Finite Group Databasebeta

← C22×C3×Dic3⋊Dic3 #13175079

Subgroups — C22×C3×Dic3⋊Dic3 #13175079

Order 1728.

2514 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C23×C32×Dic3C211yesmaximal, characteristic
3C23×C32×Dic3C211yesmaximal, characteristic
4C2×C3×Dic3⋊Dic3C2112yesmaximal, direct product factor
5C22×C3×C4⋊Dic3—31nomaximal
6C22×C3×Dic3⋊C4—31nomaximal
7C22×Dic3⋊Dic3C311yesmaximal, characteristic, direct product factor
8C24×C33C2211yescharacteristic
9C22×C3×C3⋊Dic3—24no—
10C22×C3×C3⋊Dic3C2216yes—
11C22×C32×Dic3—24no—
12C22×C32×Dic3C418yescomplemented
13C22×C32×Dic3C2216yes—
14C22×C32×Dic3C2216yes—
15C3×Dic3⋊Dic3C22116yesdirect product factor
16C23×C32×C4—31no—
17C23×C32×C4—31no—
18C23×C3⋊Dic3C2×C311yescharacteristic
19C23×C3×Dic3—21no—
20C23×C3×Dic3—21no—
21C23×C3×Dic3—31no—
22C23×C3×Dic3—31no—
23C23×C3×Dic3—61no—
24C23×C3×Dic3D311yescharacteristic
25C23×C3×Dic3D311yescharacteristic
26C23×C3×Dic3C2×C311yescharacteristic
27C23×C3×Dic3C2×C311yescharacteristic
28C2×C3×C4⋊Dic3—312no—
29C2×C3×Dic3⋊C4—312no—
30C2×Dic3⋊Dic3C2×C3112yesdirect product factor
31C23×C33C2×C414yes—
32C23×C33D414yes—
33C23×C33Q814yes—
34C23×C33C2313yes—
35C2×C3×C3⋊Dic3—212no—
36C2×C3×C3⋊Dic3C2314yes—
37C2×C32×Dic3—212no—
38C2×C32×Dic3C2×C4124yescomplemented
39C2×C32×Dic3C2314yes—
40C2×C32×Dic3C2314yes—
41C22×C3×C4⋊C4—91no—
42C22×C4⋊Dic3—31no—
43C22×Dic3⋊C4—31no—
44C24×C32—21no—
45C24×C32—21no—
46C24×C32—21no—
47C24×C32—41no—
48C24×C32C2×D311yescharacteristic
49C24×C32C2×D311yescharacteristic
50C24×C32C22×C311yescharacteristic
51C22×C32×C4—38no—
52C22×C32×C4—36no—
53C22×C32×C4—36no—
54C22×C32×C4—64no—
55C22×C3⋊Dic3—24no—
56C22×C3⋊Dic3C22×C316yes—
57C22×C3×Dic3—28no—
58C22×C3×Dic3—28no—
59C22×C3×Dic3—26no—
60C22×C3×Dic3—26no—
61C22×C3×Dic3—24no—
62C22×C3×Dic3—24no—
63C22×C3×Dic3—36no—
64C22×C3×Dic3—36no—
65C22×C3×Dic3—66no—
66C22×C3×Dic3—68no—
67C22×C3×Dic3—64no—
68C22×C3×Dic3—64no—
69C22×C3×Dic3Dic318yescomplemented
70C22×C3×Dic3C3×C418yescomplemented
71C22×C3×Dic3C2×D316yes—
72C22×C3×Dic3C2×D316yes—
73C22×C3×Dic3C22×C316yes—
74C22×C3×Dic3C22×C316yes—
75C3×C4⋊Dic3—316no—
76C3×Dic3⋊C4—316no—
77Dic3⋊Dic3C22×C3116yesdirect product factor
78C22×C33C4⋊C4116yescomplemented
79C22×C33C22×C416yes—
80C22×C33C2×D416yes—
81C22×C33C2×Q816yes—
82C22×C33C2411yescharacteristic
83C3×C3⋊Dic3—24no—
84C32×Dic3—24no—
85C32×Dic3C22×C418yescomplemented
86C23×C3×C4—31no—
87C23×C3×C4—31no—
88C23×C3×C4—31no—
89C23×C3×C4—31no—
90C23×C3×C4—61no—
91C23×C3×C4—61no—
92C23×C3×C4—91no—
93C23×Dic3—31no—
94C23×Dic3—31no—
95C23×Dic3—61no—
96C23×Dic3C3×D311yescharacteristic
97C23×Dic3C3×D311yescharacteristic
98C2×C3×C4⋊C4—912no—
99C2×C4⋊Dic3—312no—
100C2×Dic3⋊C4—312no—
101C23×C32—24no—
102C23×C32—24no—
103C23×C32—24no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32—24no—
107C23×C32—24no—
108C23×C32—24no—
109C23×C32—24no—
110C23×C32—23no—
111C23×C32—23no—
112C23×C32—23no—
113C23×C32—44no—
114C23×C32—44no—
115C23×C32—44no—
116C23×C32—43no—
117C23×C32Dic614yes—
118C23×C32Dic614yes—
119C23×C32C4×D314yes—
120C23×C32D1214yes—
121C23×C32C2×Dic314yes—
122C23×C32#6714yes—
123C23×C32C2×C3×C414yes—
124C23×C32C3×D414yes—
125C23×C32C3×Q814yes—
126C23×C32C22×D313yes—
127C23×C32C22×D313yes—
128C23×C32C23×C313yes—
129C2×C32×C4—324no—
130C2×C32×C4—34no—
131C2×C32×C4—34no—
132C2×C32×C4—612no—
133C2×C3⋊Dic3—212no—
134C2×C3⋊Dic3C23×C314yes—
135C2×C3×Dic3—224no—
136C2×C3×Dic3—212no—
137C2×C3×Dic3—212no—
138C2×C3×Dic3—224no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—624no—
145C2×C3×Dic3—612no—
146C2×C3×Dic3—612no—
147C2×C3×Dic3C2×Dic3124yescomplemented
148C2×C3×Dic3C2×C3×C4124yescomplemented
149C2×C3×Dic3C22×D314yes—
150C2×C3×Dic3C22×D314yes—
151C2×C3×Dic3C23×C314yes—
152C2×C3×Dic3C23×C314yes—
153C22×C4⋊C4—91noSylow
154C2×C33C2×C4⋊C4112yescomplemented
155C2×C33C23×C411yescharacteristic
156C2×C33C22×D411yescharacteristic
157C2×C33C22×Q811yescharacteristic
158C24×C3—21no—
159C24×C3—21no—
160C24×C3—21no—
161C24×C3—41no—
162C24×C3D3211yescharacteristic
163C24×C3C2×C3×D311yescharacteristic
164C24×C3C2×C3×D311yescharacteristic
165C22×C3×C4—38no—
166C22×C3×C4—38no—
167C22×C3×C4—36no—
168C22×C3×C4—36no—
169C22×C3×C4—36no—
170C22×C3×C4—36no—
171C22×C3×C4—68no—
172C22×C3×C4—66no—
173C22×C3×C4—66no—
174C22×C3×C4—68no—
175C22×C3×C4—64no—
176C22×C3×C4—64no—
177C22×C3×C4—96no—
178C22×C3×C4—184no—
179C22×Dic3—24no—
180C22×Dic3—36no—
181C22×Dic3—36no—
182C22×Dic3—66no—
183C22×Dic3—68no—
184C22×Dic3—64no—
185C22×Dic3—64no—
186C22×Dic3C3×Dic318yescomplemented
187C22×Dic3C2×C3×D316yes—
188C22×Dic3C2×C3×D316yes—
189C3×C4⋊C4—916no—
190C4⋊Dic3—316no—
191Dic3⋊C4—316no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—216no—
196C22×C32—216no—
197C22×C32—216no—
198C22×C32—26no—
199C22×C32—26no—
200C22×C32—26no—
201C22×C32—26no—
202C22×C32—26no—
203C22×C32—26no—
204C22×C32—26no—
205C22×C32—26no—
206C22×C32—26no—
207C22×C32—416no—
208C22×C32—46no—
209C22×C32—46no—
210C22×C32—46no—
211C22×C32—41no—
212C22×C32Dic3⋊C4116yescomplemented
213C22×C32C4⋊Dic3116yescomplemented
214C22×C32C3×C4⋊C4116yescomplemented
215C22×C32C2×Dic616yes—
216C22×C32C2×Dic616yes—
217C22×C32C2×C4×D316yes—
218C22×C32C2×D1216yes—
219C22×C32C22×Dic316yes—
220C22×C32C2×#6716yes—
221C22×C32C22×C3×C416yes—
222C22×C32C2×C3×D416yes—
223C22×C32C2×C3×Q816yes—
224C22×C32C23×D311yescharacteristic
225C22×C32C23×D311yescharacteristic
226C22×C32C24×C311yescharacteristic
227C32×C4—38no—
228C32×C4—64no—
229C3⋊Dic3—24no—
230C3×Dic3—28no—
231C3×Dic3—24no—
232C3×Dic3—24no—
233C3×Dic3—28no—
234C3×Dic3—68no—
235C3×Dic3—64no—
236C3×Dic3—64no—
237C3×Dic3C22×Dic318yescomplemented
238C3×Dic3C22×C3×C418yescomplemented
239C23×C4—31no—
240C23×C4—31no—
241C23×C4—91no—
242C2×C4⋊C4—912no—
243C33C22×C4⋊C411yescharacteristic, complemented, Sylow
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—23no—
254C23×C3—23no—
255C23×C3—23no—
256C23×C3—44no—
257C23×C3—44no—
258C23×C3—44no—
259C23×C3—43no—
260C23×C3D3×Dic314yes—
261C23×C3#62614yes—
262C23×C3#62714yes—
263C23×C3C3×Dic614yes—
264C23×C3C3×Dic614yes—
265C23×C3C3×C4×D314yes—
266C23×C3C3×D1214yes—
267C23×C3C2×C3×Dic314yes—
268C23×C3C3×#6714yes—
269C23×C3C2×D3213yes—
270C23×C3C22×C3×D313yes—
271C23×C3C22×C3×D313yes—
272C2×C3×C4—34no—
273C2×C3×C4—34no—
274C2×C3×C4—324no—
275C2×C3×C4—324no—
276C2×C3×C4—34no—
277C2×C3×C4—34no—
278C2×C3×C4—64no—
279C2×C3×C4—624no—
280C2×C3×C4—64no—
281C2×C3×C4—624no—
282C2×C3×C4—612no—
283C2×C3×C4—612no—
284C2×C3×C4—94no—
285C2×C3×C4—1812no—
286C2×Dic3—212no—
287C2×Dic3—34no—
288C2×Dic3—34no—
289C2×Dic3—64no—
290C2×Dic3—624no—
291C2×Dic3—612no—
292C2×Dic3—612no—
293C2×Dic3C2×C3×Dic3124yescomplemented
294C2×Dic3C22×C3×D314yes—
295C2×Dic3C22×C3×D314yes—
296C2×C32—212no—
297C2×C32—212no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—212no—
308C2×C32—412no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32C2×Dic3⋊C4112yescomplemented
313C2×C32C2×C4⋊Dic3112yescomplemented
314C2×C32C2×C3×C4⋊C4112yescomplemented
315C2×C32C22×Dic611yescharacteristic
316C2×C32C22×Dic611yescharacteristic
317C2×C32C22×C4×D311yescharacteristic
318C2×C32C22×D1211yescharacteristic
319C2×C32C23×Dic311yescharacteristic
320C2×C32C22×#6711yescharacteristic
321C2×C32C23×C3×C411yescharacteristic
322C2×C32C22×C3×D411yescharacteristic
323C2×C32C22×C3×Q811yescharacteristic
324C24C3×D3211yescharacteristic
325C22×C4—38no—
326C22×C4—36no—
327C22×C4—36no—
328C22×C4—64no—
329C22×C4—96no—
330C22×C4—184no—
331C4⋊C4—916no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—26no—
336C22×C3—26no—
337C22×C3—26no—
338C22×C3—26no—
339C22×C3—26no—
340C22×C3—26no—
341C22×C3—26no—
342C22×C3—26no—
343C22×C3—26no—
344C22×C3—216no—
345C22×C3—216no—
346C22×C3—216no—
347C22×C3—41no—
348C22×C3—46no—
349C22×C3—46no—
350C22×C3—46no—
351C22×C3—416no—
352C22×C3Dic3⋊Dic3116yesdirect product factor
353C22×C3C3×Dic3⋊C4116yescomplemented
354C22×C3C3×C4⋊Dic3116yescomplemented
355C22×C3C2×D3×Dic316yes—
356C22×C3C2×#62616yes—
357C22×C3C2×#62716yes—
358C22×C3C2×C3×Dic616yes—
359C22×C3C2×C3×Dic616yes—
360C22×C3C2×C3×C4×D316yes—
361C22×C3C2×C3×D1216yes—
362C22×C3C22×C3×Dic316yes—
363C22×C3C2×C3×#6716yes—
364C22×C3C22×D3211yescharacteristic
365C22×C3C23×C3×D311yescharacteristic
366C22×C3C23×C3×D311yescharacteristic
367C3×C4—38no—
368C3×C4—38no—
369C3×C4—68no—
370C3×C4—68no—
371C3×C4—64no—
372C3×C4—64no—
373C3×C4—184no—
374Dic3—24no—
375Dic3—68no—
376Dic3—64no—
377Dic3—64no—
378Dic3C22×C3×Dic318yescomplemented
379C32—21no—
380C32—21no—
381C32—21no—
382C32—41no—
383C32C22×Dic3⋊C411yescharacteristic, complemented
384C32C22×C4⋊Dic311yescharacteristic, complemented
385C32C22×C3×C4⋊C411yescharacteristic, complemented
386C23C3×D3×Dic314yes—
387C23C3×#62614yes—
388C23C3×#62714yes—
389C23C2×C3×D3213yes—
390C2×C4—34no—
391C2×C4—324no—
392C2×C4—34no—
393C2×C4—612no—
394C2×C4—94no—
395C2×C4—1812no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—212no—
406C2×C3—212no—
407C2×C3—212no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—412no—
412C2×C3C2×Dic3⋊Dic3112yesdirect product factor
413C2×C3C2×C3×Dic3⋊C4112yescomplemented
414C2×C3C2×C3×C4⋊Dic3112yescomplemented
415C2×C3C22×D3×Dic311yescharacteristic
416C2×C3C22×#62611yescharacteristic
417C2×C3C22×#62711yescharacteristic
418C2×C3C22×C3×Dic611yescharacteristic
419C2×C3C22×C3×Dic611yescharacteristic
420C2×C3C22×C3×C4×D311yescharacteristic
421C2×C3C22×C3×D1211yescharacteristic
422C2×C3C23×C3×Dic311yescharacteristic
423C2×C3C22×C3×#6711yescharacteristic
424C22C3×Dic3⋊Dic3116yesdirect product factor
425C22C2×C3×D3×Dic316yes—
426C22C2×C3×#62616yes—
427C22C2×C3×#62716yes—
428C22C22×C3×D3211yescharacteristic
429C4—38no—
430C4—64no—
431C4—184no—
432C3—21no—
433C3—21no—
434C3—21no—
435C3—41no—
436C3C22×Dic3⋊Dic311yescharacteristic, direct product factor
437C3C22×C3×Dic3⋊C411yescharacteristic, complemented
438C3C22×C3×C4⋊Dic311yescharacteristic, complemented
439C2C2×C3×Dic3⋊Dic3112yesdirect product factor
440C2C22×C3×D3×Dic311yescharacteristic
441C2C22×C3×#62611yescharacteristic
442C2C22×C3×#62711yescharacteristic
subgroupG/H
center, hypercenterC24×C3D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
Fitting, socle, prime radicalC24×C33C22
cosocleC22×C32C24×C3
hypocenterC32C22×C3×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
4. C2×C3×Dic3⋊Dic3C2direct (N × H)
7. C22×Dic3⋊Dic3C3direct (N × H)
12. C22×C32×Dic3C4semidirect
15. C3×Dic3⋊Dic3C22direct (N × H)
30. C2×Dic3⋊Dic3C2×C3direct (N × H)
38. C2×C32×Dic3C2×C4semidirect
69. C22×C3×Dic3Dic3semidirect
70. C22×C3×Dic3C3×C4semidirect
77. Dic3⋊Dic3C22×C3direct (N × H)
78. C22×C33C4⋊C4semidirect
85. C32×Dic3C22×C4semidirect
147. C2×C3×Dic3C2×Dic3semidirect
148. C2×C3×Dic3C2×C3×C4semidirect
154. C2×C33C2×C4⋊C4semidirect
186. C22×Dic3C3×Dic3semidirect
212. C22×C32Dic3⋊C4semidirect
213. C22×C32C4⋊Dic3semidirect
214. C22×C32C3×C4⋊C4semidirect
237. C3×Dic3C22×Dic3semidirect
238. C3×Dic3C22×C3×C4semidirect
243. C33C22×C4⋊C4semidirect
293. C2×Dic3C2×C3×Dic3semidirect
312. C2×C32C2×Dic3⋊C4semidirect
313. C2×C32C2×C4⋊Dic3semidirect
314. C2×C32C2×C3×C4⋊C4semidirect
352. C22×C3Dic3⋊Dic3direct (N × H)
353. C22×C3C3×Dic3⋊C4semidirect
354. C22×C3C3×C4⋊Dic3semidirect
378. Dic3C22×C3×Dic3semidirect
383. C32C22×Dic3⋊C4semidirect
384. C32C22×C4⋊Dic3semidirect
385. C32C22×C3×C4⋊C4semidirect
412. C2×C3C2×Dic3⋊Dic3direct (N × H)
413. C2×C3C2×C3×Dic3⋊C4semidirect
414. C2×C3C2×C3×C4⋊Dic3semidirect
424. C22C3×Dic3⋊Dic3direct (N × H)
436. C3C22×Dic3⋊Dic3direct (N × H)
437. C3C22×C3×Dic3⋊C4semidirect
438. C3C22×C3×C4⋊Dic3semidirect
439. C2C2×C3×Dic3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C23×C32×Dic32
C2×C3×Dic3⋊Dic32
C22×C3×C4⋊Dic33
C22×C3×Dic3⋊C43
C22×Dic3⋊Dic33