Finite Group Databasebeta

← #13161483

Subgroups — #13161483

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#11722080C211yesmaximal, characteristic, complemented
5#11722079C211yesmaximal, characteristic, complemented
6#11722075C211yesmaximal, characteristic
7#11722075C211yesmaximal, characteristic
8#10590308—31nomaximal
9#10590271—31nomaximal
10#10590270—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C33×C42C2211yescharacteristic
14C3×#3752—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C3×#3750—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C4×C4⋊(C3⋊D3)D311yescharacteristic
23#64079D311yescharacteristic, complemented
24C3×C4×D12—31no—
25C3×C4×D12—31no—
26C3×C4×D12—61no—
27C3×#870—31no—
28C3×#870—31no—
29C3×#870—61no—
30C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
31C3×C4⋊(C3⋊C8)—31no—
32C3×C4⋊(C3⋊C8)—31no—
33C3×C4⋊(C3⋊C8)—61no—
34#63576—31no—
35#63575—31no—
36#63575—31no—
37#63574—31no—
38#63573—31no—
39#63573—31no—
40#63564—31no—
41#63564—31no—
42#63563—31no—
43#63563—31no—
44#63562—31no—
45#63562—31no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4D411yescharacteristic
48C2×C33×C4D411yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)Q812yes—
55C3×C4×C3⋊D3—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5066—91no—
59#5009—91no—
60#4847—91no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C4×C3⋊D3—21no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×C3×C4×D3—62no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C32×C42—21no—
70C32×C42—21no—
71C32×C42—21no—
72C32×C42—41no—
73C32×C42C2×D311yescharacteristic
74C32×C42C2×D311yescharacteristic
75C32×C42C2×D311yescharacteristic
76#3752—21no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80#3750—21no—
81#3750—21no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×(C2×D3)⋊C4—62no—
85C3×(C2×D3)⋊C4—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×Dic3⋊C4—62no—
97C3×Dic3⋊C4—62no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×Dic3⋊C4—61no—
101C3×Dic3⋊C4—61no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8—61no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C33×C4—21no—
113C33×C4—21no—
114C33×C4C2×Q811yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C3×C3⋊Dic3—41no—
119C3×#125—91no—
120C3×C4×D4—91no—
121C4×D12—31no—
122C4×D12—31no—
123C4×D12—61no—
124#870—31no—
125#870—31no—
126#870—61no—
127C3×C4⋊C8—91no—
128#832—91no—
129#832—91no—
130#821—91no—
131#821—91no—
132#818—91no—
133#817—91no—
134#809—91no—
135#809—91no—
136#808—91no—
137#808—91no—
138#807—91no—
139#807—91no—
140C4⋊(C3⋊C8)—31no—
141C4⋊(C3⋊C8)—31no—
142C4⋊(C3⋊C8)—61no—
143C22×C3⋊D3—21no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4—41no—
158C2×C32×C4—41no—
159C2×C32×C4D1211yescharacteristic
160C2×C32×C4D1211yescharacteristic
161C2×C32×C4#6711yescharacteristic
162C2×C32×C4#6711yescharacteristic
163C2×C32×C4#6711yescharacteristic
164C2×C32×C4#6711yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)Dic612yes—
173C4×C3⋊D3—42no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C3×D12—32no—
184C3×D12—32no—
185C3×D12—62no—
186C3×D12—62no—
187C3×D12—61no—
188C3×D12—61no—
189C3×C4×D3—124no—
190C3×C4×D3—122no—
191C3×C4×D3—122no—
192C3⋊(C3⋊C8)—21no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—61no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—121no—
199C3×C3⋊C8—121no—
200#478—271noSylow
201C2×C33C22⋊Q811yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33AGL(1,ℤ/8)11yescharacteristic
204C3×C3⋊D3—42no—
205C2×C3×D4—91no—
206C22×C3×C4—181no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—61no—
210C2×C4×D3—62no—
211C2×C4×D3—61no—
212C2×C4×D3—61no—
213C2×C3×C8—91no—
214C2×C3×C8—91no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—181no—
219C3×C4⋊C4—181no—
220C3×C22⋊C4—181no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—41no—
225C3×C42D3211yescharacteristic
226C3×C42D3211yescharacteristic
227C3×C42D3211yescharacteristic
228(C2×D3)⋊C4—62no—
229(C2×D3)⋊C4—61no—
230(C2×D3)⋊C4—61no—
231C4⋊Dic3—31no—
232C4⋊Dic3—31no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—61no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240Dic3⋊C4—62no—
241Dic3⋊C4—62no—
242Dic3⋊C4—61no—
243Dic3⋊C4—61no—
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8—61no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32C2×D1211yescharacteristic
257C22×C32C2×#6711yescharacteristic
258C22×C32C2×#6711yescharacteristic
259C2×C3⋊D3—22no—
260C2×C3⋊D3—41no—
261C2×C3⋊D3—41no—
262C2×C3×D3—64no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C2×C3×D3—122no—
266C2×C3×D3—122no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—41no—
291C32×C4—42no—
292C32×C4—42no—
293C32×C4C2×Dic611yescharacteristic
294C32×C4#23511yescharacteristic
295C32×C4#23711yescharacteristic
296C32×C4D3×Q811yescharacteristic
297C32×C4D3×Q811yescharacteristic
298C32×C4#23911yescharacteristic
299C3⋊Dic3—41no—
300C3⋊Dic3—41no—
301C3⋊Dic3—41no—
302C3×Dic3—122no—
303C3×Dic3—122no—
304C3×Dic3—122no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311#125—91no—
312C4×D4—91no—
313#107—271no—
314#106—271no—
315C4⋊C8—91no—
316#102—271no—
317#102—271no—
318C33#47811yescharacteristic, complemented, Sylow
319C23×C3—181no—
320C22×D3—62no—
321C22×D3—61no—
322C22×D3—61no—
323C3×D4—92no—
324C3×D4—181no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4—362no—
341C2×C3×C4#62511yescharacteristic
342C2×C3×C4#62511yescharacteristic
343C2×C3×C4#62611yescharacteristic
344C2×C3×C4#62611yescharacteristic
345C2×C3×C4#62611yescharacteristic
346C2×C3×C4#62611yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×Dic3—62no—
351C2×Dic3—62no—
352C2×Dic3—62no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359D12—32no—
360D12—32no—
361D12—62no—
362D12—62no—
363D12—61no—
364D12—61no—
365C4×D3—124no—
366C4×D3—122no—
367C4×D3—122no—
368C3×C8—181no—
369C3×C8—181no—
370C3⋊C8—61no—
371C3⋊C8—61no—
372C3⋊C8—61no—
373C3⋊C8—61no—
374C3⋊C8—121no—
375C3⋊C8—121no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32#89711yescharacteristic
389C2×C32#90411yescharacteristic
390C2×C32#90811yescharacteristic
391C2×C32#92411yescharacteristic
392C2×C32#93211yescharacteristic
393C2×C32#94611yescharacteristic
394C2×C32#94911yescharacteristic
395C2×C32#95011yescharacteristic
396C2×C32#95011yescharacteristic
397C3⋊D3—42no—
398C3×D3—124no—
399C3×D3—122no—
400C3×D3—122no—
401C2×D4—91no—
402C22×C4—181no—
403C2×C8—91no—
404C2×C8—91no—
405C4⋊C4—91no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—181no—
409C4⋊C4—181no—
410C22⋊C4—181no—
411C42#111311yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—182no—
417C22×C3—361no—
418C22×C3—361no—
419C22×C3C2×#62511yescharacteristic
420C22×C3C2×#62611yescharacteristic
421C22×C3C2×#62611yescharacteristic
422C2×D3—64no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—122no—
426C2×D3—122no—
427C2×D3—121no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—361no—
454C3×C4—361no—
455C3×C4—361no—
456C3×C4D3×Dic611yescharacteristic
457C3×C4D3×Dic611yescharacteristic
458C3×C4#379511yescharacteristic
459C3×C4#379611yescharacteristic
460C3×C4#379711yescharacteristic
461C3×C4#379911yescharacteristic
462Dic3—122no—
463Dic3—122no—
464Dic3—122no—
465Dic3—121no—
466Dic3—121no—
467Dic3—121no—
468Dic3—121no—
469Dic3—121no—
470Dic3—121no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32#484711yescharacteristic, complemented
476C32#500911yescharacteristic, complemented
477C32#506611yescharacteristic, complemented
478C23—181no—
479D4—92no—
480D4—181no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—362no—
485C2×C4#629511yescharacteristic
486C2×C4#629511yescharacteristic
487C2×C4C2×#111311yescharacteristic
488C8—181no—
489C8—181no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—362no—
503C2×C3#6382211yescharacteristic
504C2×C3#6382311yescharacteristic
505C2×C3#6382511yescharacteristic
506C2×C3#6382911yescharacteristic
507C2×C3#6382911yescharacteristic
508C2×C3#6383011yescharacteristic
509C2×C3#6385611yescharacteristic
510C2×C3#6385711yescharacteristic
511C2×C3#6387111yescharacteristic
512D3—124no—
513D3—122no—
514D3—122no—
515C22—182no—
516C22—361no—
517C22—361no—
518C22C2×#629511yescharacteristic
519C4—21no—
520C4—21no—
521C4—361no—
522C4—361no—
523C4—361no—
524C4#8904011yescharacteristic
525C4#8904111yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059027011yescharacteristic, complemented
531C3#1059027111yescharacteristic, complemented
532C3#1059030811yescharacteristic, complemented
533C2—362no—
534C2#1172366011yescharacteristic
535C2#1172366111yescharacteristic
536C2#1172367411yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42#1113
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64079C2semidirect
3. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. #11722080C2semidirect
5. #11722079C2semidirect
23. #64079D3semidirect
30. C4⋊(C3⋊(C3⋊C8))D3semidirect
318. C33#478semidirect
475. C32#4847semidirect
476. C32#5009semidirect
477. C32#5066semidirect
530. C3#10590270semidirect
531. C3#10590271semidirect
532. C3#10590308semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C3×#640792
C3×C4⋊(C3⋊(C3⋊C8))2
#117220802
#117220792
#117220752
#105903083
#105902713
#105902703