Finite Group Databasebeta

← #13166351

Subgroups — #13166351

Order 1728.

512 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C4×D3×C3⋊Dic3C211yesmaximal, characteristic
2C4×C3⋊#622C211yesmaximal, characteristic, complemented
3C3×C4×#622C211yesmaximal, characteristic, complemented
4#11722732C212yesmaximal, complemented
5(C2×D3)⋊#622C212yesmaximal
6#10593801—31nomaximal
7C2×C32×C4×D3C411yescharacteristic
8C2×D3×C3⋊Dic3C411yescharacteristic
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12C4×C3⋊(C3⋊Dic3)C411yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14Dic3×C3⋊Dic3C411yescharacteristic
15Dic3×C3⋊Dic3C2211yescharacteristic
16C2×C4×C3⋊Dic3—31no—
17C4×D3×Dic3—62no—
18C22⋊#622—32no—
19#63779—32no—
20C4×#622—31no—
21C4×#622D311yescharacteristic, complemented
22C22×C32×D3C2×C411yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3C2×C411yescharacteristic
29C32×C4×D3—22no—
30D3×C3⋊Dic3—22no—
31D3×C3⋊Dic3C814yescomplemented
32C3⋊#622—22no—
33C3×#622—22no—
34#4685—91nomaximal
35C22×C32×C4—31no—
36C22×C3⋊Dic3—31no—
37C22×C3⋊Dic3—31no—
38C2×C3×C4×D3—22no—
39C2×D3×Dic3—62no—
40C2×D3×Dic3—62no—
41C2×#622—32no—
42C2×#622C2×D312yes—
43C4×C3⋊Dic3—31no—
44C4×C3⋊Dic3—31no—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—62no—
47C4×C3⋊Dic3—122no—
48C4×C3⋊Dic3C2×D311yescharacteristic
49C3×C4×Dic3—62no—
50Dic32—62no—
51Dic32—62no—
52C22×C33C22×C411yescharacteristic
53C2×C32×D3—21no—
54C2×C32×D3—21no—
55C2×C32×D3C2×C812yes—
56C33×C4M1612yes—
57C3⋊(C3⋊Dic3)—21no—
58C3⋊(C3⋊Dic3)C2×C812yes—
59C3×C3⋊Dic3C2×C812yes—
60C3×C3⋊Dic3#4112yes—
61C32×Dic3—21no—
62C2×C4×Dic3—182no—
63C42×D3—91no—
64C3×C4×C8—91no—
65(C2×D3)⋊C8—92no—
66Dic3⋊C8—92no—
67C4×C3⋊C8—91no—
68C23×C32—31no—
69C22×C3×D3—22no—
70C2×C32×C4—22no—
71C2×C32×C4—31no—
72C2×C32×C4—42no—
73C2×C32×C4—62no—
74C2×C32×C4C4×D311yescharacteristic
75C2×C3⋊Dic3—31no—
76C2×C3⋊Dic3—34no—
77C2×C3⋊Dic3—31no—
78C2×C3⋊Dic3—62no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—62no—
81C2×C3⋊Dic3—122no—
82C2×C3⋊Dic3—122no—
83C2×C3⋊Dic3C4×D311yescharacteristic
84C2×C3⋊Dic3C22×D311yescharacteristic
85C2×C3×Dic3—22no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C3×C4×D3—28no—
89D3×Dic3—68no—
90D3×Dic3—68no—
91#622—22no—
92#622—62no—
93#431—271noSylow
94C2×C33#11711yescharacteristic
95C2×C33C22×C811yescharacteristic
96C2×C33C2×M1611yescharacteristic
97C32×D3—22no—
98C22×C3×C4—62no—
99C22×Dic3—182no—
100C22×Dic3—182no—
101C2×C4×D3—91no—
102C2×C4×D3—91no—
103C2×C4×D3#15711yescharacteristic
104C2×C3×C8—92no—
105C3×C42—91no—
106C4×Dic3—62no—
107C4×Dic3—91no—
108C4×Dic3—91no—
109C4×Dic3—91no—
110C4×Dic3—182no—
111C4×Dic3—182no—
112C4×Dic3—182no—
113C4×Dic3—362no—
114C2×C3⋊C8—92no—
115C22×C32—22no—
116C22×C32—32no—
117C22×C32—42no—
118C22×C32—61no—
119C22×C32—61no—
120C22×C32C2×C4×D311yescharacteristic
121C2×C3×D3—24no—
122C2×C3×D3—24no—
123C2×C3×D3—24no—
124C32×C4—24no—
125C32×C4—44no—
126C32×C4—61no—
127C32×C4ΓL(1,𝔽25)12yes—
128C3⋊Dic3—32no—
129C3⋊Dic3—64no—
130C3⋊Dic3—61no—
131C3⋊Dic3—64no—
132C3⋊Dic3—124no—
133C3⋊Dic3—124no—
134C3⋊Dic3D3×C812yes—
135C3⋊Dic3#23512yes—
136C3×Dic3—24no—
137C3×Dic3—64no—
138C3×Dic3—64no—
139C2×C42—271no—
140C4⋊C8—272no—
141C22⋊C8—272no—
142C4×C8—91no—
143C4×C8—271no—
144C33#43111yescharacteristic, complemented, Sylow
145C23×C3—62no—
146C22×D3C2×#15711yescharacteristic
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—62no—
150C2×C3×C4—68no—
151C2×C3×C4—91no—
152C2×C3×C4—91no—
153C2×C3×C4C2×#15711yescharacteristic
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—91no—
157C2×Dic3—91no—
158C2×Dic3—188no—
159C2×Dic3—182no—
160C2×Dic3—182no—
161C2×Dic3—188no—
162C2×Dic3—362no—
163C2×Dic3—362no—
164C2×Dic3C2×#15711yescharacteristic
165C4×D3—22no—
166C4×D3—94no—
167C4×D3—182no—
168C3×C8—182no—
169C3⋊C8—182no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—22no—
173C2×C32—42no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32—62no—
177C2×C32#87211yescharacteristic
178C2×C32C2×D3×C811yescharacteristic
179C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
180C3×D3—28no—
181C22×C4—31no—
182C22×C4—271no—
183C22×C4—271no—
184C2×C8—92no—
185C2×C8—272no—
186C42—91no—
187C42—271no—
188C42—271no—
189C42—271no—
190C22×C3—22no—
191C22×C3—42no—
192C22×C3—64no—
193C22×C3—64no—
194C22×C3—64no—
195C22×C3C22×#15711yescharacteristic
196C2×D3—21no—
197C2×D3—21no—
198C2×D3#378712yes—
199C3×C4—24no—
200C3×C4—44no—
201C3×C4—64no—
202C3×C4—92no—
203C3×C4—92no—
204C3×C4#379212yes—
205Dic3—21no—
206Dic3—64no—
207Dic3—64no—
208Dic3—92no—
209Dic3—184no—
210Dic3—184no—
211Dic3—181no—
212Dic3—364no—
213Dic3—364no—
214C32—22no—
215C32—42no—
216C32#468511yescharacteristic, complemented
217C23—31no—
218C2×C4—31no—
219C2×C4—62no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—274no—
223C2×C4—271no—
224C2×C4—271no—
225C2×C4—542no—
226C2×C4D3×#15711yescharacteristic
227C8—182no—
228C8—542no—
229C2×C3—22no—
230C2×C3—22no—
231C2×C3—22no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—68no—
236C2×C3C2×#378711yescharacteristic
237C2×C3#6428611yescharacteristic
238C2×C3C2×#379211yescharacteristic
239D3—22no—
240C22—32no—
241C22—61no—
242C22—61no—
243C22C2×D3×#15711yescharacteristic
244C4—61no—
245C4—92no—
246C4—92no—
247C4—272no—
248C4—541no—
249C4#8892512yes—
250C3—22no—
251C3—42no—
252C3#1059380111yescharacteristic, complemented
253C2—62no—
254C2#1172424611yescharacteristic
255C2D3×#378711yescharacteristic
256C2C2×#8892511yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutatorC2×C33C22×C8
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×C3⋊#622C2semidirect
3. C3×C4×#622C2semidirect
4. #11722732C2semidirect
21. C4×#622D3semidirect
31. D3×C3⋊Dic3C8semidirect
144. C33#431semidirect
216. C32#4685semidirect
252. C3#10593801semidirect

Maximal subgroups

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

subgroupindex
C4×D3×C3⋊Dic32
C4×C3⋊#6222
C3×C4×#6222
#117227322
(C2×D3)⋊#6222
#105938013
#46859