Finite Group Databasebeta

← #13166396

Subgroups — #13166396

Order 1728.

408 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 78 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
1#11724511C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722717C212yesmaximal, complemented
5#11722716C212yesmaximal, complemented
6#10593771—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#88993—21no—
9#88989C2212yes—
10#88988—21no—
11C3×#3792—21no—
12C3×#3788C2212yes—
13C3×#3787—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C411yescharacteristic
15C2×C32×C3⋊C8C411yescharacteristic
16C3⋊D3×C3⋊C8C411yescharacteristic
17C3⋊D3×C3⋊C8C411yescharacteristic
18C3⋊D3×C3⋊C8C2211yescharacteristic
19C3⋊D3×C3⋊C8C2211yescharacteristic
20#64283—31no—
21#64283D311yescharacteristic
22C2×C8×C3⋊D3—31no—
23C2×D3×C3⋊C8—62no—
24#63771—32no—
25#63770—32no—
26C22×C3×C3⋊D3Q811yescharacteristic
27C2×C33×C4C2×C411yescharacteristic
28C2×C3×C3⋊Dic3D411yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3D411yescharacteristic
31C3×C4×C3⋊D3Q811yescharacteristic
32C3×C4×C3⋊D3C2311yescharacteristic
33C3⋊#622—22no—
34C3×#622—22no—
35C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
36C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
37C32×C3⋊C8C2×C411yescharacteristic
38C32×C3⋊C8C2×C411yescharacteristic
39C2×#822—91nomaximal
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—62no—
42#3792—21no—
43#3792—61no—
44#3788—32no—
45#3788C2×D312yes—
46#3787—21no—
47#3787—61no—
48C2×C32×C8—31no—
49C2×C3⋊(C3⋊C8)—31no—
50C2×C3⋊(C3⋊C8)—62no—
51C2×C3⋊(C3⋊C8)—122no—
52C8×C3⋊D3—31no—
53C8×C3⋊D3—31no—
54C8×C3⋊D3—31no—
55C8×C3⋊D3—31no—
56C2×C3×C3⋊C8—22no—
57D3×C3⋊C8—62no—
58D3×C3⋊C8—62no—
59D3×C3⋊C8—62no—
60D3×C3⋊C8—62no—
61C22×C33C4⋊C411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3C4⋊C411yescharacteristic
64C2×C3×C3⋊D3C2×Q811yescharacteristic
65C33×C4C4⋊C411yescharacteristic
66C33×C4C22×C411yescharacteristic
67C3×C3⋊Dic3C4⋊C411yescharacteristic
68C3×C3⋊Dic3C2×D411yescharacteristic
69C2×C3×M16—91no—
70C2×#208—91no—
71C22×C3⋊C8—91no—
72C2×D3×C8—182no—
73#822—92no—
74#822—92no—
75C22×C3⋊D3Dic611yescharacteristic
76C22×C3×D3—62no—
77C2×C32×C4—22no—
78C2×C32×C4—42no—
79C2×C32×C4C4×D311yescharacteristic
80C2×C3⋊Dic3#6711yescharacteristic
81C4×C3⋊D3Dic611yescharacteristic
82C4×C3⋊D3C4×D311yescharacteristic
83C4×C3⋊D3#6711yescharacteristic
84C4×C3⋊D3C22×D311yescharacteristic
85C2×C3×Dic3—62no—
86C3×C4×D3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90#622—22no—
91#622—62no—
92C32×C8—31no—
93C32×C8—31no—
94C3⋊(C3⋊C8)—31no—
95C3⋊(C3⋊C8)—31no—
96C3⋊(C3⋊C8)—62no—
97C3⋊(C3⋊C8)—62no—
98C3⋊(C3⋊C8)—122no—
99C3⋊(C3⋊C8)—122no—
100C3×C3⋊C8—22no—
101C3×C3⋊C8—22no—
102C2×#108—271noSylow
103C2×C33—21no—
104C2×C33C2×C4⋊C411yescharacteristic
105C3×C3⋊D3—21no—
106C3×C3⋊D3#10812yescomplemented
107C22×C3×C4—91no—
108C2×C4×D3—62no—
109C3×M16—92no—
110C3×M16—181no—
111C2×C3×C8—62no—
112C2×C3×C8—181no—
113#208—92no—
114#208—181no—
115C2×C3⋊C8—91no—
116C2×C3⋊C8—91no—
117C2×C3⋊C8—91no—
118C2×C3⋊C8—91no—
119C2×C3⋊C8—91no—
120C2×C3⋊C8—182no—
121C2×C3⋊C8—181no—
122C2×C3⋊C8—362no—
123C2×C3⋊C8#15711yescharacteristic
124D3×C8—182no—
125D3×C8—182no—
126D3×C8—182no—
127D3×C8—182no—
128C22×C32—22no—
129C22×C32—42no—
130C22×C32Dic3⋊C411yescharacteristic
131C2×C3⋊D3—22no—
132C2×C3⋊D3Dic3⋊C411yescharacteristic
133C2×C3⋊D3C2×Dic611yescharacteristic
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—68no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—42no—
140C32×C4—42no—
141C32×C4Dic3⋊C411yescharacteristic
142C32×C4C2×C4×D311yescharacteristic
143C3⋊Dic3Dic3⋊C411yescharacteristic
144C3⋊Dic3C2×#6711yescharacteristic
145C3×Dic3—62no—
146C3×Dic3—62no—
147C2×M16—91no—
148C2×M16—271no—
149C22×C8—271no—
150#108—272no—
151#108—272no—
152C33C2×#10811yescharacteristic, complemented, Sylow
153C23×C3—91no—
154C22×D3—62no—
155C2×C3×C4—22no—
156C2×C3×C4—42no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4C2×#15711yescharacteristic
163C2×Dic3—62no—
164C4×D3—62no—
165C4×D3—62no—
166C4×D3—62no—
167C4×D3—62no—
168C3×C8—62no—
169C3×C8—62no—
170C3×C8—182no—
171C3⋊C8—91no—
172C3⋊C8—91no—
173C3⋊C8—182no—
174C3⋊C8—182no—
175C3⋊C8—182no—
176C3⋊C8—362no—
177C3⋊C8—362no—
178C3⋊C8C2×#15711yescharacteristic
179C3⋊C8C2×#15711yescharacteristic
180C2×C32—21no—
181C2×C32—24no—
182C2×C32—22no—
183C2×C32—44no—
184C2×C32—42no—
185C2×C32C2×Dic3⋊C411yescharacteristic
186C3⋊D3—21no—
187C3⋊D3#82212yescomplemented
188C3×D3—64no—
189C3×D3—64no—
190C22×C4—91no—
191M16—92no—
192M16—181no—
193M16—272no—
194M16—541no—
195C2×C8—31no—
196C2×C8—181no—
197C2×C8—271no—
198C2×C8—271no—
199C2×C8—271no—
200C2×C8—271no—
201C2×C8—271no—
202C2×C8—541no—
203C22×C3—22no—
204C22×C3—42no—
205C22×C3—91no—
206C22×C3—91no—
207C22×C3—182no—
208C22×C3C4⋊#15711yescharacteristic
209C2×D3—62no—
210C2×D3—62no—
211C2×D3—68no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—42no—
215C3×C4—42no—
216C3×C4—91no—
217C3×C4—91no—
218C3×C4C4⋊#15711yescharacteristic
219C3×C4C22×#15711yescharacteristic
220Dic3—62no—
221Dic3—62no—
222C32—22no—
223C32—42no—
224C32C2×#82211yescharacteristic, complemented
225C23—91no—
226C2×C4—91no—
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4D3×#15711yescharacteristic
232C8—31no—
233C8—31no—
234C8—182no—
235C8—271no—
236C8—271no—
237C8—542no—
238C2×C3—24no—
239C2×C3—21no—
240C2×C3—22no—
241C2×C3—44no—
242C2×C3—42no—
243C2×C3—92no—
244C2×C3—181no—
245C2×C3C2×C4⋊#15711yescharacteristic
246D3—64no—
247D3—64no—
248C22—91no—
249C22—91no—
250C22—182no—
251C22Dic3⋊#15711yescharacteristic
252C4—91no—
253C4—91no—
254C4Dic3⋊#15711yescharacteristic
255C4C2×D3×#15711yescharacteristic
256C3—22no—
257C3—42no—
258C3#1059377111yescharacteristic, complemented
259C2—21no—
260C2—92no—
261C2—181no—
262C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
centerC4Dic3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#108
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722717C2semidirect
5. #11722716C2semidirect
106. C3×C3⋊D3#108semidirect
152. C33C2×#108semidirect
187. C3⋊D3#822semidirect
224. C32C2×#822semidirect
258. C3#10593771semidirect

Maximal subgroups

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

subgroupindex
#117245112
C3×#642832
C2×C3⋊D3×C3⋊C82
#117227172
#117227162
#105937713
C2×#8229