Finite Group Databasebeta

← #149429

Subgroups — #149429

Order 512.

1156 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2563C211yesmaximal, characteristic
2C2×#1393C214yesmaximal
3#10761C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic, complemented
5C22×C4×D4C2211yescharacteristic
6C22×#99C2214yes—
7#2563—28no—
8C2×#393—22no—
9C2×#390C2212yes—
10C2×#382C412yescomplemented
11C2×#380—22no—
12C2×C4×C4⋊C4C411yescharacteristic, complemented
13C2×#342—22no—
14C22⋊#95—22no—
15#1393—28no—
16C23×Q8—21no—
17C23×D4C2311yescharacteristic
18C24×C4D412yes—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—42no—
21C2×C4⋊Q8—42no—
22C2×C4×D4—24no—
23C2×C4×D4—42no—
24C2×C4×D4—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C22⋊C4—24no—
30C22×C22⋊C4—24no—
31C22×C22⋊C4D412yes—
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C412yes—
34C2×#99—28no—
35C2×#99—28no—
36C2×#99—24no—
37C2×#99Q818yescomplemented
38#393—216no—
39#390—24no—
40#390D418yes—
41#382—24no—
42#382—24no—
43#380—216no—
44C4×C4⋊C4—24no—
45C2×#95—21no—
46C2×#95—24no—
47C2×#95—44no—
48#342—44no—
49C25C2×D412yes—
50C22×Q8—24no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8—84no—
54C22×D4—41no—
55C22×D4—41no—
56C22×D4—42no—
57C22×D4C2×D414yes—
58C22×D4C2×Q814yes—
59C22×D4#4114yes—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—42no—
65C23×C4—48no—
66C23×C4—44no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C4⋊Q8—416no—
72C4×D4—416no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4C22⋊C414yes—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—28no—
88C2×C22⋊C4—216no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4—44no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—44no—
94C2×C22⋊C4—44no—
95C2×C22⋊C4—88no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—44no—
101C2×C42C22⋊C414yes—
102#99—216no—
103#99—48no—
104#95—24no—
105#95—216no—
106#95—88no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—44no—
112C24—44no—
113C24C2×C22⋊C411yescharacteristic
114C24#12014yes—
115C24C22⋊Q814yes—
116C24#12412yes—
117C24C4⋊Q812yes—
118C2×Q8—48no—
119C2×Q8—88no—
120C2×Q8—88no—
121C2×Q8—88no—
122C2×D4—44no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4C22⋊Q818yes—
129C2×D4#12414yes—
130C2×D4C4⋊Q814yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—48no—
145C22×C4—416no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—48no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—48no—
155C22×C4—48no—
156C22×C4—84no—
157C22×C4—88no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4—88no—
161C22×C4—88no—
162C22×C4#9912yes—
163C22×C4#9912yes—
164C22×C4C2×C22⋊C412yes—
165C22×C4C4×Q814yes—
166C4⋊C4—44no—
167C4⋊C4—48no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C22⋊C4—48no—
172C22⋊C4—48no—
173C22⋊C4—48no—
174C22⋊C4—816no—
175C22⋊C4—84no—
176C22⋊C4—84no—
177C42—28no—
178C42—44no—
179C42—44no—
180C23—22no—
181C23—22no—
182C23—24no—
183C23—22no—
184C23—22no—
185C23—24no—
186C23—24no—
187C23—22no—
188C23—22no—
189C23—24no—
190C23—44no—
191C23—44no—
192C23—48no—
193C23—44no—
194C23—44no—
195C23—44no—
196C23—44no—
197C23—82no—
198C23—82no—
199C23—84no—
200C23—84no—
201C23#37911yescharacteristic
202C23#39112yes—
203C23#39314yes—
204C23#39514yes—
205C23C2×#9911yescharacteristic
206C23C2×#9911yescharacteristic
207C23#41012yes—
208Q8—816no—
209D4—48no—
210D4—44no—
211D4—44no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—816no—
233C2×C4—88no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—82no—
237C2×C4—88no—
238C2×C4—816no—
239C2×C4—164no—
240C2×C4—164no—
241C2×C4—168no—
242C2×C4#35414yes—
243C2×C4#39114yes—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—24no—
252C22—44no—
253C22—44no—
254C22—44no—
255C22—44no—
256C22—48no—
257C22—44no—
258C22—44no—
259C22—84no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22#144611yescharacteristic
264C22#178111yescharacteristic
265C22#178211yescharacteristic
266C22#199612yes—
267C22#199612yes—
268C22C2×#35412yes—
269C22#212812yes—
270C4—24no—
271C4—24no—
272C4—42no—
273C4—84no—
274C4—82no—
275C4—84no—
276C4—168no—
277C2—21no—
278C2—21no—
279C2—22no—
280C2—44no—
281C2—44no—
282C2#1138111yescharacteristic
283C2#1269411yescharacteristic
284C2#1269711yescharacteristic
285C2#1329914yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊Q8C4semidirect
10. C2×#382C4semidirect
12. C2×C4×C4⋊C4C4semidirect
37. C2×#99Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#13932
#107612