Finite Group Databasebeta

← #13157224

Subgroups — #13157224

Order 1728.

560 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×C3⋊#622C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3#10590207—31nomaximal
4C22×C33×C4C411yescharacteristic
5C22×C3×C3⋊Dic3C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3⋊#622—22no—
8C2×C3⋊#622—22no—
9C2×C3⋊#622C414yescomplemented
10C3×C4×C3⋊Dic3—22no—
11C22×#622—32no—
12C2×C4×C3⋊Dic3D311yescharacteristic
13C2×C3×C4×Dic3—62no—
14C23×C33C2×C411yescharacteristic
15C2×C33×C4—21no—
16C2×C33×C4—21no—
17C2×C33×C4C812yescomplemented
18C2×C3×C3⋊Dic3—21no—
19C2×C3×C3⋊Dic3—21no—
20C2×C3×C3⋊Dic3C812yescomplemented
21C2×C3×C3⋊Dic3C2×C412yes—
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C3×C3⋊Dic3D412yes—
24C2×C3×C3⋊Dic3Q811yescharacteristic
25C3⋊#622—24no—
26#4524—91nomaximal
27C22×C32×C4—22no—
28C22×C32×C4—42no—
29C22×C32×C4Dic311yescharacteristic
30C22×C3⋊Dic3Dic311yescharacteristic
31C22×C3⋊Dic3C2×D311yescharacteristic
32C22×C3×Dic3—62no—
33C22×C3×Dic3—62no—
34C2×#622—34no—
35C2×#622—62no—
36C2×#622—62no—
37C4×C3⋊Dic3—22no—
38C3×C4×Dic3—68no—
39C22×C33C4211yescharacteristic
40C22×C33C22⋊C411yescharacteristic
41C22×C33C4⋊C411yescharacteristic
42C22×C33C2×C811yescharacteristic
43C22×C33C2×C811yescharacteristic
44C22×C33M1611yescharacteristic
45C22×C33M1611yescharacteristic
46C33×C4—22no—
47C3×C3⋊Dic3—22no—
48C3×C3⋊Dic3C22⋊C412yes—
49C3×C3⋊Dic3C4⋊C412yes—
50C2×C3×C42—91no—
51C2×C4×Dic3—62no—
52C22×C3⋊C8—92no—
53C23×C32—22no—
54C23×C32—42no—
55C23×C32C2×Dic311yescharacteristic
56C2×C32×C4—21no—
57C2×C32×C4—21no—
58C2×C32×C4—24no—
59C2×C32×C4—24no—
60C2×C32×C4—24no—
61C2×C32×C4—44no—
62C2×C32×C4—44no—
63C2×C32×C4—44no—
64C2×C32×C4C3⋊C812yescomplemented
65C2×C3⋊Dic3—21no—
66C2×C3⋊Dic3—21no—
67C2×C3⋊Dic3C3⋊C812yescomplemented
68C2×C3⋊Dic3Dic611yescharacteristic
69C2×C3⋊Dic3C4×D312yes—
70C2×C3⋊Dic3D1211yescharacteristic
71C2×C3⋊Dic3#6712yes—
72C2×C3×Dic3—64no—
73C2×C3×Dic3—64no—
74C2×C3×Dic3—64no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—64no—
78C2×C3×Dic3—64no—
79#622—64no—
80#336—271noSylow
81C2×C33#9511yescharacteristic
82C2×C33C4×C811yescharacteristic
83C2×C33#9711yescharacteristic
84C2×C33C22⋊C811yescharacteristic
85C2×C33C22⋊C811yescharacteristic
86C2×C33C4⋊C811yescharacteristic
87C2×C33C4⋊C811yescharacteristic
88C22×C3×C4—22no—
89C22×C3×C4—42no—
90C22×C3×C4—91no—
91C22×C3×C4—91no—
92C22×C3×C4#15711yescharacteristic
93C22×Dic3—62no—
94C22×Dic3—62no—
95C3×C42—182no—
96C4×Dic3—68no—
97C2×C3⋊C8—94no—
98C2×C3⋊C8—182no—
99C2×C3⋊C8—182no—
100C22×C32—22no—
101C22×C32—22no—
102C22×C32—22no—
103C22×C32—22no—
104C22×C32—22no—
105C22×C32—22no—
106C22×C32—22no—
107C22×C32—42no—
108C22×C32—42no—
109C22×C32—42no—
110C22×C32—42no—
111C22×C32—42no—
112C22×C32—42no—
113C22×C32—42no—
114C22×C32C2×C3⋊C811yescharacteristic
115C22×C32C2×C3⋊C811yescharacteristic
116C22×C32#20811yescharacteristic
117C22×C32#20811yescharacteristic
118C22×C32C4×Dic311yescharacteristic
119C22×C32C4⋊Dic311yescharacteristic
120C22×C32#21711yescharacteristic
121C32×C4—22no—
122C32×C4—28no—
123C32×C4—48no—
124C3⋊Dic3—22no—
125C3⋊Dic3Dic3⋊C412yes—
126C3⋊Dic3(C2×D3)⋊C412yes—
127C3×Dic3—64no—
128C3×Dic3—64no—
129C3×Dic3—68no—
130C22×C8—272no—
131C2×C42—91no—
132C33#33611yescharacteristic, complemented, Sylow
133C23×C3—22no—
134C23×C3—42no—
135C23×C3C2×#15711yescharacteristic
136C2×C3×C4—24no—
137C2×C3×C4—24no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—24no—
141C2×C3×C4—44no—
142C2×C3×C4—44no—
143C2×C3×C4—44no—
144C2×C3×C4—92no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—92no—
148C2×C3×C4—92no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4#62212yescomplemented
152C2×Dic3—64no—
153C2×Dic3—62no—
154C2×Dic3—64no—
155C2×Dic3—62no—
156C2×Dic3—64no—
157C2×Dic3—64no—
158C2×Dic3—64no—
159C3⋊C8—184no—
160C2×C32—22no—
161C2×C32—22no—
162C2×C32—22no—
163C2×C32—22no—
164C2×C32—22no—
165C2×C32—22no—
166C2×C32—22no—
167C2×C32—42no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32C4×C3⋊C811yescharacteristic
175C2×C32#80311yescharacteristic
176C2×C32C4⋊(C3⋊C8)11yescharacteristic
177C2×C32C4⋊(C3⋊C8)11yescharacteristic
178C2×C32#83011yescharacteristic
179C2×C32#83011yescharacteristic
180C2×C32C3⋊#9511yescharacteristic
181C22×C4—91no—
182C22×C4—91no—
183C22×C4C3⋊#15711yescharacteristic
184C2×C8—274no—
185C2×C8—542no—
186C2×C8—542no—
187C42—182no—
188C22×C3—22no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—22no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3#378711yescharacteristic
203C22×C3#378811yescharacteristic
204C22×C3C4×#15711yescharacteristic
205C22×C3C4⋊#15711yescharacteristic
206C22×C3C2×#62211yescharacteristic
207C22×C3#379211yescharacteristic
208C22×C3C22⋊#15711yescharacteristic
209C3×C4—28no—
210C3×C4—22no—
211C3×C4—48no—
212C3×C4—92no—
213C3×C4—92no—
214C3×C4—182no—
215Dic3—64no—
216Dic3—64no—
217Dic3—68no—
218C32—22no—
219C32—42no—
220C32#452411yescharacteristic, complemented
221C23C2×C3⋊#15711yescharacteristic
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—92no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—92no—
228C2×C4—92no—
229C2×C4—181no—
230C2×C4—181no—
231C2×C4C3⋊#62212yescomplemented
232C8—544no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—22no—
236C2×C3—22no—
237C2×C3—22no—
238C2×C3—22no—
239C2×C3—22no—
240C2×C3—42no—
241C2×C3—42no—
242C2×C3—42no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—42no—
246C2×C3—42no—
247C2×C3C4×#62211yescharacteristic
248C2×C3C4⋊#62211yescharacteristic
249C2×C3#6377711yescharacteristic
250C2×C3#6377811yescharacteristic
251C2×C3#6377911yescharacteristic
252C2×C3#6378111yescharacteristic
253C2×C3C22⋊#62211yescharacteristic
254C22#8898811yescharacteristic
255C22#8898911yescharacteristic
256C22C4×C3⋊#15711yescharacteristic
257C22C4⋊(C3⋊#157)11yescharacteristic
258C22C2×C3⋊#62211yescharacteristic
259C22#8899311yescharacteristic
260C22#8899411yescharacteristic
261C4—22no—
262C4—92no—
263C4—92no—
264C4—182no—
265C3—22no—
266C3—42no—
267C3#1059020711yescharacteristic, complemented
268C2C4×C3⋊#62211yescharacteristic
269C2C4⋊(C3⋊#622)11yescharacteristic
270C2#1172310211yescharacteristic
271C2#1172310311yescharacteristic
272C2#1172310411yescharacteristic
273C2#1172310611yescharacteristic
274C2#1172311211yescharacteristic
subgroupG/H
center, FrattiniC23C2×C3⋊#157
commutatorC2×C33C4×C8
FittingC22×C33×C4C4
socle, prime radicalC23×C33C2×C4
cosocleC22×C3×C3⋊Dic3C22
hypercenterC22×C4C3⋊#157
hypocenterC33#336
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
9. C2×C3⋊#622C4semidirect
17. C2×C33×C4C8semidirect
20. C2×C3×C3⋊Dic3C8semidirect
64. C2×C32×C4C3⋊C8semidirect
67. C2×C3⋊Dic3C3⋊C8semidirect
132. C33#336semidirect
151. C2×C3×C4#622semidirect
220. C32#4524semidirect
231. C2×C4C3⋊#622semidirect
267. C3#10590207semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×C3×C4×C3⋊Dic32
#105902073
#45249