Finite Group Databasebeta

← C3×#71856 #11792035

Subgroups — C3×#71856 #11792035

Order 1152.

1123 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#535C211yesmaximal, characteristic, complemented
2C2×C3×#949C211yesmaximal, characteristic, complemented
3C2×C3×#940C211yesmaximal, characteristic, complemented
4C2×C3×#837C211yesmaximal, characteristic, complemented
5C2×C3×#833C211yesmaximal, characteristic, complemented
6C3×#5001C211yesmaximal, characteristic, complemented
7C3×#4532C211yesmaximal, characteristic, complemented
8C3×#2002—31nomaximal
9#71856C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C22×C3×#67—22no—
13C22×C3×D12C2211yescharacteristic
14C32×#127—21no—
15C32×#124—21no—
16C32×#121—22no—
17C32×#120—22no—
18C32×#117C2211yescharacteristic
19C3×#949—22no—
20C3×#949—22no—
21C2×C3×#215—21no—
22C3×#940—22no—
23C3×#940—22no—
24C2×C3×#213—21no—
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27C2×C3×#208C2211yescharacteristic
28C3×#837—22no—
29C3×#833—22no—
30C3×#809—22no—
31C2×C3×AGL(1,ℤ/8)—31no—
32C3×#535—21no—
33C3×#535D311yescharacteristic, complemented
34C3×#535C2×C311yescharacteristic, complemented
35C2×C3×#127—31no—
36C2×#949C2×C311yescharacteristic, complemented
37C2×#940C2×C311yescharacteristic, complemented
38C2×C3×#104—31no—
39C3×#418—31no—
40C2×C3×#100—31no—
41C2×#837C2×C311yescharacteristic, complemented
42C2×#833C2×C311yescharacteristic, complemented
43#5001C2×C311yescharacteristic, complemented
44C3×#339—31no—
45#4532C2×C311yescharacteristic, complemented
46C24×C32D412yes—
47C23×C3×D3—21no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C32×#41—21no—
52C2×C32×Q8D411yescharacteristic, complemented
53C2×C32×D4—21no—
54C2×C32×D4—24no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4—21no—
61C2×C32×D4D411yescharacteristic, complemented
62C22×C32×C4—21no—
63C22×C32×C4C2311yescharacteristic
64C2×C3×#67—28no—
65C2×C3×#67—48no—
66C22×C3×Dic3—21no—
67C2×C3×D12—21no—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12D412yescomplemented
73C32×C4⋊C4D412yescomplemented
74C32×C22⋊C4—24no—
75C32×C22⋊C4—21no—
76C32×C42—21no—
77C3×#215—24no—
78C3×#213—24no—
79C3×C4×Dic3—21no—
80C3×C4×Dic3D412yescomplemented
81C3×#208—21no—
82C3×#208—21no—
83C3×#208—21no—
84C3×#208D412yescomplemented
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8—21no—
87#2002—31noSylow
88C2×C3×#41—21no—
89C2×C3×#41C2×D311yescharacteristic
90C2×C3×#41C22×C311yescharacteristic
91C22×C3×D4—21no—
92C22×C3×D4—31no—
93C22×C3×D4—62no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C22×#67—22no—
97C22×D12C22×C311yescharacteristic
98C3×AGL(1,ℤ/8)—62no—
99C3×AGL(1,ℤ/8)—62no—
100C2×C3×SD16—61no—
101C2×C3×D8—61no—
102C2×C3×M16—31no—
103C2×C3×M16—31no—
104C3×#127—22no—
105C3×#127—21no—
106C3×#127—21no—
107C3×#127—62no—
108C3×#127—62no—
109C3×#124—22no—
110C3×#124—21no—
111C3×#124—21no—
112C3×#121—24no—
113C3×#121—22no—
114C3×#121—22no—
115C3×#120—24no—
116C3×#120—22no—
117C3×#120—22no—
118C3×#117—21no—
119C3×#117C2×D311yescharacteristic
120C3×#117C22×C311yescharacteristic
121C2×C3×C42—31no—
122#949—22no—
123#949—22no—
124C2×#215—21no—
125#940—22no—
126#940—22no—
127C2×#213—21no—
128C2×C4×Dic3C22×C311yescharacteristic
129C2×#208C22×C311yescharacteristic
130C2×#208C22×C311yescharacteristic
131C3×#104—62no—
132C3×#102—62no—
133C3×#100—62no—
134#837—22no—
135#833—22no—
136#809—22no—
137C23×C32—22no—
138C23×C32—21no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—44no—
142C23×C32C2×D411yescharacteristic
143C22×C3×D3—22no—
144C22×C3×D3—44no—
145C22×C3×D3—41no—
146C22×C3×D3—41no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×D4—21no—
150C32×D4—21no—
151C32×D4—41no—
152C32×D4—41no—
153C32×D4—41no—
154C32×D4—41no—
155C32×D4—41no—
156C32×D4—41no—
157C32×D4—84no—
158C2×C32×C4—22no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4—41no—
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4#4111yescharacteristic
168C3×#67—416no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—22no—
171C3×D12—22no—
172C3×D12—22no—
173C3×D12—41no—
174C3×D12—41no—
175C3×C3⋊C8—22no—
176C3×C3⋊C8—41no—
177C2×AGL(1,ℤ/8)—31no—
178#535C3×D311yescharacteristic, complemented
179C2×#127—31no—
180C2×#104—31no—
181#418—31no—
182C2×#100—31no—
183#339—31no—
184C24×C3—22no—
185C24×C3—61no—
186C24×C3#6712yes—
187C24×C3C3×D412yes—
188C23×D3—21no—
189C3×#41—21no—
190C3×#41—21no—
191C3×#41—21no—
192C3×#41—21no—
193C3×#41—21no—
194C3×#41—21no—
195C3×#41—21no—
196C3×#41—21no—
197C3×#41—41no—
198C3×#41—41no—
199C3×#41—41no—
200C3×#41—41no—
201C2×C3×Q8—21no—
202C2×C3×Q8#6711yescharacteristic, complemented
203C2×C3×Q8C3×D411yescharacteristic, complemented
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—22no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—28no—
210C2×C3×D4—24no—
211C2×C3×D4—24no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—32no—
226C2×C3×D4—41no—
227C2×C3×D4—41no—
228C2×C3×D4—41no—
229C2×C3×D4—41no—
230C2×C3×D4—68no—
231C2×C3×D4—61no—
232C2×C3×D4—61no—
233C2×C3×D4—61no—
234C2×C3×D4—61no—
235C2×C3×D4—61no—
236C2×C3×D4—128no—
237C2×C3×D4#6711yescharacteristic, complemented
238C2×C3×D4C3×D411yescharacteristic, complemented
239C22×C3×C4—22no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4—61no—
244C22×C3×C4C22×D311yescharacteristic
245C22×C3×C4C23×C311yescharacteristic
246C2×#67—28no—
247C2×#67—48no—
248C22×Dic3—21no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12—21no—
252C2×D12—21no—
253C2×D12—21no—
254C2×D12C3×D412yescomplemented
255C3×SD16—64no—
256C3×D8—64no—
257C3×M16—32no—
258C3×M16—61no—
259C3×M16—61no—
260C3×M16—61no—
261C2×C3×C8—61no—
262C2×C3×C8—61no—
263C3×C4⋊C4—22no—
264C3×C4⋊C4#6712yescomplemented
265C3×C4⋊C4C3×D412yescomplemented
266C3×C22⋊C4—22no—
267C3×C22⋊C4—28no—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—24no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C42—22no—
273C3×C42—21no—
274C3×C42—21no—
275C3×C42—32no—
276C3×C42—61no—
277#215—24no—
278#213—24no—
279C4×Dic3—21no—
280C4×Dic3C3×D412yescomplemented
281#208—21no—
282#208—21no—
283#208—21no—
284#208C3×D412yescomplemented
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C22×C32—22no—
288C22×C32—41no—
289C22×C32—42no—
290C22×C32—42no—
291C22×C32—44no—
292C22×C32—44no—
293C22×C32—82no—
294C22×C32#12011yescharacteristic
295C22×C32#12011yescharacteristic
296C22×C32#12111yescharacteristic
297C2×C3×D3—44no—
298C2×C3×D3—42no—
299C2×C3×D3—82no—
300C2×C3×D3—82no—
301C32×C4—41no—
302C32×C4—42no—
303C32×C4#12011yescharacteristic
304C32×C4#12111yescharacteristic
305C32×C4#12111yescharacteristic
306C32×C4#12711yescharacteristic
307C3×Dic3—24no—
308C2×#41C2×C3×D311yescharacteristic
309C22×D4—31no—
310C22×D4—62no—
311C22×D4C2×C3×D311yescharacteristic
312AGL(1,ℤ/8)—62no—
313AGL(1,ℤ/8)—62no—
314C2×SD16—61no—
315C2×D8—61no—
316C2×M16—31no—
317C2×M16—31no—
318#127—21no—
319#127—62no—
320#127—62no—
321#124—21no—
322#121—22no—
323#120—22no—
324#117C2×C3×D311yescharacteristic
325C2×C42—31no—
326#104—62no—
327#102—62no—
328#100—62no—
329C23×C3—24no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—44no—
343C23×C3—44no—
344C23×C3—62no—
345C23×C3—84no—
346C23×C3—124no—
347C23×C3—121no—
348C23×C3—121no—
349C23×C3C2×#6711yescharacteristic
350C23×C3C2×C3×D411yescharacteristic
351C22×D3—22no—
352C22×D3—44no—
353C22×D3—41no—
354C22×D3—41no—
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—42no—
368C3×D4—42no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—62no—
386C3×D4—62no—
387C3×D4—88no—
388C3×D4—84no—
389C3×D4—84no—
390C3×D4—1216no—
391C3×D4—121no—
392C3×D4—121no—
393C2×C3×C4—24no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—64no—
412C2×C3×C4—62no—
413C2×C3×C4D3×D411yescharacteristic
414C2×C3×C4D3×D411yescharacteristic
415C2×C3×C4D3×D411yescharacteristic
416C2×C3×C4#23711yescharacteristic
417C2×C3×C4C2×#6711yescharacteristic
418C2×C3×C4C2×#6711yescharacteristic
419C2×C3×C4C2×C3×D411yescharacteristic
420C2×C3×C4C2×C3×D411yescharacteristic
421C2×C3×C4C2×C3×D411yescharacteristic
422C2×C3×C4C2×C3×D411yescharacteristic
423C2×C3×C4C2×C3×D411yescharacteristic
424C2×C3×C4C3×#4111yescharacteristic
425#67—416no—
426C2×Dic3—24no—
427C2×Dic3—22no—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C3×C8—62no—
433C3×C8—121no—
434C3⋊C8—22no—
435C3⋊C8—41no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—81no—
439C2×C32—82no—
440C2×C32#39211yescharacteristic
441C2×C32#45312yes—
442C3×D3—82no—
443C24—61no—
444C24C3×#6712yes—
445#41—21no—
446#41—21no—
447#41—21no—
448#41—21no—
449C2×Q8C3×#6711yescharacteristic, complemented
450C2×D4—21no—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—32no—
459C2×D4—68no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—128no—
466C2×D4C3×#6711yescharacteristic, complemented
467C22×C4—21no—
468C22×C4—61no—
469C22×C4C22×C3×D311yescharacteristic
470SD16—64no—
471D8—64no—
472M16—32no—
473M16—61no—
474M16—61no—
475M16—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4C3×#6712yescomplemented
479C22⋊C4—24no—
480C22⋊C4—21no—
481C42—21no—
482C42—32no—
483C42—61no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—42no—
490C22×C3—44no—
491C22×C3—42no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—48no—
495C22×C3—44no—
496C22×C3—41no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—44no—
500C22×C3—41no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—44no—
504C22×C3—84no—
505C22×C3—82no—
506C22×C3—82no—
507C22×C3—84no—
508C22×C3—124no—
509C22×C3—122no—
510C22×C3—242no—
511C22×C3—242no—
512C22×C3#93711yescharacteristic
513C22×C3#93911yescharacteristic
514C22×C3#95311yescharacteristic
515C22×C3C3×#12011yescharacteristic
516C22×C3C3×#12011yescharacteristic
517C22×C3C3×#12111yescharacteristic
518C2×D3—44no—
519C2×D3—42no—
520C2×D3—82no—
521C2×D3—82no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—42no—
527C3×C4—44no—
528C3×C4—41no—
529C3×C4—42no—
530C3×C4—41no—
531C3×C4—42no—
532C3×C4—64no—
533C3×C4#93711yescharacteristic
534C3×C4#93811yescharacteristic
535C3×C4#93911yescharacteristic
536C3×C4#94011yescharacteristic
537C3×C4C3×#12011yescharacteristic
538C3×C4C3×#12111yescharacteristic
539C3×C4C3×#12111yescharacteristic
540C3×C4C3×#12711yescharacteristic
541Dic3—24no—
542C32#200211yescharacteristic, complemented, Sylow
543C23—22no—
544C23—22no—
545C23—21no—
546C23—22no—
547C23—44no—
548C23—62no—
549C23—124no—
550C23—121no—
551C23—121no—
552C23C2×C3×#6711yescharacteristic
553Q8—21no—
554Q8—21no—
555D4—21no—
556D4—21no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—62no—
564D4—62no—
565D4—84no—
566D4—1216no—
567D4—121no—
568D4—121no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4C3×D3×D411yescharacteristic
576C2×C4C3×D3×D411yescharacteristic
577C2×C4C3×D3×D411yescharacteristic
578C2×C4C3×#23711yescharacteristic
579C2×C4C2×C3×#6711yescharacteristic
580C2×C4C2×C3×#6711yescharacteristic
581C8—62no—
582C8—121no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—22no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—84no—
592C2×C3—82no—
593C2×C3—82no—
594C2×C3—81no—
595C2×C3—82no—
596C2×C3—81no—
597C2×C3—242no—
598C2×C3#522311yescharacteristic
599C2×C3#524112yes—
600C2×C3C3×#39211yescharacteristic
601C2×C3C3×#45312yes—
602D3—82no—
603C22—22no—
604C22—44no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—82no—
610C22—124no—
611C22—122no—
612C22—242no—
613C22—242no—
614C22C3×#93711yescharacteristic
615C22C3×#93911yescharacteristic
616C22C3×#95311yescharacteristic
617C4—41no—
618C4—42no—
619C4—64no—
620C4C3×#93711yescharacteristic
621C4C3×#93811yescharacteristic
622C4C3×#93911yescharacteristic
623C4C3×#94011yescharacteristic
624C3—21no—
625C3#7185611yescharacteristic, direct product factor
626C3C3×#200211yescharacteristic, complemented
627C2—21no—
628C2—21no—
629C2—82no—
630C2—81no—
631C2—242no—
632C2C3×#522311yescharacteristic
633C2C3×#524112yes—
subgroupG/H
centerC22×C3#953
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#535C2
socleC22×C32#120
hypercenterC3×#535D3
hypocenterC3C3×#2002
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#535

Complements

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

NH ≅ G/Ntype
1. C32×#535C2semidirect
2. C2×C3×#949C2semidirect
3. C2×C3×#940C2semidirect
4. C2×C3×#837C2semidirect
5. C2×C3×#833C2semidirect
6. C3×#5001C2semidirect
7. C3×#4532C2semidirect
9. #71856C3direct (N × H)
33. C3×#535D3semidirect
34. C3×#535C2×C3semidirect
36. C2×#949C2×C3semidirect
37. C2×#940C2×C3semidirect
41. C2×#837C2×C3semidirect
42. C2×#833C2×C3semidirect
43. #5001C2×C3semidirect
45. #4532C2×C3semidirect
52. C2×C32×Q8D4semidirect
61. C2×C32×D4D4semidirect
72. C2×C3×D12D4semidirect
73. C32×C4⋊C4D4semidirect
80. C3×C4×Dic3D4semidirect
84. C3×#208D4semidirect
178. #535C3×D3semidirect
202. C2×C3×Q8#67semidirect
203. C2×C3×Q8C3×D4semidirect
237. C2×C3×D4#67semidirect
238. C2×C3×D4C3×D4semidirect
254. C2×D12C3×D4semidirect
264. C3×C4⋊C4#67semidirect
265. C3×C4⋊C4C3×D4semidirect
280. C4×Dic3C3×D4semidirect
284. #208C3×D4semidirect
449. C2×Q8C3×#67semidirect
466. C2×D4C3×#67semidirect
478. C4⋊C4C3×#67semidirect
542. C32#2002semidirect
625. C3#71856direct (N × H)
626. C3C3×#2002semidirect

Maximal subgroups

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

subgroupindex
C32×#5352
C2×C3×#9492
C2×C3×#9402
C2×C3×#8372
C2×C3×#8332
C3×#50012
C3×#45322
C3×#20023
#718563