Finite Group Databasebeta

← #13161831

Subgroups — #13161831

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3748C212yesmaximal
2C2×C3×#3748C211yesmaximal, characteristic
3#11722081C214yesmaximal, complemented
4#10591143—31nomaximal
5#10591141—32nomaximal
6C22×C33×C4C2211yescharacteristic
7C3×#3748—22no—
8C3×#3748C414yes—
9C3×#3748C2214yes—
10C2×C3×C3⋊(C3⋊C8)—22no—
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C2×#3748D312yes—
13C2×#3748D311yescharacteristic
14C2×C3×#208—32no—
15C2×C3×#208—32no—
16C2×C3×#208—32no—
17C2×C3×#208—61no—
18C2×C3×#208—62no—
19#63578—34no—
20#63577—38no—
21C23×C33Q811yescharacteristic
22C2×C33×C4C2×C412yes—
23C2×C33×C4D412yes—
24C2×C33×C4Q811yescharacteristic
25C2×C33×C4C2311yescharacteristic
26C3×C3⋊(C3⋊C8)—24no—
27C3×C3⋊(C3⋊C8)C2×C414yes—
28#5125—91no—
29#5124—92no—
30C22×C32×C4—22no—
31C22×C32×C4—21no—
32C22×C32×C4—41no—
33C22×C32×C4C2×D312yes—
34C22×C32×C4C2×D311yescharacteristic
35#3748—22no—
36#3748Dic314yes—
37#3748C2×D314yes—
38C2×C3⋊(C3⋊C8)—22no—
39C2×C3⋊(C3⋊C8)C2×D312yes—
40C3×#208—34no—
41C3×#208—34no—
42C3×#208—34no—
43C3×#208—34no—
44C3×#208—64no—
45C3×#208—64no—
46C3×#208—64no—
47C3×#208—62no—
48C3×#208—62no—
49C2×C3×C3⋊C8—34no—
50C2×C3×C3⋊C8—62no—
51C2×C3×C3⋊C8—64no—
52C2×C3×C3⋊C8—62no—
53C2×C3×C3⋊C8—62no—
54C22×C33—41no—
55C22×C33C4⋊C412yes—
56C22×C33C2×Q811yescharacteristic
57C33×C4C4⋊C412yes—
58C33×C4C22×C411yescharacteristic
59C33×C4C2×D411yescharacteristic
60C2×C3×M16—91no—
61C2×C3×M16—92no—
62C2×#208—32no—
63C2×#208—32no—
64C2×#208—32no—
65C2×#208—61no—
66C2×#208—62no—
67#822—98no—
68#819—94no—
69C23×C32—22no—
70C23×C32—21no—
71C23×C32—41no—
72C23×C32Dic612yes—
73C23×C32Dic611yescharacteristic
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—22no—
84C2×C32×C4—42no—
85C2×C32×C4—42no—
86C2×C32×C4—41no—
87C2×C32×C4—41no—
88C2×C32×C4Dic612yes—
89C2×C32×C4Dic611yescharacteristic
90C2×C32×C4C4×D312yes—
91C2×C32×C4C4×D312yes—
92C2×C32×C4D1212yes—
93C2×C32×C4C2×Dic312yes—
94C2×C32×C4#6712yes—
95C2×C32×C4#6712yes—
96C2×C32×C4C22×D312yes—
97C2×C32×C4C22×D311yescharacteristic
98C3⋊(C3⋊C8)—24no—
99C3⋊(C3⋊C8)C2×Dic314yes—
100C3×C3⋊C8—38no—
101C3×C3⋊C8—64no—
102C3×C3⋊C8—68no—
103C3×C3⋊C8—64no—
104C3×C3⋊C8—64no—
105#430—271noSylow
106C2×C33—22no—
107C2×C33—21no—
108C2×C33C2×C4⋊C411yescharacteristic
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4D3212yes—
113C22×C3×C4D3211yescharacteristic
114C3×M16—94no—
115C3×M16—94no—
116C3×M16—182no—
117C2×C3×C8—92no—
118C2×C3×C8—182no—
119#208—34no—
120#208—34no—
121#208—34no—
122#208—34no—
123#208—64no—
124#208—64no—
125#208—64no—
126#208—62no—
127#208—62no—
128C2×C3⋊C8—34no—
129C2×C3⋊C8—62no—
130C2×C3⋊C8—64no—
131C2×C3⋊C8—62no—
132C2×C3⋊C8—62no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32—22no—
136C22×C32—21no—
137C22×C32—22no—
138C22×C32—42no—
139C22×C32—41no—
140C22×C32—42no—
141C22×C32—41no—
142C22×C32—44no—
143C22×C32—42no—
144C22×C32—44no—
145C22×C32Dic3⋊C412yes—
146C22×C32Dic3⋊C412yes—
147C22×C32C4⋊Dic312yes—
148C22×C32C2×Dic612yes—
149C22×C32C2×Dic611yescharacteristic
150C32×C4—22no—
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—22no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—22no—
157C32×C4—42no—
158C32×C4—41no—
159C32×C4—41no—
160C32×C4Dic3⋊C412yes—
161C32×C4Dic3⋊C412yes—
162C32×C4C4⋊Dic312yes—
163C32×C4C2×C4×D312yes—
164C32×C4C2×D1211yescharacteristic
165C32×C4C22×Dic311yescharacteristic
166C32×C4C2×#6712yes—
167C2×M16—91no—
168C2×M16—92no—
169#108—274no—
170C33#43011yescharacteristic, complemented, Sylow
171C23×C3—21no—
172C23×C3—22no—
173C23×C3—41no—
174C23×C3#62712yes—
175C23×C3#62711yescharacteristic
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4D3×Dic312yes—
191C2×C3×C4D3×Dic312yes—
192C2×C3×C4#62412yes—
193C2×C3×C4#62512yes—
194C2×C3×C4#62612yes—
195C2×C3×C4#62612yes—
196C2×C3×C4#62712yes—
197C2×C3×C4#62711yescharacteristic
198C2×C3×C4C2×D3212yes—
199C2×C3×C4C2×D3211yescharacteristic
200C3×C8—94no—
201C3×C8—184no—
202C3⋊C8—38no—
203C3⋊C8—64no—
204C3⋊C8—68no—
205C3⋊C8—64no—
206C3⋊C8—64no—
207C2×C32—22no—
208C2×C32—21no—
209C2×C32—24no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—22no—
213C2×C32—22no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—41no—
217C2×C32—42no—
218C2×C32—42no—
219C2×C32—42no—
220C2×C32—44no—
221C2×C32—42no—
222C2×C32C2×Dic3⋊C412yes—
223C2×C32C2×C4⋊Dic311yescharacteristic
224C22×C4#111311yescharacteristic
225M16—94no—
226M16—94no—
227M16—182no—
228C2×C8—92no—
229C2×C8—182no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—42no—
236C22×C3—41no—
237C22×C3—42no—
238C22×C3—44no—
239C22×C3—44no—
240C22×C3—41no—
241C22×C3—42no—
242C22×C3Dic3⋊Dic312yes—
243C22×C3Dic3⋊Dic312yes—
244C22×C3#372412yes—
245C22×C3C2×#62712yes—
246C22×C3C2×#62711yescharacteristic
247C3×C4—21no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—21no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—42no—
257C3×C4Dic3⋊Dic312yes—
258C3×C4Dic3⋊Dic312yes—
259C3×C4#372412yes—
260C3×C4C2×D3×Dic312yes—
261C3×C4C2×#62411yescharacteristic
262C3×C4C2×#62511yescharacteristic
263C3×C4C2×#62612yes—
264C32—22no—
265C32—21no—
266C32—41no—
267C32#512412yescomplemented
268C32#512511yescharacteristic, complemented
269C23#629611yescharacteristic
270C2×C4#629412yes—
271C2×C4#629512yes—
272C2×C4#629611yescharacteristic
273C2×C4C2×#111311yescharacteristic
274C8—94no—
275C8—184no—
276C2×C3—22no—
277C2×C3—24no—
278C2×C3—22no—
279C2×C3—22no—
280C2×C3—22no—
281C2×C3—22no—
282C2×C3—21no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—42no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—42no—
289C2×C3—44no—
290C2×C3—41no—
291C2×C3C2×Dic3⋊Dic312yes—
292C2×C3C2×#372411yescharacteristic
293C22—41no—
294C22#8881512yes—
295C22C2×#629611yescharacteristic
296C4#8881512yes—
297C4C2×#629411yescharacteristic
298C4C2×#629511yescharacteristic
299C3—22no—
300C3—21no—
301C3—41no—
302C3#1059114112yescomplemented
303C3#1059114311yescharacteristic, complemented
304C2—21no—
305C2—22no—
306C2C2×#8881511yescharacteristic
subgroupG/H
centerC4C2×#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC2×C33C2×C4⋊C4
prime radicalC23×C33Q8
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#430
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11722081C2semidirect
170. C33#430semidirect
267. C32#5124semidirect
268. C32#5125semidirect
302. C3#10591141semidirect
303. C3#10591143semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
#117220812
#105911433
#105911413