Finite Group Databasebeta

← #13154809

Subgroups — #13154809

Order 1728.

376 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 68 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×#88989C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#10590185—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C2×C3⋊#622—21no—
7#88989—21no—
8#88989C412yes—
9C2×C3×#622—21no—
10C3×#3788—21no—
11C3×#3788C412yes—
12C2×C3⋊(C3⋊(C3⋊C8))C411yescharacteristic
13C2×C32×C3⋊C8C411yescharacteristic
14C3⋊D3×C3⋊C8—21no—
15C3⋊D3×C3⋊C8—21no—
16C2×#3788—31no—
17C2×#3788D311yescharacteristic
18C2×C8×C3⋊D3—31no—
19C2×D3×C3⋊C8—62no—
20C22×C3×C3⋊D3Q811yescharacteristic
21C2×C33×C4C2×C411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3⋊#622—22no—
28C3×#622—22no—
29C3⋊(C3⋊(C3⋊C8))—21no—
30C32×C3⋊C8—21no—
31#4558—91nomaximal
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×C3×C4×D3—62no—
34C2×#622—21no—
35C2×#622—61no—
36#3788—21no—
37#3788—32no—
38#3788—61no—
39#3788Dic312yes—
40C2×C32×C8—31no—
41C2×C3⋊(C3⋊C8)—31no—
42C2×C3⋊(C3⋊C8)—62no—
43C2×C3⋊(C3⋊C8)—122no—
44C8×C3⋊D3—61no—
45C8×C3⋊D3—61no—
46C2×C3×C3⋊C8—22no—
47D3×C3⋊C8—64no—
48D3×C3⋊C8—64no—
49C22×C33C4⋊C411yescharacteristic
50C2×C3×C3⋊D3—21no—
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3C4⋊C411yescharacteristic
53C2×C3×C3⋊D3C4⋊C411yescharacteristic
54C33×C4C4211yescharacteristic
55C33×C4C22⋊C411yescharacteristic
56C3×C3⋊Dic3C22⋊C411yescharacteristic
57C3×C3⋊Dic3C22⋊C411yescharacteristic
58C2×C3×M16—91no—
59C2×#208—91no—
60C22×C3⋊C8—91no—
61C2×D3×C8—182no—
62C22×C3⋊D3Dic611yescharacteristic
63C22×C3×D3—62no—
64C2×C32×C4—22no—
65C2×C32×C4—42no—
66C2×C32×C4C4×D311yescharacteristic
67C2×C3⋊Dic3#6711yescharacteristic
68C4×C3⋊D3C4×D311yescharacteristic
69C4×C3⋊D3D1211yescharacteristic
70C4×C3⋊D3C2×Dic311yescharacteristic
71C4×C3⋊D3#6711yescharacteristic
72C2×C3×Dic3—62no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C3×C4×D3—62no—
76C3×C4×D3—62no—
77#622—22no—
78#622—62no—
79C32×C8—61no—
80C3⋊(C3⋊C8)—61no—
81C3⋊(C3⋊C8)—64no—
82C3⋊(C3⋊C8)—124no—
83C3×C3⋊C8—24no—
84#341—271noSylow
85C2×C33#9511yescharacteristic
86C2×C33#10811yescharacteristic
87C2×C33#10811yescharacteristic
88C3×C3⋊D3—21no—
89C3×C3⋊D3—21no—
90C22×C3×C4—91no—
91C2×C4×D3—62no—
92C3×M16—92no—
93C3×M16—181no—
94C2×C3×C8—62no—
95C2×C3×C8—181no—
96#208—92no—
97#208—181no—
98C2×C3⋊C8—91no—
99C2×C3⋊C8—182no—
100C2×C3⋊C8—181no—
101C2×C3⋊C8—181no—
102C2×C3⋊C8—181no—
103C2×C3⋊C8—362no—
104C2×C3⋊C8#15711yescharacteristic
105D3×C8—184no—
106D3×C8—184no—
107C22×C32—22no—
108C22×C32—42no—
109C22×C32Dic3⋊C411yescharacteristic
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3Dic3⋊C411yescharacteristic
113C2×C3⋊D3C4⋊Dic311yescharacteristic
114C2×C3×D3—62no—
115C2×C3×D3—62no—
116C2×C3×D3—64no—
117C2×C3×D3—64no—
118C32×C4—22no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—42no—
122C32×C4C4×Dic311yescharacteristic
123C32×C4(C2×D3)⋊C411yescharacteristic
124C3⋊Dic3(C2×D3)⋊C411yescharacteristic
125C3⋊Dic3#21711yescharacteristic
126C3×Dic3—62no—
127C3×Dic3—62no—
128C2×M16—91no—
129C2×M16—271no—
130C22×C8—271no—
131C33#34111yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—62no—
134C2×C3×C4—22no—
135C2×C3×C4—42no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4C2×#15711yescharacteristic
142C2×Dic3—62no—
143C4×D3—62no—
144C4×D3—62no—
145C4×D3—62no—
146C4×D3—62no—
147C3×C8—64no—
148C3×C8—182no—
149C3⋊C8—21no—
150C3⋊C8—184no—
151C3⋊C8—182no—
152C3⋊C8—181no—
153C3⋊C8—364no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—22no—
157C2×C32—42no—
158C2×C32—42no—
159C2×C32—42no—
160C2×C32#82211yescharacteristic
161C2×C32#82211yescharacteristic
162C2×C32C3⋊#9511yescharacteristic
163C3⋊D3—21no—
164C3⋊D3—21no—
165C3×D3—64no—
166C3×D3—64no—
167C22×C4—91no—
168M16—92no—
169M16—181no—
170M16—272no—
171M16—541no—
172C2×C8—31no—
173C2×C8—181no—
174C2×C8—271no—
175C2×C8—541no—
176C2×C8—541no—
177C2×C8—541no—
178C22×C3—22no—
179C22×C3—42no—
180C22×C3—91no—
181C22×C3—91no—
182C22×C3—181no—
183C22×C3—181no—
184C22×C3C4⋊#15711yescharacteristic
185C2×D3—62no—
186C2×D3—62no—
187C2×D3—64no—
188C2×D3—64no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—42no—
192C3×C4—42no—
193C3×C4—91no—
194C3×C4—91no—
195C3×C4C4×#15711yescharacteristic
196C3×C4C22⋊#15711yescharacteristic
197Dic3—62no—
198Dic3—62no—
199C32—22no—
200C32—42no—
201C32#455811yescharacteristic, complemented
202C23—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4D3×#15711yescharacteristic
209C8—61no—
210C8—182no—
211C8—542no—
212C8—541no—
213C2×C3—22no—
214C2×C3—22no—
215C2×C3—22no—
216C2×C3—42no—
217C2×C3—42no—
218C2×C3—42no—
219C2×C3—181no—
220C2×C3—181no—
221C2×C3#6377011yescharacteristic
222C2×C3#6377111yescharacteristic
223C2×C3#6378111yescharacteristic
224D3—64no—
225D3—64no—
226C22—91no—
227C22—91no—
228C22—181no—
229C22—181no—
230C22Dic3⋊#15711yescharacteristic
231C4—91no—
232C4—91no—
233C4Dic3×#15711yescharacteristic
234C4(C2×D3)⋊#15711yescharacteristic
235C3—22no—
236C3—42no—
237C3#1059018511yescharacteristic, complemented
238C2—181no—
239C2—181no—
240C2#1172271611yescharacteristic
241C2#1172271711yescharacteristic
242C2#1172272211yescharacteristic
subgroupG/H
centerC22Dic3⋊#157
commutatorC33×C4C42
Frattini, hypercenterC2×C4D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#341
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
131. C33#341semidirect
201. C32#4558semidirect
237. C3#10590185semidirect

Maximal subgroups

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

subgroupindex
C2×#889892
C2×C3×#37882
C2×C3⋊D3×C3⋊C82
#105901853
#45589