Finite Group Databasebeta

← #13156407

Subgroups — #13156407

Order 1728.

648 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723515C211yesmaximal, characteristic
2C3×D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic, complemented
4#11722054C211yesmaximal, characteristic, complemented
5#11722052C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591125—31nomaximal
9#10590676—31nomaximal
10#10590500—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6290—21no—
14C2×C3×D3×Dic3—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—21no—
21C3×Dic3⋊Dic3—21no—
22#64139—31no—
23C2×C3×C4⋊Dic3—31no—
24C3×D3×C4⋊C4—31no—
25#63908—31no—
26#63908—31no—
27#63908—61no—
28D3×C4⋊Dic3D311yescharacteristic, complemented
29#63644—31no—
30#63635—31no—
31C3×#830—31no—
32C3×#821—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#809—31no—
35#63573—31no—
36#63573—31no—
37#63573—31no—
38#63573—31no—
39#63573—61no—
40#63573—61no—
41#63564—31no—
42#63564D311yescharacteristic, complemented
43#63563—31no—
44#63562—31no—
45(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C22×C32×D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57#6290—42no—
58C3×D3×Dic3—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#4968—91no—
62#4777—91no—
63#4735—91no—
64C22×C32×C4—31no—
65C2×#638—61no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C22×C3×Dic3—61no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—61no—
73C2×C3×C4×D3C2×D311yescharacteristic
74C2×C3×C4×D3C2×D311yescharacteristic
75C2×#625—62no—
76C2×#625—61no—
77C2×#625—61no—
78C2×D3×Dic3—21no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81#3752—61no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—61no—
86C4⋊(C3⋊Dic3)—61no—
87C4⋊(C3⋊Dic3)—61no—
88C4⋊(C3⋊Dic3)—121no—
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×Dic3⋊C4—61no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105Dic3⋊Dic3—21no—
106(C2×D3)⋊Dic3—62no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—61no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C32×D3—21no—
114C2×C32×D3—21no—
115C2×C32×D3SD1612yes—
116C33×C4#4111yescharacteristic
117C33×C4#4111yescharacteristic
118C3⋊(C3⋊Dic3)—41no—
119C3×C3⋊Dic3—41no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C2×C3×C4⋊C4—91no—
123#938—91no—
124#930—91no—
125#930—91no—
126#930—181no—
127C2×C4⋊Dic3—31no—
128D3×C4⋊C4—31no—
129C3×#102—91no—
130C3×C22⋊C8—91no—
131#832—91no—
132#830—31no—
133#821—31no—
134#821—91no—
135(C2×D3)⋊C8—31no—
136#817—91no—
137#817—91no—
138#817—91no—
139#817—181no—
140#809—31no—
141#809—91no—
142#808—91no—
143#808—91no—
144#808—91no—
145#808—181no—
146C23×C32—31no—
147C22×C3⋊D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—62no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3D1211yescharacteristic
153C22×C3×D3#6711yescharacteristic
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—31no—
158C2×C32×C4—41no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165#638—122no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—122no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C2×C3×Dic3—122no—
184C2×C3×Dic3D1211yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C3×D12—61no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—121no—
191C3×D12—121no—
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—41no—
197C3×C4×D3—41no—
198C3×C4×D3—122no—
199#625—124no—
200#625—122no—
201#625—122no—
202D3×Dic3—42no—
203C32×C8—61no—
204C3⋊(C3⋊C8)—21no—
205C3×C3⋊C8—22no—
206C3×C3⋊C8—21no—
207C3×C3⋊C8—21no—
208C3×C3⋊C8—62no—
209C3×C3⋊C8—61no—
210C3×C3⋊C8—61no—
211#481—271noSylow
212C2×C33#12311yescharacteristic
213C2×C33C2×SD1611yescharacteristic
214C2×C33AGL(1,ℤ/8)11yescharacteristic
215C3×C3⋊D3—81no—
216C32×D3—22no—
217C2×C3×D4—91no—
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—61no—
221C22×C3×C4—181no—
222C2×#67—181no—
223C2×#67—181no—
224C2×#67—182no—
225C2×#67—181no—
226C22×Dic3—61no—
227C2×D12—31no—
228C2×D12—31no—
229C2×D12—61no—
230C2×C4×D3—61no—
231C2×C4×D3D3211yescharacteristic
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—61no—
235C2×C3×C8—91no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—181no—
241#217—181no—
242(C2×D3)⋊C4—181no—
243(C2×D3)⋊C4—181no—
244(C2×D3)⋊C4—182no—
245C4⋊Dic3—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—61no—
248C4⋊Dic3—61no—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—181no—
253C4⋊Dic3—181no—
254C4⋊Dic3—181no—
255C4⋊Dic3—361no—
256C4⋊Dic3D3211yescharacteristic
257Dic3⋊C4—61no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8D3211yescharacteristic
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—32no—
266C22×C32—41no—
267C22×C32—61no—
268C22×C32—61no—
269C22×C32C2×D1211yescharacteristic
270C22×C32D3×D411yescharacteristic
271C22×C32C2×#6711yescharacteristic
272C2×C3⋊D3—41no—
273C2×C3⋊D3—41no—
274C2×C3⋊D3—41no—
275C2×C3×D3—21no—
276C2×C3×D3—21no—
277C2×C3×D3—21no—
278C2×C3×D3—21no—
279C2×C3×D3—22no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—122no—
283C2×C3×D3—122no—
284C2×C3×D3—122no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3#20412yes—
292C2×C3×D3#21512yes—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—61no—
302C32×C4—121no—
303C32×C4#23511yescharacteristic
304C32×C4#23511yescharacteristic
305C32×C4#23511yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4#23711yescharacteristic
308C32×C4#23711yescharacteristic
309C3⋊Dic3—41no—
310C3⋊Dic3—121no—
311C3⋊Dic3—122no—
312C3⋊Dic3—121no—
313C3⋊Dic3—121no—
314C3⋊Dic3—241no—
315C3⋊Dic3—241no—
316C3⋊Dic3—242no—
317C3×Dic3—21no—
318C3×Dic3—21no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—81no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C3×Dic3—241no—
326#121—271no—
327C2×C4⋊C4—91no—
328#106—271no—
329#106—271no—
330#102—91no—
331#102—271no—
332C22⋊C8—91no—
333C33#48111yescharacteristic, complemented, Sylow
334C23×C3—31no—
335C23×C3—31no—
336C23×C3—61no—
337C23×C3—181no—
338C22×D3—62no—
339C22×D3—61no—
340C22×D3—61no—
341C22×D3#62611yescharacteristic
342C3×D4—181no—
343C3×D4—181no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—41no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—121no—
358C2×C3×C4—121no—
359C2×C3×C4—121no—
360C2×C3×C4—181no—
361C2×C3×C4—362no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365#67—362no—
366#67—362no—
367#67—364no—
368#67—362no—
369C2×Dic3—21no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—121no—
373C2×Dic3—122no—
374C2×Dic3—182no—
375C2×Dic3—181no—
376C2×Dic3—181no—
377C2×Dic3—181no—
378C2×Dic3—361no—
379C2×Dic3—361no—
380C2×Dic3—362no—
381C2×Dic3#62611yescharacteristic
382D12—61no—
383D12—61no—
384D12—61no—
385D12—61no—
386D12—121no—
387D12—121no—
388C4×D3—21no—
389C4×D3—21no—
390C4×D3—122no—
391C3×C8—62no—
392C3×C8—61no—
393C3×C8—61no—
394C3×C8—181no—
395C3⋊C8—21no—
396C3⋊C8—62no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—62no—
412C2×C32#88311yescharacteristic
413C2×C32#88611yescharacteristic
414C2×C32C2×#20411yescharacteristic
415C2×C32#90811yescharacteristic
416C2×C32#91111yescharacteristic
417C2×C32D3×SD1611yescharacteristic
418C2×C32#92911yescharacteristic
419C2×C32#93211yescharacteristic
420C2×C32C2×#21511yescharacteristic
421C3⋊D3—81no—
422C3×D3—22no—
423C3×D3—22no—
424C3×D3—42no—
425C3×D3—242no—
426C3×D3—241no—
427C3×D3—241no—
428C2×D4—91no—
429C2×D4—541no—
430C22×C4—31no—
431C22×C4—181no—
432C2×C8—31no—
433C2×C8—91no—
434C4⋊C4—31no—
435C4⋊C4—91no—
436C4⋊C4—181no—
437C4⋊C4—271no—
438C22⋊C4—541no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—32no—
443C22×C3—32no—
444C22×C3—41no—
445C22×C3—62no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—121no—
451C22×C3—121no—
452C22×C3—361no—
453C22×C3—361no—
454C22×C3—361no—
455C22×C3D3×D1211yescharacteristic
456C22×C3C2×#62611yescharacteristic
457C22×C3D3×#6711yescharacteristic
458C2×D3—21no—
459C2×D3—21no—
460C2×D3—122no—
461C2×D3—122no—
462C2×D3—122no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3#371712yes—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—61no—
479C3×C4—61no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—241no—
484C3×C4—361no—
485C3×C4#379511yescharacteristic
486C3×C4#379511yescharacteristic
487C3×C4#379811yescharacteristic
488C3×C4#380411yescharacteristic
489C3×C4#380411yescharacteristic
490C3×C4#380511yescharacteristic
491Dic3—21no—
492Dic3—41no—
493Dic3—121no—
494Dic3—121no—
495Dic3—241no—
496Dic3—362no—
497Dic3—361no—
498Dic3—361no—
499Dic3—361no—
500Dic3—721no—
501Dic3—721no—
502Dic3—722no—
503C32—21no—
504C32—21no—
505C32—21no—
506C32—41no—
507C32#473511yescharacteristic, complemented
508C32#477711yescharacteristic, complemented
509C32#496811yescharacteristic, complemented
510C23—31no—
511C23—181no—
512D4—181no—
513D4—181no—
514D4—1082no—
515C2×C4—31no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—181no—
520C2×C4—362no—
521C2×C4—541no—
522C2×C4D3311yescharacteristic
523C8—61no—
524C8—181no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—41no—
537C2×C3—62no—
538C2×C3—62no—
539C2×C3—122no—
540C2×C3—721no—
541C2×C3D3×#20411yescharacteristic
542C2×C3#6379511yescharacteristic
543C2×C3#6383011yescharacteristic
544C2×C3C2×#371711yescharacteristic
545C2×C3#6389111yescharacteristic
546C2×C3#6390511yescharacteristic
547C2×C3#6393211yescharacteristic
548C2×C3D3×#21511yescharacteristic
549C2×C3#6396611yescharacteristic
550D3—22no—
551D3—242no—
552D3—241no—
553D3—241no—
554C22—32no—
555C22—61no—
556C22—61no—
557C22—361no—
558C22—361no—
559C22—361no—
560C22D3×#62611yescharacteristic
561C4—61no—
562C4—121no—
563C4—361no—
564C4—1081no—
565C4#8896011yescharacteristic
566C4#8896111yescharacteristic
567C3—21no—
568C3—21no—
569C3—21no—
570C3—41no—
571C3#1059050011yescharacteristic, complemented
572C3#1059067611yescharacteristic, complemented
573C3#1059112511yescharacteristic, complemented
574C2—62no—
575C2—721no—
576C2D3×#371711yescharacteristic
577C2#1172286911yescharacteristic
578C2#1172298811yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723515C2
hypocenterC33#481
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×D3×C4⋊Dic3C2semidirect
3. #11722075C2semidirect
4. #11722054C2semidirect
5. #11722052C2semidirect
6. C3×#63564C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
28. D3×C4⋊Dic3D3semidirect
42. #63564D3semidirect
45. (C2×D3)⋊(C3⋊C8)D3semidirect
333. C33#481semidirect
507. C32#4735semidirect
508. C32#4777semidirect
509. C32#4968semidirect
571. C3#10590500semidirect
572. C3#10590676semidirect
573. C3#10591125semidirect

Maximal subgroups

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

subgroupindex
#117235152
C3×D3×C4⋊Dic32
#117220752
#117220542
#117220522
C3×#635642
C3×(C2×D3)⋊(C3⋊C8)2
#105911253
#105906763
#105905003