Finite Group Databasebeta

← C2×C3×#5125 #11858639

Subgroups — C2×C3×#5125 #11858639

Order 1152.

1069 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×M16C211yesmaximal, characteristic
2C22×C3×#208C212yesmaximal
3C3×#5125C218yesmaximal, direct product factor
4C2×C3×#819C214yesmaximal, complemented
5C2×C3×#430—31nomaximal
6C2×#5125C311yesmaximal, characteristic, direct product factor
7C23×C32×C4C2211yescharacteristic
8C2×C32×M16C414yes—
9C2×C32×M16C414yes—
10C2×C32×M16C2214yes—
11C22×C32×C8C2212yes—
12C2×C3×#208—22no—
13C2×C3×#208—24no—
14C2×C3×#208C2214yes—
15C2×C3×#208C2218yes—
16C22×C3×C3⋊C8—22no—
17C3×#819C22116yescomplemented
18C22×C3×M16—21no—
19C22×C3×M16—32no—
20C22×C3×M16D311yescharacteristic
21C22×C3×M16C2×C311yescharacteristic
22C22×#208C2×C312yes—
23C3×#430—38no—
24C2×C3×#108—34no—
25#5125C2×C318yesdirect product factor
26C2×#819C2×C314yescomplemented
27C24×C32Q811yescharacteristic
28C22×C32×C4C2×C412yes—
29C22×C32×C4C2×C412yes—
30C22×C32×C4D412yes—
31C22×C32×C4D412yes—
32C22×C32×C4Q811yescharacteristic
33C22×C32×C4Q812yes—
34C22×C32×C4C2311yescharacteristic
35C22×C32×C4C2312yes—
36C32×M16C2×C4116yes—
37C2×C32×C8C2×C414yes—
38C2×C32×C8C2×C414yes—
39C2×C32×C8C2314yes—
40C3×#208—28no—
41C3×#208—24no—
42C3×#208C2318yes—
43C2×C3×C3⋊C8—28no—
44C2×C3×C3⋊C8—24no—
45C2×#430—31noSylow
46C23×C3×C4—21no—
47C23×C3×C4C2×D311yescharacteristic
48C23×C3×C4C22×C311yescharacteristic
49C2×C3×M16—24no—
50C2×C3×M16—24no—
51C2×C3×M16—24no—
52C2×C3×M16—34no—
53C2×C3×M16—38no—
54C2×C3×M16—64no—
55C2×C3×M16—62no—
56C2×C3×M16Dic314yes—
57C2×C3×M16Dic314yes—
58C2×C3×M16C3×C414yes—
59C2×C3×M16C3×C414yes—
60C2×C3×M16C2×D314yes—
61C2×C3×M16C22×C314yes—
62C22×C3×C8—22no—
63C22×C3×C8—62no—
64C22×C3×C8C2×D312yes—
65C22×C3×C8C22×C312yes—
66C2×#208—22no—
67C2×#208—24no—
68C2×#208C22×C318yes—
69C2×#208C22×C314yes—
70C22×C3⋊C8—22no—
71C3×#108—316no—
72#819C22×C3116yescomplemented
73C23×C32—42no—
74C23×C32C4⋊C412yes—
75C23×C32C4⋊C412yes—
76C23×C32C2×Q811yescharacteristic
77C23×C32C2×Q812yes—
78C2×C32×C4C4⋊C414yes—
79C2×C32×C4C4⋊C412yes—
80C2×C32×C4C4⋊C414yes—
81C2×C32×C4C4⋊C412yes—
82C2×C32×C4C22×C411yescharacteristic
83C2×C32×C4C22×C411yescharacteristic
84C2×C32×C4C22×C414yes—
85C2×C32×C4C2×D414yes—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4C2×D411yescharacteristic
88C2×C32×C4C2×Q811yescharacteristic
89C2×C32×C4C2×Q812yes—
90C2×C32×C4C2411yescharacteristic
91C32×C8C22×C418yes—
92C3×C3⋊C8—28no—
93C22×M16—32no—
94C22×M16C3×D311yescharacteristic
95#430—38no—
96C2×#108—34no—
97C24×C3—21no—
98C24×C3Dic611yescharacteristic
99C24×C3C3×Q811yescharacteristic
100C22×C3×C4—21no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4Dic611yescharacteristic
109C22×C3×C4Dic612yes—
110C22×C3×C4D1212yes—
111C22×C3×C4D1212yes—
112C22×C3×C4C2×Dic312yes—
113C22×C3×C4C2×Dic312yes—
114C22×C3×C4C2×C3×C412yes—
115C22×C3×C4C2×C3×C412yes—
116C22×C3×C4C3×D412yes—
117C22×C3×C4C3×D412yes—
118C22×C3×C4C3×Q811yescharacteristic
119C22×C3×C4C3×Q812yes—
120C22×C3×C4C22×D312yes—
121C22×C3×C4C22×D311yescharacteristic
122C22×C3×C4C23×C311yescharacteristic
123C22×C3×C4C23×C312yes—
124C3×M16—216no—
125C3×M16—38no—
126C3×M16—68no—
127C3×M16—64no—
128C3×M16C2×Dic3116yes—
129C3×M16C2×C3×C4116yes—
130C2×C3×C8—24no—
131C2×C3×C8—24no—
132C2×C3×C8—24no—
133C2×C3×C8—68no—
134C2×C3×C8—64no—
135C2×C3×C8C2×Dic314yes—
136C2×C3×C8C2×Dic314yes—
137C2×C3×C8C2×C3×C414yes—
138C2×C3×C8C2×C3×C414yes—
139C2×C3×C8C22×D314yes—
140C2×C3×C8C23×C314yes—
141#208—28no—
142#208—24no—
143#208C23×C318yes—
144C2×C3⋊C8—24no—
145C2×C3⋊C8—28no—
146C22×C32—22no—
147C22×C32—24no—
148C22×C32—42no—
149C22×C32—42no—
150C22×C32C2×C4⋊C411yescharacteristic
151C22×C32C2×C4⋊C414yes—
152C22×C32C2×C4⋊C411yescharacteristic
153C22×C32C22×Q811yescharacteristic
154C32×C4C2×C4⋊C411yescharacteristic
155C32×C4C2×C4⋊C414yes—
156C32×C4C2×C4⋊C411yescharacteristic
157C32×C4C23×C411yescharacteristic
158C32×C4C22×D411yescharacteristic
159C23×C4C2×C3×D311yescharacteristic
160C2×M16—34no—
161C2×M16—38no—
162C2×M16—62no—
163C2×M16—64no—
164C2×M16C3×Dic314yes—
165C2×M16C3×Dic314yes—
166C2×M16C2×C3×D314yes—
167C22×C8—62no—
168C22×C8C2×C3×D312yes—
169#108—316no—
170C23×C3—22no—
171C23×C3—22no—
172C23×C3—21no—
173C23×C3—22no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—44no—
177C23×C3C4⋊Dic312yes—
178C23×C3C4⋊Dic312yes—
179C23×C3C3×C4⋊C412yes—
180C23×C3C3×C4⋊C412yes—
181C23×C3C2×Dic611yescharacteristic
182C23×C3C2×Dic612yes—
183C23×C3C2×C3×Q811yescharacteristic
184C23×C3C2×C3×Q812yes—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—21no—
193C2×C3×C4—24no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—24no—
198C2×C3×C4C4⋊Dic312yes—
199C2×C3×C4C4⋊Dic314yes—
200C2×C3×C4C4⋊Dic312yes—
201C2×C3×C4C4⋊Dic314yes—
202C2×C3×C4C3×C4⋊C414yes—
203C2×C3×C4C3×C4⋊C414yes—
204C2×C3×C4C3×C4⋊C412yes—
205C2×C3×C4C3×C4⋊C412yes—
206C2×C3×C4C2×Dic612yes—
207C2×C3×C4C2×Dic611yescharacteristic
208C2×C3×C4C2×D1214yes—
209C2×C3×C4C2×D1211yescharacteristic
210C2×C3×C4C2×D1211yescharacteristic
211C2×C3×C4C22×Dic311yescharacteristic
212C2×C3×C4C22×Dic311yescharacteristic
213C2×C3×C4C22×Dic314yes—
214C2×C3×C4C22×C3×C411yescharacteristic
215C2×C3×C4C22×C3×C411yescharacteristic
216C2×C3×C4C22×C3×C414yes—
217C2×C3×C4C2×C3×D411yescharacteristic
218C2×C3×C4C2×C3×D414yes—
219C2×C3×C4C2×C3×D411yescharacteristic
220C2×C3×C4C2×C3×Q811yescharacteristic
221C2×C3×C4C2×C3×Q812yes—
222C2×C3×C4C23×D311yescharacteristic
223C2×C3×C4C24×C311yescharacteristic
224C3×C8—28no—
225C3×C8—68no—
226C3×C8C22×Dic318yes—
227C3×C8C22×C3×C418yes—
228C3⋊C8—28no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—24no—
232C2×C32#43012yescomplemented
233C2×C32C22×C4⋊C411yescharacteristic
234C24C3×Dic611yescharacteristic
235C22×C4C3×Dic611yescharacteristic
236C22×C4C3×Dic612yes—
237C22×C4C3×D1212yes—
238C22×C4C3×D1212yes—
239C22×C4C2×C3×Dic312yes—
240C22×C4C2×C3×Dic312yes—
241C22×C4C22×C3×D311yescharacteristic
242C22×C4C22×C3×D312yes—
243M16—38no—
244M16—68no—
245M16—64no—
246M16C2×C3×Dic3116yes—
247C2×C8—64no—
248C2×C8—68no—
249C2×C8C2×C3×Dic314yes—
250C2×C8C2×C3×Dic314yes—
251C2×C8C22×C3×D314yes—
252C22×C3—24no—
253C22×C3—24no—
254C22×C3—22no—
255C22×C3—24no—
256C22×C3—22no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—24no—
260C22×C3—21no—
261C22×C3—44no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3C2×C4⋊Dic314yes—
269C22×C3C2×C4⋊Dic311yescharacteristic
270C22×C3C2×C4⋊Dic311yescharacteristic
271C22×C3C2×C3×C4⋊C411yescharacteristic
272C22×C3C2×C3×C4⋊C414yes—
273C22×C3C2×C3×C4⋊C411yescharacteristic
274C22×C3C22×Dic611yescharacteristic
275C22×C3C22×C3×Q811yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—24no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4C2×C4⋊Dic314yes—
282C3×C4C2×C4⋊Dic311yescharacteristic
283C3×C4C2×C4⋊Dic311yescharacteristic
284C3×C4C2×C3×C4⋊C411yescharacteristic
285C3×C4C2×C3×C4⋊C411yescharacteristic
286C3×C4C2×C3×C4⋊C414yes—
287C3×C4C22×D1211yescharacteristic
288C3×C4C23×Dic311yescharacteristic
289C3×C4C23×C3×C411yescharacteristic
290C3×C4C22×C3×D411yescharacteristic
291C32C2×#43011yescharacteristic, complemented, Sylow
292C23—42no—
293C23C3×C4⋊Dic312yes—
294C23C3×C4⋊Dic312yes—
295C23C2×C3×Dic611yescharacteristic
296C23C2×C3×Dic612yes—
297C2×C4C3×C4⋊Dic314yes—
298C2×C4C3×C4⋊Dic312yes—
299C2×C4C3×C4⋊Dic314yes—
300C2×C4C3×C4⋊Dic312yes—
301C2×C4C2×C3×Dic611yescharacteristic
302C2×C4C2×C3×Dic612yes—
303C2×C4C2×C3×D1211yescharacteristic
304C2×C4C2×C3×D1214yes—
305C2×C4C2×C3×D1211yescharacteristic
306C2×C4C22×C3×Dic311yescharacteristic
307C2×C4C22×C3×Dic311yescharacteristic
308C2×C4C22×C3×Dic314yes—
309C2×C4C23×C3×D311yescharacteristic
310C8—68no—
311C8C22×C3×Dic318yes—
312C2×C3—22no—
313C2×C3—22no—
314C2×C3—21no—
315C2×C3—24no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—24no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—44no—
323C2×C3#512512yesdirect product factor
324C2×C3C3×#43012yescomplemented
325C2×C3C22×C4⋊Dic311yescharacteristic
326C2×C3C22×C3×C4⋊C411yescharacteristic
327C22—24no—
328C22—22no—
329C22—42no—
330C22—42no—
331C22C2×C3×C4⋊Dic311yescharacteristic
332C22C2×C3×C4⋊Dic314yes—
333C22C2×C3×C4⋊Dic311yescharacteristic
334C22C22×C3×Dic611yescharacteristic
335C4C2×C3×C4⋊Dic311yescharacteristic
336C4C2×C3×C4⋊Dic314yes—
337C4C2×C3×C4⋊Dic311yescharacteristic
338C4C22×C3×D1211yescharacteristic
339C4C23×C3×Dic311yescharacteristic
340C3—21no—
341C3C2×#512511yescharacteristic, direct product factor
342C3C2×C3×#43011yescharacteristic, complemented
343C2—21no—
344C2—24no—
345C2—21no—
346C2C3×#512512yesdirect product factor
347C2C22×C3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D12
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC22×C32×M16C2
socleC22×C32C2×C4⋊C4
prime radicalC24×C32Q8
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×M16D3
hypocenterC3C2×C3×#430
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C22×C3×M16

Complements

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

NH ≅ G/Ntype
3. C3×#5125C2direct (N × H)
4. C2×C3×#819C2semidirect
6. C2×#5125C3direct (N × H)
17. C3×#819C22semidirect
25. #5125C2×C3direct (N × H)
26. C2×#819C2×C3semidirect
72. #819C22×C3semidirect
232. C2×C32#430semidirect
291. C32C2×#430semidirect
323. C2×C3#5125direct (N × H)
324. C2×C3C3×#430semidirect
341. C3C2×#5125direct (N × H)
342. C3C2×C3×#430semidirect
346. C2C3×#5125direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×M162
C22×C3×#2082
C3×#51252
C2×C3×#8192
C2×C3×#4303
C2×#51253