Finite Group Databasebeta

← #294887

Subgroups — #294887

Order 512.

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

normal subgroups only

This group has 786 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#39641C211yesmaximal, characteristic
2#33310C211yesmaximal, characteristic
3#29866C211yesmaximal, characteristic
4#28755C211yesmaximal, characteristic
5#10075C214yesmaximal
6#9871C211yesmaximal, characteristic
7#9703C211yesmaximal, characteristic
8#9059C214yesmaximal
9#8176C211yesmaximal, characteristic
10C2×#498C2214yes—
11C2×#491—21no—
12C2×#475C2212yes—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2791—24no—
16#2647—21no—
17#2546—22no—
18#2546C2212yes—
19#2507—22no—
20#2446—22no—
21#2365C2212yes—
22#2330—21no—
23C4×#125C2211yescharacteristic
24#2295C2211yescharacteristic
25#2084—22no—
26#2072—22no—
27#2043—22no—
28#1988C2212yes—
29#1981C2214yes—
30#1960—21no—
31#1844—24no—
32#1831C2211yescharacteristic
33#1767C2211yescharacteristic
34C42×C8C2211yescharacteristic
35C2×#332C2214yes—
36C2×Q8⋊C8C2212yes—
37#1279C2214yes—
38#1279C2212yes—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2312yes—
44C2×C4⋊Q8C2312yes—
45C2×#125D411yescharacteristic
46C2×C4×Q8—22no—
47C2×C4×Q8C2311yescharacteristic
48#498—28no—
49#498—44no—
50#491—44no—
51#475—44no—
52C2×#106—22no—
53C2×#106—28no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#103—22no—
57C2×#103—24no—
58C2×#103—24no—
59C2×#97—21no—
60C2×C4×C8—22no—
61C2×C4×C8C2312yes—
62#400—24no—
63#398—24no—
64#398—24no—
65#398—24no—
66#398—24no—
67#391—22no—
68#391—22no—
69#391D412yes—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389D412yes—
76#389D412yes—
77#387—24no—
78#387—24no—
79#387—24no—
80#387—22no—
81#387—22no—
82#384—21no—
83#384—21no—
84#384—22no—
85#384—22no—
86#384D414yes—
87#382D411yescharacteristic
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2312yes—
92C4×C4⋊C4C2311yescharacteristic
93#376—21no—
94C43C2311yescharacteristic
95#340—24no—
96#336—22no—
97#336—21no—
98#332—44no—
99Q8⋊C8—24no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8C2×D411yescharacteristic
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—84no—
113C4⋊Q8—82no—
114C4⋊Q8—84no—
115#125—22no—
116#125SD1614yes—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—84no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D412yes—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—24no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2411yescharacteristic
148#106—416no—
149#106—44no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—24no—
153#103—44no—
154#103—48no—
155#103—84no—
156#97—42no—
157C4×C8—28no—
158C4×C8—22no—
159C4×C8#4114yes—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—21no—
170#95—22no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—24no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—44no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—42no—
189C22×C4#12012yes—
190C22×C4#12012yes—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C2×C8—28no—
195C2×C8—24no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—44no—
201C4⋊C4—22no—
202C4⋊C4—28no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—28no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C4⋊C4—84no—
222C4⋊C4C2×SD1614yes—
223C42—24no—
224C42—22no—
225C42—24no—
226C42—42no—
227C42—42no—
228C42—44no—
229C42—84no—
230C42C2×SD1612yes—
231C42#13712yes—
232C42C2×#4112yes—
233C42H(5,𝔽2)12yes—
234C23C2×#12011yescharacteristic
235Q8—42no—
236Q8—42no—
237Q8—84no—
238Q8—84no—
239Q8—84no—
240Q8—84no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#44814yes—
266C2×C4#45614yes—
267C2×C4#53211yescharacteristic
268C2×C4#53511yescharacteristic
269C2×C4#54612yes—
270C2×C4#55012yes—
271C2×C4C22×SD1611yescharacteristic
272C2×C4C2×#13711yescharacteristic
273C2×C4#57712yes—
274C2×C4#57714yes—
275C8—28no—
276C8—82no—
277C8—82no—
278C22#243111yescharacteristic
279C22C2×#44811yescharacteristic
280C22#300912yes—
281C22#301011yescharacteristic
282C22C2×#45612yes—
283C4—24no—
284C4—24no—
285C4—22no—
286C4—44no—
287C4—81no—
288C4—84no—
289C4—82no—
290C4—82no—
291C4—81no—
292C4—162no—
293C4#317412yes—
294C4#322112yes—
295C4#339212yes—
296C4#340012yes—
297C2#1865212yes—
298C2#1870012yes—
299C2#2558911yescharacteristic
300C2#2560611yescharacteristic
301C2#2563011yescharacteristic
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
#396412
#333102
#298662
#287552
#100752
#98712
#97032
#90592
#81762