Finite Group Databasebeta

← #11723626

Subgroups — #11723626

Order 864.

780 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C22×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic
3C2×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×#217C211yesmaximal, characteristic, complemented
5#88807C211yesmaximal, characteristic, complemented
6#88803C211yesmaximal, characteristic, complemented
7Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#64090—31nomaximal
9#67⋊Dic3—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)—22no—
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#638C414yescomplemented
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C2×C32×Dic3C2211yescharacteristic
19Dic3×C3⋊D3—22no—
20C2×#638D311yescharacteristic
21C22×C3⋊Dic3—31no—
22C22×C3⋊Dic3—34no—
23C22×C3⋊Dic3—64no—
24C2×C4×C3⋊D3—31no—
25C2×C3×#67—34no—
26C2×D3×Dic3—34no—
27C32×C22⋊C4—31no—
28#3752—31no—
29#3750—31no—
30C4×C3⋊Dic3—31no—
31C3×#217D314yescomplemented
32Dic3⋊Dic3—34no—
33(C2×D3)⋊Dic3—34no—
34Dic32—34no—
35C22×C33—21no—
36C22×C33—21no—
37C22×C33C2×C412yescomplemented
38C22×C33C2311yescharacteristic
39C2×C3×C3⋊D3—21no—
40C2×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3C2×C412yescomplemented
42C3⋊(C3⋊Dic3)—21no—
43C3⋊(C3⋊Dic3)D412yescomplemented
44C3×C3⋊Dic3C2×C412yescomplemented
45C32×Dic3—21no—
46C32×Dic3—21no—
47D4×Dic3—91no—
48#67⋊C4—94no—
49C23×C32—24no—
50C23×C32C2×D314yes—
51C23×C32C2×D311yescharacteristic
52C22×C3⋊D3C2×D311yescharacteristic
53C22×C3×D3—34no—
54C2×C32×C4—31no—
55C2×C32×C4—31no—
56#638Dic314yescomplemented
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—31no—
59C2×C3⋊Dic3—34no—
60C2×C3⋊Dic3—34no—
61C2×C3⋊Dic3—64no—
62C2×C3⋊Dic3—64no—
63C2×C3⋊Dic3—62no—
64C2×C3⋊Dic3—68no—
65C2×C3⋊Dic3—616no—
66C2×C3⋊Dic3C2×D311yescharacteristic
67C4×C3⋊D3—62no—
68C3×#67—316no—
69C2×C3×Dic3—34no—
70C2×C3×Dic3C2×D314yes—
71C2×C3×Dic3C2×D314yes—
72D3×Dic3—68no—
73C2×C33—22no—
74C2×C33C22×C411yescharacteristic
75C2×C33C2×D411yescharacteristic
76C2×C33#4111yescharacteristic
77C3×C3⋊D3—22no—
78C2×C3×D4—91no—
79C2×#67—34no—
80C22×Dic3—94no—
81C22×Dic3—91no—
82C22×Dic3—91no—
83C22×Dic3—184no—
84C2×C4×D3—94no—
85C3×C22⋊C4—34no—
86#217—91no—
87#217C3⋊D311yescharacteristic, complemented
88(C2×D3)⋊C4—94no—
89C4⋊Dic3—91no—
90Dic3⋊C4—94no—
91C4×Dic3—94no—
92C4×Dic3—91no—
93C22×C32—28no—
94C22×C32—28no—
95C22×C32—24no—
96C22×C32—21no—
97C22×C32—24no—
98C22×C32—21no—
99C22×C32—28no—
100C22×C32—24no—
101C22×C32C4×D318yescomplemented
102C22×C32C2×Dic312yescomplemented
103C22×C32C22×D314yes—
104C22×C32C22×D311yescharacteristic
105C2×C3⋊D3—21no—
106C2×C3⋊D3—21no—
107C2×C3⋊D3C2×Dic312yescomplemented
108C2×C3×D3—38no—
109C2×C3×D3—64no—
110C2×C3×D3—64no—
111C32×C4—61no—
112C32×C4—61no—
113C3⋊Dic3—32no—
114C3⋊Dic3—38no—
115C3⋊Dic3—68no—
116C3⋊Dic3—68no—
117C3⋊Dic3—64no—
118C3⋊Dic3—61no—
119C3⋊Dic3C2×Dic312yescomplemented
120C3×Dic3—24no—
121C3×Dic3—24no—
122C3×Dic3—38no—
123C4×D4—271noSylow
124C33C4×D411yescharacteristic, complemented, Sylow
125C23×C3—24no—
126C23×C3—91no—
127C23×C3D3214yes—
128C23×C3C2×C3⋊D311yescharacteristic
129C22×D3—34no—
130C3×D4—94no—
131C2×C3×C4—34no—
132C2×C3×C4—34no—
133C2×C3×C4—91no—
134#67—316no—
135C2×Dic3—34no—
136C2×Dic3—94no—
137C2×Dic3—91no—
138C2×Dic3—94no—
139C2×Dic3—91no—
140C2×Dic3—184no—
141C2×Dic3—184no—
142C2×Dic3—182no—
143C2×Dic3—188no—
144C2×Dic3—182no—
145C2×Dic3—1816no—
146C2×Dic3C2×C3⋊D311yescharacteristic
147C2×Dic3C2×C3⋊D311yescharacteristic
148C4×D3—188no—
149C2×C32—216no—
150C2×C32—28no—
151C2×C32—22no—
152C2×C32—24no—
153C2×C32—24no—
154C2×C32—24no—
155C2×C32C2×C4×D314yes—
156C2×C32D3×D414yes—
157C2×C32D3×D411yescharacteristic
158C2×C32#23714yes—
159C2×C32#23711yescharacteristic
160C2×C32C22×Dic311yescharacteristic
161C3⋊D3—22no—
162C3×D3—68no—
163C2×D4—91no—
164C22×C4—271no—
165C22×C4—271no—
166C4⋊C4—271no—
167C22⋊C4—31no—
168C22⋊C4—271no—
169C42—271no—
170C22×C3—28no—
171C22×C3—28no—
172C22×C3—24no—
173C22×C3—24no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—28no—
177C22×C3—24no—
178C22×C3—92no—
179C22×C3—181no—
180C22×C3—181no—
181C22×C3D3×Dic318yescomplemented
182C22×C3C4×C3⋊D312yescomplemented
183C22×C3C2×D3214yes—
184C22×C3C22×C3⋊D311yescharacteristic
185C2×D3—38no—
186C2×D3—64no—
187C2×D3—64no—
188C3×C4—64no—
189C3×C4—64no—
190C3×C4—92no—
191Dic3—21no—
192Dic3—21no—
193Dic3—38no—
194Dic3—98no—
195Dic3—92no—
196Dic3—188no—
197Dic3—188no—
198Dic3—184no—
199Dic3—181no—
200C32—24no—
201C32#67⋊C414yescomplemented
202C32D4×Dic311yescharacteristic, complemented
203C23—91no—
204C23D3×C3⋊D311yescharacteristic
205D4—94no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—91no—
209C2×C4—271no—
210C2×C4—271no—
211C2×C4—542no—
212C2×C4—542no—
213C2×C3—216no—
214C2×C3—22no—
215C2×C3—28no—
216C2×C3—24no—
217C2×C3—24no—
218C2×C3—24no—
219C2×C3—182no—
220C2×C3C2×D3×Dic314yes—
221C2×C3#380514yes—
222C2×C3#381114yes—
223C2×C3C2×C4×C3⋊D311yescharacteristic
224C2×C3D4×C3⋊D311yescharacteristic
225C2×C3#383011yescharacteristic
226D3—68no—
227C22—21no—
228C22—21no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22Dic3×C3⋊D312yescomplemented
233C22C2×D3×C3⋊D311yescharacteristic
234C4—61no—
235C4—61no—
236C4—92no—
237C4—272no—
238C4—541no—
239C3—24no—
240C3#67⋊Dic314yescomplemented
241C3#6409011yescharacteristic, complemented
242C2—22no—
243C2—182no—
244C2#8902911yescharacteristic
245C2C2×Dic3×C3⋊D311yescharacteristic
246C2#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutatorC2×C33C22×C4
FittingC23×C33C22
socle, cosocleC22×C33C23
prime radicalC2×C3×#638C2
hypercenterC23D3×C3⋊D3
hypocenterC33C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊(C3⋊Dic3)C2semidirect
3. C2×Dic3×C3⋊D3C2semidirect
4. C32×#217C2semidirect
5. #88807C2semidirect
6. #88803C2semidirect
7. Dic3×C3⋊Dic3C2semidirect
15. C3×#638C4semidirect
31. C3×#217D3semidirect
37. C22×C33C2×C4semidirect
41. C2×C3×C3⋊D3C2×C4semidirect
43. C3⋊(C3⋊Dic3)D4semidirect
44. C3×C3⋊Dic3C2×C4semidirect
56. #638Dic3semidirect
87. #217C3⋊D3semidirect
101. C22×C32C4×D3semidirect
102. C22×C32C2×Dic3semidirect
107. C2×C3⋊D3C2×Dic3semidirect
119. C3⋊Dic3C2×Dic3semidirect
124. C33C4×D4semidirect
181. C22×C3D3×Dic3semidirect
182. C22×C3C4×C3⋊D3semidirect
201. C32#67⋊C4semidirect
202. C32D4×Dic3semidirect
232. C22Dic3×C3⋊D3semidirect
240. C3#67⋊Dic3semidirect
241. C3#64090semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊Dic3)2
C2×C3×#6382
C2×Dic3×C3⋊D32
C32×#2172
#888072
#888032
Dic3×C3⋊Dic32
#640903
#67⋊Dic33