Finite Group Databasebeta

← C3×#10590829 #13157794

Subgroups — C3×#10590829 #13157794

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#204C211yesmaximal, characteristic, complemented
2C2×C32×D3×C8C211yesmaximal, characteristic, complemented
3C3×#63908C211yesmaximal, characteristic
4C3×#63856C211yesmaximal, characteristic, complemented
5C3×#63644C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8C3×#5171—31nomaximal
9C3×#5114—31nomaximal
10#10590829C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#625—21no—
15C2×C33×C8C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C32×#204—22no—
20C32×D3×C8—22no—
21C3×#3724—21no—
22C3×(C2×D3)⋊Dic3—21no—
23C3×(C2×D3)⋊Dic3—21no—
24C2×C32×SD16—31no—
25C22×C32×C8—31no—
26C3×#946—31no—
27C3×#938—31no—
28C3×#930—31no—
29C3×#924—31no—
30C2×C3×#204—21no—
31C2×C3×#204D311yescharacteristic, complemented
32C2×C3×#204C2×C311yescharacteristic, complemented
33C2×C3×D3×C8—21no—
34C2×C3×D3×C8D311yescharacteristic, complemented
35C2×C3×D3×C8C2×C311yescharacteristic, complemented
36#63908C2×C311yescharacteristic
37#63856C2×C311yescharacteristic, complemented
38#63644C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#832—31no—
41C3×#821—31no—
42C3×#817—31no—
43C3×#817—31no—
44C3×#817—61no—
45C3×#816—31no—
46#63566C2×C311yescharacteristic, complemented
47#63563C2×C311yescharacteristic, complemented
48C22×C32×D3—21no—
49C22×C32×D3D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×#625—42no—
62C33×C8D412yescomplemented
63C32×C3⋊C8—21no—
64C3×#465—91no—
65#5171—31no—
66#5114—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C22×C32×C4—31no—
70C2×C3×#67—61no—
71C2×C3×#67—61no—
72C2×C3×D12—21no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×D12C22×C311yescharacteristic
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×C3×C4×D3C22×C311yescharacteristic
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C2×#625—21no—
82C32×SD16—62no—
83C2×C32×C8—21no—
84C2×C32×C8—21no—
85C2×C32×C8—21no—
86C2×C32×C8—31no—
87C2×C32×C8—41no—
88C2×C32×C8—62no—
89C2×C32×C8C2×D311yescharacteristic
90C2×C32×C8C2×D311yescharacteristic
91C2×C32×C8C22×C311yescharacteristic
92C4⋊(C3⋊Dic3)C22×C311yescharacteristic
93C4⋊(C3⋊Dic3)C22×C311yescharacteristic
94C3×#217—61no—
95C3×#217—61no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—61no—
103C3×C4⋊Dic3—61no—
104C3×Dic3⋊C4—61no—
105C3×Dic3⋊C4—61no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C3×#204—24no—
110C3×#204—22no—
111C3×#204—22no—
112C3×D3×C8—24no—
113C3×D3×C8—22no—
114C3×D3×C8—22no—
115#3724—21no—
116(C2×D3)⋊Dic3—21no—
117(C2×D3)⋊Dic3—21no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—21no—
120C2×C32×D3—21no—
121C2×C32×D3—41no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3SD1612yescomplemented
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—41no—
128C3×C3⋊Dic3—41no—
129C32×Dic3—21no—
130C32×Dic3—41no—
131C2×C3×SD16—31no—
132C2×C3×SD16—31no—
133C2×C3×SD16—61no—
134C22×C3×C8—31no—
135C22×C3×C8—31no—
136C22×C3×C8—61no—
137C3×C22⋊Q8—91no—
138C3×#121—91no—
139#946—31no—
140#938—31no—
141#930—31no—
142#924—31no—
143C2×#204C3×D311yescharacteristic, complemented
144C2×D3×C8C3×D311yescharacteristic, complemented
145C3×#106—91no—
146C3×#103—91no—
147C3×#102—91no—
148#835—31no—
149#832—31no—
150#821—31no—
151#817—31no—
152#817—31no—
153#817—61no—
154#816—31no—
155C23×C32—31no—
156C23×C32—61no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C22×C3×D3D1211yescharacteristic
162C22×C3×D3C3×D411yescharacteristic
163C32×Q8—61no—
164C32×Q8—61no—
165C32×D4—61no—
166C32×D4—61no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—31no—
171C2×C32×C4—41no—
172C2×C32×C4—61no—
173C2×C32×C4—61no—
174C2×C32×C4—61no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C23×C311yescharacteristic
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C3×#67—122no—
181C3×#67—122no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—41no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3—121no—
192C2×C3×Dic3D1211yescharacteristic
193C2×C3×Dic3C3×D411yescharacteristic
194C3×D12—22no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—41no—
200C3×C4×D3—22no—
201C3×C4×D3—22no—
202C3×C4×D3—21no—
203C3×C4×D3—21no—
204C3×C4×D3—21no—
205C3×C4×D3—21no—
206C3×Dic6—22no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—41no—
212#625—42no—
213C32×C8—22no—
214C32×C8—22no—
215C32×C8—22no—
216C32×C8—42no—
217C32×C8—61no—
218C32×C8#6712yescomplemented
219C32×C8#6712yescomplemented
220C32×C8C3×D412yescomplemented
221C3×C3⋊C8—22no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224#465—91noSylow
225C2×C33#12111yescharacteristic
226C2×C33C2×SD1611yescharacteristic
227C2×C33#13511yescharacteristic
228C32×D3—22no—
229C32×D3—81no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—61no—
233C2×C3×D4—31no—
234C2×C3×D4—31no—
235C2×C3×D4—61no—
236C2×C3×D4—181no—
237C22×C3×C4—31no—
238C22×C3×C4—31no—
239C22×C3×C4—61no—
240C2×#67—61no—
241C2×#67—61no—
242C2×D12C2×C3×D311yescharacteristic
243C2×C4×D3C2×C3×D311yescharacteristic
244C2×Dic6C2×C3×D311yescharacteristic
245C3×SD16—64no—
246C3×SD16—62no—
247C3×SD16—62no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—41no—
254C2×C3×C8—61no—
255C2×C3×C8—64no—
256C2×C3×C8—62no—
257C2×C3×C8—62no—
258C2×C3×C8D3211yescharacteristic
259C2×C3×C8C2×C3×D311yescharacteristic
260C2×C3×C8C2×C3×D311yescharacteristic
261C3×C4⋊C4—91no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—181no—
264C3×C22⋊C4—181no—
265C3×C22⋊C4—181no—
266#217—61no—
267#217—61no—
268(C2×D3)⋊C4—61no—
269(C2×D3)⋊C4—61no—
270C4⋊Dic3—31no—
271C4⋊Dic3—31no—
272C4⋊Dic3—31no—
273C4⋊Dic3—31no—
274C4⋊Dic3—61no—
275C4⋊Dic3—61no—
276Dic3⋊C4—61no—
277Dic3⋊C4—61no—
278C2×C3⋊C8C2×C3×D311yescharacteristic
279#204—22no—
280D3×C8—22no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—32no—
285C22×C32—41no—
286C22×C32—61no—
287C22×C32—61no—
288C22×C32—121no—
289C22×C32—121no—
290C22×C32—121no—
291C22×C32C2×D1211yescharacteristic
292C22×C32D3×D411yescharacteristic
293C22×C32C2×C3×D411yescharacteristic
294C2×C3×D3—22no—
295C2×C3×D3—22no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—21no—
300C2×C3×D3—22no—
301C2×C3×D3—42no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—81no—
309C2×C3×D3—81no—
310C2×C3×D3#20412yescomplemented
311C2×C3×D3C3×SD1612yescomplemented
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—61no—
321C32×C4—121no—
322C32×C4#23511yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4C2×#6711yescharacteristic
325C32×C4C2×#6711yescharacteristic
326C32×C4C2×C3×D411yescharacteristic
327C32×C4C3×#4111yescharacteristic
328C3⋊Dic3—41no—
329C3⋊Dic3—41no—
330C3×Dic3—22no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—42no—
334C3×Dic3—41no—
335C3×Dic3—41no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C3×Dic3—241no—
341C3×Dic3—241no—
342C2×SD16—31no—
343C22×C8—31no—
344C22⋊Q8—91no—
345#121—91no—
346#106—91no—
347#103—91no—
348#102—91no—
349C33#46511yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—61no—
353C23×C3—61no—
354C23×C3—61no—
355C23×C3—121no—
356C22×D3—21no—
357C22×D3C3×D1211yescharacteristic
358C3×Q8—62no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—121no—
364C3×D4—62no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—121no—
370C3×D4—362no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—41no—
377C2×C3×C4—61no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—121no—
387C2×C3×C4—181no—
388C2×C3×C4—181no—
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C22×C3×D311yescharacteristic
391C2×C3×C4C22×C3×D311yescharacteristic
392#67—122no—
393#67—122no—
394C2×Dic3—21no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—121no—
400C2×Dic3—121no—
401C2×Dic3C3×D1211yescharacteristic
402D12—21no—
403D12—21no—
404C4×D3—21no—
405C4×D3—21no—
406Dic6—21no—
407Dic6—21no—
408C3×C8—22no—
409C3×C8—22no—
410C3×C8—22no—
411C3×C8—42no—
412C3×C8—62no—
413C3×C8—61no—
414C3×C8—61no—
415C3×C8#62512yescomplemented
416C3×C8C3×#6712yescomplemented
417C3×C8C3×#6712yescomplemented
418C3⋊C8—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—62no—
432C2×C32—241no—
433C2×C32C2×#20411yescharacteristic
434C2×C32#90411yescharacteristic
435C2×C32D3×SD1611yescharacteristic
436C2×C32#91611yescharacteristic
437C2×C32#93011yescharacteristic
438C2×C32#93811yescharacteristic
439C2×C32C3×#12111yescharacteristic
440C2×C32C2×C3×SD1611yescharacteristic
441C2×C32C3×#13511yescharacteristic
442C3×D3—24no—
443C3×D3—22no—
444C3×D3—22no—
445C3×D3—82no—
446C3×D3—81no—
447C3×D3—81no—
448C2×Q8—31no—
449C2×D4—31no—
450C2×D4—181no—
451C22×C4—31no—
452SD16—62no—
453C2×C8—31no—
454C2×C8—62no—
455C2×C8C3×D3211yescharacteristic
456C4⋊C4—91no—
457C4⋊C4—91no—
458C4⋊C4—181no—
459C22⋊C4—181no—
460C22⋊C4—181no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—32no—
465C22×C3—32no—
466C22×C3—41no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—122no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—241no—
482C22×C3—241no—
483C22×C3D3×D1211yescharacteristic
484C22×C3C2×C3×D1211yescharacteristic
485C22×C3C3×D3×D411yescharacteristic
486C2×D3—21no—
487C2×D3—21no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3C3×#20412yescomplemented
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—62no—
501C3×C4—61no—
502C3×C4—61no—
503C3×C4—122no—
504C3×C4—121no—
505C3×C4—121no—
506C3×C4—361no—
507C3×C4—361no—
508C3×C4#379511yescharacteristic
509C3×C4C2×#62511yescharacteristic
510C3×C4C3×#23511yescharacteristic
511C3×C4C3×#23711yescharacteristic
512C3×C4C2×C3×#6711yescharacteristic
513C3×C4C2×C3×#6711yescharacteristic
514Dic3—21no—
515Dic3—41no—
516Dic3—121no—
517Dic3—121no—
518Dic3—121no—
519Dic3—121no—
520Dic3—241no—
521Dic3—241no—
522C32—21no—
523C32—21no—
524C32—21no—
525C32—41no—
526C32#511411yescharacteristic, complemented
527C32#517111yescharacteristic, complemented
528C32C3×#46511yescharacteristic, complemented
529C23—31no—
530C23—61no—
531Q8—61no—
532Q8—61no—
533D4—61no—
534D4—61no—
535D4—362no—
536C2×C4—31no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—181no—
541C2×C4—181no—
542C2×C4C2×C3×D3211yescharacteristic
543C8—61no—
544C8C3×#62512yescomplemented
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—41no—
557C2×C3—64no—
558C2×C3—62no—
559C2×C3—62no—
560C2×C3—242no—
561C2×C3—241no—
562C2×C3—241no—
563C2×C3D3×#20411yescharacteristic
564C2×C3#6380611yescharacteristic
565C2×C3#6390811yescharacteristic
566C2×C3C2×C3×#20411yescharacteristic
567C2×C3C3×#90411yescharacteristic
568C2×C3C3×D3×SD1611yescharacteristic
569C2×C3C3×#91611yescharacteristic
570C2×C3C3×#93011yescharacteristic
571C2×C3C3×#93811yescharacteristic
572D3—22no—
573D3—81no—
574C22—32no—
575C22—61no—
576C22—61no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22C3×D3×D1211yescharacteristic
581C4—61no—
582C4—121no—
583C4—361no—
584C4—361no—
585C4C3×#379511yescharacteristic
586C4C2×C3×#62511yescharacteristic
587C3—21no—
588C3—21no—
589C3—21no—
590C3—41no—
591C3#1059082911yescharacteristic, direct product factor
592C3C3×#511411yescharacteristic, complemented
593C3C3×#517111yescharacteristic, complemented
594C2—62no—
595C2—241no—
596C2C3×D3×#20411yescharacteristic
597C2C3×#6380611yescharacteristic
598C2C3×#6390811yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC3×#63908C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C32×#204C2semidirect
2. C2×C32×D3×C8C2semidirect
4. C3×#63856C2semidirect
5. C3×#63644C2semidirect
6. C3×#63566C2semidirect
7. C3×#63563C2semidirect
10. #10590829C3direct (N × H)
31. C2×C3×#204D3semidirect
32. C2×C3×#204C2×C3semidirect
34. C2×C3×D3×C8D3semidirect
35. C2×C3×D3×C8C2×C3semidirect
37. #63856C2×C3semidirect
38. #63644C2×C3semidirect
46. #63566C2×C3semidirect
47. #63563C2×C3semidirect
62. C33×C8D4semidirect
124. C2×C32×D3SD16semidirect
143. C2×#204C3×D3semidirect
144. C2×D3×C8C3×D3semidirect
218. C32×C8#67semidirect
219. C32×C8#67semidirect
220. C32×C8C3×D4semidirect
310. C2×C3×D3#204semidirect
311. C2×C3×D3C3×SD16semidirect
349. C33#465semidirect
415. C3×C8#625semidirect
416. C3×C8C3×#67semidirect
417. C3×C8C3×#67semidirect
491. C2×D3C3×#204semidirect
526. C32#5114semidirect
527. C32#5171semidirect
528. C32C3×#465semidirect
544. C8C3×#625semidirect
591. C3#10590829direct (N × H)
592. C3C3×#5114semidirect
593. C3C3×#5171semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2042
C2×C32×D3×C82
C3×#639082
C3×#638562
C3×#636442
C3×#635662
C3×#635632
C3×#51713
C3×#51143
#105908293