Finite Group Databasebeta

← #10589734

Subgroups — #10589734

Order 576.

420 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 139 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×Q8×D9C211yesmaximal, characteristic, complemented
2C22×Dic18C211yesmaximal, characteristic, complemented
3C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4#63506C211yesmaximal, characteristic, complemented
5C4×#611C211yesmaximal, characteristic
6#63459C211yesmaximal, characteristic, complemented
7#63458C212yesmaximal, complemented
8#63450C212yesmaximal, complemented
9#63449C211yesmaximal, characteristic, complemented
10#611⋊C4C212yesmaximal
11#63440C212yesmaximal, complemented
12#5632—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C2×#611C2211yescharacteristic
16C22×Dic9C2212yes—
17Q8×D9—24no—
18C2×C4×D9C2211yescharacteristic
19C2×C4×D9C2212yes—
20C2×Dic18—24no—
21C2×Dic18C2211yescharacteristic
22C2×Dic18C2212yes—
23C2×Dic18C2212yes—
24C2×Dic18C2211yescharacteristic
25C9×C4⋊C4C2211yescharacteristic
26C9×C4⋊C4C2212yes—
27C9×C22⋊C4C2212yes—
28#3676C2211yescharacteristic
29(C2×D9)⋊C4C2211yescharacteristic
30(C2×D9)⋊C4C2212yes—
31C4⋊Dic9C2212yes—
32Dic9⋊C4C2211yescharacteristic
33Dic9⋊C4C2212yes—
34Dic9⋊C4C2212yes—
35Dic9⋊C4C2212yes—
36C4×Dic9C2211yescharacteristic
37C4×Dic9C2212yes—
38C2×D3×Q8—31no—
39C22×Dic6—31no—
40C3×C22⋊Q8D311yescharacteristic
41#944—31no—
42C4×#67—31no—
43#897—31no—
44#896—32no—
45#888—32no—
46#887—31no—
47#67⋊C4—32no—
48#878—32no—
49C23×C9C2311yescharacteristic
50C22×D9C2311yescharacteristic
51Q8×C9—22no—
52C2×C4×C9—22no—
53C2×C4×C9C2312yes—
54C2×C4×C9C2312yes—
55C2×C4×C9C2311yescharacteristic
56C2×C4×C9C2311yescharacteristic
57#611Q814yescomplemented
58C2×Dic9—24no—
59C2×Dic9C2311yescharacteristic
60C2×Dic9C2312yes—
61C2×Dic9C2311yescharacteristic
62C2×Dic9C2312yes—
63C2×Dic9C2311yescharacteristic
64C4×D9—22no—
65C4×D9—24no—
66Dic18—22no—
67Dic18—24no—
68Dic18—24no—
69Dic18D414yescomplemented
70D4×Q8—91noSylow
71C2×C3×Q8C2×D311yescharacteristic
72C22×C3×C4C2×D311yescharacteristic
73C2×#67—31no—
74C22×Dic3—32no—
75D3×Q8—64no—
76C2×C4×D3—32no—
77C2×C4×D3—31no—
78C2×Dic6—31no—
79C2×Dic6—32no—
80C2×Dic6—32no—
81C2×Dic6—31no—
82C2×Dic6—64no—
83C3×C4⋊C4C2×D311yescharacteristic
84C3×C4⋊C4C2×D312yes—
85C3×C22⋊C4C2×D312yes—
86#217—31no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—31no—
89C4⋊Dic3—32no—
90Dic3⋊C4—32no—
91Dic3⋊C4—32no—
92Dic3⋊C4—32no—
93Dic3⋊C4—31no—
94C4×Dic3—32no—
95C4×Dic3—31no—
96C22×C9—21no—
97C22×C9—21no—
98C22×C9C2×Q812yescomplemented
99C22×C9C2411yescharacteristic
100C2×D9—21no—
101C2×D9—21no—
102C2×D9C2×Q812yescomplemented
103C4×C9—21no—
104C4×C9—22no—
105C4×C9—22no—
106C4×C9C2×D412yes—
107Dic9—21no—
108Dic9—22no—
109Dic9—21no—
110Dic9C2×D414yes—
111Dic9C2×Q812yescomplemented
112C22×Q8—91no—
113C22×Q8—91no—
114C4⋊Q8—91no—
115C4⋊Q8—92no—
116C22⋊Q8—92no—
117C22⋊Q8—91no—
118C22⋊Q8—92no—
119C22⋊Q8D911yescharacteristic, complemented
120C4×Q8—91no—
121C4×D4—91no—
122C4×D4—92no—
123C23×C3C22×D311yescharacteristic
124C22×D3—31no—
125C3×Q8—22no—
126C2×C3×C4—22no—
127C2×C3×C4C22×D312yes—
128C2×C3×C4C22×D312yes—
129C2×C3×C4C22×D311yescharacteristic
130C2×C3×C4C22×D311yescharacteristic
131#67—34no—
132C2×Dic3—31no—
133C2×Dic3—32no—
134C2×Dic3—31no—
135C2×Dic3—32no—
136C2×Dic3—31no—
137C2×Dic3—64no—
138C4×D3—64no—
139C4×D3—62no—
140Dic6—34no—
141Dic6—64no—
142Dic6—64no—
143Dic6—62no—
144C2×C9—22no—
145C2×C9C22×D411yescharacteristic
146C2×C9C22×Q811yescharacteristic
147C2×C92−1+411yescharacteristic
148D9—22no—
149C2×Q8—91no—
150C2×Q8—92no—
151C2×Q8—92no—
152C2×Q8—91no—
153C2×Q8—184no—
154C2×Q8—184no—
155C2×Q8C2×D911yescharacteristic
156C2×D4—91no—
157C22×C4—92no—
158C22×C4—92no—
159C22×C4—91no—
160C22×C4C2×D911yescharacteristic
161C4⋊C4—91no—
162C4⋊C4—92no—
163C4⋊C4—92no—
164C4⋊C4—92no—
165C4⋊C4—92no—
166C4⋊C4C2×D911yescharacteristic
167C4⋊C4C2×D912yes—
168C22⋊C4—91no—
169C22⋊C4—91no—
170C22⋊C4—92no—
171C22⋊C4C2×D912yes—
172C42—92no—
173C42—91no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3D3×Q812yes—
177C22×C3C23×D311yescharacteristic
178C2×D3—32no—
179C2×D3—61no—
180C2×D3—61no—
181C3×C4—21no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4D3×D412yes—
185Dic3—34no—
186Dic3—32no—
187Dic3—61no—
188Dic3—62no—
189Dic3—61no—
190C9D4×Q811yescharacteristic, complemented, Sylow
191C23—91no—
192C23C22×D911yescharacteristic
193Q8—22no—
194Q8—94no—
195Q8—184no—
196Q8—184no—
197Q8—182no—
198D4—94no—
199C2×C4—22no—
200C2×C4—91no—
201C2×C4—92no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—92no—
205C2×C4—184no—
206C2×C4—184no—
207C2×C4—182no—
208C2×C4C22×D912yes—
209C2×C4C22×D911yescharacteristic
210C2×C4C22×D912yes—
211C2×C4C22×D911yescharacteristic
212C2×C3—22no—
213C2×C3C2×D3×D411yescharacteristic
214C2×C3C2×D3×Q811yescharacteristic
215C2×C3#101011yescharacteristic
216D3—62no—
217C22—21no—
218C22—21no—
219C22—92no—
220C22—181no—
221C22—181no—
222C22Q8×D912yescomplemented
223C22C23×D911yescharacteristic
224C4—21no—
225C4—22no—
226C4—22no—
227C4—94no—
228C4—92no—
229C4—182no—
230C4—181no—
231C4—181no—
232C4D4×D912yes—
233C3#563211yescharacteristic
234C2—22no—
235C2—182no—
236C2C2×D4×D911yescharacteristic
237C2C2×Q8×D911yescharacteristic
238C2#6371611yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
prime radicalC2×#611C22
hypercenterC22⋊Q8D9
hypocenterC9D4×Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D9C2semidirect
2. C22×Dic18C2semidirect
3. C9×C22⋊Q8C2semidirect
4. #63506C2semidirect
6. #63459C2semidirect
7. #63458C2semidirect
8. #63450C2semidirect
9. #63449C2semidirect
11. #63440C2semidirect
57. #611Q8semidirect
69. Dic18D4semidirect
98. C22×C9C2×Q8semidirect
102. C2×D9C2×Q8semidirect
111. Dic9C2×Q8semidirect
119. C22⋊Q8D9semidirect
190. C9D4×Q8semidirect
222. C22Q8×D9semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D92
C22×Dic182
C9×C22⋊Q82
#635062
C4×#6112
#634592
#634582
#634502
#634492
#611⋊C42
#634402
#56323