Finite Group Databasebeta

← #393×Dic61 #29601819

Subgroups — #393×Dic61 #29601819

Order 15616.

2158 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic61C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×Dic61C213yesmaximal
3#21444090C211yesmaximal, characteristic, complemented
4#21444076C213yesmaximal, complemented
5#21444064C213yesmaximal
6#95×Dic61C213yesmaximal, complemented
7C2×C61×#393C211yesmaximal, characteristic
8C24×Dic61C2211yescharacteristic
9C2×Q8×Dic61—212no—
10C22×C4⋊Dic61C2213yes—
11C22×C22⋊Dic61C2213yes—
12C22×C4×Dic61C2213yes—
13C22×C4×Dic61C2213yes—
14C22⋊C4×Dic61—212no—
15C2×#14519684C2216yes—
16C2×#14519684C2213yes—
17C2×#14519684C2213yes—
18C2×#14519684C2213yes—
19C23×Q8×C61C2211yescharacteristic
20C22×C22⋊C4×C61C2213yes—
21C61×#393C418yescomplemented
22C2×#95×C61C2213yes—
23C23×Dic61—26no—
24C23×Dic61—26no—
25C23×Dic61Q814yescomplemented
26C23×Dic61C2313yes—
27C23×Dic61C2311yescharacteristic
28C23×Dic61C2313yes—
29C23×Dic61C2311yescharacteristic
30Q8×Dic61—232no—
31C2×C4⋊Dic61—26no—
32C2×C4⋊Dic61—212no—
33C2×C22⋊Dic61—26no—
34C2×C22⋊Dic61—26no—
35C2×C22⋊Dic61—26no—
36C2×C4×Dic61—212no—
37C2×C4×Dic61—26no—
38C2×C4×Dic61—212no—
39C2×C4×Dic61D4112yes—
40#14519684—26no—
41#14519684—26no—
42#14519684—26no—
43#14519684—212no—
44C25×C61C2311yescharacteristic
45C22×Q8×C61—26no—
46C22×Q8×C61—26no—
47C22×Q8×C61C2×C414yescomplemented
48C23×C4×C61C2313yes—
49C23×C4×C61C2313yes—
50C2×C22⋊C4×C61—26no—
51C2×C22⋊C4×C61—26no—
52C2×C22⋊C4×C61C2×C4112yes—
53#95×C61C2×C4112yescomplemented
54C24×C61—23no—
55C24×C61—23no—
56C24×C61—23no—
57C24×C61—23no—
58C24×C61C22×C412yes—
59C24×C61C2×Q812yes—
60C24×C61#4112yes—
61C24×C61C2411yescharacteristic
62C2×Q8×C61—224no—
63C2×Q8×C61—28no—
64C2×Q8×C61—224no—
65C22×C4×C61—26no—
66C22×C4×C61—26no—
67C22×C4×C61—23no—
68C22×C4×C61—23no—
69C22×C4×C61—26no—
70C22×C4×C61—26no—
71C22×C4×C61C22×C416yes—
72C22×C4×C61C22×C416yes—
73C22×C4×C61C2×D416yes—
74C22×C4×C61#4116yes—
75C22×Dic61—23no—
76C22×Dic61—23no—
77C22×Dic61—26no—
78C22×Dic61—23no—
79C22×Dic61—26no—
80C22×Dic61—26no—
81C22×Dic61—26no—
82C22×Dic61—23no—
83C22×Dic61—23no—
84C22×Dic61—23no—
85C22×Dic61—224no—
86C22×Dic61—44no—
87C22×Dic61—412no—
88C22×Dic61C2×D416yes—
89C22×Dic61C2×D416yes—
90C22×Dic61C2×Q812yes—
91C22×Dic61C2×Q812yes—
92C22×Dic61#4116yes—
93C22×Dic61#4116yes—
94C22⋊C4×C61—224no—
95C22⋊Dic61—224no—
96C4⋊Dic61—224no—
97C4×Dic61—224no—
98C4×Dic61—412no—
99C23×C61—26no—
100C23×C61—26no—
101C23×C61—26no—
102C23×C61—26no—
103C23×C61—26no—
104C23×C61—26no—
105C23×C61—41no—
106C23×C61—43no—
107C23×C61—43no—
108C23×C61—46no—
109C23×C61—43no—
110C23×C61C4×Q814yescomplemented
111C23×C61C23×C411yescharacteristic
112C23×C61C22×D413yes—
113C23×C61C22×Q811yescharacteristic
114C23×C61C2×#4113yes—
115C23×C61C2×#4111yescharacteristic
116C23×C61C2×#4113yes—
117C23×C61C2×#4113yes—
118Q8×C61—232no—
119C2×C4×C61—212no—
120C2×C4×C61—26no—
121C2×C4×C61—26no—
122C2×C4×C61—212no—
123C2×C4×C61—26no—
124C2×C4×C61—212no—
125C2×C4×C61—43no—
126C2×C4×C61—46no—
127C2×C4×C61—43no—
128C2×C4×C61C4×D4112yes—
129C2×Dic61—26no—
130C2×Dic61—26no—
131C2×Dic61—26no—
132C2×Dic61—26no—
133C2×Dic61—212no—
134C2×Dic61—26no—
135C2×Dic61—41no—
136C2×Dic61—46no—
137C2×Dic61—43no—
138C2×Dic61—43no—
139C2×Dic61—43no—
140C2×Dic61—46no—
141C2×Dic61—46no—
142C2×Dic61—416no—
143C2×Dic61#12014yes—
144C2×Dic61C22⋊Q8112yes—
145C2×Dic61#124112yes—
146C4×#393—611nomaximal, Sylow
147C22×C61—212no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61—46no—
151C22×C61—46no—
152C22×C61—42no—
153C22×C61—46no—
154C22×C61C2×C4×D413yes—
155C22×C61C2×C4×Q811yescharacteristic
156C22×C61C4×#4113yes—
157C22×C61C2×#12011yescharacteristic
158C22×C61C2×C22⋊Q813yes—
159C22×C61#52513yes—
160C22×C61#52513yes—
161C22×C61C2×#12413yes—
162C22×C61#52916yes—
163C22×C61#52913yes—
164C22×C61#53213yes—
165C22×C61#53313yes—
166C4×C61—212no—
167C4×C61—46no—
168Dic61—42no—
169Dic61—46no—
170Dic61—46no—
171Dic61#39318yesdirect product factor
172C22×C4×Q8—611no—
173C2×#393D6111yescharacteristic
174C2×C4×C22⋊C4—613no—
175#1446—611no—
176#1442—613no—
177#1439—613no—
178C4×#95—613no—
179C2×C61—44no—
180C2×C61C4×#12011yescharacteristic
181C2×C61C4×C22⋊Q813yes—
182C2×C61C4×#12413yes—
183C2×C61C2×#39311yescharacteristic
184C2×C61#238811yescharacteristic
185C2×C61#239613yes—
186C2×C61#239813yes—
187C23×Q8C2×D6111yescharacteristic
188C24×C4—611no—
189C2×C4×Q8—12212no—
190C22×C4⋊C4—613no—
191C22×C22⋊C4—613no—
192C22×C22⋊C4C2×D6113yes—
193C22×C42—613no—
194C22×C42—613no—
195#393Dic6118yesdirect product factor
196C4×C22⋊C4—12212no—
197C2×#95—616no—
198C2×#95—613no—
199C2×#95—613no—
200C2×#95—613no—
201C2×#95C2×D6113yes—
202C61C4×#39311yescharacteristic, complemented, Sylow
203C25C22×D6111yescharacteristic
204C22×Q8—26no—
205C22×Q8—26no—
206C22×Q8C2×Dic6114yescomplemented
207C23×C4—611no—
208C23×C4—614no—
209C23×C4—613no—
210C23×C4—613no—
211C23×C4—611no—
212C23×C4—1226no—
213C23×C4—1226no—
214C23×C4C22×D6113yes—
215C23×C4C22×D6113yes—
216C4×Q8—12232no—
217C2×C4⋊C4—12212no—
218C2×C4⋊C4—1226no—
219C2×C22⋊C4—26no—
220C2×C22⋊C4—26no—
221C2×C22⋊C4—1226no—
222C2×C22⋊C4—1226no—
223C2×C22⋊C4—1226no—
224C2×C22⋊C4C2×Dic61112yes—
225C2×C42—6112no—
226C2×C42—12212no—
227C2×C42—1226no—
228C2×C42—12212no—
229#95—1226no—
230#95—12212no—
231#95—1226no—
232#95—1226no—
233#95C2×Dic61112yescomplemented
234C24—23no—
235C24—23no—
236C24—23no—
237C24—23no—
238C24Q8×D6112yes—
239C24#1175822612yes—
240C24C22×Dic6112yes—
241C24C23×D6111yescharacteristic
242C2×Q8—28no—
243C2×Q8—224no—
244C2×Q8—224no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—23no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—23no—
251C22×C4—616no—
252C22×C4—612no—
253C22×C4—616no—
254C22×C4—616no—
255C22×C4—612no—
256C22×C4—616no—
257C22×C4—12224no—
258C22×C4—1223no—
259C22×C4—1223no—
260C22×C4—1226no—
261C22×C4—1226no—
262C22×C4—1226no—
263C22×C4—1223no—
264C22×C4—1223no—
265C22×C4—1226no—
266C22×C4—1223no—
267C22×C4—1223no—
268C22×C4—2444no—
269C22×C4—24412no—
270C22×C4D4×D6116yes—
271C22×C4#1175822416yes—
272C22×C4C22×Dic6116yes—
273C22×C4C22×Dic6116yes—
274C4⋊C4—12224no—
275C22⋊C4—224no—
276C22⋊C4—12224no—
277C42—12224no—
278C42—24412no—
279C23—26no—
280C23—26no—
281C23—26no—
282C23—26no—
283C23—26no—
284C23—26no—
285C23—41no—
286C23—43no—
287C23—43no—
288C23—46no—
289C23—43no—
290C23Q8×Dic6114yescomplemented
291C23C23×Dic6111yescharacteristic
292C23C2×D4×D6113yes—
293C23C2×Q8×D6111yescharacteristic
294C23#41×D6113yes—
295C23#41×D6113yes—
296C23C2×#1175822413yes—
297C23C2×#1175822611yescharacteristic
298Q8—232no—
299C2×C4—212no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—212no—
303C2×C4—26no—
304C2×C4—212no—
305C2×C4—43no—
306C2×C4—46no—
307C2×C4—43no—
308C2×C4—614no—
309C2×C4—6112no—
310C2×C4—6112no—
311C2×C4—1226no—
312C2×C4—1226no—
313C2×C4—1226no—
314C2×C4—1226no—
315C2×C4—12212no—
316C2×C4—1226no—
317C2×C4—24416no—
318C2×C4—2441no—
319C2×C4—2443no—
320C2×C4—2446no—
321C2×C4—2443no—
322C2×C4—2443no—
323C2×C4—2446no—
324C2×C4—2446no—
325C2×C4D4×Dic61112yes—
326C22—212no—
327C22—42no—
328C22—42no—
329C22—46no—
330C22—42no—
331C22—46no—
332C22—46no—
333C22C2×D4×Dic6113yes—
334C22C2×Q8×Dic6111yescharacteristic
335C22#41×Dic6113yes—
336C22#120×D6111yescharacteristic
337C22C22⋊Q8×D6113yes—
338C22#1684026813yes—
339C22#1684027513yes—
340C22#124×D6113yes—
341C22#1684029116yes—
342C22#1684029813yes—
343C22#1684031113yes—
344C22#1684031413yes—
345C4—212no—
346C4—46no—
347C4—618no—
348C4—2442no—
349C4—2446no—
350C4—2446no—
351C2—44no—
352C2#120×Dic6111yescharacteristic
353C2C22⋊Q8×Dic6113yes—
354C2#124×Dic6113yes—
355C2#393×D6111yescharacteristic
356C2#2144903611yescharacteristic
357C2#2144908613yes—
358C2#2144910113yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#393C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#393D61
hypocenterC61C4×#393
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic61C2semidirect
3. #21444090C2semidirect
4. #21444076C2semidirect
6. #95×Dic61C2semidirect
21. C61×#393C4semidirect
25. C23×Dic61Q8semidirect
47. C22×Q8×C61C2×C4semidirect
53. #95×C61C2×C4semidirect
110. C23×C61C4×Q8semidirect
171. Dic61#393direct (N × H)
195. #393Dic61direct (N × H)
202. C61C4×#393semidirect
206. C22×Q8C2×Dic61semidirect
233. #95C2×Dic61semidirect
290. C23Q8×Dic61semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic612
C2×C22⋊C4×Dic612
#214440902
#214440762
#214440642
#95×Dic612
C2×C61×#3932
C4×#39361