Finite Group Databasebeta

← #133832

Subgroups — #133832

Order 512.

1502 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2434C211yesmaximal, characteristic
2#12700C214yesmaximal, complemented
3#8550C212yesmaximal
4C24×Q8C411yescharacteristic
5C22×C4⋊Q8C2212yes—
6C22×C22⋊Q8C2211yescharacteristic
7#2434—28no—
8C2×#395—22no—
9C2×#391C412yes—
10C2×#385—22no—
11C2×C4×C22⋊C4C411yescharacteristic
12C2×#356—28no—
13#1784C2214yes—
14#1463—22no—
15#1462—22no—
16C23×Q8—24no—
17C23×Q8—21no—
18C23×Q8—42no—
19C23×Q8—42no—
20C23×Q8—48no—
21C23×Q8C2×C412yes—
22C23×Q8C2311yescharacteristic
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C2×C4⋊Q8—24no—
26C2×C4⋊Q8—48no—
27C2×C4⋊Q8D418yes—
28C2×C22⋊Q8—24no—
29C2×C22⋊Q8—42no—
30C2×C22⋊Q8—42no—
31C22×C4⋊C4—24no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4D412yes—
36C22×C42C2×C412yes—
37C2×#101—28no—
38#407—24no—
39#395—216no—
40#391—24no—
41#391—24no—
42#385—216no—
43C4×C22⋊C4—24no—
44C2×#95—21no—
45C2×#95—24no—
46#356—416no—
47#324—44no—
48#323—44no—
49C25C22⋊C411yescharacteristic
50C22×Q8—28no—
51C22×Q8—24no—
52C22×Q8—22no—
53C22×Q8—41no—
54C22×Q8—41no—
55C22×Q8—44no—
56C22×Q8—44no—
57C22×Q8—48no—
58C22×Q8—416no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—42no—
62C22×Q8—416no—
63C22×Q8—48no—
64C22×Q8—432no—
65C22×Q8—432no—
66C22×Q8—48no—
67C22×Q8—42no—
68C22×Q8C22⋊C414yes—
69C22×Q8C2×D414yes—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—24no—
76C23×C4—48no—
77C23×C4—44no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C22⋊C412yes—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D412yes—
84C2×M16—44no—
85C4⋊Q8—216no—
86C4⋊Q8—48no—
87C4⋊Q8—416no—
88C22⋊Q8—416no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—216no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—416no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—42no—
102C2×C42—22no—
103C2×C42—44no—
104C2×C42C22⋊C414yes—
105#101—416no—
106C22⋊C8—44no—
107C22⋊C8—44no—
108#95—24no—
109#95—216no—
110C24—24no—
111C24—21no—
112C24—21no—
113C24—42no—
114C24—42no—
115C24#9912yes—
116C24C2×C22⋊C411yescharacteristic
117C2×Q8—28no—
118C2×Q8—24no—
119C2×Q8—28no—
120C2×Q8—44no—
121C2×Q8—44no—
122C2×Q8—416no—
123C2×Q8—416no—
124C2×Q8—44no—
125C2×Q8—416no—
126C2×Q8—48no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—464no—
130C2×Q8—464no—
131C2×Q8—416no—
132C2×Q8—416no—
133C2×Q8—44no—
134C2×Q8—84no—
135C2×Q8—84no—
136C2×Q8#9918yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—28no—
141C22×C4—28no—
142C22×C4—28no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—48no—
155C22×C4—44no—
156C22×C4—416no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—48no—
161C22×C4—416no—
162C22×C4—416no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4#9912yes—
169C22×C4C2×C22⋊C412yes—
170C22×C4#12014yes—
171M16—88no—
172C2×C8—84no—
173C4⋊C4—48no—
174C4⋊C4—432no—
175C4⋊C4—44no—
176C22⋊C4—216no—
177C22⋊C4—48no—
178C42—28no—
179C42—44no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—24no—
184C23—24no—
185C23—22no—
186C23—22no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—416no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23#35614yescomplemented
195C23#37911yescharacteristic
196C23C2×#9911yescharacteristic
197C23C2×#9911yescharacteristic
198Q8—48no—
199Q8—416no—
200Q8—48no—
201Q8—432no—
202Q8—88no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—28no—
208C2×C4—28no—
209C2×C4—28no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—416no—
213C2×C4—416no—
214C2×C4—44no—
215C2×C4—48no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—432no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4C2×#9914yes—
231C8—164no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—24no—
240C22—44no—
241C22—48no—
242C22—44no—
243C22—48no—
244C22—42no—
245C22—42no—
246C22#178112yes—
247C22#178111yescharacteristic
248C22C2×#35612yes—
249C22#213512yes—
250C4—24no—
251C4—48no—
252C4—44no—
253C4—48no—
254C4—42no—
255C4—82no—
256C2—22no—
257C2—22no—
258C2—22no—
259C2—44no—
260C2#854111yescharacteristic
261C2#1270011yescharacteristic
262C2#1270411yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12700C2semidirect
194. C23#356semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
#127002
#85502