Finite Group Databasebeta

← #13166397

Subgroups — #13166397

Order 1728.

380 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 74 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
2C2×C3×#3788C211yesmaximal, characteristic
3C3⋊D3×#208C211yesmaximal, characteristic
4#11722717C212yesmaximal, complemented
5#11722716C212yesmaximal, complemented
6#10593776—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#88993—21no—
9#88989C2211yescharacteristic
10#88989C2211yescharacteristic
11#88988—21no—
12C2×C3×#622—21no—
13C3×#3788—21no—
14C3×#3788C2212yes—
15#88855C411yescharacteristic
16C32×#208C411yescharacteristic
17#88787C412yes—
18C3⋊D3×C3⋊C8C2212yes—
19#64283—31no—
20C2×#3788D311yescharacteristic
21M16×C3⋊D3—31no—
22D3×#208—62no—
23#63771—32no—
24#63770—32no—
25C22×C3×C3⋊D3Q811yescharacteristic
26C2×C33×C4C2×C411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C3×C4×C3⋊D3C2×C411yescharacteristic
29C3×C4×C3⋊D3D411yescharacteristic
30C3×C4×C3⋊D3Q811yescharacteristic
31C3×C4×C3⋊D3C2311yescharacteristic
32C3⋊#622—21no—
33C3⋊#622—21no—
34C3×#622—22no—
35C3⋊(C3⋊(C3⋊C8))C2×C412yes—
36C32×C3⋊C8C2×C412yes—
37#5124—91nomaximal
38C2×C4×C3⋊D3C2×D311yescharacteristic
39C2×C3×C4×D3—62no—
40#3792—61no—
41C2×#622—21no—
42#3788—21no—
43#3788—31no—
44#3788—31no—
45#3788C2×D312yes—
46#3787—61no—
47C32×M16—31no—
48#3748—31no—
49#3748—62no—
50#3748—122no—
51#3743—32no—
52C8×C3⋊D3—32no—
53C3×#208—22no—
54#3711—64no—
55D3×C3⋊C8—64no—
56C22×C33C4⋊C411yescharacteristic
57C2×C3×C3⋊D3—41no—
58C2×C3×C3⋊D3C4⋊C411yescharacteristic
59C2×C3×C3⋊D3C2×Q811yescharacteristic
60C33×C4C4⋊C411yescharacteristic
61C33×C4C22×C411yescharacteristic
62C3×C3⋊Dic3C4⋊C411yescharacteristic
63C3×C3⋊Dic3C2×D411yescharacteristic
64C2×C3×M16—91no—
65C2×#208—91no—
66C2×#208—91no—
67D3×M16—182no—
68#822—92no—
69#822—92no—
70C22×C3⋊D3Dic611yescharacteristic
71C22×C3×D3—62no—
72C2×C32×C4—22no—
73C2×C32×C4—42no—
74C2×C32×C4C4×D311yescharacteristic
75C2×C3⋊Dic3#6711yescharacteristic
76C4×C3⋊D3Dic611yescharacteristic
77C4×C3⋊D3C4×D311yescharacteristic
78C4×C3⋊D3#6711yescharacteristic
79C4×C3⋊D3C22×D311yescharacteristic
80C2×C3×Dic3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85#622—22no—
86#622—61no—
87#622—61no—
88C32×C8—32no—
89C3⋊(C3⋊C8)—32no—
90C3⋊(C3⋊C8)—64no—
91C3⋊(C3⋊C8)—124no—
92C3×C3⋊C8—24no—
93#430—271noSylow
94C2×C33—21no—
95C2×C33C2×C4⋊C411yescharacteristic
96C3×C3⋊D3—21no—
97C3×C3⋊D3—21no—
98C22×C3×C4—91no—
99C2×C4×D3—62no—
100C3×M16—62no—
101C3×M16—92no—
102C3×M16—181no—
103C2×C3×C8—181no—
104#208—91no—
105#208—92no—
106#208—91no—
107#208—91no—
108#208—182no—
109#208—181no—
110#208—362no—
111#208#15711yescharacteristic
112C2×C3⋊C8—92no—
113C2×C3⋊C8—181no—
114ΓL(1,𝔽25)—184no—
115D3×C8—184no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32Dic3⋊C411yescharacteristic
119C2×C3⋊D3—41no—
120C2×C3⋊D3Dic3⋊C411yescharacteristic
121C2×C3⋊D3C2×Dic611yescharacteristic
122C2×C3×D3—62no—
123C2×C3×D3—62no—
124C2×C3×D3—124no—
125C32×C4—22no—
126C32×C4—22no—
127C32×C4—42no—
128C32×C4—42no—
129C32×C4Dic3⋊C411yescharacteristic
130C32×C4C2×C4×D311yescharacteristic
131C3⋊Dic3Dic3⋊C411yescharacteristic
132C3⋊Dic3C2×#6711yescharacteristic
133C3×Dic3—62no—
134C3×Dic3—62no—
135C2×M16—91no—
136C2×M16—271no—
137C2×M16—271no—
138#108—272no—
139#108—272no—
140C33#43011yescharacteristic, complemented, Sylow
141C23×C3—91no—
142C22×D3—62no—
143C2×C3×C4—22no—
144C2×C3×C4—42no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4C2×#15711yescharacteristic
151C2×Dic3—62no—
152C4×D3—62no—
153C4×D3—62no—
154C4×D3—62no—
155C4×D3—62no—
156C3×C8—64no—
157C3×C8—182no—
158C3⋊C8—92no—
159C3⋊C8—184no—
160C3⋊C8—181no—
161C3⋊C8—181no—
162C3⋊C8—364no—
163C3⋊C8C2×#15712yes—
164C2×C32—22no—
165C2×C32—21no—
166C2×C32—42no—
167C2×C32—42no—
168C2×C32—82no—
169C2×C32C2×Dic3⋊C411yescharacteristic
170C3⋊D3—21no—
171C3⋊D3—21no—
172C3×D3—64no—
173C3×D3—122no—
174C22×C4—91no—
175M16—31no—
176M16—92no—
177M16—181no—
178M16—272no—
179M16—271no—
180M16—271no—
181M16—271no—
182M16—541no—
183C2×C8—181no—
184C2×C8—272no—
185C2×C8—541no—
186C22×C3—22no—
187C22×C3—42no—
188C22×C3—91no—
189C22×C3—91no—
190C22×C3—361no—
191C22×C3C4⋊#15711yescharacteristic
192C2×D3—62no—
193C2×D3—62no—
194C2×D3—124no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—42no—
198C3×C4—42no—
199C3×C4—91no—
200C3×C4—91no—
201C3×C4C4⋊#15711yescharacteristic
202C3×C4C22×#15711yescharacteristic
203Dic3—62no—
204Dic3—62no—
205C32—22no—
206C32—42no—
207C32#512411yescharacteristic, complemented
208C23—91no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4D3×#15711yescharacteristic
215C8—32no—
216C8—182no—
217C8—272no—
218C8—541no—
219C8—541no—
220C2×C3—21no—
221C2×C3—22no—
222C2×C3—42no—
223C2×C3—42no—
224C2×C3—82no—
225C2×C3—181no—
226C2×C3—181no—
227C2×C3C2×C4⋊#15711yescharacteristic
228D3—64no—
229D3—122no—
230C22—91no—
231C22—91no—
232C22—361no—
233C22Dic3⋊#15711yescharacteristic
234C4—91no—
235C4—91no—
236C4Dic3⋊#15711yescharacteristic
237C4C2×D3×#15711yescharacteristic
238C3—22no—
239C3—42no—
240C3#1059377611yescharacteristic, complemented
241C2—21no—
242C2—181no—
243C2—181no—
244C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
centerC2C2×Dic3⋊#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
hypocenterC33#430
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722717C2semidirect
5. #11722716C2semidirect
140. C33#430semidirect
207. C32#5124semidirect
240. C3#10593776semidirect

Maximal subgroups

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

subgroupindex
#117245112
C2×C3×#37882
C3⋊D3×#2082
#117227172
#117227162
#105937763
#51249