Finite Group Databasebeta

← C3×#10590512 #13157482

Subgroups — C3×#10590512 #13157482

Order 1728.

612 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64101C211yesmaximal, characteristic, complemented
2C3×#63916C211yesmaximal, characteristic
3C32×#809C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic, complemented
8C3×#4885—31nomaximal
9C3×#4779—31nomaximal
10#10590512C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#626—21no—
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—21no—
22#64101C2×C311yescharacteristic, complemented
23C3×#938—31no—
24C3×#895—31no—
25C3×#892—31no—
26C3×#892—31no—
27C3×#892—61no—
28#63916C2×C311yescharacteristic
29C32×#107—31no—
30C32×#102—31no—
31C3×#832—31no—
32C3×#821—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×(C2×D3)⋊C8—31no—
35C3×#817—31no—
36C3×#809—21no—
37C3×#809D311yescharacteristic, complemented
38C3×#809C2×C311yescharacteristic, complemented
39C3×#808—31no—
40C3×#807—21no—
41C3×#807D311yescharacteristic, complemented
42C3×#807C2×C311yescharacteristic, complemented
43#63573C2×C311yescharacteristic, complemented
44#63563C2×C311yescharacteristic, complemented
45#63557C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57C3×#626—42no—
58C32×C3⋊C8—21no—
59C32×C3⋊C8—21no—
60C3×#482—91no—
61#4885—31no—
62#4779—31no—
63C2×C32×D4—31no—
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×D12—21no—
67C2×C3×D12—61no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C22×C311yescharacteristic
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×#626—21no—
74C2×C32×C8—31no—
75C2×C32×C8—31no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—41no—
81C32×C4⋊C4C2×D311yescharacteristic
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C22×C311yescharacteristic
84#3752—21no—
85C4⋊(C3⋊Dic3)C22×C311yescharacteristic
86C4×C3⋊Dic3—21no—
87C3×#217—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×(C2×D3)⋊C4—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×(C2×D3)⋊C4—62no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C22×C311yescharacteristic
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—62no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107(C2×D3)⋊Dic3—21no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3—21no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C33×C4—41no—
116C33×C4#4111yescharacteristic
117C33×C4#4111yescharacteristic
118C3×C3⋊Dic3—21no—
119C3×C3⋊Dic3—41no—
120C32×Dic3—41no—
121C3×#121—91no—
122C3×#117—91no—
123#938—31no—
124#895—31no—
125#892—31no—
126#892—31no—
127#892—61no—
128C3×#107—31no—
129C3×#107—31no—
130C3×#107—61no—
131C3×#106—91no—
132C3×#102—31no—
133C3×#102—31no—
134C3×#102—61no—
135C3×#102—91no—
136C3×C22⋊C8—91no—
137#832—31no—
138#821—31no—
139(C2×D3)⋊C8—31no—
140(C2×D3)⋊C8—31no—
141#817—31no—
142#809C3×D311yescharacteristic, complemented
143#808—31no—
144#807C3×D311yescharacteristic, complemented
145C23×C32—61no—
146C22×C3⋊D3C3×D411yescharacteristic
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—31no—
150C22×C3×D3—31no—
151C22×C3×D3—41no—
152C22×C3×D3—61no—
153C32×D4—61no—
154C32×D4—61no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4—42no—
166C2×C32×C4—61no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C23×C311yescharacteristic
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3C3×D411yescharacteristic
172C4×C3⋊D3—21no—
173C4×C3⋊D3—21no—
174C3×#67—122no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C3×D12—22no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—41no—
190C3×D12—122no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197#626—42no—
198C32×C8—61no—
199C32×C8—61no—
200C3×C3⋊C8—22no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—41no—
206#482—91noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33AGL(1,ℤ/8)11yescharacteristic
210C3×C3⋊D3—41no—
211C32×D3—81no—
212C2×C3×D4—31no—
213C2×C3×D4—31no—
214C2×C3×D4—61no—
215C2×C3×D4—181no—
216C22×C3×C4—91no—
217C2×#67—61no—
218C2×D12—61no—
219C2×D12C2×C3×D311yescharacteristic
220C2×C4×D3—31no—
221C2×C4×D3—31no—
222C2×C4×D3—61no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4D3211yescharacteristic
238C3×C4⋊C4C2×C3×D311yescharacteristic
239C3×C4⋊C4C2×C3×D311yescharacteristic
240C3×C22⋊C4—181no—
241C3×C22⋊C4—181no—
242C3×C42—181no—
243#217—61no—
244(C2×D3)⋊C4—61no—
245(C2×D3)⋊C4—61no—
246(C2×D3)⋊C4—61no—
247(C2×D3)⋊C4—62no—
248C4⋊Dic3—31no—
249C4⋊Dic3—31no—
250C4⋊Dic3—61no—
251C4⋊Dic3C2×C3×D311yescharacteristic
252C4×Dic3—61no—
253C4×Dic3—61no—
254C4×Dic3—62no—
255C2×C3⋊C8C2×C3×D311yescharacteristic
256C2×C3⋊C8C2×C3×D311yescharacteristic
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32—121no—
262C22×C32—121no—
263C22×C32—121no—
264C22×C32D3×D411yescharacteristic
265C22×C32D3×D411yescharacteristic
266C22×C32C2×C3×D411yescharacteristic
267C2×C3⋊D3—21no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3×D3—42no—
271C2×C3×D3—41no—
272C2×C3×D3—41no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—62no—
278C2×C3×D3—61no—
279C2×C3×D3—61no—
280C2×C3×D3—61no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—81no—
285C2×C3×D3—81no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—41no—
297C32×C4—42no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—81no—
301C32×C4—81no—
302C32×C4—82no—
303C32×C4—121no—
304C32×C4#23511yescharacteristic
305C32×C4#23511yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4#23911yescharacteristic
308C32×C4C3×#4111yescharacteristic
309C32×C4C3×#4111yescharacteristic
310C3⋊Dic3—21no—
311C3⋊Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—62no—
315C3×Dic3—61no—
316C3×Dic3—61no—
317C3×Dic3—81no—
318C3×Dic3—122no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321#121—91no—
322#117—91no—
323#107—31no—
324#106—91no—
325#102—31no—
326#102—91no—
327C22⋊C8—91no—
328C33#48211yescharacteristic, complemented, Sylow
329C23×C3—61no—
330C23×C3—61no—
331C23×C3—91no—
332C23×C3—121no—
333C22×D3—21no—
334C22×D3—31no—
335C22×D3—31no—
336C22×D3—61no—
337C3×D4—62no—
338C3×D4—61no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—121no—
343C3×D4—362no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—91no—
358C2×C3×C4—121no—
359C2×C3×C4—181no—
360C2×C3×C4—181no—
361C2×C3×C4—181no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D311yescharacteristic
365#67—122no—
366C2×Dic3—21no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—61no—
370C2×Dic3—62no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373D12—21no—
374D12—21no—
375D12—122no—
376C4×D3—62no—
377C4×D3—62no—
378C4×D3—61no—
379C4×D3—61no—
380C4×D3—61no—
381C4×D3—61no—
382C3×C8—62no—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—61no—
386C3×C8—61no—
387C3×C8—121no—
388C3⋊C8—21no—
389C3⋊C8—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—241no—
403C2×C32#88311yescharacteristic
404C2×C32#89411yescharacteristic
405C2×C32#91111yescharacteristic
406C2×C32#91111yescharacteristic
407C2×C32#91611yescharacteristic
408C2×C32#91911yescharacteristic
409C2×C32C3×#12311yescharacteristic
410C2×C32C3×#13511yescharacteristic
411C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
412C3⋊D3—41no—
413C3×D3—82no—
414C3×D3—81no—
415C3×D3—81no—
416C3×D3—122no—
417C3×D3—121no—
418C3×D3—121no—
419C2×D4—31no—
420C2×D4—181no—
421C22×C4—91no—
422C2×C8—31no—
423C2×C8—31no—
424C4⋊C4—31no—
425C4⋊C4—91no—
426C4⋊C4C3×D3211yescharacteristic
427C22⋊C4—181no—
428C22⋊C4—181no—
429C42—181no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—122no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3—121no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—181no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—241no—
445C22×C3—241no—
446C22×C3#381111yescharacteristic
447C22×C3C3×D3×D411yescharacteristic
448C22×C3C3×D3×D411yescharacteristic
449C2×D3—41no—
450C2×D3—41no—
451C2×D3—41no—
452C2×D3—62no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—121no—
460C2×D3—121no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—82no—
476C3×C4—121no—
477C3×C4—121no—
478C3×C4—181no—
479C3×C4—241no—
480C3×C4—361no—
481C3×C4#379611yescharacteristic
482C3×C4#379811yescharacteristic
483C3×C4C3×#23511yescharacteristic
484C3×C4C3×#23511yescharacteristic
485C3×C4C3×#23711yescharacteristic
486C3×C4C3×#23911yescharacteristic
487Dic3—41no—
488Dic3—62no—
489Dic3—61no—
490Dic3—61no—
491Dic3—122no—
492Dic3—121no—
493Dic3—121no—
494C32—21no—
495C32—21no—
496C32—21no—
497C32—41no—
498C32#477911yescharacteristic, complemented
499C32#488511yescharacteristic, complemented
500C32C3×#48211yescharacteristic, complemented
501C23—61no—
502C23—91no—
503D4—61no—
504D4—61no—
505D4—362no—
506C2×C4—21no—
507C2×C4—61no—
508C2×C4—91no—
509C2×C4—181no—
510C2×C4—181no—
511C2×C4—181no—
512C2×C4C2×C3×D3211yescharacteristic
513C8—61no—
514C8—61no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—242no—
528C2×C3—241no—
529C2×C3—241no—
530C2×C3—361no—
531C2×C3#6385311yescharacteristic
532C2×C3#6392711yescharacteristic
533C2×C3#6395011yescharacteristic
534C2×C3C3×#88311yescharacteristic
535C2×C3C3×#89411yescharacteristic
536C2×C3C3×#91111yescharacteristic
537C2×C3C3×#91111yescharacteristic
538C2×C3C3×#91611yescharacteristic
539C2×C3C3×#91911yescharacteristic
540D3—81no—
541D3—122no—
542D3—121no—
543D3—121no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22—181no—
548C22—181no—
549C22—181no—
550C22C3×#381111yescharacteristic
551C4—41no—
552C4—121no—
553C4—181no—
554C4—361no—
555C4C3×#379611yescharacteristic
556C4C3×#379811yescharacteristic
557C3—21no—
558C3—21no—
559C3—21no—
560C3—41no—
561C3#1059051211yescharacteristic, direct product factor
562C3C3×#477911yescharacteristic, complemented
563C3C3×#488511yescharacteristic, complemented
564C2—241no—
565C2—361no—
566C2C3×#6385311yescharacteristic
567C2C3×#6392711yescharacteristic
568C2C3×#6395011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63916C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#482
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. C3×#64101C2semidirect
3. C32×#809C2semidirect
4. C32×#807C2semidirect
5. C3×#63573C2semidirect
6. C3×#63563C2semidirect
7. C3×#63557C2semidirect
10. #10590512C3direct (N × H)
22. #64101C2×C3semidirect
37. C3×#809D3semidirect
38. C3×#809C2×C3semidirect
41. C3×#807D3semidirect
42. C3×#807C2×C3semidirect
43. #63573C2×C3semidirect
44. #63563C2×C3semidirect
45. #63557C2×C3semidirect
142. #809C3×D3semidirect
144. #807C3×D3semidirect
328. C33#482semidirect
498. C32#4779semidirect
499. C32#4885semidirect
500. C32C3×#482semidirect
561. C3#10590512direct (N × H)
562. C3C3×#4779semidirect
563. C3C3×#4885semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
C3×#639162
C32×#8092
C32×#8072
C3×#635732
C3×#635632
C3×#635572
C3×#48853
C3×#47793
#105905123