Finite Group Databasebeta

← #13145737

Subgroups — #13145737

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C3×#3748C212yesmaximal
3#10589168—31nomaximal
4#10589167—32nomaximal
5C22×C33×C4C2211yescharacteristic
6C22×C3×C3⋊Dic3C412yes—
7C3×C4⋊(C3⋊Dic3)—22no—
8C3×#3748—22no—
9C3×#3748C414yescomplemented
10C2×C3×C3⋊(C3⋊C8)—22no—
11C2×C4⋊(C3⋊Dic3)D311yescharacteristic
12C2×#3748D312yes—
13C2×C3×C4⋊Dic3—32no—
14C2×C3×C4⋊Dic3—61no—
15C2×C3×#208—32no—
16C2×C3×#208—32no—
17C2×C3×#208—62no—
18C23×C33C2×C411yescharacteristic
19C2×C33×C4C2×C412yes—
20C2×C33×C4D411yescharacteristic
21C2×C33×C4D412yes—
22C2×C33×C4Q811yescharacteristic
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3—42no—
25C3×C3⋊(C3⋊C8)—24no—
26#4554—92no—
27#4530—91no—
28C22×C32×C4—22no—
29C22×C32×C4—21no—
30C22×C32×C4—41no—
31C22×C32×C4C2×D311yescharacteristic
32C22×C32×C4C2×D312yes—
33C22×C3⋊Dic3Dic312yes—
34C22×C3×Dic3—34no—
35C22×C3×Dic3—62no—
36C4⋊(C3⋊Dic3)—22no—
37#3748—22no—
38#3748Dic314yescomplemented
39C2×C3⋊(C3⋊C8)—22no—
40C3×C4⋊Dic3—64no—
41C3×C4⋊Dic3—62no—
42C3×C4⋊Dic3—62no—
43C3×#208—34no—
44C3×#208—34no—
45C3×#208—64no—
46C3×#208—64no—
47C3×#208—62no—
48C3×#208—62no—
49C2×C3×C3⋊C8—64no—
50C2×C3×C3⋊C8—62no—
51C2×C3×C3⋊C8—62no—
52C22×C33—21no—
53C22×C33—21no—
54C22×C33C4211yescharacteristic
55C22×C33C22⋊C411yescharacteristic
56C22×C33C4⋊C411yescharacteristic
57C33×C4C22⋊C412yes—
58C33×C4C4⋊C412yes—
59C3×C3⋊Dic3—42no—
60C2×C3×M16—92no—
61C2×C3×C4⋊C4—91no—
62C2×C4⋊Dic3—32no—
63C2×C4⋊Dic3—61no—
64C2×#208—32no—
65C2×#208—32no—
66C2×#208—62no—
67C23×C32—22no—
68C23×C32—21no—
69C23×C32—41no—
70C23×C32C4×D312yes—
71C23×C32C2×Dic311yescharacteristic
72C2×C32×C4—22no—
73C2×C32×C4—22no—
74C2×C32×C4—22no—
75C2×C32×C4—21no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4—22no—
82C2×C32×C4—42no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4Dic611yescharacteristic
87C2×C32×C4Dic612yes—
88C2×C32×C4C4×D312yes—
89C2×C32×C4C4×D312yes—
90C2×C32×C4D1212yes—
91C2×C32×C4D1211yescharacteristic
92C2×C32×C4C2×Dic312yes—
93C2×C32×C4#6712yes—
94C2×C32×C4#6712yes—
95C2×C32×C4#6712yes—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—42no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—124no—
101C2×C3×Dic3—124no—
102C3⋊(C3⋊C8)—24no—
103C3×C3⋊C8—68no—
104C3×C3⋊C8—64no—
105C3×C3⋊C8—64no—
106#343—271noSylow
107C2×C33—21no—
108C2×C33—21no—
109C2×C33#9511yescharacteristic
110C2×C33#10011yescharacteristic
111C2×C33#10111yescharacteristic
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—92no—
116C22×C3×C4D3211yescharacteristic
117C22×C3×C4D3212yes—
118C22×Dic3—34no—
119C22×Dic3—62no—
120C3×M16—94no—
121C3×M16—182no—
122C2×C3×C8—182no—
123C3×C4⋊C4—182no—
124C4⋊Dic3—64no—
125C4⋊Dic3—62no—
126C4⋊Dic3—62no—
127#208—34no—
128#208—34no—
129#208—64no—
130#208—64no—
131#208—62no—
132#208—62no—
133C2×C3⋊C8—64no—
134C2×C3⋊C8—62no—
135C2×C3⋊C8—62no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—21no—
143C22×C32—22no—
144C22×C32—22no—
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—41no—
151C22×C32—42no—
152C22×C32—42no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32C4×Dic311yescharacteristic
156C22×C32C4×Dic312yes—
157C22×C32Dic3⋊C412yes—
158C22×C32C4⋊Dic311yescharacteristic
159C22×C32(C2×D3)⋊C412yes—
160C22×C32#21711yescharacteristic
161C32×C4—22no—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4Dic3⋊C412yes—
170C32×C4Dic3⋊C412yes—
171C32×C4C4⋊Dic312yes—
172C32×C4(C2×D3)⋊C412yes—
173C32×C4(C2×D3)⋊C412yes—
174C32×C4#21712yes—
175C3⋊Dic3—42no—
176C3×Dic3—124no—
177C3×Dic3—124no—
178C2×M16—92no—
179C2×C4⋊C4—91no—
180C33#34311yescharacteristic, complemented, Sylow
181C23×C3—21no—
182C23×C3—22no—
183C23×C3—41no—
184C23×C3D3×Dic312yes—
185C23×C3#62411yescharacteristic
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—182no—
201C2×C3×C4—362no—
202C2×C3×C4D3×Dic312yes—
203C2×C3×C4D3×Dic312yes—
204C2×C3×C4#62412yes—
205C2×C3×C4#62511yescharacteristic
206C2×C3×C4#62512yes—
207C2×C3×C4#62612yes—
208C2×C3×C4#62612yes—
209C2×C3×C4#62612yes—
210C2×C3×C4#62711yescharacteristic
211C2×C3×C4#62712yes—
212C2×Dic3—64no—
213C2×Dic3—64no—
214C2×Dic3—124no—
215C2×Dic3—124no—
216C3×C8—184no—
217C3⋊C8—68no—
218C3⋊C8—64no—
219C3⋊C8—64no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—22no—
231C2×C32—21no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—42no—
236C2×C32—42no—
237C2×C32—42no—
238C2×C32—42no—
239C2×C32#82312yes—
240C2×C32#82412yes—
241C2×C32C3⋊#9511yescharacteristic
242C2×C32C3⋊#9512yes—
243C2×C32#83311yescharacteristic
244C2×C32#83611yescharacteristic
245C22×C4—92no—
246C22×C4#111311yescharacteristic
247M16—94no—
248M16—182no—
249C2×C8—182no—
250C4⋊C4—182no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—21no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—22no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3Dic3211yescharacteristic
271C22×C3Dic3212yes—
272C22×C3(C2×D3)⋊Dic312yes—
273C22×C3#372211yescharacteristic
274C22×C3Dic3⋊Dic312yes—
275C22×C3#372411yescharacteristic
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—42no—
283C3×C4—42no—
284C3×C4—362no—
285C3×C4(C2×D3)⋊Dic312yes—
286C3×C4(C2×D3)⋊Dic312yes—
287C3×C4#372212yes—
288C3×C4Dic3⋊Dic312yes—
289C3×C4Dic3⋊Dic312yes—
290C3×C4#372412yes—
291Dic3—124no—
292Dic3—124no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#453011yescharacteristic, complemented
297C32#455412yescomplemented
298C23#629411yescharacteristic
299C2×C4—182no—
300C2×C4—362no—
301C2×C4#629412yes—
302C2×C4#629511yescharacteristic
303C2×C4#629512yes—
304C2×C4#629611yescharacteristic
305C8—184no—
306C2×C3—22no—
307C2×C3—22no—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—21no—
315C2×C3—22no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—42no—
319C2×C3—42no—
320C2×C3—42no—
321C2×C3—42no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3#6355812yes—
326C2×C3#6355911yescharacteristic
327C2×C3#6356012yes—
328C2×C3#6356111yescharacteristic
329C2×C3#6357912yes—
330C2×C3#6357911yescharacteristic
331C22—21no—
332C22—21no—
333C22#8881311yescharacteristic
334C22#8881411yescharacteristic
335C22#8881511yescharacteristic
336C4—362no—
337C4#8881412yes—
338C4#8881512yes—
339C3—21no—
340C3—22no—
341C3—41no—
342C3#1058916712yescomplemented
343C3#1058916811yescharacteristic, complemented
344C2—21no—
345C2—21no—
346C2#1172207311yescharacteristic
347C2#1172207411yescharacteristic
348C2#1172208211yescharacteristic
subgroupG/H
centerC22#88814
commutatorC22×C33C42
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
hypocenterC33#343
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
9. C3×#3748C4semidirect
38. #3748Dic3semidirect
180. C33#343semidirect
296. C32#4530semidirect
297. C32#4554semidirect
342. C3#10589167semidirect
343. C3#10589168semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37482
#105891683
#105891673