Finite Group Databasebeta

← #13166490

Subgroups — #13166490

Order 1728.

560 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 98 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×#88925C212yesmaximal
2C4×D3×C3⋊Dic3C211yesmaximal, characteristic
3#11723102C211yesmaximal, characteristic, complemented
4C3×#63777C211yesmaximal, characteristic, complemented
5Dic3×#622C212yesmaximal, complemented
6C4×#3792—31nomaximal
7C2×C32×C4×D3C411yescharacteristic
8C2×D3×C3⋊Dic3C411yescharacteristic
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12#88925C418yescomplemented
13C4×C3⋊(C3⋊Dic3)C411yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17C2×#3792—32no—
18C2×C4×C3⋊Dic3—31no—
19C4×D3×Dic3—62no—
20#63777—31no—
21#63777D311yescharacteristic, complemented
22C4×#622—32no—
23C22×C32×D3C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3C2×C411yescharacteristic
30C32×C4×D3—22no—
31D3×C3⋊Dic3—22no—
32D3×C3⋊Dic3C2×C414yes—
33C3⋊#622C2×C414yescomplemented
34C3×#622C2×C414yescomplemented
35#4707—91nomaximal
36C22×C32×C4—31no—
37C22×C3⋊Dic3—31no—
38C22×C3⋊Dic3—31no—
39C2×C3×C4×D3—22no—
40C2×D3×Dic3—62no—
41C2×D3×Dic3—62no—
42#3792—38no—
43C2×#622—32no—
44C2×#622C2×D312yes—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3—62no—
49C4×C3⋊Dic3—122no—
50C4×C3⋊Dic3C2×D311yescharacteristic
51C3×C4×Dic3—62no—
52Dic32—62no—
53Dic32—62no—
54C22×C33C22×C411yescharacteristic
55C2×C32×D3—21no—
56C2×C32×D3—21no—
57C2×C32×D3C4212yes—
58C33×C4—21no—
59C3⋊(C3⋊Dic3)C4212yes—
60C3⋊(C3⋊Dic3)M1612yescomplemented
61C3×C3⋊Dic3—21no—
62C3×C3⋊Dic3C22×C412yes—
63C32×Dic3M1612yescomplemented
64C2×C4×Dic3—182no—
65C2×ΓL(1,𝔽25)—92no—
66C42×D3—91no—
67C3×#97—91no—
68C8×Dic3—92no—
69#803—91no—
70C23×C32—31no—
71C22×C3×D3—22no—
72C2×C32×C4—22no—
73C2×C32×C4—31no—
74C2×C32×C4—42no—
75C2×C32×C4—62no—
76C2×C32×C4C4×D311yescharacteristic
77C2×C3⋊Dic3—31no—
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3—34no—
80C2×C3⋊Dic3—62no—
81C2×C3⋊Dic3—62no—
82C2×C3⋊Dic3—62no—
83C2×C3⋊Dic3—122no—
84C2×C3⋊Dic3—122no—
85C2×C3⋊Dic3C4×D311yescharacteristic
86C2×C3⋊Dic3C22×D311yescharacteristic
87C2×C3×Dic3—22no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C3×C4×D3—28no—
91D3×Dic3—68no—
92D3×Dic3—68no—
93#622—34no—
94#622C4×D314yescomplemented
95C4×M16—271noSylow
96C2×C33C2×C4211yescharacteristic
97C2×C33C2×M1611yescharacteristic
98C2×C33C2×M1611yescharacteristic
99C32×D3—22no—
100C22×C3×C4—62no—
101C22×Dic3—182no—
102C22×Dic3—182no—
103C2×C4×D3—91no—
104C2×C4×D3—91no—
105C2×C4×D3#15711yescharacteristic
106C2×C3×C8—92no—
107C3×C42—91no—
108C4×Dic3—62no—
109C4×Dic3—91no—
110C4×Dic3—91no—
111C4×Dic3—91no—
112C4×Dic3—182no—
113C4×Dic3—182no—
114C4×Dic3—182no—
115C4×Dic3—362no—
116C2×C3⋊C8—92no—
117ΓL(1,𝔽25)—98no—
118C22×C32—22no—
119C22×C32—32no—
120C22×C32—42no—
121C22×C32—61no—
122C22×C32—61no—
123C22×C32C2×C4×D311yescharacteristic
124C2×C3×D3—24no—
125C2×C3×D3—24no—
126C2×C3×D3—24no—
127C32×C4—21no—
128C32×C4—24no—
129C32×C4—32no—
130C32×C4—44no—
131C3⋊Dic3—21no—
132C3⋊Dic3—32no—
133C3⋊Dic3—32no—
134C3⋊Dic3—64no—
135C3⋊Dic3—64no—
136C3⋊Dic3—124no—
137C3⋊Dic3—124no—
138C3⋊Dic3C2×C4×D312yes—
139C3×Dic3—24no—
140C3×Dic3—64no—
141C3×Dic3—64no—
142C2×M16—272no—
143C2×C42—271no—
144#97—91no—
145#97—271no—
146C4×C8—272no—
147C33C4×M1611yescharacteristic, complemented, Sylow
148C23×C3—62no—
149C22×D3C2×#15711yescharacteristic
150C2×C3×C4—22no—
151C2×C3×C4—42no—
152C2×C3×C4—62no—
153C2×C3×C4—68no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4C2×#15711yescharacteristic
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—91no—
160C2×Dic3—91no—
161C2×Dic3—188no—
162C2×Dic3—182no—
163C2×Dic3—182no—
164C2×Dic3—188no—
165C2×Dic3—362no—
166C2×Dic3—362no—
167C2×Dic3C2×#15711yescharacteristic
168C4×D3—22no—
169C4×D3—94no—
170C4×D3—182no—
171C3×C8—94no—
172C3⋊C8—94no—
173C2×C32—22no—
174C2×C32—22no—
175C2×C32—22no—
176C2×C32—42no—
177C2×C32—42no—
178C2×C32—42no—
179C2×C32—62no—
180C2×C32C42×D311yescharacteristic
181C2×C32D3×M1611yescharacteristic
182C2×C32D3×M1611yescharacteristic
183C3×D3—28no—
184C22×C4—31no—
185C22×C4—271no—
186C22×C4—271no—
187M16—278no—
188C2×C8—92no—
189C2×C8—272no—
190C42—91no—
191C42—271no—
192C42—271no—
193C42—271no—
194C22×C3—22no—
195C22×C3—42no—
196C22×C3—64no—
197C22×C3—64no—
198C22×C3—64no—
199C22×C3C22×#15711yescharacteristic
200C2×D3—21no—
201C2×D3—21no—
202C2×D3C4×#15712yes—
203C3×C4—24no—
204C3×C4—21no—
205C3×C4—44no—
206C3×C4—64no—
207C3×C4—92no—
208C3×C4—181no—
209Dic3—64no—
210Dic3—64no—
211Dic3—92no—
212Dic3—92no—
213Dic3—184no—
214Dic3—184no—
215Dic3—364no—
216Dic3—364no—
217Dic3#379212yescomplemented
218C32—22no—
219C32—42no—
220C32#470711yescharacteristic, complemented
221C23—31no—
222C2×C4—31no—
223C2×C4—62no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4—274no—
227C2×C4—271no—
228C2×C4—271no—
229C2×C4—542no—
230C2×C4D3×#15711yescharacteristic
231C8—94no—
232C8—274no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—22no—
236C2×C3—42no—
237C2×C3—42no—
238C2×C3—42no—
239C2×C3—68no—
240C2×C3#6428311yescharacteristic
241C2×C3C2×C4×#15711yescharacteristic
242C2×C3C2×#379211yescharacteristic
243D3—22no—
244C22—32no—
245C22—61no—
246C22—61no—
247C22C2×D3×#15711yescharacteristic
248C4—21no—
249C4—32no—
250C4—92no—
251C4—181no—
252C4—272no—
253C4—272no—
254C3—22no—
255C3—42no—
256C3C4×#379211yescharacteristic, complemented
257C2—62no—
258C2#1172425511yescharacteristic
259C2C4×D3×#15711yescharacteristic
260C2D3×#379211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C2×C42
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33C4×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723102C2semidirect
4. C3×#63777C2semidirect
5. Dic3×#622C2semidirect
12. #88925C4semidirect
21. #63777D3semidirect
33. C3⋊#622C2×C4semidirect
34. C3×#622C2×C4semidirect
60. C3⋊(C3⋊Dic3)M16semidirect
63. C32×Dic3M16semidirect
94. #622C4×D3semidirect
147. C33C4×M16semidirect
217. Dic3#3792semidirect
220. C32#4707semidirect
256. C3C4×#3792semidirect

Maximal subgroups

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

subgroupindex
C2×#889252
C4×D3×C3⋊Dic32
#117231022
C3×#637772
Dic3×#6222
C4×#37923
#47079