Finite Group Databasebeta

← #13161500

Subgroups — #13161500

Order 1728.

802 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2C2×#88808C212yesmaximal
3C3×#63630C212yesmaximal, complemented
4#11722071C212yesmaximal, complemented
5#10590322—31nomaximal
6D3×#803—32nomaximal
7C2×C3×C4×C3⋊D3C412yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C33×C42C2211yescharacteristic
10C3×C4×C3⋊Dic3C412yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2214yes—
13#88808C418yescomplemented
14C42×C3⋊D3D311yescharacteristic
15C3×C42×D3—32no—
16C3×C42×D3—61no—
17C2×#3710—32no—
18C2×D3×C3⋊C8—34no—
19#63630D312yescomplemented
20C3×#803—32no—
21C3×#803—32no—
22C3×#803—62no—
23#63555—32no—
24#63554—34no—
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4C2311yescharacteristic
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C3×C4×C3⋊D3—24no—
31C3×C4×C3⋊D3C2×C414yes—
32C3×C3⋊(C3⋊C8)C2×C418yescomplemented
33C2×#803—91no—
34D3×#97—92no—
35C2×C4×C3⋊D3Dic312yes—
36C2×C4×C3⋊D3C2×D311yescharacteristic
37C2×C3×C4×D3—32no—
38C2×C3×C4×D3—34no—
39C2×C3×C4×D3—62no—
40C2×C3×C4×D3—61no—
41C32×C42—22no—
42C32×C42—21no—
43C32×C42—41no—
44C32×C42C2×D312yes—
45C32×C42C2×D311yescharacteristic
46C4×C3⋊Dic3Dic312yes—
47C4×C3⋊Dic3C2×D311yescharacteristic
48C2×C3⋊(C3⋊C8)C2×D314yes—
49C3×C4×Dic3—34no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—62no—
52C3×C4×Dic3—61no—
53C2×C3×C3⋊C8—34no—
54C2×C3×C3⋊C8—34no—
55C2×C3×C3⋊C8—64no—
56#3710—38no—
57D3×C3⋊C8—316no—
58C22×C33C22×C411yescharacteristic
59C2×C3×C3⋊D3C4212yes—
60C2×C3×C3⋊D3M1614yes—
61C33×C4—22no—
62C33×C4C22×C412yes—
63C3×C3⋊Dic3—22no—
64C3×C3⋊Dic3C4212yes—
65C2×C3×C42—91no—
66C22×C3⋊C8—92no—
67C2×D3×C8—94no—
68C42×D3—32no—
69C42×D3—61no—
70C3×#97—92no—
71#815—94no—
72#803—32no—
73#803—32no—
74#803—62no—
75#803—92no—
76C22×C3⋊D3C2×Dic311yescharacteristic
77C22×C3×D3—32no—
78C22×C3×D3—61no—
79C2×C32×C4—24no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C2×C32×C4C4×D314yes—
86C2×C32×C4C2×Dic312yes—
87C2×C32×C4C22×D312yes—
88C2×C32×C4C22×D311yescharacteristic
89C2×C3⋊Dic3C2×Dic312yes—
90C2×C3⋊Dic3C2×Dic311yescharacteristic
91C4×C3⋊D3—24no—
92C4×C3⋊D3C4×D314yes—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—62no—
97C3×C4×D3—38no—
98C3×C4×D3—64no—
99C3×C4×D3—68no—
100C3×C4×D3—68no—
101C3⋊(C3⋊C8)C4×D318yescomplemented
102C3×C3⋊C8—38no—
103C3×C3⋊C8—38no—
104C3×C3⋊C8—68no—
105C2×#97—271noSylow
106C2×C33C2×C4211yescharacteristic
107C2×C33C2×M1612yes—
108C3×C3⋊D3#9714yescomplemented
109C22×C3×C4—91no—
110C22×C3×C4—92no—
111C2×C4×D3—32no—
112C2×C4×D3—34no—
113C2×C4×D3—62no—
114C2×C4×D3—61no—
115C2×C3×C8—94no—
116C3×C42—22no—
117C3×C42—21no—
118C3×C42—41no—
119C3×C42—92no—
120C3×C42—91no—
121C3×C42D3212yes—
122C3×C42D3211yescharacteristic
123C4×Dic3—34no—
124C4×Dic3—32no—
125C4×Dic3—62no—
126C4×Dic3—61no—
127C2×C3⋊C8—34no—
128C2×C3⋊C8—34no—
129C2×C3⋊C8—64no—
130C2×C3⋊C8—98no—
131D3×C8—916no—
132C22×C32—22no—
133C22×C32—21no—
134C22×C32—41no—
135C22×C32C2×C4×D312yes—
136C22×C32C22×Dic311yescharacteristic
137C2×C3⋊D3#20814yes—
138C2×C3⋊D3C4×Dic312yes—
139C2×C3×D3—34no—
140C2×C3×D3—38no—
141C2×C3×D3—62no—
142C2×C3×D3—64no—
143C32×C4—22no—
144C32×C4—24no—
145C32×C4—24no—
146C32×C4—24no—
147C32×C4—22no—
148C32×C4—42no—
149C32×C4—44no—
150C32×C4—44no—
151C32×C4C2×C4×D314yes—
152C32×C4C2×C4×D312yes—
153C3⋊Dic3—22no—
154C3⋊Dic3C4×Dic312yes—
155C3×Dic3—34no—
156C3×Dic3—62no—
157C3×Dic3—64no—
158C3×Dic3—64no—
159C22×C8—272no—
160C2×C42—91no—
161#97—92no—
162#97—272no—
163C33C2×#9711yescharacteristic, complemented, Sylow
164C23×C3—91no—
165C22×D3—32no—
166C22×D3—61no—
167C2×C3×C4—22no—
168C2×C3×C4—24no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—91no—
174C2×C3×C4—94no—
175C2×C3×C4—92no—
176C2×C3×C4—184no—
177C2×C3×C4D3×Dic314yes—
178C2×C3×C4#62412yes—
179C2×C3×C4C2×D3212yes—
180C2×C3×C4C2×D3211yescharacteristic
181C2×Dic3—32no—
182C2×Dic3—34no—
183C2×Dic3—61no—
184C2×Dic3—62no—
185C4×D3—38no—
186C4×D3—64no—
187C4×D3—68no—
188C4×D3—68no—
189C3×C8—98no—
190C3⋊C8—38no—
191C3⋊C8—38no—
192C3⋊C8—68no—
193C2×C32—24no—
194C2×C32—22no—
195C2×C32—22no—
196C2×C32—21no—
197C2×C32—41no—
198C2×C32—42no—
199C2×C32C42×D312yes—
200C2×C32D3×M1614yes—
201C2×C32C2×#20812yes—
202C2×C32C2×C4×Dic311yescharacteristic
203C3⋊D3#80314yescomplemented
204C3×D3—38no—
205C3×D3—64no—
206C22×C4—91no—
207C22×C4—92no—
208C2×C8—94no—
209C2×C8—278no—
210C42—92no—
211C42—91no—
212C42#111311yescharacteristic
213C22×C3—21no—
214C22×C3—22no—
215C22×C3—41no—
216C22×C3—94no—
217C22×C3—92no—
218C22×C3C2×D3×Dic312yes—
219C22×C3C2×#62411yescharacteristic
220C2×D3—38no—
221C2×D3—34no—
222C2×D3—64no—
223C2×D3—62no—
224C3×C4—24no—
225C3×C4—24no—
226C3×C4—22no—
227C3×C4—24no—
228C3×C4—22no—
229C3×C4—44no—
230C3×C4—44no—
231C3×C4—42no—
232C3×C4—92no—
233C3×C4—182no—
234C3×C4C4×D3214yes—
235C3×C4C4×D3212yes—
236Dic3—34no—
237Dic3—62no—
238Dic3—64no—
239Dic3—64no—
240C32—22no—
241C32—21no—
242C32—41no—
243C32D3×#9712yescomplemented
244C32C2×#80311yescharacteristic, complemented
245C23—91no—
246C2×C4—91no—
247C2×C4—94no—
248C2×C4—92no—
249C2×C4—184no—
250C2×C4#629412yes—
251C2×C4C2×#111311yescharacteristic
252C8—98no—
253C2×C3—22no—
254C2×C3—24no—
255C2×C3—21no—
256C2×C3—22no—
257C2×C3—42no—
258C2×C3—41no—
259C2×C3—94no—
260C2×C3D3×#20814yes—
261C2×C3#6381812yes—
262C2×C3C4×D3×Dic312yes—
263C2×C3C4×#62411yescharacteristic
264D3—38no—
265D3—64no—
266C22—94no—
267C22—92no—
268C22C2×#629411yescharacteristic
269C4—22no—
270C4—92no—
271C4—182no—
272C4C4×#111312yes—
273C3—21no—
274C3—22no—
275C3—41no—
276C3D3×#80312yescomplemented
277C3#1059032211yescharacteristic, complemented
278C2—94no—
279C2#1172365612yes—
280C2C4×#629411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutatorC2×C33C2×C42
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42#1113
hypocenterC33C2×#97
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. C3×#63630C2semidirect
4. #11722071C2semidirect
13. #88808C4semidirect
19. #63630D3semidirect
32. C3×C3⋊(C3⋊C8)C2×C4semidirect
101. C3⋊(C3⋊C8)C4×D3semidirect
108. C3×C3⋊D3#97semidirect
163. C33C2×#97semidirect
203. C3⋊D3#803semidirect
243. C32D3×#97semidirect
244. C32C2×#803semidirect
276. C3D3×#803semidirect
277. C3#10590322semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
C2×#888082
C3×#636302
#117220712
#105903223
D3×#8033