Finite Group Databasebeta

← #13179652

Subgroups — #13179652

Order 1728.

4620 conjugacy classes of subgroups; only the 895 normal subgroups are shown.

normal subgroups only

This group has 4620 conjugacy classes of subgroups — too many to draw. Only its 895 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#11723938C212yesmaximal, complemented
5#11723936C212yesmaximal, complemented
6#11723927C212yesmaximal, complemented
7#11723923C212yesmaximal, complemented
8C4×C4⋊(C3⋊(C3⋊D3))C214yesmaximal, complemented
9#10595995—313nomaximal
10C23×C3⋊(C3⋊D3)C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C2×#6312C2212yes—
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14#89076—24no—
15C2×C4⋊(C3⋊(C3⋊D3))C2214yes—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
17C2×C4×C3⋊(C3⋊D3)—24no—
18C2×C4×C3⋊(C3⋊D3)C2212yes—
19C2×C4×C3⋊(C3⋊D3)C2212yes—
20C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
21C2×#6308C2211yescharacteristic, complemented
22C33×C4⋊C4C2212yescomplemented
23C33×C22⋊C4C2212yes—
24C33×C42C2212yescomplemented
25#88859C2214yes—
26#88859C2214yescomplemented
27C4⋊(C3⋊(C3⋊Dic3))C2214yescomplemented
28C2×#3828—313no—
29C22×C4×C3⋊D3—313no—
30C32×#117D3113yescomplemented
31#64106—326no—
32#64104—326no—
33#64095—326no—
34#64091—326no—
35C4×C4⋊(C3⋊D3)—352no—
36C23×C33C2311yescharacteristic
37C22×C3⋊(C3⋊D3)—22no—
38C22×C3⋊(C3⋊D3)—24no—
39C22×C3⋊(C3⋊D3)C2312yes—
40C22×C3⋊(C3⋊D3)C2312yes—
41C2×C33×C4D414yescomplemented
42C2×C33×C4C2314yes—
43C2×C33×C4C2311yescharacteristic
44C2×C33×C4C2311yescharacteristic
45#6312—24no—
46C2×C3⋊(C3⋊Dic3)—22no—
47C2×C3⋊(C3⋊Dic3)C2312yes—
48C2×C3⋊(C3⋊Dic3)C2312yes—
49C4⋊(C3⋊(C3⋊D3))—28no—
50C4⋊(C3⋊(C3⋊D3))—22no—
51C4×C3⋊(C3⋊D3)—24no—
52C4×C3⋊(C3⋊D3)—24no—
53C4×C3⋊(C3⋊D3)—24no—
54#6308—22no—
55#5524—913no—
56C23×C3⋊D3—313no—
57C22×C32×C4C2×D3113yes—
58C2×#638—326no—
59C22×C3⋊Dic3—313no—
60#3828—652no—
61C2×C4⋊(C3⋊D3)—352no—
62C2×C4⋊(C3⋊D3)—313no—
63C2×C4×C3⋊D3—326no—
64C2×C4×C3⋊D3—326no—
65C2×C4×C3⋊D3—326no—
66C2×C4×C3⋊D3—652no—
67C2×#634—313no—
68C32×C4⋊C4C2×D3126yescomplemented
69C32×C22⋊C4C2×D3126yes—
70C32×C42C2×D3126yescomplemented
71#3752—352no—
72#3752—352no—
73C4⋊(C3⋊Dic3)—352no—
74C22×C33—22no—
75C22×C33C2×D412yes—
76C22×C33C2411yescharacteristic
77C2×C3⋊(C3⋊D3)—28no—
78C2×C3⋊(C3⋊D3)—22no—
79C2×C3⋊(C3⋊D3)—22no—
80C2×C3⋊(C3⋊D3)—22no—
81C2×C3⋊(C3⋊D3)—24no—
82C2×C3⋊(C3⋊D3)#4114yescomplemented
83C33×C4—24no—
84C33×C4C2×D412yes—
85C33×C4C2×D412yes—
86C3⋊(C3⋊Dic3)—22no—
87C3⋊(C3⋊Dic3)—22no—
88C2×#235—913no—
89C22×C4×D3—913no—
90C3×#117C3⋊D3113yescomplemented
91#897—926no—
92#895—926no—
93#886—926no—
94#882—926no—
95C4×D12—952no—
96C23×C32C22×D3113yes—
97C22×C3⋊D3—326no—
98C22×C3⋊D3—326no—
99C22×C3⋊D3—652no—
100C22×C3⋊D3—626no—
101C2×C32×C4D12152yescomplemented
102C2×C32×C4C22×D3152yes—
103C2×C32×C4C22×D3113yes—
104C2×C32×C4C22×D3113yes—
105#638—652no—
106C2×C3⋊Dic3—326no—
107C2×C3⋊Dic3—326no—
108C2×C3⋊Dic3—626no—
109C4⋊(C3⋊D3)—6104no—
110C4⋊(C3⋊D3)—626no—
111C4×C3⋊D3—652no—
112C4×C3⋊D3—652no—
113C4×C3⋊D3—652no—
114#634—626no—
115#525—271noSylow
116C2×C33—22no—
117C2×C33C22×D411yescharacteristic
118C2×C33C2×#4112yes—
119C3⋊(C3⋊D3)—24no—
120C3⋊(C3⋊D3)—42no—
121C23×D3—913no—
122C22×C3×C4C2×C3⋊D3113yes—
123C2×#67—926no—
124C22×Dic3—913no—
125#235—1852no—
126C2×D12—952no—
127C2×D12—913no—
128C2×C4×D3—926no—
129C2×C4×D3—926no—
130C2×C4×D3—926no—
131C2×C4×D3—1852no—
132C2×Dic6—913no—
133C3×C4⋊C4C2×C3⋊D3126yescomplemented
134C3×C22⋊C4C2×C3⋊D3126yes—
135C3×C42C2×C3⋊D3126yescomplemented
136(C2×D3)⋊C4—952no—
137(C2×D3)⋊C4—952no—
138C4⋊Dic3—952no—
139C22×C32—226no—
140C22×C32C2×D12126yes—
141C22×C32C23×D3113yes—
142C2×C3⋊D3—352no—
143C2×C3⋊D3—6104no—
144C2×C3⋊D3—626no—
145C2×C3⋊D3—626no—
146C2×C3⋊D3—626no—
147C2×C3⋊D3—652no—
148C32×C4—252no—
149C32×C4C2×D12126yes—
150C32×C4C2×D12126yes—
151C3⋊Dic3—626no—
152C3⋊Dic3—626no—
153C2×#41—271no—
154C23×C4—271no—
155#123—272no—
156C22⋊Q8—272no—
157#121—272no—
158#120—272no—
159C4×D4—274no—
160#117C3⋊(C3⋊D3)11yescharacteristic, complemented
161C33#52511yescharacteristic, complemented, Sylow
162C23×C3C22×C3⋊D3113yes—
163C22×D3—926no—
164C22×D3—926no—
165C22×D3—1852no—
166C22×D3—1826no—
167C2×C3×C4C4⋊(C3⋊D3)152yescomplemented
168C2×C3×C4C22×C3⋊D3152yes—
169C2×C3×C4C22×C3⋊D3113yes—
170C2×C3×C4C22×C3⋊D3113yes—
171#67—1852no—
172C2×Dic3—926no—
173C2×Dic3—926no—
174C2×Dic3—1826no—
175D12—18104no—
176D12—1826no—
177C4×D3—1852no—
178C4×D3—1852no—
179C4×D3—1852no—
180Dic6—1826no—
181C2×C32—226no—
182C2×C32C22×D12113yes—
183C2×C32D3×#41126yes—
184C3⋊D3—652no—
185C3⋊D3—1226no—
186C24—271no—
187#41—544no—
188C2×Q8—271no—
189C2×D4—272no—
190C2×D4—274no—
191C2×D4—271no—
192C22×C4—271no—
193C22×C4—272no—
194C22×C4—272no—
195C22×C4—272no—
196C22×C4—544no—
197C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
198C4⋊C4—274no—
199C4⋊C4C2×C3⋊(C3⋊D3)12yescomplemented
200C22⋊C4—274no—
201C22⋊C4—274no—
202C22⋊C4C2×C3⋊(C3⋊D3)12yes—
203C42C2×C3⋊(C3⋊D3)12yescomplemented
204C22×C3—226no—
205C22×C3C2×C4⋊(C3⋊D3)126yes—
206C22×C3C23×C3⋊D3113yes—
207C2×D3—952no—
208C2×D3—18104no—
209C2×D3—1826no—
210C2×D3—1826no—
211C2×D3—1826no—
212C2×D3—1852no—
213C3×C4—252no—
214C3×C4C2×C4⋊(C3⋊D3)126yes—
215C3×C4C2×C4⋊(C3⋊D3)126yes—
216Dic3—1826no—
217Dic3—1826no—
218C32#5524113yescomplemented
219C23—272no—
220C23—272no—
221C23—544no—
222C23—542no—
223C23C22×C3⋊(C3⋊D3)11yescharacteristic
224Q8—542no—
225D4—548no—
226D4—544no—
227D4—542no—
228C2×C4—272no—
229C2×C4—272no—
230C2×C4—542no—
231C2×C4—544no—
232C2×C4—544no—
233C2×C4—544no—
234C2×C4C4⋊(C3⋊(C3⋊D3))14yescomplemented
235C2×C4C22×C3⋊(C3⋊D3)14yes—
236C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
237C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
238C2×C3—226no—
239C2×C3C22×C4⋊(C3⋊D3)113yes—
240C2×C3#41×C3⋊D3126yes—
241D3—1852no—
242D3—3626no—
243C22—22no—
244C22—274no—
245C22—548no—
246C22—542no—
247C22—542no—
248C22—542no—
249C22—544no—
250C22C2×C4⋊(C3⋊(C3⋊D3))12yes—
251C22C23×C3⋊(C3⋊D3)11yescharacteristic
252C4—24no—
253C4—542no—
254C4—542no—
255C4C2×C4⋊(C3⋊(C3⋊D3))12yes—
256C4C2×C4⋊(C3⋊(C3⋊D3))12yes—
257C3#10595995113yescomplemented
258C2—22no—
259C2—544no—
260C2—1082no—
261C2C22×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
262C2#41×C3⋊(C3⋊D3)12yes—
subgroupG/H
centerC2×C4C22×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FrattiniC22C23×C3⋊(C3⋊D3)
FittingC33×#117C2
hypercenter#117C3⋊(C3⋊D3)
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ #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. #11723938C2semidirect
5. #11723936C2semidirect
6. #11723927C2semidirect
7. #11723923C2semidirect
8. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
20. C2×C4×C3⋊(C3⋊D3)C22semidirect
21. C2×#6308C22semidirect
22. C33×C4⋊C4C22semidirect
24. C33×C42C22semidirect
26. #88859C22semidirect
27. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. C32×#117D3semidirect
41. C2×C33×C4D4semidirect
68. C32×C4⋊C4C2×D3semidirect
70. C32×C42C2×D3semidirect
82. C2×C3⋊(C3⋊D3)#41semidirect
90. C3×#117C3⋊D3semidirect
101. C2×C32×C4D12semidirect
133. C3×C4⋊C4C2×C3⋊D3semidirect
135. C3×C42C2×C3⋊D3semidirect
160. #117C3⋊(C3⋊D3)semidirect
161. C33#525semidirect
167. C2×C3×C4C4⋊(C3⋊D3)semidirect
199. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
203. C42C2×C3⋊(C3⋊D3)semidirect
218. C32#5524semidirect
234. C2×C4C4⋊(C3⋊(C3⋊D3))semidirect
257. C3#10595995semidirect

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
#117239382
#117239362
#117239272
#117239232
C4×C4⋊(C3⋊(C3⋊D3))2
#105959953