Finite Group Databasebeta

← #304727

Subgroups — #304727

Order 512.

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

normal subgroups only

This group has 772 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
1Q8×#125C211yesmaximal, characteristic
2#33163C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic
4#29225C211yesmaximal, characteristic
5#10357C214yesmaximal
6#9902C211yesmaximal, characteristic
7#9872C211yesmaximal, characteristic
8#8987C214yesmaximal
9#8175C211yesmaximal, characteristic
10C2×#462—22no—
11C2×#462C2212yes—
12#2868C2211yescharacteristic
13#2859—21no—
14#2859C2211yescharacteristic
15#2552—22no—
16#2552C2212yes—
17#2507—24no—
18#2507—22no—
19Q8×C4⋊C4—22no—
20Q8×C4⋊C4C2212yes—
21C4×#125—21no—
22C4×#125C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#2100—24no—
25#2100—22no—
26#1992—21no—
27#1990C2214yes—
28#1988C2214yes—
29#1981—22no—
30#1981C2212yes—
31#1842C2211yescharacteristic
32#1836—21no—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×Q8⋊C8C2212yes—
36#1296C2214yes—
37#1279C2214yes—
38#1279C2212yes—
39C22×Q16—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2312yes—
42C2×#125—22no—
43C2×#125—21no—
44C2×#125D411yescharacteristic
45C2×C4×Q8—22no—
46C2×C4×Q8D412yes—
47C2×C4×Q8C2311yescharacteristic
48#462—28no—
49#462—48no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—22no—
54C2×#103—24no—
55C2×#97—21no—
56C2×C4×C8C2312yes—
57#398—24no—
58#398—24no—
59#398—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—24no—
67#389—22no—
68#389D412yes—
69#389D414yes—
70#387—24no—
71#387—22no—
72#384—24no—
73#384—22no—
74#384—22no—
75#384—21no—
76#384—21no—
77#384—22no—
78#384C2312yes—
79#382—21no—
80#382D411yescharacteristic
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4D412yes—
87C4×C4⋊C4C2312yes—
88C4×C4⋊C4C2311yescharacteristic
89C43—21no—
90C43C2311yescharacteristic
91#340—24no—
92#340—28no—
93#340—24no—
94#336—22no—
95#336—21no—
96Q8⋊C8—24no—
97C22×Q8—22no—
98C22×Q8C2×D411yescharacteristic
99C2×Q16—44no—
100C2×Q16—48no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—44no—
107C4⋊Q8—44no—
108#125—22no—
109#125—82no—
110#125—84no—
111#125Q1614yescomplemented
112C4×Q8—22no—
113C4×Q8—24no—
114C4×Q8—42no—
115C4×Q8—44no—
116C4×Q8—44no—
117C4×Q8#4114yes—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
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—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D414yes—
139C2×C42—21no—
140C2×C42—24no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42C2411yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—24no—
152C4⋊C8—22no—
153C4⋊C8—44no—
154#103—48no—
155#103—44no—
156#97—42no—
157C4×C8—22no—
158C4×C8—22no—
159#95—24no—
160#95—24no—
161#95—22no—
162#95—24no—
163#95—22no—
164#95—22no—
165#95—21no—
166#95—22no—
167C2×Q8—21no—
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8#12412yes—
174C2×Q8AGL(1,ℤ/8)14yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4#12012yes—
183C22×C4#12012yes—
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D411yescharacteristic
187Q16—88no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—44no—
193C2×C8—42no—
194C2×C8—42no—
195C4⋊C4—21no—
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—28no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—24no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C4⋊C4—84no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4#12412yes—
219C4⋊C4C2×Q1614yes—
220C42—24no—
221C42—22no—
222C42—22no—
223C42—22no—
224C42—42no—
225C42—44no—
226C42—48no—
227C42—42no—
228C42—42no—
229C42—42no—
230C42—82no—
231C42C2×Q1612yes—
232C42#13712yes—
233C42C2×#4112yes—
234C422−1+412yes—
235C23C2×#12011yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—82no—
239Q8—82no—
240Q8—84no—
241Q8#49014yes—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—84no—
266C2×C4#45114yes—
267C2×C4C2×#12411yescharacteristic
268C2×C4#53611yescharacteristic
269C2×C4#54812yes—
270C2×C4#54812yes—
271C2×C4C22×Q1611yescharacteristic
272C2×C4C2×AGL(1,ℤ/8)12yes—
273C2×C4C2×#13711yescharacteristic
274C2×C4#57512yes—
275C2×C4#57712yes—
276C8—82no—
277C8—82no—
278C8—82no—
279C22#243611yescharacteristic
280C22C2×#45111yescharacteristic
281C22#300612yes—
282C22#300911yescharacteristic
283C22#302212yes—
284C4—24no—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—44no—
289C4—42no—
290C4—82no—
291C4—82no—
292C4—84no—
293C4—162no—
294C4C2×#49011yescharacteristic
295C4#314211yescharacteristic
296C4#323412yes—
297C4#336312yes—
298C4#336912yes—
299C2#1880112yes—
300C2#1885812yes—
301C2#2591311yescharacteristic
302C2#2592711yescharacteristic
303C2#2593611yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
111. #125Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×#1252
#331632
#302302
#292252
#103572
#99022
#98722
#89872
#81752