Finite Group Databasebeta

← C3×#10591102 #13158130

Subgroups — C3×#10591102 #13158130

Order 1728.

638 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×#926C211yesmaximal, characteristic
2C32×#924C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic
4C3×#63576C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#5045—31nomaximal
9C3×#4996—31nomaximal
10#10591102C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×#217—21no—
16C32×#217—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4—21no—
21C32×C4×Dic3—21no—
22C32×C22⋊Q8—31no—
23C32×#117—31no—
24C3×#926—21no—
25C3×#926D311yescharacteristic
26C3×#926C2×C311yescharacteristic
27C3×#924—21no—
28C3×#924D311yescharacteristic
29C3×#924C2×C311yescharacteristic
30#63657C2×C311yescharacteristic
31C3×#835—31no—
32C3×#835—31no—
33C3×#830—31no—
34C3×#830—31no—
35C3×#830—61no—
36C3×#810—31no—
37C3×#810—31no—
38C3×#808—31no—
39C3×#808—31no—
40C3×#807—31no—
41C3×#807—31no—
42#63576C2×C311yescharacteristic, complemented
43#63574C2×C311yescharacteristic, complemented
44#63565C2×C311yescharacteristic, complemented
45#63565C2×C311yescharacteristic, complemented
46C23×C33D411yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#485—91no—
60#5045—31no—
61#4996—31no—
62C2×C32×Q8—31no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—41no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C22×C311yescharacteristic
73C32×C4⋊C4—31no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—61no—
77C32×C22⋊C4—61no—
78C32×C22⋊C4—61no—
79C32×C42—61no—
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×#217—22no—
83C3×#217—22no—
84C3×#217—21no—
85C3×#217—21no—
86C3×#217—21no—
87C3×#217—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8—61no—
109C22×C33—21no—
110C22×C33—21no—
111C22×C33—21no—
112C22×C33C2×D411yescharacteristic
113C33×C4—21no—
114C33×C4#4111yescharacteristic
115C33×C4#4111yescharacteristic
116C32×Dic3—41no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3—41no—
120C3×C22⋊Q8—31no—
121C3×C22⋊Q8—31no—
122C3×C22⋊Q8—61no—
123C3×#117—31no—
124C3×#117—31no—
125C3×#117—61no—
126#926C3×D311yescharacteristic
127#924C3×D311yescharacteristic
128C3×#107—91no—
129C3×#106—91no—
130C3×#103—91no—
131C3×#103—91no—
132C3×C22⋊C8—91no—
133#835—31no—
134#835—31no—
135#830—31no—
136#830—31no—
137#830—61no—
138#810—31no—
139#810—31no—
140#808—31no—
141#808—31no—
142#807—31no—
143#807—31no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—41no—
148C23×C32#6711yescharacteristic
149C23×C32#6711yescharacteristic
150C23×C32C3×D411yescharacteristic
151C32×Q8—61no—
152C32×Q8—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
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—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4—42no—
174C2×C32×C4—42no—
175C2×C32×C4—61no—
176C2×C32×C4—61no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4#6711yescharacteristic
180C2×C32×C4#6711yescharacteristic
181C2×C32×C4C3×D411yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—41no—
202C3×Dic6—41no—
203C3⋊(C3⋊C8)—21no—
204C3⋊(C3⋊C8)—21no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210C3×C3⋊C8—121no—
211#485—91noSylow
212C2×C33—41no—
213C2×C33#12311yescharacteristic
214C2×C33#13511yescharacteristic
215C2×C33#13711yescharacteristic
216C2×C3×Q8—31no—
217C2×C3×Q8—31no—
218C2×C3×Q8—61no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C22×C3×C4D3211yescharacteristic
224C22×C3×C4C2×C3×D311yescharacteristic
225C22×C3×C4C2×C3×D311yescharacteristic
226C2×Dic6C2×C3×D311yescharacteristic
227C2×C3×C8—91no—
228C2×C3×C8—91no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4—61no—
241C3×C22⋊C4—62no—
242C3×C22⋊C4—62no—
243C3×C22⋊C4—61no—
244C3×C22⋊C4—61no—
245C3×C22⋊C4—61no—
246C3×C22⋊C4—61no—
247C3×C42—62no—
248C3×C42—61no—
249C3×C42—61no—
250#217—21no—
251#217—21no—
252C4⋊Dic3C2×C3×D311yescharacteristic
253C4⋊Dic3C2×C3×D311yescharacteristic
254C4⋊Dic3C2×C3×D311yescharacteristic
255Dic3⋊C4—21no—
256C4×Dic3—21no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—31no—
261C2×C3⋊C8—61no—
262C2×C3⋊C8—61no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—41no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—42no—
287C22×C32—42no—
288C22×C32C2×#6711yescharacteristic
289C22×C32C2×#6711yescharacteristic
290C22×C32C2×C3×D411yescharacteristic
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—42no—
306C32×C4—121no—
307C32×C4—121no—
308C32×C4—121no—
309C32×C4—121no—
310C32×C4#23511yescharacteristic
311C32×C4#23511yescharacteristic
312C32×C4#23711yescharacteristic
313C32×C4#23711yescharacteristic
314C32×C4C3×#4111yescharacteristic
315C32×C4C3×#4111yescharacteristic
316C3×Dic3—42no—
317C3×Dic3—42no—
318C3×Dic3—42no—
319C3×Dic3—42no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—41no—
328C22⋊Q8—31no—
329#117—31no—
330#107—91no—
331#106—91no—
332#103—91no—
333#103—91no—
334C22⋊C8—91no—
335C33#48511yescharacteristic, complemented, Sylow
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—41no—
340C23×C3#62511yescharacteristic
341C23×C3C3×#6711yescharacteristic
342C23×C3C3×#6711yescharacteristic
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—121no—
348C3×Q8—121no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
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—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4#62511yescharacteristic
384C2×C3×C4C3×#6711yescharacteristic
385C2×C3×C4C3×#6711yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C22×C3×D311yescharacteristic
388C2×C3×C4C22×C3×D311yescharacteristic
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393Dic6—21no—
394Dic6—21no—
395C3×C8—181no—
396C3×C8—181no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—61no—
401C3⋊C8—121no—
402C3⋊C8—121no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—41no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—42no—
416C2×C32—42no—
417C2×C32—42no—
418C2×C32—44no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32#92911yescharacteristic
423C2×C32#93511yescharacteristic
424C2×C32#94211yescharacteristic
425C2×C32#95011yescharacteristic
426C2×C32#95011yescharacteristic
427C2×C32#95111yescharacteristic
428C2×C32C3×#12311yescharacteristic
429C2×C32C3×#13511yescharacteristic
430C2×C32C3×#13711yescharacteristic
431C2×Q8—31no—
432C22×C4C3×D3211yescharacteristic
433C2×C8—91no—
434C2×C8—91no—
435C4⋊C4—31no—
436C4⋊C4—31no—
437C4⋊C4—31no—
438C4⋊C4—61no—
439C22⋊C4—61no—
440C22⋊C4—61no—
441C42—61no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—41no—
467C22×C3C2×#62511yescharacteristic
468C22×C3C2×C3×#6711yescharacteristic
469C22×C3C2×C3×#6711yescharacteristic
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—122no—
486C3×C4—122no—
487C3×C4—122no—
488C3×C4—122no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—121no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—121no—
495C3×C4—121no—
496C3×C4—121no—
497C3×C4#380411yescharacteristic
498C3×C4#380411yescharacteristic
499C3×C4C3×#23511yescharacteristic
500C3×C4C3×#23511yescharacteristic
501C3×C4C3×#23711yescharacteristic
502C3×C4C3×#23711yescharacteristic
503Dic3—41no—
504Dic3—41no—
505Dic3—41no—
506Dic3—41no—
507C32—21no—
508C32—21no—
509C32—21no—
510C32—41no—
511C32#499611yescharacteristic, complemented
512C32#504511yescharacteristic, complemented
513C32C3×#48511yescharacteristic, complemented
514C23C3×#62511yescharacteristic
515Q8—61no—
516Q8—61no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4C3×#62511yescharacteristic
524C2×C4C2×C3×D3211yescharacteristic
525C8—181no—
526C8—181no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—42no—
537C2×C3—42no—
538C2×C3—42no—
539C2×C3—44no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3#6382211yescharacteristic
547C2×C3#6382711yescharacteristic
548C2×C3#6396111yescharacteristic
549C2×C3C3×#92911yescharacteristic
550C2×C3C3×#93511yescharacteristic
551C2×C3C3×#94211yescharacteristic
552C2×C3C3×#95011yescharacteristic
553C2×C3C3×#95011yescharacteristic
554C2×C3C3×#95111yescharacteristic
555C22—21no—
556C22—21no—
557C22—21no—
558C22C2×C3×#62511yescharacteristic
559C4—21no—
560C4—121no—
561C4—121no—
562C4—121no—
563C4—121no—
564C4C3×#380411yescharacteristic
565C4C3×#380411yescharacteristic
566C3—21no—
567C3—21no—
568C3—21no—
569C3—41no—
570C3#1059110211yescharacteristic, direct product factor
571C3C3×#499611yescharacteristic, complemented
572C3C3×#504511yescharacteristic, complemented
573C2—41no—
574C2C3×#6382211yescharacteristic
575C2C3×#6382711yescharacteristic
576C2C3×#6396111yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63576C2semidirect
5. C3×#63574C2semidirect
6. C3×#63565C2semidirect
7. C3×#63565C2semidirect
10. #10591102C3direct (N × H)
42. #63576C2×C3semidirect
43. #63574C2×C3semidirect
44. #63565C2×C3semidirect
45. #63565C2×C3semidirect
335. C33#485semidirect
511. C32#4996semidirect
512. C32#5045semidirect
513. C32C3×#485semidirect
570. C3#10591102direct (N × H)
571. C3C3×#4996semidirect
572. C3C3×#5045semidirect

Maximal subgroups

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

subgroupindex
C32×#9262
C32×#9242
C3×#636572
C3×#635762
C3×#635742
C3×#635652
C3×#50453
C3×#49963
#105911023