Finite Group Databasebeta

← #13179653

Subgroups — #13179653

Order 1728.

4676 conjugacy classes of subgroups; only the 891 normal subgroups are shown.

normal subgroups only

This group has 4676 conjugacy classes of subgroups — too many to draw. Only its 891 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×#89076C211yesmaximal, characteristic, complemented
2C22×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
3C33×#117C211yesmaximal, characteristic, complemented
4#11723936C214yesmaximal, complemented
5#11723923C214yesmaximal, complemented
6#11723916C212yesmaximal, complemented
7#11723915C212yesmaximal, complemented
8#10595996—313nomaximal
9C23×C3⋊(C3⋊D3)C2212yes—
10C22×C33×C4C2211yescharacteristic
11C2×#6312C2212yes—
12#89076—24no—
13C2×C4⋊(C3⋊(C3⋊D3))—24no—
14C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
17C2×C4⋊(C3⋊(C3⋊D3))C2218yescomplemented
18C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
19C2×#6308C2211yescharacteristic, complemented
20C33×C4⋊C4C2212yescomplemented
21C33×C22⋊C4C2212yes—
22C33×C42C2212yescomplemented
23#88859C2218yescomplemented
24C2×#3828—313no—
25C22×C4⋊(C3⋊D3)—313no—
26C32×#117D3113yescomplemented
27#64104—352no—
28#64091—352no—
29#64084—326no—
30#64083—326no—
31C23×C33C2311yescharacteristic
32C22×C3⋊(C3⋊D3)—24no—
33C22×C3⋊(C3⋊D3)—44no—
34C22×C3⋊(C3⋊D3)C2314yes—
35C22×C3⋊(C3⋊D3)C2312yes—
36C2×C33×C4D414yescomplemented
37C2×C33×C4C2314yes—
38C2×C33×C4C2311yescharacteristic
39C2×C33×C4C2311yescharacteristic
40#6312—24no—
41C2×C3⋊(C3⋊Dic3)C2312yes—
42C4⋊(C3⋊(C3⋊D3))—24no—
43C4⋊(C3⋊(C3⋊D3))—24no—
44C4⋊(C3⋊(C3⋊D3))—22no—
45C4⋊(C3⋊(C3⋊D3))—48no—
46C4×C3⋊(C3⋊D3)—24no—
47#6308—22no—
48#5525—913no—
49C23×C3⋊D3—326no—
50C22×C32×C4C2×D3113yes—
51C2×#638—326no—
52#3828—652no—
53C2×C4⋊(C3⋊D3)—313no—
54C2×C4⋊(C3⋊D3)—326no—
55C2×C4⋊(C3⋊D3)—326no—
56C2×C4⋊(C3⋊D3)—3104no—
57C2×C4⋊(C3⋊D3)—652no—
58C2×C4×C3⋊D3—326no—
59C2×#634—313no—
60C32×C4⋊C4C2×D3126yescomplemented
61C32×C22⋊C4C2×D3126yes—
62C32×C42C2×D3126yescomplemented
63#3752—3104no—
64C22×C33—22no—
65C22×C33C2×D412yes—
66C22×C33C2411yescharacteristic
67C2×C3⋊(C3⋊D3)—22no—
68C2×C3⋊(C3⋊D3)—24no—
69C2×C3⋊(C3⋊D3)—24no—
70C2×C3⋊(C3⋊D3)—28no—
71C2×C3⋊(C3⋊D3)—48no—
72C33×C4—24no—
73C33×C4C2×D412yes—
74C33×C4C2×D412yes—
75C3⋊(C3⋊Dic3)—22no—
76C2×#235—913no—
77C22×D12—913no—
78C3×#117C3⋊D3113yescomplemented
79#895—952no—
80#882—952no—
81#875—926no—
82#874—926no—
83C23×C32C22×D3113yes—
84C22×C3⋊D3—352no—
85C22×C3⋊D3—326no—
86C22×C3⋊D3—652no—
87C22×C3⋊D3—1252no—
88C2×C32×C4D12152yescomplemented
89C2×C32×C4C22×D3152yes—
90C2×C32×C4C22×D3113yes—
91C2×C32×C4C22×D3113yes—
92#638—652no—
93C2×C3⋊Dic3—326no—
94C4⋊(C3⋊D3)—652no—
95C4⋊(C3⋊D3)—652no—
96C4⋊(C3⋊D3)—626no—
97C4⋊(C3⋊D3)—12104no—
98C4×C3⋊D3—652no—
99#634—626no—
100#535—271noSylow
101C2×C33—22no—
102C2×C33C22×D411yescharacteristic
103C2×C33H(5,𝔽2)12yes—
104C3⋊(C3⋊D3)—42no—
105C3⋊(C3⋊D3)—44no—
106C23×D3—926no—
107C22×C3×C4C2×C3⋊D3113yes—
108C2×#67—926no—
109#235—1852no—
110C2×D12—913no—
111C2×D12—926no—
112C2×D12—926no—
113C2×D12—9104no—
114C2×D12—1852no—
115C2×C4×D3—926no—
116C2×Dic6—913no—
117C3×C4⋊C4C2×C3⋊D3126yescomplemented
118C3×C22⋊C4C2×C3⋊D3126yes—
119C3×C42C2×C3⋊D3126yescomplemented
120(C2×D3)⋊C4—9104no—
121C22×C32—226no—
122C22×C32C2×D12126yes—
123C22×C32C23×D3113yes—
124C2×C3⋊D3—626no—
125C2×C3⋊D3—652no—
126C2×C3⋊D3—652no—
127C2×C3⋊D3—6104no—
128C2×C3⋊D3—12104no—
129C32×C4—252no—
130C32×C4C2×D12126yes—
131C32×C4C2×D12126yes—
132C3⋊Dic3—626no—
133C2×#41—271no—
134C22×D4—271no—
135#127—272no—
136#124—272no—
137#121—274no—
138#120—274no—
139#117C3⋊(C3⋊D3)11yescharacteristic, complemented
140C33#53511yescharacteristic, complemented, Sylow
141C23×C3C22×C3⋊D3113yes—
142C22×D3—952no—
143C22×D3—926no—
144C22×D3—1852no—
145C22×D3—3652no—
146C2×C3×C4C4⋊(C3⋊D3)152yescomplemented
147C2×C3×C4C22×C3⋊D3152yes—
148C2×C3×C4C22×C3⋊D3113yes—
149C2×C3×C4C22×C3⋊D3113yes—
150#67—1852no—
151C2×Dic3—926no—
152D12—1852no—
153D12—1852no—
154D12—1826no—
155D12—36104no—
156C4×D3—1852no—
157Dic6—1826no—
158C2×C32—226no—
159C2×C32C22×D12113yes—
160C2×C32#1009126yes—
161C3⋊D3—1226no—
162C3⋊D3—1252no—
163C24—272no—
164#41—544no—
165C2×Q8—271no—
166C2×D4—272no—
167C2×D4—272no—
168C2×D4—272no—
169C2×D4—271no—
170C2×D4—278no—
171C2×D4—544no—
172C22×C4—272no—
173C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
174C4⋊C4C2×C3⋊(C3⋊D3)12yescomplemented
175C22⋊C4—278no—
176C22⋊C4C2×C3⋊(C3⋊D3)12yes—
177C42C2×C3⋊(C3⋊D3)12yescomplemented
178C22×C3—226no—
179C22×C3C2×C4⋊(C3⋊D3)126yes—
180C22×C3C23×C3⋊D3113yes—
181C2×D3—1826no—
182C2×D3—1852no—
183C2×D3—1852no—
184C2×D3—18104no—
185C2×D3—36104no—
186C3×C4—252no—
187C3×C4C2×C4⋊(C3⋊D3)126yes—
188C3×C4C2×C4⋊(C3⋊D3)126yes—
189Dic3—1826no—
190C32#5525113yescomplemented
191C23—274no—
192C23—272no—
193C23—544no—
194C23—1084no—
195C23C22×C3⋊(C3⋊D3)11yescharacteristic
196Q8—542no—
197D4—544no—
198D4—544no—
199D4—544no—
200D4—542no—
201D4—1088no—
202C2×C4—272no—
203C2×C4—544no—
204C2×C4C4⋊(C3⋊(C3⋊D3))14yescomplemented
205C2×C4C22×C3⋊(C3⋊D3)14yes—
206C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
207C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
208C2×C3—226no—
209C2×C3C22×C4⋊(C3⋊D3)113yes—
210C2×C3#64366126yes—
211D3—3652no—
212D3—3626no—
213C22—22no—
214C22—542no—
215C22—544no—
216C22—544no—
217C22—548no—
218C22—1088no—
219C22C2×C4⋊(C3⋊(C3⋊D3))12yes—
220C22C23×C3⋊(C3⋊D3)11yescharacteristic
221C4—24no—
222C4—542no—
223C4C2×C4⋊(C3⋊(C3⋊D3))12yes—
224C4C2×C4⋊(C3⋊(C3⋊D3))12yes—
225C3#10595996113yescomplemented
226C2—22no—
227C2—1084no—
228C2—1082no—
229C2C22×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
230C2#1172468412yes—
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×#117C2
hypercenter#117C3⋊(C3⋊D3)
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C22×C4⋊(C3⋊(C3⋊D3))C2semidirect
3. C33×#117C2semidirect
4. #11723936C2semidirect
5. #11723923C2semidirect
6. #11723916C2semidirect
7. #11723915C2semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C2×C4×C3⋊(C3⋊D3)C22semidirect
19. C2×#6308C22semidirect
20. C33×C4⋊C4C22semidirect
22. C33×C42C22semidirect
23. #88859C22semidirect
26. C32×#117D3semidirect
36. C2×C33×C4D4semidirect
60. C32×C4⋊C4C2×D3semidirect
62. C32×C42C2×D3semidirect
78. C3×#117C3⋊D3semidirect
88. C2×C32×C4D12semidirect
117. C3×C4⋊C4C2×C3⋊D3semidirect
119. C3×C42C2×C3⋊D3semidirect
139. #117C3⋊(C3⋊D3)semidirect
140. C33#535semidirect
146. C2×C3×C4C4⋊(C3⋊D3)semidirect
174. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
177. C42C2×C3⋊(C3⋊D3)semidirect
190. C32#5525semidirect
204. C2×C4C4⋊(C3⋊(C3⋊D3))semidirect
225. C3#10595996semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C22×C4⋊(C3⋊(C3⋊D3))2
C33×#1172
#117239362
#117239232
#117239162
#117239152
#105959963