Finite Group Databasebeta

← #13167498

Subgroups — #13167498

Order 1728.

648 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 71 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×#88941C214yesmaximal, complemented
2#11723697C212yesmaximal, complemented
3C3×C4×#622C211yesmaximal, characteristic, complemented
4#10593418—31nomaximal
5C2×C3×C4⋊(C3⋊D3)—22no—
6C2×#6295—22no—
7C2×#6295C2214yes—
8C2×C3×#622C2212yes—
9#88941—28no—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×#3774—34no—
12#63914—32no—
13#63911—62no—
14C4×#622D311yescharacteristic, complemented
15C22×C3×C3⋊D3—24no—
16C2×C33×C4D411yescharacteristic
17C2×C3×C3⋊Dic3D411yescharacteristic
18C2×C3×C3⋊Dic3C2311yescharacteristic
19C3×C4⋊(C3⋊D3)—44no—
20#6295—28no—
21#6295—44no—
22C3×#622D414yescomplemented
23#4695—91nomaximal
24C2×C4⋊(C3⋊D3)—22no—
25C2×C3×D12—62no—
26C2×C3×D12—62no—
27C2×C3×D12—122no—
28C2×#626—64no—
29C2×#626—64no—
30C2×#625—34no—
31C2×#625—62no—
32C2×#622C2×D312yes—
33#3774—68no—
34C4×C3⋊Dic3C2×D311yescharacteristic
35C3×C4×Dic3—62no—
36C22×C33C2×D411yescharacteristic
37C2×C3×C3⋊D3—44no—
38C2×C3×C3⋊D3—44no—
39C2×C3×C3⋊D3—44no—
40C33×C4D812yescomplemented
41C3×C3⋊Dic3D812yescomplemented
42C3×C3⋊Dic3C2×D412yes—
43#940—182no—
44C2×D24—94no—
45#874—92no—
46C3×C4×C8—91no—
47C22×C3⋊D3—24no—
48C22×C3×D3—64no—
49C22×C3×D3—64no—
50C22×C3×D3—124no—
51C2×C32×C4—22no—
52C2×C32×C4—42no—
53C2×C32×C4D1211yescharacteristic
54C2×C3⋊Dic3D1211yescharacteristic
55C2×C3⋊Dic3C22×D311yescharacteristic
56C4⋊(C3⋊D3)—44no—
57C2×C3×Dic3—62no—
58C2×C3×Dic3—62no—
59C3×D12—124no—
60C3×D12—128no—
61C3×D12—124no—
62#626—128no—
63#626—128no—
64#625—68no—
65#625—124no—
66#622D1214yescomplemented
67#493—271noSylow
68C2×C33#12711yescharacteristic
69C2×C33C2×D811yescharacteristic
70C2×C33C2×D811yescharacteristic
71C3×C3⋊D3—84no—
72C2×C3×D4—182no—
73C2×#67—184no—
74C2×#67—184no—
75C2×D12—62no—
76C2×D12—62no—
77C2×D12—94no—
78C2×D12—122no—
79C2×D12—182no—
80C2×C3×C8—92no—
81C3×C42—91no—
82C4×Dic3—62no—
83D24—188no—
84C22×C32—22no—
85C22×C32—42no—
86C22×C32C2×D1211yescharacteristic
87C2×C3⋊D3—44no—
88C2×C3⋊D3—44no—
89C2×C3⋊D3—44no—
90C2×C3×D3—124no—
91C2×C3×D3—124no—
92C2×C3×D3—124no—
93C2×C3×D3—128no—
94C2×C3×D3—124no—
95C2×C3×D3—124no—
96C2×C3×D3—124no—
97C2×C3×D3—244no—
98C2×C3×D3—244no—
99C32×C4—24no—
100C32×C4—44no—
101C32×C4D2412yescomplemented
102C3⋊Dic3D2412yescomplemented
103C3⋊Dic3C2×D1212yes—
104C3×Dic3—64no—
105C3×Dic3—64no—
106C2×D8—274no—
107#127—272no—
108C4×C8—91no—
109C33#49311yescharacteristic, complemented, Sylow
110C23×C3—184no—
111C22×D3—64no—
112C22×D3—64no—
113C22×D3—124no—
114C3×D4—364no—
115C2×C3×C4—22no—
116C2×C3×C4—42no—
117C2×C3×C4—91no—
118C2×C3×C4—91no—
119C2×C3×C4GO+(4,𝔽2)11yescharacteristic
120#67—368no—
121#67—368no—
122C2×Dic3—62no—
123C2×Dic3—62no—
124D12—124no—
125D12—128no—
126D12—124no—
127D12—188no—
128D12—364no—
129C3×C8—94no—
130C2×C32—22no—
131C2×C32—22no—
132C2×C32—22no—
133C2×C32—42no—
134C2×C32—42no—
135C2×C32—42no—
136C2×C32#87411yescharacteristic
137C2×C32C2×D2411yescharacteristic
138C2×C32C2×D2411yescharacteristic
139C3⋊D3—84no—
140C3×D3—244no—
141C3×D3—248no—
142C3×D3—244no—
143C2×D4—182no—
144C2×D4—274no—
145C2×D4—542no—
146D8—548no—
147C2×C8—92no—
148C42—91no—
149C22×C3—22no—
150C22×C3—42no—
151C22×C3—364no—
152C22×C3—364no—
153C22×C3—364no—
154C22×C3C2×GO+(4,𝔽2)11yescharacteristic
155C2×D3—124no—
156C2×D3—124no—
157C2×D3—124no—
158C2×D3—128no—
159C2×D3—124no—
160C2×D3—124no—
161C2×D3—124no—
162C2×D3—244no—
163C2×D3—244no—
164C3×C4—24no—
165C3×C4—44no—
166C3×C4—92no—
167C3×C4—92no—
168C3×C4#377412yescomplemented
169Dic3—64no—
170Dic3—64no—
171C32—22no—
172C32—42no—
173C32#469511yescharacteristic, complemented
174C23—184no—
175D4—364no—
176D4—548no—
177D4—1084no—
178C2×C4—91no—
179C2×C4—91no—
180C2×C4#157⋊D311yescharacteristic
181C8—94no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—22no—
185C2×C3—42no—
186C2×C3—42no—
187C2×C3—42no—
188C2×C3—724no—
189C2×C3#6422511yescharacteristic
190C2×C3#6423111yescharacteristic
191C2×C3C2×#377411yescharacteristic
192D3—244no—
193D3—248no—
194D3—244no—
195C22—364no—
196C22—364no—
197C22—364no—
198C22C2×#157⋊D311yescharacteristic
199C4—92no—
200C4—92no—
201C4#8894112yescomplemented
202C3—22no—
203C3—42no—
204C3#1059341811yescharacteristic, complemented
205C2—724no—
206C2#1172433611yescharacteristic
207C2#1172434411yescharacteristic
208C2C2×#8894111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#493
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88941C2semidirect
2. #11723697C2semidirect
3. C3×C4×#622C2semidirect
14. C4×#622D3semidirect
22. C3×#622D4semidirect
40. C33×C4D8semidirect
41. C3×C3⋊Dic3D8semidirect
66. #622D12semidirect
101. C32×C4D24semidirect
102. C3⋊Dic3D24semidirect
109. C33#493semidirect
168. C3×C4#3774semidirect
173. C32#4695semidirect
201. C4#88941semidirect
204. C3#10593418semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
#117236972
C3×C4×#6222
#105934183
#46959