Finite Group Databasebeta

← #294905

Subgroups — #294905

Order 512.

779 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 162 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#39513C211yesmaximal, characteristic
2#33225C211yesmaximal, characteristic, complemented
3#29868C211yesmaximal, characteristic
4#29226C211yesmaximal, characteristic, complemented
5#10058C214yesmaximal
6#9871C211yesmaximal, characteristic, complemented
7#9703C211yesmaximal, characteristic, complemented
8#9065C214yesmaximal, complemented
9#8177C211yesmaximal, characteristic
10C2×#487C2214yes—
11C2×#476C2212yes—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14#2793—24no—
15#2629—21no—
16#2547C2212yes—
17#2545—22no—
18#2497—22no—
19#2446—22no—
20#2365C2212yes—
21#2314—21no—
22C4×#125C2211yescharacteristic
23#2293C2211yescharacteristic
24#2099—22no—
25#2084—22no—
26#2071—22no—
27#1992—21no—
28#1987C2214yes—
29#1987C2212yes—
30#1956—24no—
31#1836—21no—
32#1831C2211yescharacteristic
33#1767C2211yescharacteristic
34C42×C8C2211yescharacteristic
35C2×#332C2214yes—
36C2×D4⋊C8C2212yes—
37#1279C2214yes—
38#1279C2212yes—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2312yes—
43C2×#125D411yescharacteristic
44C2×#124—22no—
45C2×C4×Q8—22no—
46C2×C4×D4C2311yescharacteristic
47C2×#117—21no—
48#487—28no—
49#487—44no—
50#476—44no—
51C2×#106—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—28no—
55C2×#102—22no—
56C2×#97—21no—
57C2×C4×C8—22no—
58C2×C4×C8C2312yes—
59#400—24no—
60#398—24no—
61#397—24no—
62#396—24no—
63#396—24no—
64#391—22no—
65#391—22no—
66#390D412yes—
67#389—22no—
68#389D412yes—
69#389D412yes—
70#388—24no—
71#388—24no—
72#388—22no—
73#387—24no—
74#387—24no—
75#387—24no—
76#387—22no—
77#384—21no—
78#384—21no—
79#384—22no—
80#384—21no—
81#384—21no—
82#384D414yes—
83#382D411yescharacteristic
84#380—22no—
85#380—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2312yes—
89C4×C4⋊C4C2311yescharacteristic
90C4×C22⋊C4—21no—
91C43C2311yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#336—22no—
96#336—21no—
97#332—44no—
98D4⋊C8—24no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×D4C2×D411yescharacteristic
102C23×C4—21no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—48no—
111C4⋊Q8—84no—
112#125—22no—
113#125—22no—
114#124—84no—
115C4×Q8—84no—
116C4×D4—22no—
117C4×D4—22no—
118#117—42no—
119#117—42no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×D412yes—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—24no—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2411yescharacteristic
149#106—44no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—24no—
153#103—416no—
154#102—44no—
155#97—42no—
156C4×C8—22no—
157C4×C8—44no—
158C4×C8#4114yes—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—21no—
167#95—22no—
168C24—21no—
169C2×Q8—48no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—24no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4#12012yes—
189C22×C4#12012yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C2×C8—24no—
194C2×C8—44no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—44no—
199C2×C8—42no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—28no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C22⋊C4—42no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—84no—
223C22⋊C4—84no—
224C42—24no—
225C42—22no—
226C42—24no—
227C42—41no—
228C42—41no—
229C42—44no—
230C42—84no—
231C42#13512yes—
232C42#13712yes—
233C42C2×#4112yes—
234C42H(5,𝔽2)12yes—
235C23—41no—
236C23—41no—
237C23—42no—
238C23—41no—
239C23—42no—
240C23C2×#12011yescharacteristic
241Q8—88no—
242Q8—84no—
243Q8—84no—
244D4—42no—
245D4—42no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—84no—
270C2×C4#45214yes—
271C2×C4#45414yes—
272C2×C4#53211yescharacteristic
273C2×C4#53511yescharacteristic
274C2×C4#54612yes—
275C2×C4#55012yes—
276C2×C4C2×#13511yescharacteristic
277C2×C4C2×#13711yescharacteristic
278C2×C4#57714yes—
279C2×C4#57812yes—
280C8—44no—
281C8—82no—
282C8—82no—
283C22—81no—
284C22—82no—
285C22—81no—
286C22—82no—
287C22—81no—
288C22#243111yescharacteristic
289C22C2×#45211yescharacteristic
290C22#300912yes—
291C22#301111yescharacteristic
292C22C2×#45412yes—
293C4—24no—
294C4—24no—
295C4—22no—
296C4—84no—
297C4—82no—
298C4—81no—
299C4—82no—
300C4—82no—
301C4—162no—
302C4#317412yes—
303C4#322112yes—
304C4#339312yes—
305C4#339812yes—
306C2—161no—
307C2#1864712yes—
308C2#1869512yes—
309C2#2559111yescharacteristic
310C2#2560611yescharacteristic
311C2#2563211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #33225C2semidirect
4. #29226C2semidirect
6. #9871C2semidirect
7. #9703C2semidirect
8. #9065C2semidirect

Maximal subgroups

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

subgroupindex
#395132
#332252
#298682
#292262
#100582
#98712
#97032
#90652
#81772