Finite Group Databasebeta

← #13166497

Subgroups — #13166497

Order 1728.

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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×#88925C212yesmaximal, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
4C3×C4×#622C211yesmaximal, characteristic, complemented
5(C2×D3)⋊#622C212yesmaximal, complemented
6#10593821—31nomaximal
7C2×C32×D12C411yescharacteristic
8C2×D3×C3⋊Dic3C2212yes—
9C2×C3⋊#622C2212yes—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88925—24no—
13C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15(C2×D3)⋊(C3⋊Dic3)C412yes—
16C2×#3792—32no—
17D4×C3⋊Dic3—31no—
18Dic3×D12—62no—
19C22⋊#622—32no—
20C4⋊#622—31no—
21C4×#622D311yescharacteristic, complemented
22C22×C32×D3C2×C412yes—
23C2×C33×C4C2×C411yescharacteristic
24C2×C3⋊(C3⋊Dic3)C2×C412yes—
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C32×D12—22no—
28D3×C3⋊Dic3—22no—
29D3×C3⋊Dic3—22no—
30C3⋊#622—22no—
31C3×#622—21no—
32C3×#622D412yescomplemented
33#4688—91nomaximal
34C2×C32×D4—31no—
35C22×C3⋊Dic3—32no—
36C2×C3×D12—22no—
37C2×D3×Dic3—64no—
38#3792—64no—
39C2×#622—32no—
40C2×#622C2×D311yescharacteristic
41C2×#622C2×D311yescharacteristic
42#3757—32no—
43C4⋊(C3⋊Dic3)—31no—
44C4⋊(C3⋊Dic3)—62no—
45C4⋊(C3⋊Dic3)—122no—
46C4×C3⋊Dic3C2×D311yescharacteristic
47C3×C4×Dic3—62no—
48(C2×D3)⋊Dic3—64no—
49C22×C33C22×C411yescharacteristic
50C2×C32×D3—22no—
51C2×C32×D3—22no—
52C2×C32×D3—22no—
53C33×C4M1612yescomplemented
54C3⋊(C3⋊Dic3)—22no—
55C3×C3⋊Dic3—21no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3#4111yescharacteristic
58D4×Dic3—182no—
59C2×ΓL(1,𝔽25)—92no—
60C4×D12—91no—
61C3×C4×C8—91no—
62(C2×D3)⋊C8—92no—
63C4⋊(C3⋊C8)—91no—
64C23×C32—32no—
65C22×C3×D3—24no—
66C32×D4—62no—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4C4×D311yescharacteristic
70C2×C3⋊Dic3—32no—
71C2×C3⋊Dic3—64no—
72C2×C3⋊Dic3—62no—
73C2×C3⋊Dic3—62no—
74C2×C3⋊Dic3—124no—
75C2×C3⋊Dic3C4×D311yescharacteristic
76C2×C3⋊Dic3C22×D311yescharacteristic
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C3×D12—28no—
80D3×Dic3—124no—
81D3×Dic3—124no—
82#622—21no—
83#622—62no—
84#622D1212yescomplemented
85#436—271noSylow
86C2×C33C4×D411yescharacteristic
87C2×C33C2×M1611yescharacteristic
88C2×C33#13111yescharacteristic
89C32×D3—42no—
90C2×C3×D4—62no—
91C22×Dic3—184no—
92C2×D12#15711yescharacteristic
93C2×C4×D3—92no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C3×C42—91no—
97#217—184no—
98(C2×D3)⋊C4—92no—
99C4⋊Dic3—91no—
100C4⋊Dic3—182no—
101C4⋊Dic3—362no—
102C4×Dic3—62no—
103C2×C3⋊C8—92no—
104ΓL(1,𝔽25)—184no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32—62no—
108C22×C32—62no—
109C22×C32—62no—
110C22×C32C2×C4×D311yescharacteristic
111C2×C3×D3—28no—
112C2×C3×D3—44no—
113C2×C3×D3—44no—
114C32×C4—24no—
115C32×C4—44no—
116C32×C4ΓL(1,𝔽25)12yescomplemented
117C3⋊Dic3—21no—
118C3⋊Dic3—62no—
119C3⋊Dic3—124no—
120C3⋊Dic3—244no—
121C3⋊Dic3C2×D1211yescharacteristic
122C3⋊Dic3#23511yescharacteristic
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—122no—
126C2×M16—272no—
127C4×D4—271no—
128C4⋊C8—271no—
129C22⋊C8—272no—
130C4×C8—91no—
131C33#43611yescharacteristic, complemented, Sylow
132C23×C3—64no—
133C22×D3C2×#15712yes—
134C3×D4—68no—
135C2×C3×C4—22no—
136C2×C3×C4—42no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4C2×#15711yescharacteristic
140C2×Dic3—62no—
141C2×Dic3—62no—
142C2×Dic3—92no—
143C2×Dic3—184no—
144C2×Dic3—364no—
145C2×Dic3—364no—
146C2×Dic3—364no—
147D12—22no—
148C4×D3—182no—
149C4×D3—182no—
150C3×C8—92no—
151C3×C8—181no—
152C3⋊C8—182no—
153C2×C32—22no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—42no—
157C2×C32—42no—
158C2×C32—42no—
159C2×C32—122no—
160C2×C32C4×D1211yescharacteristic
161C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
162C2×C32#90111yescharacteristic
163C3×D3—48no—
164C2×D4—31no—
165C22×C4—272no—
166M16—544no—
167C2×C8—91no—
168C2×C8—91no—
169C2×C8—272no—
170C4⋊C4—271no—
171C22⋊C4—272no—
172C42—91no—
173C22×C3—22no—
174C22×C3—42no—
175C22×C3—68no—
176C22×C3—124no—
177C22×C3—124no—
178C22×C3C22×#15711yescharacteristic
179C2×D3—22no—
180C2×D3—22no—
181C2×D3—22no—
182C3×C4—24no—
183C3×C4—44no—
184C3×C4—91no—
185C3×C4—91no—
186C3×C4—181no—
187C3×C4#379212yescomplemented
188Dic3—62no—
189Dic3—62no—
190Dic3—122no—
191Dic3—182no—
192Dic3—364no—
193Dic3—724no—
194C32—22no—
195C32—42no—
196C32#468811yescharacteristic, complemented
197C23—32no—
198D4—62no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—272no—
202C2×C4—542no—
203C2×C4—542no—
204C2×C4D3×#15711yescharacteristic
205C8—92no—
206C8—181no—
207C8—542no—
208C2×C3—22no—
209C2×C3—22no—
210C2×C3—22no—
211C2×C3—42no—
212C2×C3—42no—
213C2×C3—42no—
214C2×C3—128no—
215C2×C3#6428711yescharacteristic
216C2×C3D4×#15711yescharacteristic
217C2×C3C2×#379211yescharacteristic
218D3—42no—
219C22—62no—
220C22—62no—
221C22—62no—
222C22C2×D3×#15711yescharacteristic
223C4—91no—
224C4—91no—
225C4—181no—
226C4—542no—
227C4#8892512yescomplemented
228C3—22no—
229C3—42no—
230C3#1059382111yescharacteristic, complemented
231C2—122no—
232C2#1172425011yescharacteristic
233C2D12×#15711yescharacteristic
234C2C2×#8892511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×C32×D12C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88925C2semidirect
3. C4⋊(C3⋊#622)C2semidirect
4. C3×C4×#622C2semidirect
5. (C2×D3)⋊#622C2semidirect
21. C4×#622D3semidirect
32. C3×#622D4semidirect
53. C33×C4M16semidirect
84. #622D12semidirect
116. C32×C4ΓL(1,𝔽25)semidirect
131. C33#436semidirect
187. C3×C4#3792semidirect
196. C32#4688semidirect
227. C4#88925semidirect
230. C3#10593821semidirect

Maximal subgroups

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

subgroupindex
C2×#889252
D12×C3⋊Dic32
C4⋊(C3⋊#622)2
C3×C4×#6222
(C2×D3)⋊#6222
#105938213
#46889