Finite Group Databasebeta

← C3×#10590548 #13157531

Subgroups — C3×#10590548 #13157531

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#888C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic, complemented
5C3×#63564C211yesmaximal, characteristic, complemented
6C3×#63562C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5088—31nomaximal
9C3×#4760—31nomaximal
10#10590548C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6—21no—
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C33×C4⋊C4C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×C4⋊Q8—31no—
23C3×#940—31no—
24C3×#888—21no—
25C3×#888D311yescharacteristic, complemented
26C3×#888C2×C311yescharacteristic, complemented
27C3×#874—31no—
28#63911C2×C311yescharacteristic
29C32×#102—31no—
30#63636C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#821—31no—
33C3×C8×Dic3—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×C4×C3⋊C8—31no—
43#63564C2×C311yescharacteristic, complemented
44#63562C2×C311yescharacteristic, complemented
45Dic3×C3⋊C8C2×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—
57C32×Dic6—42no—
58C3×#626—42no—
59C3×#626—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8—21no—
62C3×#486—91no—
63#5088—31no—
64#4760—31no—
65C2×C32×Q8—61no—
66C2×C32×D4—31no—
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
68C2×C3×#67—61no—
69C2×C3×#67—61no—
70C2×C3×D12—21no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12—61no—
75C2×C3×D12—61no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×D12C22×C311yescharacteristic
78C2×C3×Dic6—22no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6—21no—
81C2×#626—21no—
82C2×#626—21no—
83C2×C32×C8—31no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—21no—
87C32×C4⋊C4—31no—
88C32×C4⋊C4—41no—
89C32×C4⋊C4—61no—
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C4⋊C4C22×C311yescharacteristic
93C32×C42—31no—
94C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C22×C311yescharacteristic
98C3×Dic3⋊C4—22no—
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3C2×D311yescharacteristic
103C3×C4×Dic3C22×C311yescharacteristic
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C2×C32×D3—41no—
117C33×C4—41no—
118C33×C4#4111yescharacteristic
119C33×C4#4111yescharacteristic
120C32×Dic3—41no—
121C32×Dic3D812yes—
122C32×Dic3SD1612yes—
123C3×C4⋊Q8—31no—
124C3×C4⋊Q8—31no—
125C3×C4⋊Q8—61no—
126C3×#127—91no—
127#940—31no—
128#888C3×D311yescharacteristic, complemented
129#874—31no—
130C3×#102—31no—
131C3×#102—31no—
132C3×#102—61no—
133C3×#102—91no—
134C3×#102—91no—
135C3×#102—91no—
136C3×C4×C8—91no—
137#832—31no—
138#821—31no—
139C8×Dic3—31no—
140#809—31no—
141#809—31no—
142#809—31no—
143#809—31no—
144#809—61no—
145#809C3×D311yescharacteristic, complemented
146C4×C3⋊C8—31no—
147C23×C32—61no—
148C22×C3⋊D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—21no—
151C22×C3×D3—41no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—121no—
155C32×Q8—122no—
156C32×D4—61no—
157C32×D4—61no—
158C2×C32×C4—21no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—31no—
166C2×C32×C4—31no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—42no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177C3×#67—122no—
178C3×#67—122no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C2×C3×Dic3C3×D411yescharacteristic
187C2×C3×Dic3C3×D411yescharacteristic
188C3×D12—22no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—41no—
194C3×D12—62no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—121no—
200C3×D12—122no—
201C3×D12—122no—
202C3×Dic6—44no—
203C3×Dic6—42no—
204C3×Dic6—42no—
205#626—42no—
206#626—42no—
207C32×C8—61no—
208C3⋊(C3⋊C8)—21no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—41no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—121no—
215#486—91noSylow
216C2×C33#12411yescharacteristic
217C2×C33C2×D811yescharacteristic
218C2×C33C2×SD1611yescharacteristic
219C3×C3⋊D3—81no—
220C32×D3—81no—
221C2×C3×Q8—61no—
222C2×C3×Q8—61no—
223C2×C3×Q8—62no—
224C2×C3×D4—31no—
225C2×C3×D4—31no—
226C2×C3×D4—61no—
227C2×C3×D4—91no—
228C2×C3×D4—181no—
229C2×C3×D4—181no—
230C2×#67—61no—
231C2×#67—61no—
232C2×D12—31no—
233C2×D12—31no—
234C2×D12—61no—
235C2×D12—61no—
236C2×D12—61no—
237C2×D12C2×C3×D311yescharacteristic
238C2×Dic6—21no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—91no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4—61no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4D3211yescharacteristic
254C3×C4⋊C4C2×C3×D311yescharacteristic
255C3×C4⋊C4C2×C3×D311yescharacteristic
256C3×C42—31no—
257C3×C42—31no—
258C3×C42—61no—
259C4⋊Dic3C2×C3×D311yescharacteristic
260Dic3⋊C4—21no—
261C4×Dic3C2×C3×D311yescharacteristic
262C2×C3⋊C8—31no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—61no—
265C2×C3⋊C8C2×C3×D311yescharacteristic
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—41no—
270C22×C32—121no—
271C22×C32—121no—
272C22×C32—121no—
273C22×C32D3×D411yescharacteristic
274C22×C32C2×#6711yescharacteristic
275C22×C32C2×C3×D411yescharacteristic
276C2×C3⋊D3—41no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—42no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C2×C3×D3—241no—
296C2×C3×D3—241no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—32no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—42no—
309C32×C4—41no—
310C32×C4—41no—
311C32×C4—81no—
312C32×C4—81no—
313C32×C4—82no—
314C32×C4—121no—
315C32×C4#23511yescharacteristic
316C32×C4#23711yescharacteristic
317C32×C4#23911yescharacteristic
318C32×C4#23911yescharacteristic
319C32×C4C3×#4111yescharacteristic
320C32×C4C3×#4111yescharacteristic
321C3×Dic3—22no—
322C3×Dic3—22no—
323C3×Dic3—42no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3#21312yes—
327C3×Dic3#21512yes—
328C3×Dic3C3×D812yes—
329C3×Dic3C3×SD1612yes—
330C4⋊Q8—31no—
331#127—91no—
332#102—31no—
333#102—91no—
334#102—91no—
335#102—91no—
336C4×C8—91no—
337C33#48611yescharacteristic, complemented, Sylow
338C23×C3—61no—
339C23×C3—61no—
340C23×C3—121no—
341C23×C3—181no—
342C22×D3—21no—
343C22×D3—61no—
344C22×D3—61no—
345C22×D3—121no—
346C3×Q8—122no—
347C3×Q8—122no—
348C3×Q8—124no—
349C3×D4—62no—
350C3×D4—61no—
351C3×D4—61no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—121no—
355C3×D4—181no—
356C3×D4—181no—
357C3×D4—362no—
358C3×D4—362no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—41no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—62no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C22×C3×D311yescharacteristic
381C2×C3×C4C22×C3×D311yescharacteristic
382#67—122no—
383#67—122no—
384C2×Dic3—21no—
385C2×Dic3C3×#6711yescharacteristic
386C2×Dic3C3×#6711yescharacteristic
387D12—21no—
388D12—21no—
389D12—62no—
390D12—61no—
391D12—61no—
392D12—61no—
393D12—61no—
394D12—121no—
395D12—122no—
396D12—122no—
397Dic6—42no—
398C3×C8—61no—
399C3×C8—61no—
400C3×C8—121no—
401C3×C8—181no—
402C3⋊C8—21no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—121no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—241no—
419C2×C32#88511yescharacteristic
420C2×C32D3×D811yescharacteristic
421C2×C32D3×SD1611yescharacteristic
422C2×C32C2×#21311yescharacteristic
423C2×C32C2×#21511yescharacteristic
424C2×C32#94711yescharacteristic
425C2×C32C3×#12411yescharacteristic
426C2×C32C2×C3×D811yescharacteristic
427C2×C32C2×C3×SD1611yescharacteristic
428C3⋊D3—81no—
429C3×D3—82no—
430C3×D3—81no—
431C3×D3—81no—
432C3×D3—242no—
433C3×D3—241no—
434C3×D3—241no—
435C2×Q8—61no—
436C2×D4—31no—
437C2×D4—91no—
438C2×D4—181no—
439C2×D4—181no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—31no—
443C4⋊C4—61no—
444C4⋊C4C3×D3211yescharacteristic
445C42—31no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—122no—
451C22×C3—121no—
452C22×C3—121no—
453C22×C3—121no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—121no—
457C22×C3—241no—
458C22×C3—241no—
459C22×C3—361no—
460C22×C3—361no—
461C22×C3—361no—
462C22×C3D3×#6711yescharacteristic
463C22×C3C3×D3×D411yescharacteristic
464C22×C3C2×C3×#6711yescharacteristic
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—122no—
469C2×D3—121no—
470C2×D3—121no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—241no—
476C2×D3—241no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—32no—
486C3×C4—32no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—62no—
494C3×C4—62no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—82no—
498C3×C4—122no—
499C3×C4—121no—
500C3×C4—121no—
501C3×C4#379611yescharacteristic
502C3×C4#379911yescharacteristic
503C3×C4C3×#23511yescharacteristic
504C3×C4C3×#23711yescharacteristic
505C3×C4C3×#23911yescharacteristic
506C3×C4C3×#23911yescharacteristic
507Dic3—41no—
508Dic3C3×#21312yes—
509Dic3C3×#21512yes—
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32#476011yescharacteristic, complemented
515C32#508811yescharacteristic, complemented
516C32C3×#48611yescharacteristic, complemented
517C23—61no—
518C23—181no—
519Q8—122no—
520D4—61no—
521D4—61no—
522D4—181no—
523D4—181no—
524D4—362no—
525D4—362no—
526C2×C4—21no—
527C2×C4—31no—
528C2×C4—31no—
529C2×C4—61no—
530C2×C4C2×C3×D3211yescharacteristic
531C8—61no—
532C8—181no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—242no—
546C2×C3—241no—
547C2×C3—241no—
548C2×C3—721no—
549C2×C3#6390711yescharacteristic
550C2×C3D3×#21311yescharacteristic
551C2×C3D3×#21511yescharacteristic
552C2×C3C3×#88511yescharacteristic
553C2×C3C3×D3×D811yescharacteristic
554C2×C3C3×D3×SD1611yescharacteristic
555C2×C3C2×C3×#21311yescharacteristic
556C2×C3C2×C3×#21511yescharacteristic
557C2×C3C3×#94711yescharacteristic
558D3—81no—
559D3—242no—
560D3—241no—
561D3—241no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—361no—
566C22—361no—
567C22—361no—
568C22C3×D3×#6711yescharacteristic
569C4—32no—
570C4—32no—
571C4—41no—
572C4—121no—
573C4C3×#379611yescharacteristic
574C4C3×#379911yescharacteristic
575C3—21no—
576C3—21no—
577C3—21no—
578C3—41no—
579C3#1059054811yescharacteristic, direct product factor
580C3C3×#476011yescharacteristic, complemented
581C3C3×#508811yescharacteristic, complemented
582C2—241no—
583C2—721no—
584C2C3×#6390711yescharacteristic
585C2C3×D3×#21311yescharacteristic
586C2C3×D3×#21511yescharacteristic
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×#486
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×#888C2semidirect
3. C3×#63636C2semidirect
4. C32×#809C2semidirect
5. C3×#63564C2semidirect
6. C3×#63562C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10590548C3direct (N × H)
25. C3×#888D3semidirect
26. C3×#888C2×C3semidirect
30. #63636C2×C3semidirect
40. C3×#809D3semidirect
41. C3×#809C2×C3semidirect
43. #63564C2×C3semidirect
44. #63562C2×C3semidirect
45. Dic3×C3⋊C8C2×C3semidirect
128. #888C3×D3semidirect
145. #809C3×D3semidirect
337. C33#486semidirect
514. C32#4760semidirect
515. C32#5088semidirect
516. C32C3×#486semidirect
579. C3#10590548direct (N × H)
580. C3C3×#4760semidirect
581. C3C3×#5088semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
C3×#639112
C3×#636362
C32×#8092
C3×#635642
C3×#635622
C3×Dic3×C3⋊C82
C3×#50883
C3×#47603
#105905483