Finite Group Databasebeta

← #11723622

Subgroups — #11723622

Order 864.

780 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 140 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×#6312C211yesmaximal, characteristic
2C22×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#6289C211yesmaximal, characteristic, complemented
4C32×#217C211yesmaximal, characteristic, complemented
5#88806C211yesmaximal, characteristic, complemented
6#88804C211yesmaximal, characteristic, complemented
7Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#64090—31nomaximal
9#63952—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C3⋊(C3⋊D3)C2211yescharacteristic
12#6312C414yescomplemented
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊Dic3—22no—
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C2×C32×Dic3C2211yescharacteristic
19#6289—22no—
20C2×#638—31no—
21C2×#638—34no—
22C2×#638—64no—
23C22×C3⋊Dic3D311yescharacteristic, complemented
24C2×C4×C3⋊D3—31no—
25C22×C3×Dic3—34no—
26C2×#624—34no—
27C32×C22⋊C4—31no—
28#3752—31no—
29#3750—31no—
30C4×C3⋊Dic3—31no—
31C3×#217D314yescomplemented
32#3724—34no—
33#3722—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⋊D3)C2×C412yescomplemented
42C3⋊(C3⋊Dic3)C2×C412yescomplemented
43C3×C3⋊Dic3—21no—
44C3×C3⋊Dic3D412yescomplemented
45C32×Dic3—21no—
46C32×Dic3—21no—
47C4×#67—91no—
48#67⋊C4—94no—
49C23×C32—24no—
50C23×C32C2×D314yes—
51C23×C32C2×D311yescharacteristic
52C22×C3⋊D3—31no—
53C22×C3⋊D3—34no—
54C22×C3⋊D3—64no—
55C2×C32×C4—31no—
56C2×C32×C4—31no—
57#638—34no—
58#638—316no—
59#638—616no—
60C2×C3⋊Dic3—22no—
61C2×C3⋊Dic3—31no—
62C2×C3⋊Dic3—34no—
63C2×C3⋊Dic3—64no—
64C2×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊Dic3C2×D311yescharacteristic
66C4×C3⋊D3—62no—
67C2×C3×Dic3—34no—
68C2×C3×Dic3—34no—
69C2×C3×Dic3—68no—
70C2×C3×Dic3C2×D314yes—
71C2×C3×Dic3C2×D314yes—
72#624—68no—
73C2×C33—22no—
74C2×C33C22×C411yescharacteristic
75C2×C33C2×D411yescharacteristic
76C2×C33#4111yescharacteristic
77C3⋊(C3⋊D3)—22no—
78C22×C3×C4—91no—
79C2×#67—94no—
80C2×#67—91no—
81C2×#67—184no—
82C22×Dic3—34no—
83C2×C4×D3—94no—
84C2×C4×D3—91no—
85C3×C22⋊C4—34no—
86#217C3⋊D311yescharacteristic, complemented
87(C2×D3)⋊C4—94no—
88(C2×D3)⋊C4—91no—
89Dic3⋊C4—94no—
90Dic3⋊C4—91no—
91C4×Dic3—94no—
92C4×Dic3—91no—
93C22×C32—21no—
94C22×C32—21no—
95C22×C32—28no—
96C22×C32—24no—
97C22×C32—24no—
98C22×C32—24no—
99C22×C32—44no—
100C22×C32—44no—
101C22×C32C4×D318yescomplemented
102C22×C32C4×D312yescomplemented
103C22×C32C22×D314yes—
104C22×C32C22×D311yescharacteristic
105C2×C3⋊D3—32no—
106C2×C3⋊D3—38no—
107C2×C3⋊D3—68no—
108C2×C3⋊D3—61no—
109C2×C3⋊D3—61no—
110C2×C3⋊D3—64no—
111C2×C3⋊D3—64no—
112C2×C3⋊D3—124no—
113C2×C3⋊D3—124no—
114C32×C4—61no—
115C32×C4—61no—
116C3⋊Dic3—21no—
117C3⋊Dic3—32no—
118C3⋊Dic3—38no—
119C3⋊Dic3—68no—
120C3⋊Dic3#6712yescomplemented
121C3×Dic3—24no—
122C3×Dic3—24no—
123C3×Dic3—38no—
124C3×Dic3—64no—
125C4×D4—271noSylow
126C33C4×D411yescharacteristic, complemented, Sylow
127C23×C3—24no—
128C23×C3D3214yes—
129C23×C3C2×C3⋊D311yescharacteristic
130C22×D3—94no—
131C22×D3—91no—
132C22×D3—184no—
133C2×C3×C4—34no—
134C2×C3×C4—34no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—182no—
138#67—916no—
139#67—94no—
140#67—1816no—
141C2×Dic3—34no—
142C2×Dic3—34no—
143C2×Dic3—68no—
144C2×Dic3—94no—
145C2×Dic3—91no—
146C2×Dic3—184no—
147C2×Dic3C2×C3⋊D311yescharacteristic
148C2×Dic3C2×C3⋊D311yescharacteristic
149C4×D3—182no—
150C4×D3—188no—
151C2×C32—22no—
152C2×C32—24no—
153C2×C32—24no—
154C2×C32—24no—
155C2×C32—28no—
156C2×C32—48no—
157C2×C32C2×C4×D314yes—
158C2×C32C2×C4×D311yescharacteristic
159C2×C32#23511yescharacteristic
160C2×C32D3×D414yes—
161C2×C32#23714yes—
162C2×C32C2×#6711yescharacteristic
163C3⋊D3—62no—
164C3⋊D3—68no—
165C3⋊D3—128no—
166C2×D4—271no—
167C22×C4—91no—
168C22×C4—271no—
169C4⋊C4—271no—
170C22⋊C4—31no—
171C22⋊C4—271no—
172C42—271no—
173C22×C3—24no—
174C22×C3—24no—
175C22×C3—21no—
176C22×C3—21no—
177C22×C3—28no—
178C22×C3—24no—
179C22×C3—44no—
180C22×C3—44no—
181C22×C3#62418yescomplemented
182C22×C3C4×C3⋊D312yescomplemented
183C22×C3C2×D3214yes—
184C22×C3C22×C3⋊D311yescharacteristic
185C2×D3—98no—
186C2×D3—92no—
187C2×D3—188no—
188C2×D3—184no—
189C2×D3—184no—
190C2×D3—181no—
191C2×D3—181no—
192C2×D3—364no—
193C2×D3—364no—
194C3×C4—64no—
195C3×C4—64no—
196C3×C4—92no—
197C3×C4—181no—
198Dic3—21no—
199Dic3—21no—
200Dic3—38no—
201Dic3—64no—
202Dic3—98no—
203Dic3—92no—
204Dic3—188no—
205C32—24no—
206C32#67⋊C414yescomplemented
207C32C4×#6711yescharacteristic, complemented
208C23—271no—
209C23D3×C3⋊D311yescharacteristic
210D4—274no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—182no—
216C2×C4—271no—
217C2×C4—542no—
218C2×C3—22no—
219C2×C3—28no—
220C2×C3—24no—
221C2×C3—24no—
222C2×C3—24no—
223C2×C3—48no—
224C2×C3#380414yes—
225C2×C3C2×#62414yes—
226C2×C3D3×#6714yes—
227C2×C3C2×C4×C3⋊D311yescharacteristic
228C2×C3D4×C3⋊D311yescharacteristic
229C2×C3#383011yescharacteristic
230D3—182no—
231D3—188no—
232D3—368no—
233C22—21no—
234C22—21no—
235C22—272no—
236C22—541no—
237C22—541no—
238C22#628912yescomplemented
239C22C2×D3×C3⋊D311yescharacteristic
240C4—61no—
241C4—61no—
242C4—92no—
243C4—181no—
244C4—272no—
245C3—24no—
246C3#6395214yescomplemented
247C3#6409011yescharacteristic, complemented
248C2—22no—
249C2—542no—
250C2#8902811yescharacteristic
251C2C2×#628911yescharacteristic
252C2C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutatorC2×C33C22×C4
FittingC23×C33C22
socle, cosocleC22×C33C23
prime radicalC2×#6312C2
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
2. C22×C3×C3⋊Dic3C2semidirect
3. C2×#6289C2semidirect
4. C32×#217C2semidirect
5. #88806C2semidirect
6. #88804C2semidirect
7. Dic3×C3⋊Dic3C2semidirect
12. #6312C4semidirect
23. C22×C3⋊Dic3D3semidirect
31. C3×#217D3semidirect
37. C22×C33C2×C4semidirect
41. C2×C3⋊(C3⋊D3)C2×C4semidirect
42. C3⋊(C3⋊Dic3)C2×C4semidirect
44. C3×C3⋊Dic3D4semidirect
86. #217C3⋊D3semidirect
101. C22×C32C4×D3semidirect
102. C22×C32C4×D3semidirect
120. C3⋊Dic3#67semidirect
126. C33C4×D4semidirect
181. C22×C3#624semidirect
182. C22×C3C4×C3⋊D3semidirect
206. C32#67⋊C4semidirect
207. C32C4×#67semidirect
238. C22#6289semidirect
246. C3#63952semidirect
247. C3#64090semidirect

Maximal subgroups

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

subgroupindex
C2×#63122
C22×C3×C3⋊Dic32
C2×#62892
C32×#2172
#888062
#888042
Dic3×C3⋊Dic32
#640903
#639523