Finite Group Databasebeta

← C3×#10590550 #13157533

Subgroups — C3×#10590550 #13157533

Order 1728.

622 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#890C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic, complemented
5C3×#63564C211yesmaximal, characteristic, complemented
6C3×#63562C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5067—31nomaximal
9C3×#4764—31nomaximal
10#10590550C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C3×#626—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×#125—31no—
23C3×#940—31no—
24C3×#890—21no—
25C3×#890D311yescharacteristic, complemented
26C3×#890C2×C311yescharacteristic, complemented
27C3×#874—31no—
28#63911C2×C311yescharacteristic
29C32×#102—31no—
30#63636C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#821—31no—
33C3×#815—31no—
34C3×#809—21no—
35C3×#809—31no—
36C3×#809—31no—
37C3×#809—31no—
38C3×#809—31no—
39C3×#809—61no—
40C3×#809D311yescharacteristic, complemented
41C3×#809C2×C311yescharacteristic, complemented
42C3×#803—31no—
43#63564C2×C311yescharacteristic, complemented
44#63562C2×C311yescharacteristic, complemented
45#63554C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C22×C32×D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C3×#626—42no—
58C3×#626—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61C3×#490—91no—
62#5067—31no—
63#4764—31no—
64C2×C32×D4—31no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C3×#67—61no—
67C2×C3×#67—61no—
68C2×C3×D12—21no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×D12—61no—
73C2×C3×D12—61no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C22×C311yescharacteristic
76C2×#626—21no—
77C2×#626—21no—
78C2×C32×C8—31no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—31no—
83C32×C4⋊C4—41no—
84C32×C4⋊C4—61no—
85C32×C4⋊C4—61no—
86C32×C4⋊C4C2×D311yescharacteristic
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C22×C311yescharacteristic
89C32×C42—31no—
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91C3×C4⋊Dic3—22no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C22×C311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C33×C4—41no—
117C33×C4#4111yescharacteristic
118C33×C4#4111yescharacteristic
119C32×Dic3—21no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C3×#127—91no—
123C3×#125—31no—
124C3×#125—31no—
125C3×#125—61no—
126#940—31no—
127#890C3×D311yescharacteristic, complemented
128#874—31no—
129C3×#102—31no—
130C3×#102—31no—
131C3×#102—61no—
132C3×#102—91no—
133C3×#102—91no—
134C3×#102—91no—
135C3×#97—91no—
136#832—31no—
137#821—31no—
138#815—31no—
139#809—31no—
140#809—31no—
141#809—31no—
142#809—31no—
143#809—61no—
144#809C3×D311yescharacteristic, complemented
145#803—31no—
146C23×C32—61no—
147C22×C3⋊D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—41no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C22×C3×D3—121no—
154C32×D4—61no—
155C32×D4—61no—
156C2×C32×C4—21no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—31no—
164C2×C32×C4—31no—
165C2×C32×C4—41no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—42no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C23×C311yescharacteristic
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)—21no—
175C3×#67—122no—
176C3×#67—122no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3#6711yescharacteristic
183C2×C3×Dic3#6711yescharacteristic
184C2×C3×Dic3C3×D411yescharacteristic
185C2×C3×Dic3C3×D411yescharacteristic
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—41no—
192C3×D12—62no—
193C3×D12—61no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—121no—
198C3×D12—122no—
199C3×D12—122no—
200#626—42no—
201#626—42no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210#490—91noSylow
211C2×C33#12411yescharacteristic
212C2×C33AGL(1,ℤ/8)11yescharacteristic
213C2×C33AGL(1,ℤ/8)11yescharacteristic
214C3×C3⋊D3—81no—
215C32×D3—81no—
216C2×C3×D4—31no—
217C2×C3×D4—31no—
218C2×C3×D4—61no—
219C2×C3×D4—91no—
220C2×C3×D4—181no—
221C2×C3×D4—181no—
222C2×#67—61no—
223C2×#67—61no—
224C2×D12—31no—
225C2×D12—31no—
226C2×D12—61no—
227C2×D12—61no—
228C2×D12—61no—
229C2×D12C2×C3×D311yescharacteristic
230C2×C3×C8—31no—
231C2×C3×C8—31no—
232C2×C3×C8—61no—
233C2×C3×C8—91no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4—61no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—62no—
246C3×C4⋊C4—62no—
247C3×C4⋊C4D3211yescharacteristic
248C3×C4⋊C4C2×C3×D311yescharacteristic
249C3×C4⋊C4C2×C3×D311yescharacteristic
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253C4⋊Dic3—21no—
254C4⋊Dic3C2×C3×D311yescharacteristic
255Dic3⋊C4—21no—
256C4×Dic3C2×C3×D311yescharacteristic
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8C2×C3×D311yescharacteristic
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—121no—
266C22×C32—121no—
267C22×C32—121no—
268C22×C32D3×D411yescharacteristic
269C22×C32C2×#6711yescharacteristic
270C22×C32C2×C3×D411yescharacteristic
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—41no—
274C2×C3×D3—42no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C2×C3×D3—81no—
282C2×C3×D3—81no—
283C2×C3×D3—122no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—121no—
290C2×C3×D3—241no—
291C2×C3×D3—241no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—42no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—61no—
305C32×C4—61no—
306C32×C4—81no—
307C32×C4—81no—
308C32×C4—82no—
309C32×C4—121no—
310C32×C4#23511yescharacteristic
311C32×C4#23711yescharacteristic
312C32×C4#23911yescharacteristic
313C32×C4#23911yescharacteristic
314C32×C4C3×#4111yescharacteristic
315C32×C4C3×#4111yescharacteristic
316C3×Dic3—21no—
317C3×Dic3—21no—
318C3×Dic3—21no—
319C3×Dic3—21no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325#127—91no—
326#125—31no—
327#102—31no—
328#102—91no—
329#102—91no—
330#102—91no—
331#97—91no—
332C33#49011yescharacteristic, complemented, Sylow
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—121no—
336C23×C3—181no—
337C22×D3—21no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—121no—
341C3×D4—62no—
342C3×D4—61no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—121no—
347C3×D4—181no—
348C3×D4—181no—
349C3×D4—362no—
350C3×D4—362no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×C3×C4C22×C3×D311yescharacteristic
374#67—122no—
375#67—122no—
376C2×Dic3—21no—
377C2×Dic3C3×#6711yescharacteristic
378C2×Dic3C3×#6711yescharacteristic
379D12—21no—
380D12—21no—
381D12—62no—
382D12—61no—
383D12—61no—
384D12—61no—
385D12—61no—
386D12—121no—
387D12—122no—
388D12—122no—
389C3×C8—61no—
390C3×C8—61no—
391C3×C8—121no—
392C3×C8—181no—
393C3⋊C8—21no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—121no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—41no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—241no—
410C2×C32#88511yescharacteristic
411C2×C32#91111yescharacteristic
412C2×C32#91411yescharacteristic
413C2×C32#94711yescharacteristic
414C2×C32#94911yescharacteristic
415C2×C32#94911yescharacteristic
416C2×C32C3×#12411yescharacteristic
417C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
418C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
419C3⋊D3—81no—
420C3×D3—82no—
421C3×D3—81no—
422C3×D3—81no—
423C3×D3—242no—
424C3×D3—241no—
425C3×D3—241no—
426C2×D4—31no—
427C2×D4—91no—
428C2×D4—181no—
429C2×D4—181no—
430C2×C8—31no—
431C2×C8—91no—
432C4⋊C4—31no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4C3×D3211yescharacteristic
436C42—31no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—122no—
442C22×C3—121no—
443C22×C3—121no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—241no—
449C22×C3—241no—
450C22×C3—361no—
451C22×C3—361no—
452C22×C3—361no—
453C22×C3D3×#6711yescharacteristic
454C22×C3C3×D3×D411yescharacteristic
455C22×C3C2×C3×#6711yescharacteristic
456C2×D3—41no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—122no—
460C2×D3—121no—
461C2×D3—121no—
462C2×D3—121no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—241no—
467C2×D3—241no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—61no—
483C3×C4—61no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4—82no—
487C3×C4—121no—
488C3×C4—121no—
489C3×C4—122no—
490C3×C4—121no—
491C3×C4—121no—
492C3×C4#379611yescharacteristic
493C3×C4#379911yescharacteristic
494C3×C4C3×#23511yescharacteristic
495C3×C4C3×#23711yescharacteristic
496C3×C4C3×#23911yescharacteristic
497C3×C4C3×#23911yescharacteristic
498Dic3—21no—
499Dic3—21no—
500Dic3—41no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#476411yescharacteristic, complemented
506C32#506711yescharacteristic, complemented
507C32C3×#49011yescharacteristic, complemented
508C23—61no—
509C23—181no—
510D4—61no—
511D4—61no—
512D4—181no—
513D4—181no—
514D4—362no—
515D4—362no—
516C2×C4—21no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—61no—
520C2×C4C2×C3×D3211yescharacteristic
521C8—61no—
522C8—181no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—41no—
533C2×C3—41no—
534C2×C3—41no—
535C2×C3—242no—
536C2×C3—241no—
537C2×C3—241no—
538C2×C3—721no—
539C2×C3#6390711yescharacteristic
540C2×C3#6392511yescharacteristic
541C2×C3#6393911yescharacteristic
542C2×C3C3×#88511yescharacteristic
543C2×C3C3×#91111yescharacteristic
544C2×C3C3×#91411yescharacteristic
545C2×C3C3×#94711yescharacteristic
546C2×C3C3×#94911yescharacteristic
547C2×C3C3×#94911yescharacteristic
548D3—81no—
549D3—242no—
550D3—241no—
551D3—241no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22—361no—
556C22—361no—
557C22—361no—
558C22C3×D3×#6711yescharacteristic
559C4—41no—
560C4—61no—
561C4—61no—
562C4—121no—
563C4C3×#379611yescharacteristic
564C4C3×#379911yescharacteristic
565C3—21no—
566C3—21no—
567C3—21no—
568C3—41no—
569C3#1059055011yescharacteristic, direct product factor
570C3C3×#476411yescharacteristic, complemented
571C3C3×#506711yescharacteristic, complemented
572C2—241no—
573C2—721no—
574C2C3×#6390711yescharacteristic
575C2C3×#6392511yescharacteristic
576C2C3×#6393911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#890C2semidirect
3. C3×#63636C2semidirect
4. C32×#809C2semidirect
5. C3×#63564C2semidirect
6. C3×#63562C2semidirect
7. C3×#63554C2semidirect
10. #10590550C3direct (N × H)
25. C3×#890D3semidirect
26. C3×#890C2×C3semidirect
30. #63636C2×C3semidirect
40. C3×#809D3semidirect
41. C3×#809C2×C3semidirect
43. #63564C2×C3semidirect
44. #63562C2×C3semidirect
45. #63554C2×C3semidirect
127. #890C3×D3semidirect
144. #809C3×D3semidirect
332. C33#490semidirect
505. C32#4764semidirect
506. C32#5067semidirect
507. C32C3×#490semidirect
569. C3#10590550direct (N × H)
570. C3C3×#4764semidirect
571. C3C3×#5067semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
C3×#639112
C3×#636362
C32×#8092
C3×#635642
C3×#635622
C3×#635542
C3×#50673
C3×#47643
#105905503