Finite Group Databasebeta

← #13166294

Subgroups — #13166294

Order 1728.

460 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 106 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×C4×C3⋊#157C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic
3C3⋊D3×#208C211yesmaximal, characteristic
4#11722705C212yesmaximal, complemented
5#11722703C212yesmaximal, complemented
6M16×#157—31nomaximal
7C22×C3⋊#157C412yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C4×C3⋊#157C412yes—
10C4×C3⋊#157C2211yescharacteristic
11C4×C3⋊#157C2211yescharacteristic
12C3×#3792C412yescomplemented
13C3×#3788C412yescomplemented
14C3×#3787C2212yes—
15#88855C411yescharacteristic, complemented
16C32×#208C411yescharacteristic, complemented
17#88787C412yescomplemented
18C3⋊D3×C3⋊C8C2212yes—
19C2×C4×#157—31no—
20#64283D311yescharacteristic
21M16×C3⋊D3—31no—
22D3×#208—62no—
23#63767—32no—
24C8×#157—32no—
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C2×C3⋊#157—24no—
27C2×C3⋊#157C2×C412yes—
28C2×C3⋊#157C2×C412yes—
29C2×C33×C4C2×C411yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C4×C3⋊D3C2311yescharacteristic
35C3×#622C2×C414yescomplemented
36C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
37C32×C3⋊C8C2×C412yescomplemented
38Dic3×M16—91nomaximal
39C22×#157—32no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—62no—
42#3792Dic312yescomplemented
43C4×#157—32no—
44C4×#157—31no—
45C4×#157—31no—
46#3788Dic312yescomplemented
47#3787C2×D312yes—
48C32×M16—31no—
49#3748—31no—
50#3748—62no—
51#3748—122no—
52#3743—32no—
53C8×C3⋊D3—32no—
54C3×#208—22no—
55#3711—64no—
56D3×C3⋊C8—64no—
57C22×C33C4211yescharacteristic
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3C4211yescharacteristic
60C2×C3×C3⋊D3C22×C411yescharacteristic
61C3⋊#157—22no—
62C3⋊#157M1614yescomplemented
63C33×C4C4211yescharacteristic
64C33×C4C22×C411yescharacteristic
65C3×C3⋊Dic3C4211yescharacteristic
66C3×C3⋊Dic3C22×C411yescharacteristic
67C2×C3×M16—91no—
68C2×C4×Dic3—91no—
69C2×#208—91no—
70D3×M16—182no—
71#815—92no—
72C8×Dic3—92no—
73C22×C3⋊D3C4×D311yescharacteristic
74C22×C3×D3—62no—
75C2×#157—32no—
76C2×#157—32no—
77C2×#157—64no—
78C2×C32×C4—22no—
79C2×C32×C4—42no—
80C2×C32×C4C4×D311yescharacteristic
81C2×C3⋊Dic3C2×Dic311yescharacteristic
82C4×C3⋊D3C4×D311yescharacteristic
83C4×C3⋊D3C4×D311yescharacteristic
84C4×C3⋊D3C2×Dic311yescharacteristic
85C4×C3⋊D3C22×D311yescharacteristic
86C2×C3×Dic3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91#622C2×Dic314yescomplemented
92C32×C8—32no—
93C3⋊(C3⋊C8)—32no—
94C3⋊(C3⋊C8)—64no—
95C3⋊(C3⋊C8)—124no—
96C3×C3⋊C8—24no—
97C4×M16—271noSylow
98C2×C33—21no—
99C2×C33C2×C4211yescharacteristic
100C3×C3⋊D3—21no—
101C3×C3⋊D3C2×M1612yes—
102C22×C3×C4—91no—
103C22×Dic3—92no—
104C2×C4×D3—62no—
105C3×M16—62no—
106C3×M16—92no—
107C3×M16—92no—
108C2×C3×C8—92no—
109C4×Dic3—91no—
110C4×Dic3—91no—
111C4×Dic3—92no—
112#208—91no—
113#208—92no—
114#208—182no—
115#208—362no—
116#208#15711yescharacteristic, complemented
117C2×C3⋊C8—92no—
118ΓL(1,𝔽25)—184no—
119D3×C8—184no—
120C22×C32—22no—
121C22×C32—42no—
122C22×C32C4×Dic311yescharacteristic
123C2×C3⋊D3—22no—
124C2×C3⋊D3C4×Dic311yescharacteristic
125C2×C3⋊D3C2×C4×D311yescharacteristic
126C2×C3×D3—62no—
127C2×C3×D3—62no—
128C2×C3×D3—124no—
129#157—34no—
130#157—62no—
131C32×C4—22no—
132C32×C4—22no—
133C32×C4—42no—
134C32×C4—42no—
135C32×C4C4×Dic311yescharacteristic
136C32×C4C2×C4×D311yescharacteristic
137C3⋊Dic3C4×Dic311yescharacteristic
138C3⋊Dic3C22×Dic311yescharacteristic
139C3×Dic3—62no—
140C3×Dic3—62no—
141C2×M16—91no—
142C2×M16—271no—
143C2×C42—271no—
144#97—272no—
145C4×C8—272no—
146C33C4×M1611yescharacteristic, complemented, Sylow
147C23×C3—91no—
148C22×D3—62no—
149C2×C3×C4—22no—
150C2×C3×C4—42no—
151C2×C3×C4—91no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4C2×#15711yescharacteristic
157C2×Dic3—62no—
158C2×Dic3—92no—
159C2×Dic3—92no—
160C2×Dic3—184no—
161C4×D3—62no—
162C4×D3—62no—
163C4×D3—62no—
164C4×D3—62no—
165C3×C8—64no—
166C3×C8—94no—
167C3⋊C8—92no—
168C3⋊C8—184no—
169C3⋊C8—364no—
170C3⋊C8C2×#15712yescomplemented
171C2×C32—22no—
172C2×C32—21no—
173C2×C32—42no—
174C2×C32—42no—
175C2×C32—82no—
176C2×C32C2×C4×Dic311yescharacteristic
177C3⋊D3—21no—
178C3⋊D3D3×M1612yes—
179C3×D3—64no—
180C3×D3—122no—
181C22×C4—91no—
182C22×C4—272no—
183M16—31no—
184M16—92no—
185M16—92no—
186M16—272no—
187M16—271no—
188C2×C8—92no—
189C2×C8—272no—
190C42—271no—
191C42—271no—
192C42—272no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3—91no—
196C22×C3—91no—
197C22×C3—182no—
198C22×C3C4×#15711yescharacteristic
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—124no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—42no—
205C3×C4—42no—
206C3×C4—91no—
207C3×C4—91no—
208C3×C4C4×#15711yescharacteristic
209C3×C4C22×#15711yescharacteristic
210Dic3—62no—
211Dic3—62no—
212Dic3—94no—
213Dic3—182no—
214C32—22no—
215C32—42no—
216C32Dic3×M1611yescharacteristic, complemented
217C23—91no—
218C2×C4—91no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—272no—
224C2×C4—272no—
225C2×C4—544no—
226C2×C4D3×#15711yescharacteristic
227C8—32no—
228C8—94no—
229C8—272no—
230C2×C3—21no—
231C2×C3—22no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—82no—
235C2×C3—92no—
236C2×C3—181no—
237C2×C3C2×C4×#15711yescharacteristic
238D3—64no—
239D3—122no—
240C22—91no—
241C22—91no—
242C22—182no—
243C22Dic3×#15711yescharacteristic
244C4—91no—
245C4—91no—
246C4—274no—
247C4—542no—
248C4Dic3×#15711yescharacteristic
249C4C2×D3×#15711yescharacteristic
250C3—22no—
251C3—42no—
252C3M16×#15711yescharacteristic, complemented
253C2—21no—
254C2—92no—
255C2—181no—
256C2C2×Dic3×#15711yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×#157
commutator, socleC2×C33C2×C42
FittingC2×C33×C4C2×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C4×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722705C2semidirect
5. #11722703C2semidirect
12. C3×#3792C4semidirect
13. C3×#3788C4semidirect
15. #88855C4semidirect
16. C32×#208C4semidirect
17. #88787C4semidirect
35. C3×#622C2×C4semidirect
36. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
37. C32×C3⋊C8C2×C4semidirect
42. #3792Dic3semidirect
46. #3788Dic3semidirect
62. C3⋊#157M16semidirect
91. #622C2×Dic3semidirect
116. #208#157semidirect
146. C33C4×M16semidirect
170. C3⋊C8C2×#157semidirect
216. C32Dic3×M16semidirect
252. C3M16×#157semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊#1572
C3×#642832
C3⋊D3×#2082
#117227052
#117227032
M16×#1573
Dic3×M169