Finite Group Databasebeta

← Dic3×#3830 #13177460

Subgroups — Dic3×#3830 #13177460

Order 1728.

2236 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723626C212yesmaximal, complemented
5Dic3×#638C212yesmaximal, complemented
6#11723595C212yesmaximal, complemented
7C2×Dic3×C3⋊Dic3C212yesmaximal, complemented
8C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
9#11723410C211yesmaximal, characteristic, complemented
10#11723404C211yesmaximal, characteristic, complemented
11Dic3×#634C211yesmaximal, characteristic, complemented
12C4×#3830—31nomaximal
13Dic3×#237—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C22×C3×C3⋊Dic3C2212yes—
18C3×#3830C418yescomplemented
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C3×#634C2211yescharacteristic
21C22×C32×Dic3C2212yes—
22C2×Dic3×C3⋊D3C2211yescharacteristic
23C2×Dic3×C3⋊D3C2211yescharacteristic
24#88864C2212yes—
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29C32×C4×Dic3C2211yescharacteristic
30#88807C2212yes—
31#88807C2212yes—
32#88803C2212yes—
33Dic3×C3⋊Dic3C2211yescharacteristic
34Dic3×C3⋊Dic3C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37Dic3×C3⋊Dic3C2211yescharacteristic
38C2×#3830D311yescharacteristic
39C2×C3×#237—34no—
40C32×C4×D4—31no—
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44C4×#638—32no—
45C2×C4×C3⋊Dic3—32no—
46#64101—31no—
47#64096—31no—
48#64090—32no—
49#64086—32no—
50C42×C3⋊D3—31no—
51C4×#634—31no—
52C3×D4×Dic3D314yescomplemented
53#67⋊Dic3—38no—
54Dic3×#67—38no—
55#63955—38no—
56C2×Dic32—38no—
57C4×D3×Dic3—34no—
58#63843—34no—
59Dic3×Dic6—34no—
60#63841—34no—
61C23×C33C2312yes—
62C22×C3×C3⋊D3C2311yescharacteristic
63C33×D4C2×C414yescomplemented
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)—24no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C3×#638C2×C418yescomplemented
70C2×C3×C3⋊Dic3C2×C418yescomplemented
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C3×C4×C3⋊D3C2×C414yescomplemented
74C3×#634C2×C414yescomplemented
75C2×C32×Dic3—24no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79Dic3×C3⋊D3—22no—
80Dic3×C3⋊D3—22no—
81Dic3×#41—91no—
82C4×#237—94no—
83C2×C32×D4—24no—
84C2×C32×D4C2×D314yes—
85C2×C32×D4C2×D311yescharacteristic
86C22×C32×C4—32no—
87C2×#638C2×D312yes—
88C22×C3⋊Dic3—32no—
89C22×C3⋊Dic3—38no—
90C22×C3⋊Dic3—68no—
91C22×C3⋊Dic3C2×D312yes—
92#3830Dic318yesdirect product factor
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—31no—
95C2×C4×C3⋊D3C2×D311yescharacteristic
96C2×#634C2×D311yescharacteristic
97C2×C3×#67—38no—
98C22×C3×Dic3—38no—
99C22×C3×Dic3C2×D318yes—
100C3×#237—332no—
101C2×C3×C4×D3—34no—
102C2×C3×Dic6—34no—
103C2×D3×Dic3—34no—
104C2×D3×Dic3—34no—
105C32×C4⋊C4—31no—
106C32×C22⋊C4—32no—
107C32×C42—31no—
108#3757—32no—
109#3757—38no—
110#3757—68no—
111#3752—32no—
112C4⋊(C3⋊Dic3)—31no—
113C4⋊(C3⋊Dic3)—34no—
114C4⋊(C3⋊Dic3)—64no—
115#3750—32no—
116#3750—32no—
117C4×C3⋊Dic3—31no—
118C4×C3⋊Dic3—34no—
119C4×C3⋊Dic3—32no—
120C4×C3⋊Dic3—31no—
121C4×C3⋊Dic3—31no—
122C4×C3⋊Dic3—32no—
123C4×C3⋊Dic3—32no—
124C4×C3⋊Dic3—64no—
125C3×#217C2×D318yes—
126C3×C4⋊Dic3C2×D314yes—
127C3×C4×Dic3C2×D314yes—
128Dic3⋊Dic3—38no—
129Dic3⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132Dic32—38no—
133Dic32—38no—
134Dic32—34no—
135Dic32—34no—
136C22×C33—22no—
137C22×C33—22no—
138C22×C33C22×C414yes—
139C22×C33C2411yescharacteristic
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3—21no—
142C2×C3×C3⋊D3C22×C412yes—
143C33×C4C22×C412yes—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)—22no—
146C3⋊(C3⋊Dic3)#4112yescomplemented
147C3×C3⋊Dic3C22×C414yes—
148C3×C3⋊Dic3C22×C412yes—
149C32×Dic3—21no—
150C32×Dic3—22no—
151C32×Dic3#4112yescomplemented
152C2×C3×#41—91no—
153C2×#237—34no—
154C3×C4×D4—34no—
155Q8×Dic3—91no—
156Q8×Dic3—91no—
157D4×Dic3—94no—
158D4×Dic3—91no—
159D4×Dic3—92no—
160D4×Dic3—92no—
161D4×Dic3—184no—
162D4×Dic3C3⋊D311yescharacteristic, complemented
163C4×#67—98no—
164#926—92no—
165#926—91no—
166C2×C4×Dic3—98no—
167C2×C4×Dic3—92no—
168C2×C4×Dic3—91no—
169#892—94no—
170#887—94no—
171#67⋊C4—98no—
172#877—98no—
173C42×D3—94no—
174C4×Dic6—94no—
175C23×C32—28no—
176C23×C32C22×D318yes—
177C23×C32C22×D312yes—
178C22×C3⋊D3C22×D311yescharacteristic
179C22×C3×D3—34no—
180C32×D4—216no—
181C32×D4C4×D3116yescomplemented
182C32×D4C2×Dic314yescomplemented
183C2×C32×C4—24no—
184C2×C32×C4—31no—
185C2×C32×C4—32no—
186C2×C32×C4—31no—
187C2×C32×C4—64no—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D311yescharacteristic
190#638C2×Dic318yescomplemented
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—34no—
195C2×C3⋊Dic3—38no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3—68no—
200C2×C3⋊Dic3—64no—
201C2×C3⋊Dic3—616no—
202C2×C3⋊Dic3—1216no—
203C2×C3⋊Dic3C2×Dic318yescomplemented
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D312yes—
206C4×C3⋊D3—62no—
207C4×C3⋊D3—62no—
208C4×C3⋊D3C2×Dic314yescomplemented
209#634C2×Dic314yescomplemented
210C3×#67—332no—
211C2×C3×Dic3—216no—
212C2×C3×Dic3—332no—
213C2×C3×Dic3—34no—
214C2×C3×Dic3—38no—
215C2×C3×Dic3C22×D318yes—
216C2×C3×Dic3C22×D314yes—
217C2×C3×Dic3C22×D314yes—
218C3×C4×D3—316no—
219C3×Dic6—316no—
220D3×Dic3—68no—
221D3×Dic3—68no—
222C4×#41—271noSylow
223C2×C33—24no—
224C2×C33C23×C411yescharacteristic
225C2×C33C2×#4111yescharacteristic
226C2×C33C2×#4111yescharacteristic
227C3×C3⋊D3—22no—
228C3×#41—98no—
229C2×C3×Q8—91no—
230C2×C3×D4—24no—
231C2×C3×D4—92no—
232C2×C3×D4D3214yes—
233C2×C3×D4C2×C3⋊D311yescharacteristic
234C22×C3×C4—38no—
235C22×C3×C4—92no—
236C22×C3×C4—91no—
237C2×#67—38no—
238C22×Dic3—38no—
239C22×Dic3—98no—
240C22×Dic3—92no—
241C22×Dic3—91no—
242C22×Dic3—91no—
243C22×Dic3—188no—
244C22×Dic3C2×C3⋊D312yes—
245#237—332no—
246C2×C4×D3—34no—
247C2×C4×D3—94no—
248C2×C4×D3—94no—
249C2×Dic6—34no—
250C3×C4⋊C4—34no—
251C3×C22⋊C4—38no—
252C3×C42—34no—
253#217—98no—
254#217—92no—
255#217—92no—
256#217—188no—
257#217C2×C3⋊D312yes—
258(C2×D3)⋊C4—98no—
259C4⋊Dic3—94no—
260C4⋊Dic3—91no—
261C4⋊Dic3—92no—
262C4⋊Dic3—92no—
263C4⋊Dic3—184no—
264C4⋊Dic3C2×C3⋊D311yescharacteristic
265Dic3⋊C4—98no—
266Dic3⋊C4—98no—
267C4×Dic3—98no—
268C4×Dic3—94no—
269C4×Dic3—94no—
270C4×Dic3—98no—
271C4×Dic3—98no—
272C4×Dic3—94no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—92no—
276C4×Dic3—91no—
277C4×Dic3—92no—
278C4×Dic3—92no—
279C4×Dic3—184no—
280C4×Dic3C2×C3⋊D311yescharacteristic
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—24no—
284C22×C32—216no—
285C22×C32—28no—
286C22×C32—28no—
287C22×C32—48no—
288C22×C32—48no—
289C22×C32C2×C4×D3116yes—
290C22×C32C22×Dic314yes—
291C22×C32C23×D314yes—
292C22×C32C23×D311yescharacteristic
293C2×C3⋊D3—21no—
294C2×C3⋊D3—21no—
295C2×C3⋊D3C22×Dic312yes—
296C2×C3×D3—38no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C32×C4—28no—
300C32×C4—32no—
301C32×C4—61no—
302C32×C4—62no—
303C32×C4C2×C4×D318yes—
304C32×C4C22×Dic312yes—
305C3⋊Dic3—32no—
306C3⋊Dic3—38no—
307C3⋊Dic3—68no—
308C3⋊Dic3—61no—
309C3⋊Dic3—62no—
310C3⋊Dic3—64no—
311C3⋊Dic3—68no—
312C3⋊Dic3—124no—
313C3⋊Dic3—128no—
314C3⋊Dic3C22×Dic314yes—
315C3⋊Dic3C22×Dic312yes—
316C3×Dic3—28no—
317C3×Dic3—24no—
318C3×Dic3—316no—
319C3×Dic3—38no—
320C3×Dic3#23718yescomplemented
321C2×#41—91no—
322C4×Q8—271no—
323C4×Q8—271no—
324C4×D4—31no—
325C4×D4—272no—
326C4×D4—272no—
327C4×D4—271no—
328#117—272no—
329#117—271no—
330C2×C42—271no—
331C2×C42—272no—
332C33C4×#4111yescharacteristic, complemented, Sylow
333C23×C3—28no—
334C23×C3—91no—
335C23×C3C2×D3218yes—
336C23×C3C22×C3⋊D312yes—
337C22×D3—34no—
338C3×Q8—94no—
339C3×D4—216no—
340C3×D4—98no—
341C3×D4D3×Dic3116yescomplemented
342C3×D4C4×C3⋊D314yescomplemented
343C2×C3×C4—24no—
344C2×C3×C4—34no—
345C2×C3×C4—38no—
346C2×C3×C4—34no—
347C2×C3×C4—616no—
348C2×C3×C4—91no—
349C2×C3×C4—92no—
350C2×C3×C4—98no—
351C2×C3×C4—94no—
352C2×C3×C4C2×D3214yes—
353C2×C3×C4C22×C3⋊D311yescharacteristic
354#67—332no—
355C2×Dic3—24no—
356C2×Dic3—34no—
357C2×Dic3—38no—
358C2×Dic3—332no—
359C2×Dic3—94no—
360C2×Dic3—91no—
361C2×Dic3—94no—
362C2×Dic3—98no—
363C2×Dic3—91no—
364C2×Dic3—92no—
365C2×Dic3—184no—
366C2×Dic3—184no—
367C2×Dic3—188no—
368C2×Dic3—182no—
369C2×Dic3—184no—
370C2×Dic3—1816no—
371C2×Dic3—182no—
372C2×Dic3—3616no—
373C2×Dic3C22×C3⋊D312yes—
374C2×Dic3C22×C3⋊D311yescharacteristic
375C2×Dic3C22×C3⋊D311yescharacteristic
376C4×D3—316no—
377C4×D3—188no—
378C4×D3—188no—
379Dic6—316no—
380C2×C32—24no—
381C2×C32—24no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—216no—
385C2×C32—416no—
386C2×C32C22×C4×D314yes—
387C2×C32C2×#23714yes—
388C2×C32D3×#4114yes—
389C2×C32D3×#4111yescharacteristic
390C2×C32D3×#4111yescharacteristic
391C2×C32C23×Dic311yescharacteristic
392C3⋊D3—22no—
393C3×D3—68no—
394#41—98no—
395C2×Q8—91no—
396C2×D4—92no—
397C2×D4D3×C3⋊D311yescharacteristic
398C22×C4—32no—
399C22×C4—92no—
400C22×C4—91no—
401C22×C4—271no—
402C22×C4—271no—
403C22×C4—272no—
404C4⋊C4—31no—
405C4⋊C4—272no—
406C4⋊C4—272no—
407C4⋊C4—271no—
408C22⋊C4—32no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C42—31no—
412C42—271no—
413C42—271no—
414C42—272no—
415C42—272no—
416C42—272no—
417C42—271no—
418C22×C3—28no—
419C22×C3—28no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—24no—
423C22×C3—216no—
424C22×C3—48no—
425C22×C3—48no—
426C22×C3—92no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3C2×D3×Dic3116yes—
430C22×C3C2×C4×C3⋊D314yes—
431C22×C3C22×D3214yes—
432C22×C3C23×C3⋊D311yescharacteristic
433C2×D3—38no—
434C2×D3—64no—
435C2×D3—64no—
436C3×C4—28no—
437C3×C4—38no—
438C3×C4—68no—
439C3×C4—64no—
440C3×C4—92no—
441C3×C4—94no—
442C3×C4C2×D3×Dic318yes—
443C3×C4C2×C4×C3⋊D312yes—
444Dic3—22no—
445Dic3—21no—
446Dic3—316no—
447Dic3—38no—
448Dic3—98no—
449Dic3—92no—
450Dic3—188no—
451Dic3—184no—
452Dic3—188no—
453Dic3—181no—
454Dic3—182no—
455Dic3—368no—
456Dic3—364no—
457Dic3#383012yesdirect product factor
458C32—24no—
459C32C4×#23714yescomplemented
460C32Dic3×#4111yescharacteristic, complemented
461C23—91no—
462C23C2×D3×C3⋊D312yes—
463Q8—94no—
464D4—98no—
465D4Dic3×C3⋊D314yescomplemented
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—32no—
469C2×C4—64no—
470C2×C4—91no—
471C2×C4—92no—
472C2×C4—98no—
473C2×C4—94no—
474C2×C4—271no—
475C2×C4—271no—
476C2×C4—272no—
477C2×C4—542no—
478C2×C4—542no—
479C2×C4—544no—
480C2×C4C2×D3×C3⋊D311yescharacteristic
481C2×C3—216no—
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—24no—
485C2×C3—24no—
486C2×C3—416no—
487C2×C3—182no—
488C2×C3D3×#23714yes—
489C2×C3#6431214yes—
490C2×C3C22×D3×Dic314yes—
491C2×C3C22×C4×C3⋊D311yescharacteristic
492C2×C3C2×#383011yescharacteristic
493C2×C3#41×C3⋊D311yescharacteristic
494D3—68no—
495C22—22no—
496C22—22no—
497C22—92no—
498C22—181no—
499C22—181no—
500C22C2×Dic3×C3⋊D314yes—
501C22C22×D3×C3⋊D311yescharacteristic
502C4—32no—
503C4—62no—
504C4—61no—
505C4—92no—
506C4—94no—
507C4—272no—
508C4—541no—
509C4—542no—
510C4C2×Dic3×C3⋊D312yes—
511C3—24no—
512C3Dic3×#23714yescomplemented
513C3C4×#383011yescharacteristic, complemented
514C2—24no—
515C2—182no—
516C2D3×#383011yescharacteristic
517C2#1172458611yescharacteristic
518C2C22×Dic3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC2×C33×D4C22
socle, cosocleC22×C33C24
prime radicalC2×C3×#3830C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C4×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×C3⋊(C3⋊Dic3)C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723626C2semidirect
5. Dic3×#638C2semidirect
6. #11723595C2semidirect
7. C2×Dic3×C3⋊Dic3C2semidirect
8. C4×Dic3×C3⋊D3C2semidirect
9. #11723410C2semidirect
10. #11723404C2semidirect
11. Dic3×#634C2semidirect
18. C3×#3830C4semidirect
52. C3×D4×Dic3D3semidirect
63. C33×D4C2×C4semidirect
69. C3×#638C2×C4semidirect
70. C2×C3×C3⋊Dic3C2×C4semidirect
73. C3×C4×C3⋊D3C2×C4semidirect
74. C3×#634C2×C4semidirect
92. #3830Dic3direct (N × H)
146. C3⋊(C3⋊Dic3)#41semidirect
151. C32×Dic3#41semidirect
162. D4×Dic3C3⋊D3semidirect
181. C32×D4C4×D3semidirect
182. C32×D4C2×Dic3semidirect
190. #638C2×Dic3semidirect
203. C2×C3⋊Dic3C2×Dic3semidirect
208. C4×C3⋊D3C2×Dic3semidirect
209. #634C2×Dic3semidirect
320. C3×Dic3#237semidirect
332. C33C4×#41semidirect
341. C3×D4D3×Dic3semidirect
342. C3×D4C4×C3⋊D3semidirect
457. Dic3#3830direct (N × H)
459. C32C4×#237semidirect
460. C32Dic3×#41semidirect
465. D4Dic3×C3⋊D3semidirect
512. C3Dic3×#237semidirect
513. C3C4×#3830semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
D4×C3⋊(C3⋊Dic3)2
C32×D4×Dic32
#117236262
Dic3×#6382
#117235952
C2×Dic3×C3⋊Dic32
C4×Dic3×C3⋊D32
#117234102
#117234042
Dic3×#6342
C4×#38303
Dic3×#2373