Finite Group Databasebeta

← #238118

Subgroups — #238118

Order 512.

1222 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 218 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
1C2×#2551C211yesmaximal, characteristic
2#21394C212yesmaximal
3#21322C211yesmaximal, characteristic
4#10350C218yesmaximal, complemented
5#8040C211yesmaximal, characteristic
6#7904C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4⋊Q8—22no—
9C2×#432C2212yes—
10C2×C4×M16C2211yescharacteristic
11#2551—22no—
12#2551—24no—
13#2551C2214yes—
14C2×#395—22no—
15C2×#389—22no—
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2212yes—
18#1992—22no—
19#1990C2214yes—
20#1988C2214yes—
21#1981C2214yes—
22#1872—22no—
23#1872—22no—
24#1763—24no—
25#1736—22no—
26#1296C2214yes—
27#1279C2218yes—
28C23×Q8—22no—
29C22×M16—21no—
30#533—22no—
31#533—24no—
32#533Q814yescomplemented
33C2×C4⋊Q8—28no—
34C2×C4⋊Q8—28no—
35C2×C4⋊Q8—416no—
36C2×C4⋊Q8C2312yes—
37C2×#125—21no—
38C2×C22⋊Q8—22no—
39C2×C4×Q8—22no—
40C2×#117—21no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C42—22no—
47C22×C42C2311yescharacteristic
48#432—24no—
49#432—24no—
50C2×C4⋊C8C2314yes—
51C2×#103—24no—
52C2×#103—24no—
53C2×C22⋊C8—22no—
54C4×M16—24no—
55C2×#97—21no—
56C2×#97—22no—
57C2×C4×C8C2312yes—
58#395—28no—
59#395—28no—
60#389—28no—
61#389—28no—
62#384—22no—
63#384D412yes—
64#384C2312yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4D414yes—
68C4×C4⋊C4C2314yes—
69C2×#95—22no—
70C2×#95—22no—
71#340—24no—
72#340—24no—
73#340—28no—
74C22×Q8—28no—
75C22×Q8—22no—
76C22×Q8—416no—
77C22×Q8—84no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4C2×D412yes—
82C23×C4C2×D411yescharacteristic
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—42no—
86C22×C8—22no—
87C22×C8—24no—
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C4⋊Q8—432no—
92C4⋊Q8—416no—
93C4⋊Q8—416no—
94C4⋊Q8C2×Q814yescomplemented
95#125—24no—
96#125—42no—
97C22⋊Q8—28no—
98C22⋊Q8—44no—
99C4×Q8—28no—
100C4×Q8—44no—
101#117—24no—
102#117—42no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—216no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—416no—
114C2×C4⋊C4—48no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4C2×D414yes—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D414yes—
120C2×C22⋊C4—41no—
121C2×C42—24no—
122C2×C42—24no—
123C2×C42—48no—
124C2×C42—41no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q812yes—
128C2×C42#4114yes—
129C2×C42#4112yes—
130C2×C42C2411yescharacteristic
131C4⋊C8—24no—
132C4⋊C8—24no—
133#103—48no—
134#103—48no—
135C22⋊C8—44no—
136#97—42no—
137#97—44no—
138C4×C8—24no—
139#95—28no—
140#95—24no—
141#95—24no—
142C24#12011yescharacteristic
143C2×Q8—24no—
144C2×Q8—432no—
145C2×Q8—42no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—88no—
149C2×Q8—832no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—28no—
155C22×C4—28no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4#12011yescharacteristic
167C22×C4#12012yes—
168C22×C4C22⋊Q812yes—
169C22×C4C22⋊Q814yes—
170C22×C4#12414yes—
171C22×C4#12412yes—
172C22×C4C22×D411yescharacteristic
173C22×C4C22×D412yes—
174M16—48no—
175C2×C8—24no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8—44no—
180C2×C8—44no—
181C4⋊C4—28no—
182C4⋊C4—28no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—44no—
188C4⋊C4—48no—
189C4⋊C4—432no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—48no—
193C4⋊C4—48no—
194C4⋊C4—82no—
195C4⋊C4AGL(1,ℤ/8)18yes—
196C4⋊C4#13718yes—
197C22⋊C4—44no—
198C42—48no—
199C42—48no—
200C42—44no—
201C42C22⋊Q814yes—
202C42C22⋊Q812yes—
203C42#12414yes—
204C42#12412yes—
205C42C22×Q811yescharacteristic
206C42C2×#4112yes—
207C42C2×#4111yescharacteristic
208C23—22no—
209C23—22no—
210C23#39312yes—
211C23#45512yes—
212C23#45612yes—
213C23C2×#12011yescharacteristic
214Q8—48no—
215Q8—832no—
216Q8—84no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—28no—
226C2×C4—28no—
227C2×C4—28no—
228C2×C4—28no—
229C2×C4—416no—
230C2×C4—416no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—44no—
238C2×C4—88no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4#39312yes—
242C2×C4#39314yes—
243C2×C4C2×C22⋊Q811yescharacteristic
244C2×C4C2×C22⋊Q812yes—
245C2×C4C2×#12411yescharacteristic
246C2×C4C2×#12412yes—
247C2×C4C2×AGL(1,ℤ/8)14yes—
248C2×C4C2×#13714yes—
249C8—44no—
250C8—84no—
251C22—22no—
252C22—24no—
253C22—22no—
254C22—24no—
255C22#202814yes—
256C22C2×#39311yescharacteristic
257C22#300312yes—
258C22#300412yes—
259C22C2×#45511yescharacteristic
260C22C2×#45611yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—22no—
265C4—48no—
266C4—48no—
267C4—84no—
268C4—84no—
269C4#307914yes—
270C4#314114yes—
271C2—24no—
272C2#1852911yescharacteristic
273C2#1854012yes—
274C2#1855111yescharacteristic
275C2#2345212yes—
276C2C2×#202811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
4. #10350C2semidirect
32. #533Q8semidirect
94. C4⋊Q8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#213942
#213222
#103502
#80402
#79042