Finite Group Databasebeta

← #294912

Subgroups — #294912

Order 512.

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

normal subgroups only

This group has 742 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#39769C211yesmaximal, characteristic
2#33225C211yesmaximal, characteristic
3#29869C211yesmaximal, characteristic
4#29226C211yesmaximal, characteristic
5#10061C214yesmaximal
6#9871C211yesmaximal, characteristic
7#9705C211yesmaximal, characteristic
8#9065C214yesmaximal
9#8176C211yesmaximal, characteristic
10C2×#487C2214yes—
11C2×#477C2212yes—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14#2794—24no—
15#2642—21no—
16#2570—22no—
17#2568C2212yes—
18#2511—22no—
19#2446—22no—
20#2365C2212yes—
21#2327—21no—
22C4×#125C2211yescharacteristic
23#2295C2211yescharacteristic
24#2099—22no—
25#2083—22no—
26#2072—22no—
27#1992—21no—
28#1987C2214yes—
29#1987C2212yes—
30#1956—24no—
31#1836—21no—
32#1831C2211yescharacteristic
33#1767C2211yescharacteristic
34C42×C8C2211yescharacteristic
35C2×#332C2214yes—
36C2×Q8⋊C8C2212yes—
37#1279C2214yes—
38#1279C2212yes—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2312yes—
43C2×C4⋊Q8C2312yes—
44C2×#125D411yescharacteristic
45C2×C4×Q8—22no—
46C2×C4×Q8C2311yescharacteristic
47#487—28no—
48#487—44no—
49#477—44no—
50C2×#107—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—22no—
54C2×#103—28no—
55C2×#97—21no—
56C2×C4×C8—22no—
57C2×C4×C8C2312yes—
58#400—24no—
59#400—24no—
60#400—24no—
61#400—24no—
62#398—24no—
63#391—22no—
64#391—22no—
65#391—22no—
66#391—22no—
67#391D412yes—
68#389—22no—
69#389D412yes—
70#389D412yes—
71#387—24no—
72#387—24no—
73#387—24no—
74#387—24no—
75#387—24no—
76#387—22no—
77#387—22no—
78#384—22no—
79#384—22no—
80#384—21no—
81#384—21no—
82#384D414yes—
83#382—22no—
84#382D411yescharacteristic
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2312yes—
89C4×C4⋊C4C2311yescharacteristic
90#376—21no—
91C43C2311yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#336—22no—
96#336—21no—
97#332—44no—
98Q8⋊C8—24no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8C2×D411yescharacteristic
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—48no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—82no—
113#125—22no—
114#125—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—84no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D412yes—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42—24no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2411yescharacteristic
144#107—44no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—24no—
148#103—416no—
149#103—44no—
150#97—42no—
151C4×C8—22no—
152C4×C8—44no—
153C4×C8#4114yes—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—21no—
166#95—22no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—24no—
170C2×Q8—48no—
171C2×Q8—44no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—44no—
175C2×Q8—44no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—42no—
183C22×C4#12012yes—
184C22×C4#12012yes—
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C2×C8—24no—
189C2×C8—44no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—44no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—28no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—84no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C42—22no—
217C42—24no—
218C42—24no—
219C42—42no—
220C42—42no—
221C42—44no—
222C42—84no—
223C42#13512yes—
224C42#13712yes—
225C42C2×#4112yes—
226C42H(5,𝔽2)12yes—
227C23C2×#12011yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—88no—
231Q8—84no—
232Q8—84no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4#45214yes—
257C2×C4#45414yes—
258C2×C4#53211yescharacteristic
259C2×C4#53511yescharacteristic
260C2×C4#54612yes—
261C2×C4#55012yes—
262C2×C4C2×#13511yescharacteristic
263C2×C4C2×#13711yescharacteristic
264C2×C4#57712yes—
265C2×C4#57814yes—
266C8—44no—
267C8—82no—
268C8—82no—
269C22#243111yescharacteristic
270C22C2×#45211yescharacteristic
271C22#300911yescharacteristic
272C22#301112yes—
273C22C2×#45412yes—
274C4—24no—
275C4—24no—
276C4—22no—
277C4—82no—
278C4—82no—
279C4—81no—
280C4—84no—
281C4—82no—
282C4—81no—
283C4—162no—
284C4#317712yes—
285C4#322312yes—
286C4#339412yes—
287C4#340012yes—
288C2#1864512yes—
289C2#1869512yes—
290C2#2559111yescharacteristic
291C2#2560911yescharacteristic
292C2#2562911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#397692
#332252
#298692
#292262
#100612
#98712
#97052
#90652
#81762