Finite Group Databasebeta

← #13178766

Subgroups — #13178766

Order 1728.

1306 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C3×#64156C213yesmaximal, complemented
2#11723701C216yesmaximal, complemented
3#11723672C216yesmaximal, complemented
4#10595439—33nomaximal
5C22×#1113C2212yescomplemented
6C2×C33×Q8C2211yescharacteristic, complemented
7C2×C3×C4⋊(C3⋊D3)C2213yes—
8C2×#6296C2212yescomplemented
9C2×#6295C2216yes—
10C3×#3752C22112yescomplemented
11C3×C4×C3⋊Dic3C2213yescomplemented
12#88814C2216yescomplemented
13#64156D313yescomplemented
14C3×#947—36no—
15C3×#947—63no—
16#63923—36no—
17#63922—312no—
18#63915—36no—
19#63852—312no—
20C22×C3×C3⋊D3C2316yes—
21C2×#1113—44no—
22C2×#1113—42no—
23C33×Q8—41no—
24C2×C33×C4C2313yes—
25C2×C3×C3⋊Dic3C2316yes—
26C3×C4⋊(C3⋊D3)—43no—
27#6296—42no—
28#6295—46no—
29#5683—93no—
30C22×D32—36no—
31C2×C32×Q8—23no—
32C2×C32×Q8—41no—
33C2×C32×Q8C2×D313yescomplemented
34C2×C4⋊(C3⋊D3)C2×D313yes—
35C2×C3×D12—36no—
36C2×C3×D12—63no—
37C2×#627—36no—
38C2×#626—312no—
39C2×#625—36no—
40#3752C2×D3112yescomplemented
41C4×C3⋊Dic3C2×D313yescomplemented
42C3×(C2×D3)⋊C4—312no—
43C3×(C2×D3)⋊C4—312no—
44C3×(C2×D3)⋊C4—612no—
45C3×C4×Dic3—36no—
46C3×C4×Dic3—63no—
47#3722—36no—
48(C2×D3)⋊Dic3—312no—
49C22×C33C2411yescharacteristic
50C2×C3×C3⋊D3—212no—
51C2×C3×C3⋊D3—26no—
52#1113—48no—
53C33×C4—23no—
54C3×C3⋊Dic3—26no—
55C3×#124—93no—
56#947—36no—
57#947—63no—
58#937—96no—
59#885—912no—
60#882—912no—
61#875—96no—
62C22×C3⋊D3C22×D316yes—
63C22×C3×D3—312no—
64C22×C3×D3—66no—
65C2×D32—126no—
66C2×D32—1212no—
67C32×Q8—43no—
68C32×Q8—46no—
69C32×Q8—44no—
70C2×C32×C4—23no—
71C2×C32×C4—26no—
72C2×C32×C4—43no—
73C2×C32×C4C22×D316yes—
74C2×C32×C4C22×D313yes—
75C2×C3⋊Dic3C22×D316yes—
76C4⋊(C3⋊D3)—43no—
77C2×C3×Dic3—312no—
78C2×C3×Dic3—66no—
79C3×D12—126no—
80C3×D12—126no—
81#627—126no—
82#626—1212no—
83#625—126no—
84H(3,𝔽4)—271noSylow
85C2×C33H(5,𝔽2)11yescharacteristic
86C2×C33H(5,𝔽2)12yes—
87C3×C3⋊D3—46no—
88C23×D3—96no—
89C2×C3×Q8—23no—
90C2×C3×Q8—41no—
91C2×C3×Q8D3213yescomplemented
92C2×C3×D4—93no—
93C2×#67—912no—
94C2×D12—36no—
95C2×D12—63no—
96C2×D12—96no—
97C2×Dic6—96no—
98C3×C22⋊C4—912no—
99C3×C42—93no—
100#217—96no—
101(C2×D3)⋊C4—312no—
102(C2×D3)⋊C4—312no—
103(C2×D3)⋊C4—612no—
104(C2×D3)⋊C4—912no—
105C4×Dic3—36no—
106C4×Dic3—63no—
107C22×C32—23no—
108C22×C32—41no—
109C22×C32C23×D313yes—
110C2×C3⋊D3—26no—
111C2×C3⋊D3—212no—
112C2×C3×D3—612no—
113C2×C3×D3—612no—
114C2×C3×D3—612no—
115C2×C3×D3—126no—
116C2×C3×D3—1212no—
117D32—1224no—
118C32×C4—23no—
119C32×C4—26no—
120C32×C4—26no—
121C32×C4—46no—
122C32×C4—46no—
123C3⋊Dic3—26no—
124C3×Dic3—612no—
125C3×Dic3—126no—
126#124—93no—
127#124—276no—
128#120—276no—
129C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
130C23×C3—96no—
131C22×D3—312no—
132C22×D3—66no—
133C22×D3—3612no—
134C22×D3—366no—
135C3×Q8—43no—
136C3×Q8—46no—
137C3×Q8—44no—
138C3×D4—363no—
139C2×C3×C4—23no—
140C2×C3×C4—26no—
141C2×C3×C4—43no—
142C2×C3×C4—96no—
143C2×C3×C4C2×D3216yes—
144C2×C3×C4C2×D3213yes—
145#67—3612no—
146C2×Dic3—312no—
147C2×Dic3—66no—
148D12—126no—
149D12—126no—
150D12—366no—
151Dic6—366no—
152C2×C32—26no—
153C2×C32—23no—
154C2×C32—41no—
155C2×C32—42no—
156C2×C32#100413yes—
157C2×C32#100916yes—
158C3⋊D3—46no—
159C3×D3—1212no—
160C3×D3—246no—
161C24—272no—
162C2×Q8—272no—
163C2×Q8#111311yescharacteristic, complemented
164C2×D4—93no—
165C2×D4—276no—
166C22⋊C4—912no—
167C22⋊C4—276no—
168C42—93no—
169C22×C3—23no—
170C22×C3—41no—
171C22×C3—1812no—
172C22×C3—186no—
173C22×C3C22×D3213yes—
174C2×D3—612no—
175C2×D3—612no—
176C2×D3—612no—
177C2×D3—126no—
178C2×D3—1212no—
179C2×D3—3624no—
180C3×C4—26no—
181C3×C4—26no—
182C3×C4—23no—
183C3×C4—46no—
184C3×C4—46no—
185C3×C4—186no—
186Dic3—612no—
187Dic3—126no—
188C32—23no—
189C32—41no—
190C32#568313yescomplemented
191C23—96no—
192C23—1084no—
193C23—1082no—
194Q8—41no—
195Q8—1082no—
196D4—363no—
197D4—1086no—
198C2×C4—96no—
199C2×C4C2×#111313yes—
200C2×C3—26no—
201C2×C3—23no—
202C2×C3—42no—
203C2×C3—41no—
204C2×C3—366no—
205C2×C3#6432016yes—
206C2×C3#6433013yes—
207D3—1212no—
208D3—246no—
209C22—1812no—
210C22—186no—
211C22—1088no—
212C22C22×#111311yescharacteristic
213C4—23no—
214C4—186no—
215C3—23no—
216C3—41no—
217C3#1059543913yescomplemented
218C2—366no—
219C2#1172462712yes—
220C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. #11723701C2semidirect
3. #11723672C2semidirect
5. C22×#1113C22semidirect
6. C2×C33×Q8C22semidirect
8. C2×#6296C22semidirect
10. C3×#3752C22semidirect
11. C3×C4×C3⋊Dic3C22semidirect
12. #88814C22semidirect
13. #64156D3semidirect
33. C2×C32×Q8C2×D3semidirect
40. #3752C2×D3semidirect
41. C4×C3⋊Dic3C2×D3semidirect
91. C2×C3×Q8D32semidirect
129. C33H(3,𝔽4)semidirect
163. C2×Q8#1113semidirect
190. C32#5683semidirect
217. C3#10595439semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
#117237012
#117236722
#105954393