Finite Group Databasebeta

← #13157198

Subgroups — #13157198

Order 1728.

432 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 89 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×C4×C3⋊#157C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic
3C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
4#10590181—31nomaximal
5C22×C3⋊#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3⋊#622C412yescomplemented
8C4×C3⋊#157—21no—
9C4×C3⋊#157—21no—
10#88988—21no—
11#88988—21no—
12C2×C33×C8C411yescharacteristic, complemented
13C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
14C3×C8×C3⋊D3—21no—
15C3×C8×C3⋊D3—21no—
16C2×C4×#157—31no—
17C2×#3787—31no—
18C2×C8×C3⋊D3D311yescharacteristic
19C2×C3×D3×C8—62no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C3⋊#157—22no—
22C2×C3⋊#157—22no—
23C2×C3⋊#157C814yescomplemented
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C3×C4×C3⋊D3D411yescharacteristic
29C3×C4×C3⋊D3Q811yescharacteristic
30C3⋊#622—22no—
31C33×C8—21no—
32C3×C3⋊(C3⋊C8)—21no—
33#4550—91nomaximal
34C22×#157—32no—
35C2×C4×C3⋊D3C2×D311yescharacteristic
36C2×C3×C4×D3—62no—
37C2×#622—32no—
38C4×#157—61no—
39C4×#157—61no—
40#3787—61no—
41#3787—61no—
42C2×C32×C8—22no—
43C2×C32×C8—42no—
44C2×C32×C8Dic311yescharacteristic, complemented
45C2×C3⋊(C3⋊C8)Dic311yescharacteristic, complemented
46C8×C3⋊D3—21no—
47C8×C3⋊D3—21no—
48C2×C3×C3⋊C8—62no—
49C3×D3×C8—64no—
50C3×D3×C8—64no—
51C22×C33C4211yescharacteristic
52C2×C3×C3⋊D3C22⋊C411yescharacteristic
53C2×C3×C3⋊D3C4⋊C411yescharacteristic
54C2×C3×C3⋊D3C2×C812yes—
55C2×C3×C3⋊D3M1612yes—
56C3⋊#157—24no—
57C33×C4C22⋊C411yescharacteristic
58C33×C4C4⋊C411yescharacteristic
59C3×C3⋊Dic3C22⋊C411yescharacteristic
60C3×C3⋊Dic3C4⋊C411yescharacteristic
61C22×C3×C8—91no—
62C2×C4×Dic3—91no—
63C22×C3⋊C8—91no—
64C2×D3×C8—62no—
65C22×C3⋊D3C4×D311yescharacteristic
66C22×C3×D3—62no—
67C2×#157—34no—
68C2×#157—62no—
69C2×#157—62no—
70C2×C32×C4—22no—
71C2×C32×C4—42no—
72C2×C32×C4C2×Dic311yescharacteristic
73C2×C3⋊Dic3C4×D311yescharacteristic
74C4×C3⋊D3Dic611yescharacteristic
75C4×C3⋊D3D1211yescharacteristic
76C4×C3⋊D3#6711yescharacteristic
77C4×C3⋊D3#6711yescharacteristic
78C2×C3×Dic3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83#622—62no—
84C32×C8—21no—
85C32×C8—24no—
86C32×C8—44no—
87C3⋊(C3⋊C8)—21no—
88C3×C3⋊C8—64no—
89#336—271noSylow
90C2×C33#9511yescharacteristic
91C2×C33C4×C811yescharacteristic
92C2×C33#9711yescharacteristic
93C3×C3⋊D3C22⋊C812yes—
94C3×C3⋊D3C4⋊C812yes—
95C22×C3×C4—91no—
96C22×Dic3—92no—
97C2×C4×D3—62no—
98C2×C3×C8—22no—
99C2×C3×C8—42no—
100C2×C3×C8—91no—
101C2×C3×C8—181no—
102C2×C3×C8—181no—
103C2×C3×C8#15711yescharacteristic, complemented
104C4×Dic3—181no—
105C4×Dic3—181no—
106C2×C3⋊C8—62no—
107C2×C3⋊C8—92no—
108C2×C3⋊C8—181no—
109C2×C3⋊C8—181no—
110D3×C8—64no—
111D3×C8—64no—
112C22×C32—22no—
113C22×C32—42no—
114C22×C32C4×Dic311yescharacteristic
115C2×C3⋊D3D3×C812yes—
116C2×C3⋊D3ΓL(1,𝔽25)12yes—
117C2×C3⋊D3Dic3⋊C411yescharacteristic
118C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
119C2×C3×D3—62no—
120C2×C3×D3—62no—
121C2×C3×D3—64no—
122C2×C3×D3—64no—
123#157—64no—
124C32×C4—22no—
125C32×C4—22no—
126C32×C4—42no—
127C32×C4—42no—
128C32×C4C4⋊Dic311yescharacteristic
129C32×C4#21711yescharacteristic
130C3⋊Dic3Dic3⋊C411yescharacteristic
131C3⋊Dic3(C2×D3)⋊C411yescharacteristic
132C3×Dic3—62no—
133C3×Dic3—62no—
134C22×C8—91no—
135C22×C8—271no—
136C2×C42—271no—
137C33#33611yescharacteristic, complemented, Sylow
138C23×C3—91no—
139C22×D3—62no—
140C2×C3×C4—22no—
141C2×C3×C4—42no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4C2×#15711yescharacteristic
148C2×Dic3—62no—
149C2×Dic3—94no—
150C2×Dic3—182no—
151C2×Dic3—182no—
152C4×D3—62no—
153C4×D3—62no—
154C4×D3—62no—
155C4×D3—62no—
156C3×C8—24no—
157C3×C8—21no—
158C3×C8—44no—
159C3×C8—181no—
160C3⋊C8—64no—
161C3⋊C8—182no—
162C2×C32—22no—
163C2×C32—22no—
164C2×C32—22no—
165C2×C32—42no—
166C2×C32—42no—
167C2×C32—42no—
168C2×C32C8×Dic311yescharacteristic
169C2×C32#81511yescharacteristic
170C2×C32C3⋊#9511yescharacteristic
171C3⋊D3Dic3⋊C812yes—
172C3⋊D3(C2×D3)⋊C812yes—
173C3×D3—64no—
174C3×D3—64no—
175C22×C4—91no—
176C22×C4—272no—
177C2×C8—91no—
178C2×C8—181no—
179C2×C8—181no—
180C2×C8—272no—
181C2×C8—541no—
182C2×C8—541no—
183C2×C8C3⋊#15711yescharacteristic, complemented
184C42—541no—
185C42—541no—
186C22×C3—22no—
187C22×C3—42no—
188C22×C3—92no—
189C22×C3—92no—
190C22×C3—91no—
191C22×C3—91no—
192C22×C3C4×#15711yescharacteristic
193C2×D3—62no—
194C2×D3—62no—
195C2×D3—64no—
196C2×D3—64no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4—91no—
202C3×C4—91no—
203C3×C4C4⋊#15711yescharacteristic
204C3×C4C22⋊#15711yescharacteristic
205Dic3—62no—
206Dic3—62no—
207Dic3—184no—
208C32—22no—
209C32—42no—
210C32#455011yescharacteristic, complemented
211C23—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—274no—
218C2×C4—542no—
219C2×C4—542no—
220C2×C4C2×C3⋊#15711yescharacteristic
221C8—21no—
222C8—181no—
223C8—542no—
224C2×C3—22no—
225C2×C3—22no—
226C2×C3—22no—
227C2×C3—42no—
228C2×C3—42no—
229C2×C3—42no—
230C2×C3—92no—
231C2×C3—92no—
232C2×C3C8×#15711yescharacteristic
233C2×C3#6376711yescharacteristic
234C2×C3#6378111yescharacteristic
235D3—64no—
236D3—64no—
237C22—92no—
238C22—92no—
239C22—91no—
240C22—91no—
241C22C4×C3⋊#15711yescharacteristic
242C4—91no—
243C4—91no—
244C4—544no—
245C4C4⋊(C3⋊#157)11yescharacteristic
246C4#8899411yescharacteristic
247C3—22no—
248C3—42no—
249C3#1059018111yescharacteristic, complemented
250C2—92no—
251C2—92no—
252C2C8×C3⋊#15711yescharacteristic
253C2#1172309211yescharacteristic
254C2#1172310611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊#157
commutatorC2×C33C4×C8
FittingC2×C33×C8C4
socleC22×C33C42
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypercenterC2×C8C3⋊#157
hypocenterC33#336
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
7. C2×C3⋊#622C4semidirect
12. C2×C33×C8C4semidirect
13. C2×C3×C3⋊(C3⋊C8)C4semidirect
23. C2×C3⋊#157C8semidirect
44. C2×C32×C8Dic3semidirect
45. C2×C3⋊(C3⋊C8)Dic3semidirect
103. C2×C3×C8#157semidirect
137. C33#336semidirect
183. C2×C8C3⋊#157semidirect
210. C32#4550semidirect
249. C3#10590181semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊#1572
C2×#889882
C2×C3×C8×C3⋊D32
#105901813
#45509