Finite Group Databasebeta

← C3×#10591006 #13158023

Subgroups — C3×#10591006 #13158023

Order 1728.

772 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3795C211yesmaximal, characteristic
2C32×D3×M16C211yesmaximal, characteristic, complemented
3C3×D3×#208C211yesmaximal, characteristic, complemented
4C3×#63651C211yesmaximal, characteristic, complemented
5C32×#823C211yesmaximal, characteristic, complemented
6C3×#63561C211yesmaximal, characteristic, complemented
7C3×#63558C211yesmaximal, characteristic, complemented
8C3×#5132—31nomaximal
9C3×#4745—31nomaximal
10#10591006C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C3×#625C412yes—
15C2×C3×D3×Dic3C412yes—
16C3×#3795—21no—
17C3×#3795—21no—
18C3×#3795—21no—
19C3×#3795—21no—
20C33×M16C2211yescharacteristic
21C3×#3748C2211yescharacteristic
22C32×#208C2211yescharacteristic
23C32×#208C2211yescharacteristic
24C32×ΓL(1,𝔽25)—21no—
25C32×D3×C8—21no—
26C3×#3711—21no—
27C3×D3×C3⋊C8—21no—
28C2×C3×#237—31no—
29C2×C3×#235—31no—
30C2×#3795C2×C311yescharacteristic
31C2×C32×M16—31no—
32C2×C3×#208—31no—
33C3×D3×M16—21no—
34C3×D3×M16—31no—
35C3×D3×M16D311yescharacteristic, complemented
36C3×D3×M16C2×C311yescharacteristic, complemented
37D3×#208C2×C311yescharacteristic, complemented
38C32×#100—31no—
39#63651C2×C311yescharacteristic, complemented
40C3×#833—31no—
41C3×#824—31no—
42C3×#824—31no—
43C3×#824—31no—
44C3×#824—31no—
45C3×#824—61no—
46C3×#823—21no—
47C3×#823—31no—
48C3×#823D311yescharacteristic, complemented
49C3×#823C2×C311yescharacteristic, complemented
50#63561C2×C311yescharacteristic, complemented
51#63558C2×C311yescharacteristic, complemented
52C22×C32×D3C2×C412yes—
53C22×C32×D3C2×C411yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3C2×C412yes—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3C2×C411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C32×C4×D3D411yescharacteristic
62C32×C4×D3D411yescharacteristic
63C32×C4×D3D411yescharacteristic
64C32×C4×D3D411yescharacteristic
65C3×#625—22no—
66C3×#625—22no—
67C3×D3×Dic3—22no—
68C3×D3×Dic3—22no—
69C33×C8—21no—
70C3×C3⋊(C3⋊C8)—21no—
71C32×C3⋊C8—21no—
72C32×C3⋊C8—21no—
73C3×#413—91no—
74#5132—31no—
75#4745—31no—
76C2×C32×D4—31no—
77C22×C32×C4—31no—
78C2×#634C22×C311yescharacteristic
79C2×C3×#67—32no—
80C2×C3×#67—32no—
81C22×C3×Dic3—32no—
82C3×#237—61no—
83C3×#237—61no—
84C3×#237—61no—
85C3×#237—61no—
86C3×#235—61no—
87C3×#235—61no—
88C3×#235—61no—
89C3×#235—61no—
90C2×C3×D12—21no—
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×D12C22×C311yescharacteristic
93C2×C3×C4×D3—21no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3C2×D311yescharacteristic
96C2×C3×C4×D3C22×C311yescharacteristic
97C2×C3×Dic6—31no—
98C2×C3×Dic6—31no—
99C2×C3×Dic6—61no—
100C2×#625C3×C412yes—
101C2×D3×Dic3C3×C412yes—
102#3795—21no—
103#3795—21no—
104#3795—21no—
105#3795—21no—
106C32×M16—21no—
107C32×M16—21no—
108C32×M16—21no—
109C32×M16—31no—
110C32×M16—31no—
111C32×M16—41no—
112C32×M16—61no—
113C32×M16C2×D311yescharacteristic
114C32×M16C2×D311yescharacteristic
115C32×M16C22×C311yescharacteristic
116C2×C32×C8—61no—
117#3748C22×C311yescharacteristic
118C3×#208—21no—
119C3×#208—21no—
120C3×#208—31no—
121C3×#208—31no—
122C3×#208—61no—
123C3×#208—61no—
124C3×#208C2×D311yescharacteristic
125C3×#208C2×D311yescharacteristic
126C3×#208C22×C311yescharacteristic
127C3×#208C22×C311yescharacteristic
128C2×C3×C3⋊C8—61no—
129C3×ΓL(1,𝔽25)—22no—
130C3×ΓL(1,𝔽25)—21no—
131C3×ΓL(1,𝔽25)—21no—
132C3×ΓL(1,𝔽25)—61no—
133C3×D3×C8—22no—
134C3×D3×C8—21no—
135C3×D3×C8—21no—
136C3×D3×C8—61no—
137#3711—21no—
138D3×C3⋊C8—21no—
139C22×C33C22×C411yescharacteristic
140C2×C32×D3—22no—
141C2×C32×D3—41no—
142C2×C32×D3—42no—
143C2×C32×D3C22⋊C411yescharacteristic
144C2×C32×D3C22⋊C411yescharacteristic
145C33×C4C2×D411yescharacteristic
146C33×C4C2×D411yescharacteristic
147C3×C3⋊Dic3—22no—
148C32×Dic3C22⋊C411yescharacteristic
149C32×Dic3C22⋊C411yescharacteristic
150C2×C3×#41—91no—
151C2×#237—31no—
152C2×#235—31no—
153C2×C3×M16—31no—
154C2×C3×M16—31no—
155C2×C3×M16—61no—
156C2×C3×M16—91no—
157C2×#208—31no—
158D3×M16—31no—
159D3×M16C3×D311yescharacteristic, complemented
160C3×#101—91no—
161C3×#101—91no—
162C3×#100—31no—
163C3×#100—31no—
164C3×#100—61no—
165C3×#100—91no—
166#833—31no—
167#824—31no—
168#824—31no—
169#824—31no—
170#824—31no—
171#824—61no—
172#823—31no—
173#823C3×D311yescharacteristic, complemented
174C23×C32—31no—
175C23×C32—32no—
176C22×C3×D3—21no—
177C22×C3×D3—22no—
178C22×C3×D3C4×D312yes—
179C22×C3×D3C4×D311yescharacteristic
180C22×C3×D3C2×C3×C412yes—
181C22×C3×D3C2×C3×C411yescharacteristic
182C32×D4—61no—
183C32×D4—61no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—31no—
188C2×C32×C4—31no—
189C2×C32×C4—31no—
190C2×C32×C4—31no—
191C2×C32×C4—31no—
192C2×C32×C4—41no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C23×C311yescharacteristic
196C2×C3⋊Dic3C2×C3×C412yes—
197#634—21no—
198#634—21no—
199C3×#67—62no—
200C3×#67—62no—
201C3×#67—62no—
202C3×#67—62no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3C4×D311yescharacteristic
210C2×C3×Dic3C2×C3×C411yescharacteristic
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—21no—
215C3×D12—41no—
216C3×D12—41no—
217C3×C4×D3—21no—
218C3×C4×D3—21no—
219C3×C4×D3—21no—
220C3×C4×D3—21no—
221C3×C4×D3—62no—
222C3×C4×D3—62no—
223C3×C4×D3D1211yescharacteristic
224C3×C4×D3D1211yescharacteristic
225C3×C4×D3#6711yescharacteristic
226C3×C4×D3#6711yescharacteristic
227C3×C4×D3C3×D411yescharacteristic
228C3×C4×D3C3×D411yescharacteristic
229C3×C4×D3C3×D411yescharacteristic
230C3×C4×D3C3×D411yescharacteristic
231C3×Dic6—61no—
232C3×Dic6—61no—
233C3×Dic6—61no—
234C3×Dic6—61no—
235C3×Dic6—121no—
236C3×Dic6—121no—
237#625—22no—
238#625—22no—
239D3×Dic3—22no—
240D3×Dic3—22no—
241C32×C8—21no—
242C32×C8—22no—
243C32×C8—21no—
244C32×C8—21no—
245C32×C8—41no—
246C32×C8—41no—
247C32×C8—42no—
248C32×C8—61no—
249C32×C8—61no—
250C3⋊(C3⋊C8)—21no—
251C3×C3⋊C8—22no—
252C3×C3⋊C8—21no—
253C3×C3⋊C8—21no—
254C3×C3⋊C8—21no—
255C3×C3⋊C8—21no—
256C3×C3⋊C8—41no—
257C3×C3⋊C8—61no—
258C3×C3⋊C8—61no—
259C3×C3⋊C8—121no—
260#413—91noSylow
261C2×C33—21no—
262C2×C33C2×C22⋊C411yescharacteristic
263C32×D3—21no—
264C32×D3—21no—
265C32×D3—42no—
266C3×#41—181no—
267C3×#41—181no—
268C3×#41—181no—
269C3×#41—181no—
270C2×C3×Q8—91no—
271C2×C3×D4—31no—
272C2×C3×D4—31no—
273C2×C3×D4—61no—
274C2×C3×D4—92no—
275C22×C3×C4—31no—
276C22×C3×C4—31no—
277C22×C3×C4—61no—
278C22×C3×C4—92no—
279C2×#67—32no—
280C2×#67—32no—
281C22×Dic3—32no—
282#237—61no—
283#237—61no—
284#237—61no—
285#237—61no—
286#235—61no—
287#235—61no—
288#235—61no—
289#235—61no—
290C2×D12C2×C3×D311yescharacteristic
291C2×C4×D3—32no—
292C2×C4×D3C2×C3×D311yescharacteristic
293C2×Dic6—31no—
294C2×Dic6—31no—
295C2×Dic6—61no—
296C3×M16—21no—
297C3×M16—21no—
298C3×M16—21no—
299C3×M16—31no—
300C3×M16—31no—
301C3×M16—31no—
302C3×M16—31no—
303C3×M16—41no—
304C3×M16—61no—
305C3×M16—61no—
306C3×M16—61no—
307C3×M16—61no—
308C3×M16—62no—
309C3×M16—91no—
310C3×M16—181no—
311C3×M16D3211yescharacteristic
312C3×M16C2×C3×D311yescharacteristic
313C3×M16C2×C3×D311yescharacteristic
314C2×C3×C8—61no—
315C2×C3×C8—61no—
316C2×C3×C8—62no—
317C2×C3×C8—181no—
318#208—31no—
319#208—31no—
320#208—61no—
321#208—61no—
322#208C2×C3×D311yescharacteristic
323#208C2×C3×D311yescharacteristic
324C2×C3⋊C8—61no—
325ΓL(1,𝔽25)—21no—
326ΓL(1,𝔽25)—61no—
327D3×C8—21no—
328D3×C8—61no—
329C22×C32—21no—
330C22×C32—21no—
331C22×C32—21no—
332C22×C32—31no—
333C22×C32—31no—
334C22×C32—41no—
335C22×C32—62no—
336C22×C32—121no—
337C22×C32—122no—
338C22×C32C2×C4×D311yescharacteristic
339C22×C32C2×C4×D311yescharacteristic
340C22×C32C22×C3×C411yescharacteristic
341C2×C3×D3—22no—
342C2×C3×D3—22no—
343C2×C3×D3—21no—
344C2×C3×D3—21no—
345C2×C3×D3—42no—
346C2×C3×D3—41no—
347C2×C3×D3—41no—
348C2×C3×D3—42no—
349C2×C3×D3—42no—
350C2×C3×D3—42no—
351C2×C3×D3—82no—
352C2×C3×D3(C2×D3)⋊C411yescharacteristic
353C2×C3×D3(C2×D3)⋊C411yescharacteristic
354C2×C3×D3C3×C22⋊C411yescharacteristic
355C2×C3×D3C3×C22⋊C411yescharacteristic
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—31no—
363C32×C4—31no—
364C32×C4—41no—
365C32×C4—41no—
366C32×C4C2×D1211yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4C2×#6711yescharacteristic
370C32×C4C2×C3×D411yescharacteristic
371C32×C4C2×C3×D411yescharacteristic
372C3⋊Dic3—22no—
373C3×Dic3—21no—
374C3×Dic3—21no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—122no—
378C3×Dic3(C2×D3)⋊C411yescharacteristic
379C3×Dic3(C2×D3)⋊C411yescharacteristic
380C3×Dic3C3×C22⋊C411yescharacteristic
381C3×Dic3C3×C22⋊C411yescharacteristic
382C2×#41—91no—
383C2×M16—31no—
384C2×M16—91no—
385#101—91no—
386#101—91no—
387#100—31no—
388#100—91no—
389C33#41311yescharacteristic, complemented, Sylow
390C23×C3—31no—
391C23×C3—31no—
392C23×C3—32no—
393C23×C3—32no—
394C23×C3—61no—
395C23×C3—62no—
396C22×D3C3×C4×D312yes—
397C22×D3C3×C4×D311yescharacteristic
398C3×Q8—181no—
399C3×Q8—181no—
400C3×D4—61no—
401C3×D4—61no—
402C3×D4—61no—
403C3×D4—61no—
404C3×D4—121no—
405C3×D4—121no—
406C3×D4—182no—
407C3×D4—182no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—31no—
412C2×C3×C4—31no—
413C2×C3×C4—31no—
414C2×C3×C4—31no—
415C2×C3×C4—31no—
416C2×C3×C4—31no—
417C2×C3×C4—31no—
418C2×C3×C4—31no—
419C2×C3×C4—31no—
420C2×C3×C4—31no—
421C2×C3×C4—41no—
422C2×C3×C4—61no—
423C2×C3×C4—61no—
424C2×C3×C4—61no—
425C2×C3×C4—61no—
426C2×C3×C4—61no—
427C2×C3×C4—92no—
428C2×C3×C4—182no—
429C2×C3×C4—182no—
430C2×C3×C4C2×D3211yescharacteristic
431C2×C3×C4C22×C3×D311yescharacteristic
432C2×C3×C4C22×C3×D311yescharacteristic
433#67—62no—
434#67—62no—
435#67—62no—
436#67—62no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—62no—
440C2×Dic3—62no—
441C2×Dic3—62no—
442C2×Dic3C3×C4×D311yescharacteristic
443D12—21no—
444D12—21no—
445C4×D3—62no—
446C4×D3—62no—
447C4×D3C3×D1211yescharacteristic
448C4×D3C3×D1211yescharacteristic
449C4×D3C3×#6711yescharacteristic
450C4×D3C3×#6711yescharacteristic
451Dic6—61no—
452Dic6—61no—
453Dic6—61no—
454Dic6—61no—
455Dic6—121no—
456Dic6—121no—
457C3×C8—22no—
458C3×C8—21no—
459C3×C8—21no—
460C3×C8—21no—
461C3×C8—41no—
462C3×C8—41no—
463C3×C8—42no—
464C3×C8—62no—
465C3×C8—61no—
466C3×C8—61no—
467C3×C8—61no—
468C3×C8—61no—
469C3×C8—121no—
470C3×C8—181no—
471C3⋊C8—21no—
472C3⋊C8—21no—
473C3⋊C8—61no—
474C3⋊C8—61no—
475C3⋊C8—121no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—21no—
482C2×C32—41no—
483C2×C32—41no—
484C2×C32—41no—
485C2×C32—41no—
486C2×C32—61no—
487C2×C32—61no—
488C2×C32—81no—
489C2×C32—122no—
490C2×C32D3×C22⋊C411yescharacteristic
491C2×C32C2×(C2×D3)⋊C411yescharacteristic
492C2×C32C2×C3×C22⋊C411yescharacteristic
493C3×D3—22no—
494C3×D3—21no—
495C3×D3—21no—
496C3×D3—21no—
497C3×D3—21no—
498C3×D3—41no—
499C3×D3—42no—
500C3×D3—42no—
501C3×D3—82no—
502#41—181no—
503#41—181no—
504#41—181no—
505#41—181no—
506C2×Q8—91no—
507C2×D4—31no—
508C2×D4—92no—
509C22×C4—31no—
510C22×C4—92no—
511M16—31no—
512M16—31no—
513M16—61no—
514M16—91no—
515M16—181no—
516M16C3×D3211yescharacteristic
517C2×C8—61no—
518C2×C8—181no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—31no—
523C22×C3—31no—
524C22×C3—31no—
525C22×C3—31no—
526C22×C3—41no—
527C22×C3—61no—
528C22×C3—61no—
529C22×C3—62no—
530C22×C3—62no—
531C22×C3—122no—
532C22×C3—121no—
533C22×C3—121no—
534C22×C3—122no—
535C22×C3—122no—
536C22×C3—122no—
537C22×C3—242no—
538C22×C3C4×D3211yescharacteristic
539C22×C3C2×C3×C4×D311yescharacteristic
540C22×C3C2×C3×C4×D311yescharacteristic
541C2×D3—22no—
542C2×D3—41no—
543C2×D3—42no—
544C2×D3C3×(C2×D3)⋊C411yescharacteristic
545C2×D3C3×(C2×D3)⋊C411yescharacteristic
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—31no—
553C3×C4—31no—
554C3×C4—31no—
555C3×C4—31no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—61no—
559C3×C4—61no—
560C3×C4—182no—
561C3×C4D3×D1211yescharacteristic
562C3×C4D3×#6711yescharacteristic
563C3×C4C2×C3×D1211yescharacteristic
564C3×C4C3×D3×D411yescharacteristic
565C3×C4C3×D3×D411yescharacteristic
566C3×C4C2×C3×#6711yescharacteristic
567Dic3—62no—
568Dic3—62no—
569Dic3—122no—
570Dic3C3×(C2×D3)⋊C411yescharacteristic
571Dic3C3×(C2×D3)⋊C411yescharacteristic
572C32—21no—
573C32—21no—
574C32—21no—
575C32—41no—
576C32#474511yescharacteristic, complemented
577C32#513211yescharacteristic, complemented
578C32C3×#41311yescharacteristic, complemented
579C23—31no—
580C23—32no—
581Q8—181no—
582Q8—181no—
583D4—61no—
584D4—61no—
585D4—182no—
586D4—182no—
587C2×C4—31no—
588C2×C4—31no—
589C2×C4—31no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—92no—
593C2×C4—182no—
594C2×C4—182no—
595C2×C4C2×C3×D3211yescharacteristic
596C8—21no—
597C8—61no—
598C8—61no—
599C8—181no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—62no—
611C2×C3—61no—
612C2×C3—61no—
613C2×C3—61no—
614C2×C3—61no—
615C2×C3—81no—
616C2×C3—121no—
617C2×C3—122no—
618C2×C3—122no—
619C2×C3—242no—
620C2×C3D3×(C2×D3)⋊C411yescharacteristic
621C2×C3C3×D3×C22⋊C411yescharacteristic
622C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
623D3—21no—
624D3—21no—
625D3—42no—
626C22—31no—
627C22—31no—
628C22—62no—
629C22—121no—
630C22—122no—
631C22C3×C4×D3211yescharacteristic
632C4—31no—
633C4—31no—
634C4—182no—
635C4C3×D3×D1211yescharacteristic
636C4C3×D3×#6711yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059100611yescharacteristic, direct product factor
642C3C3×#474511yescharacteristic, complemented
643C3C3×#513211yescharacteristic, complemented
644C2—21no—
645C2—61no—
646C2—61no—
647C2—122no—
648C2C3×D3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC2×C3D3×(C2×D3)⋊C4
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3795C2
cosocleC2×C32×C4C23×C3
hypercenterC3×M16D32
hypocenterC32C3×#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. C32×D3×M16C2semidirect
3. C3×D3×#208C2semidirect
4. C3×#63651C2semidirect
5. C32×#823C2semidirect
6. C3×#63561C2semidirect
7. C3×#63558C2semidirect
10. #10591006C3direct (N × H)
35. C3×D3×M16D3semidirect
36. C3×D3×M16C2×C3semidirect
37. D3×#208C2×C3semidirect
39. #63651C2×C3semidirect
48. C3×#823D3semidirect
49. C3×#823C2×C3semidirect
50. #63561C2×C3semidirect
51. #63558C2×C3semidirect
159. D3×M16C3×D3semidirect
173. #823C3×D3semidirect
389. C33#413semidirect
576. C32#4745semidirect
577. C32#5132semidirect
578. C32C3×#413semidirect
641. C3#10591006direct (N × H)
642. C3C3×#4745semidirect
643. C3C3×#5132semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37952
C32×D3×M162
C3×D3×#2082
C3×#636512
C32×#8232
C3×#635612
C3×#635582
C3×#51323
C3×#47453
#105910063