211yescharacteristic
305C22#315312yes—
306C4—42no—
307C4—41no—
308C4—161no—
309C4—161no—
310C4—162no—
311C4—162no—
312C4—161no—
313C4—161no—
314C4#319312yes—
315C4#319614yes—
316C4#325214yes—
317C4#325412yes—
318C2—81no—
319C2#1974611yescharacteristic
320C2#1975011yescharacteristic
321C2#1975112yes—
322C2#3024612yes—
323C2#3024811yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
5. #10684C2semidirect
6. #10676C2semidirect
7. #10674C2semidirect
8. #10318C2semidirect
9. #10317C2semidirect
10. #10316C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#331762
#330492
#106842
#106762
#106742
#103182
#103172
#103162
#78522