Finite Group Databasebeta

← #13166453

Subgroups — #13166453

Order 1728.

616 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 100 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×#88924C212yesmaximal
2C4×#6289C211yesmaximal, characteristic
3#11723102C211yesmaximal, characteristic, complemented
4C3×#63777C211yesmaximal, characteristic, complemented
5#11722724C212yesmaximal, complemented
6#10593723—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
8C2×#6289C411yescharacteristic
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12#88924C418yescomplemented
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4×Dic3C411yescharacteristic
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17C2×#3787—32no—
18C42×C3⋊D3—31no—
19C4×#624—62no—
20#63777—32no—
21#63777—31no—
22#63777D311yescharacteristic, complemented
23C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C4×C3⋊(C3⋊D3)—22no—
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3C2×C411yescharacteristic
30C2×C32×Dic3C2×C411yescharacteristic
31#6289—22no—
32#6289C2×C414yes—
33C3⋊#622C2×C414yescomplemented
34C3×#622C2×C414yescomplemented
35D3×#97—91nomaximal
36C2×C4×C3⋊D3—31no—
37C2×C4×C3⋊D3—31no—
38C2×C4×C3⋊D3—31no—
39C2×C4×C3⋊D3—62no—
40C2×C4×C3⋊D3—122no—
41C2×#624—62no—
42C2×#624—62no—
43C2×#622—32no—
44C2×#622C2×D312yes—
45#3787—38no—
46C32×C42—31no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3—31no—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C3×C4×Dic3—22no—
51C3×C4×Dic3—62no—
52Dic32—62no—
53Dic32—62no—
54C22×C33C22×C411yescharacteristic
55C2×C3⋊(C3⋊D3)C4212yes—
56C2×C3⋊(C3⋊D3)M1612yes—
57C2×C3⋊(C3⋊D3)M1612yes—
58C33×C4—21no—
59C3⋊(C3⋊Dic3)—21no—
60C3×C3⋊Dic3—21no—
61C3×C3⋊Dic3C22×C412yes—
62C32×Dic3—21no—
63C32×Dic3C4212yes—
64C2×D3×C8—92no—
65C42×D3—91no—
66C42×D3—182no—
67C3×#97—91no—
68#815—92no—
69#803—91no—
70C22×C3⋊D3—31no—
71C22×C3⋊D3—62no—
72C22×C3⋊D3—122no—
73C2×C32×C4—22no—
74C2×C32×C4—31no—
75C2×C32×C4—31no—
76C2×C32×C4—42no—
77C2×C32×C4C4×D311yescharacteristic
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—122no—
81C2×C3⋊Dic3C4×D311yescharacteristic
82C2×C3⋊Dic3C22×D311yescharacteristic
83C4×C3⋊D3—34no—
84C4×C3⋊D3—62no—
85C4×C3⋊D3—62no—
86C4×C3⋊D3—68no—
87C4×C3⋊D3—128no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92#624—68no—
93#624—68no—
94#622—34no—
95#622C4×D314yescomplemented
96C2×#97—271noSylow
97C2×C33C2×C4211yescharacteristic
98C2×C33C2×M1611yescharacteristic
99C2×C33C2×M1611yescharacteristic
100C3⋊(C3⋊D3)#9714yescomplemented
101C2×C4×D3—91no—
102C2×C4×D3—91no—
103C2×C4×D3—91no—
104C2×C4×D3—182no—
105C2×C4×D3—182no—
106C2×C4×D3—182no—
107C2×C4×D3—362no—
108C2×C3×C8—92no—
109C3×C42—62no—
110C3×C42—91no—
111C4×Dic3—62no—
112C4×Dic3—91no—
113C4×Dic3—91no—
114C4×Dic3—182no—
115C4×Dic3—182no—
116C4×Dic3#15711yescharacteristic
117C2×C3⋊C8—92no—
118D3×C8—98no—
119C22×C32—22no—
120C22×C32—42no—
121C22×C32C2×C4×D311yescharacteristic
122C2×C3⋊D3—32no—
123C2×C3⋊D3—32no—
124C2×C3⋊D3—32no—
125C2×C3⋊D3—64no—
126C2×C3⋊D3—64no—
127C2×C3⋊D3—64no—
128C2×C3⋊D3—124no—
129C2×C3⋊D3—124no—
130C2×C3⋊D3—124no—
131C32×C4—21no—
132C32×C4—24no—
133C32×C4—32no—
134C32×C4—44no—
135C32×C4—61no—
136C3⋊Dic3—21no—
137C3⋊Dic3—61no—
138C3⋊Dic3—64no—
139C3⋊Dic3—124no—
140C3⋊Dic3C2×C4×D312yes—
141C3×Dic3—24no—
142C3×Dic3—24no—
143C3×Dic3—64no—
144C3×Dic3—64no—
145C22×C8—272no—
146C2×C42—271no—
147#97—91no—
148#97—271no—
149#97—272no—
150C33C2×#9711yescharacteristic, complemented, Sylow
151C22×D3—91no—
152C22×D3—182no—
153C22×D3—362no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4C2×#15711yescharacteristic
161C2×Dic3—62no—
162C2×Dic3—62no—
163C2×Dic3—91no—
164C2×Dic3—182no—
165C2×Dic3—362no—
166C2×Dic3C2×#15711yescharacteristic
167C2×Dic3C2×#15711yescharacteristic
168C4×D3—94no—
169C4×D3—188no—
170C4×D3—182no—
171C4×D3—188no—
172C4×D3—182no—
173C4×D3—188no—
174C4×D3—368no—
175C3×C8—94no—
176C3⋊C8—94no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—42no—
181C2×C32—42no—
182C2×C32—42no—
183C2×C32C42×D311yescharacteristic
184C2×C32D3×M1611yescharacteristic
185C2×C32D3×M1611yescharacteristic
186C3⋊D3—34no—
187C3⋊D3—68no—
188C3⋊D3—128no—
189C22×C4—271no—
190C22×C4—271no—
191C22×C4—271no—
192C2×C8—92no—
193C2×C8—278no—
194C2×C8—272no—
195C42—31no—
196C42—91no—
197C42—271no—
198C42—271no—
199C22×C3—22no—
200C22×C3—42no—
201C22×C3C22×#15711yescharacteristic
202C2×D3—92no—
203C2×D3—92no—
204C2×D3—92no—
205C2×D3—184no—
206C2×D3—184no—
207C2×D3—184no—
208C2×D3—364no—
209C2×D3—364no—
210C2×D3—364no—
211C3×C4—24no—
212C3×C4—21no—
213C3×C4—44no—
214C3×C4—64no—
215C3×C4—64no—
216C3×C4—92no—
217C3×C4—181no—
218Dic3—21no—
219Dic3—64no—
220Dic3—64no—
221Dic3—184no—
222Dic3—181no—
223Dic3—364no—
224Dic3C4×#15712yes—
225C32—22no—
226C32—42no—
227C32D3×#9711yescharacteristic, complemented
228C23—271no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—274no—
234C2×C4—271no—
235C2×C4—542no—
236C2×C4—542no—
237C2×C4D3×#15711yescharacteristic
238C8—94no—
239C8—274no—
240C2×C3—22no—
241C2×C3—22no—
242C2×C3—22no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—42no—
246C2×C3#6428311yescharacteristic
247C2×C3#6428311yescharacteristic
248C2×C3C2×C4×#15711yescharacteristic
249D3—94no—
250D3—188no—
251D3—368no—
252C22—272no—
253C22—272no—
254C22—272no—
255C22C2×D3×#15711yescharacteristic
256C4—21no—
257C4—32no—
258C4—61no—
259C4—92no—
260C4—181no—
261C4—541no—
262C3—22no—
263C3—42no—
264C3#1059372311yescharacteristic, complemented
265C2—274no—
266C2#1172425511yescharacteristic
267C2C4×D3×#15711yescharacteristic
268C2#1172426911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C2×C42
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#97
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. #11722724C2semidirect
12. #88924C4semidirect
22. #63777D3semidirect
33. C3⋊#622C2×C4semidirect
34. C3×#622C2×C4semidirect
95. #622C4×D3semidirect
100. C3⋊(C3⋊D3)#97semidirect
150. C33C2×#97semidirect
227. C32D3×#97semidirect
264. C3#10593723semidirect

Maximal subgroups

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

subgroupindex
C2×#889242
C4×#62892
#117231022
C3×#637772
#117227242
#105937233
D3×#979