Finite Group Databasebeta

← C3×#72056 #11792541

Subgroups — C3×#72056 #11792541

Order 1152.

1097 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic, complemented
2C22×C3×#215C211yesmaximal, characteristic, complemented
3C2×C3×#940C211yesmaximal, characteristic
4C3×#5229C211yesmaximal, characteristic, complemented
5C2×C3×#832C211yesmaximal, characteristic, complemented
6C2×C3×#821C211yesmaximal, characteristic, complemented
7C3×#4550C211yesmaximal, characteristic, complemented
8C3×#2014—31nomaximal
9#72056C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C22×C3×#67—22no—
13C22×C3×D12C2211yescharacteristic
14C2×C32×SD16—22no—
15C2×C32×SD16—22no—
16C2×C32×SD16—22no—
17C22×C32×C8C2211yescharacteristic
18C2×C3×#215—22no—
19C2×C3×#215—22no—
20C2×C3×#215—22no—
21C3×#940—22no—
22C3×#940—22no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#832—22no—
27C3×C3⋊#95—22no—
28C3×#821—22no—
29C22×C3×SD16—21no—
30C22×C3×SD16—31no—
31C22×C3×SD16D311yescharacteristic, complemented
32C22×C3×SD16C2×C311yescharacteristic, complemented
33C2×C3×#127—31no—
34C22×#215C2×C311yescharacteristic, complemented
35C2×#940C2×C311yescharacteristic
36C2×C3×#102—31no—
37C2×C3×#102—31no—
38C3×#391—31no—
39#5229C2×C311yescharacteristic, complemented
40C2×#832C2×C311yescharacteristic, complemented
41C2×#821C2×C311yescharacteristic, complemented
42C3×#336—31no—
43#4550C2×C311yescharacteristic, complemented
44C24×C32—21no—
45C23×C3×D3—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8D412yescomplemented
52C2×C32×D4—21no—
53C2×C32×D4—21no—
54C2×C32×D4—21no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4D412yes—
58C22×C32×C4—21no—
59C22×C32×C4C2311yescharacteristic
60C2×C3×#67—28no—
61C2×C3×#67—44no—
62C2×C3×#67—44no—
63C22×C3×Dic3—21no—
64C22×C3×Dic3D412yes—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—21no—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12D412yes—
71C32×SD16—28no—
72C2×C32×C8—21no—
73C2×C32×C8—21no—
74C2×C32×C8D412yescomplemented
75C3×#215—28no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—21no—
78C3×C4×Dic3—21no—
79C3×C4×Dic3—21no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8D412yescomplemented
83#2014—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8C2×D311yescharacteristic
86C22×C3×Q8C22×C311yescharacteristic
87C22×C3×D4—21no—
88C22×C3×D4—31no—
89C22×C3×D4—62no—
90C22×C3×D4C2×D311yescharacteristic
91C22×C3×D4C22×C311yescharacteristic
92C22×#67—22no—
93C22×D12C22×C311yescharacteristic
94C2×C3×SD16—24no—
95C2×C3×SD16—24no—
96C2×C3×SD16—24no—
97C2×C3×SD16—22no—
98C2×C3×SD16—22no—
99C2×C3×SD16—22no—
100C2×C3×SD16—22no—
101C2×C3×SD16—22no—
102C2×C3×SD16—22no—
103C2×C3×SD16—62no—
104C2×C3×SD16—62no—
105C2×C3×SD16—62no—
106C22×C3×C8—21no—
107C22×C3×C8—31no—
108C22×C3×C8C2×D311yescharacteristic
109C22×C3×C8C22×C311yescharacteristic
110C3×#127—62no—
111C3×#127—62no—
112C2×C3×C4⋊C4—31no—
113C2×C3×C42—31no—
114C2×#215—22no—
115C2×#215—22no—
116C2×#215—22no—
117#940—22no—
118#940—22no—
119C2×C4⋊Dic3C22×C311yescharacteristic
120C2×C4×Dic3C22×C311yescharacteristic
121C22×C3⋊C8C22×C311yescharacteristic
122C3×#102—62no—
123C3×#102—62no—
124C3×#95—62no—
125#832—22no—
126C3⋊#95—22no—
127#821—22no—
128C23×C32—22no—
129C23×C32—41no—
130C23×C32—41no—
131C23×C32—42no—
132C23×C32—42no—
133C23×C32C2×D411yescharacteristic
134C22×C3×D3—22no—
135C22×C3×D3—41no—
136C22×C3×D3—41no—
137C22×C3×D3—42no—
138C22×C3×D3—42no—
139C32×Q8—22no—
140C32×Q8—22no—
141C32×Q8—41no—
142C32×Q8—41no—
143C32×D4—22no—
144C32×D4—22no—
145C32×D4—41no—
146C32×D4—41no—
147C2×C32×C4—22no—
148C2×C32×C4—41no—
149C2×C32×C4—41no—
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4#4111yescharacteristic
156C3×#67—416no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—41no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3SD1614yescomplemented
162C3×D12—22no—
163C3×D12—22no—
164C3×D12—41no—
165C3×D12—41no—
166C32×C8—22no—
167C3×C3⋊C8—22no—
168C22×SD16—31no—
169C22×SD16C3×D311yescharacteristic, complemented
170C2×#127—31no—
171C2×#102—31no—
172C2×#102—31no—
173#391—31no—
174#336—31no—
175C24×C3—21no—
176C24×C3—21no—
177C24×C3—41no—
178C24×C3—61no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—22no—
193C2×C3×Q8—41no—
194C2×C3×Q8—41no—
195C2×C3×Q8—41no—
196C2×C3×Q8#6712yescomplemented
197C2×C3×Q8C3×D412yescomplemented
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—22no—
211C2×C3×D4—32no—
212C2×C3×D4—41no—
213C2×C3×D4—41no—
214C2×C3×D4—41no—
215C2×C3×D4—68no—
216C2×C3×D4—61no—
217C2×C3×D4—61no—
218C2×C3×D4—61no—
219C2×C3×D4—61no—
220C2×C3×D4—61no—
221C2×C3×D4—124no—
222C2×C3×D4—124no—
223C2×C3×D4#6712yes—
224C2×C3×D4C3×D412yes—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—32no—
229C22×C3×C4—41no—
230C22×C3×C4—61no—
231C22×C3×C4C22×D311yescharacteristic
232C22×C3×C4C23×C311yescharacteristic
233C2×#67—28no—
234C2×#67—44no—
235C2×#67—44no—
236C22×Dic3—21no—
237C22×Dic3C3×D412yes—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12C3×D412yes—
244C3×SD16—216no—
245C3×SD16—28no—
246C3×SD16—28no—
247C3×SD16—68no—
248C2×C3×C8—22no—
249C2×C3×C8—22no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—22no—
255C2×C3×C8—32no—
256C2×C3×C8—61no—
257C2×C3×C8—61no—
258C2×C3×C8#6712yescomplemented
259C2×C3×C8C3×D412yescomplemented
260C3×C4⋊C4—61no—
261C3×C4⋊C4—61no—
262C3×C42—61no—
263C3×C42—61no—
264#215—28no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C3×D412yescomplemented
272C22×C32—42no—
273C22×C32—44no—
274C22×C32—82no—
275C22×C32—81no—
276C22×C32—81no—
277C22×C32#12011yescharacteristic
278C22×C32#12111yescharacteristic
279C22×C32#12711yescharacteristic
280C22×C32C2×SD1612yes—
281C22×C32AGL(1,ℤ/8)12yes—
282C2×C3×D3—42no—
283C2×C3×D3—44no—
284C2×C3×D3—82no—
285C2×C3×D3—81no—
286C2×C3×D3—81no—
287C32×C4—42no—
288C32×C4#12011yescharacteristic
289C32×C4#12011yescharacteristic
290C32×C4#12111yescharacteristic
291C32×C4#12111yescharacteristic
292C3×Dic3—24no—
293C3×Dic3—81no—
294C22×Q8C2×C3×D311yescharacteristic
295C22×D4—31no—
296C22×D4—62no—
297C22×D4C2×C3×D311yescharacteristic
298C2×SD16—22no—
299C2×SD16—22no—
300C2×SD16—22no—
301C2×SD16—62no—
302C2×SD16—62no—
303C2×SD16—62no—
304C22×C8—31no—
305C22×C8C2×C3×D311yescharacteristic
306#127—62no—
307#127—62no—
308C2×C4⋊C4—31no—
309C2×C42—31no—
310#102—62no—
311#102—62no—
312#95—62no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—21no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—62no—
328C23×C3—82no—
329C23×C3—82no—
330C23×C3—122no—
331C23×C3—122no—
332C23×C3—121no—
333C23×C3—121no—
334C23×C3D3×D411yescharacteristic
335C23×C3C2×C3×D411yescharacteristic
336C22×D3—22no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—42no—
340C22×D3—42no—
341C3×Q8—24no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8—42no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—41no—
353C3×D4—24no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4—42no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—62no—
366C3×D4—62no—
367C3×D4—1216no—
368C3×D4—121no—
369C3×D4—121no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—34no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—62no—
387C2×C3×C4—62no—
388C2×C3×C4—122no—
389C2×C3×C4—121no—
390C2×C3×C4D3×D411yescharacteristic
391C2×C3×C4D3×D411yescharacteristic
392C2×C3×C4#23711yescharacteristic
393C2×C3×C4C2×#6711yescharacteristic
394C2×C3×C4C2×#6711yescharacteristic
395C2×C3×C4C2×#6711yescharacteristic
396C2×C3×C4C2×C3×D411yescharacteristic
397C2×C3×C4C2×C3×D411yescharacteristic
398C2×C3×C4C2×C3×D411yescharacteristic
399C2×C3×C4C2×C3×D411yescharacteristic
400C2×C3×C4C2×C3×D411yescharacteristic
401C2×C3×C4C3×#4111yescharacteristic
402#67—416no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—41no—
406C2×Dic3—42no—
407C2×Dic3C3×SD1614yescomplemented
408D12—22no—
409D12—22no—
410D12—41no—
411D12—41no—
412C3×C8—24no—
413C3×C8—22no—
414C3×C8—22no—
415C3×C8—62no—
416C3⋊C8—22no—
417C2×C32—82no—
418C2×C32#39211yescharacteristic
419C2×C32#45012yes—
420C2×C32#46012yes—
421C2×C32#49211yescharacteristic
422C2×C32#49611yescharacteristic
423C3×D3—82no—
424C24—21no—
425C24—61no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C3×#6712yescomplemented
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—32no—
438C2×D4—68no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—124no—
445C2×D4—124no—
446C2×D4C3×#6712yes—
447C22×C4—21no—
448C22×C4—32no—
449C22×C4—61no—
450C22×C4C22×C3×D311yescharacteristic
451SD16—28no—
452SD16—68no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—32no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8C3×#6712yescomplemented
459C4⋊C4—61no—
460C4⋊C4—61no—
461C42—61no—
462C42—61no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—44no—
473C22×C3—84no—
474C22×C3—84no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—82no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—124no—
484C22×C3—122no—
485C22×C3—242no—
486C22×C3—241no—
487C22×C3—241no—
488C22×C3D3×SD1612yes—
489C22×C3#91412yes—
490C22×C3#93711yescharacteristic
491C22×C3#93911yescharacteristic
492C22×C3#94011yescharacteristic
493C22×C3C3×#12011yescharacteristic
494C22×C3C3×#12111yescharacteristic
495C22×C3C3×#12711yescharacteristic
496C22×C3C2×C3×SD1612yes—
497C22×C3C3×AGL(1,ℤ/8)12yes—
498C2×D3—42no—
499C2×D3—44no—
500C2×D3—82no—
501C2×D3—81no—
502C2×D3—81no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—44no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—64no—
511C3×C4—241no—
512C3×C4#93711yescharacteristic
513C3×C4#93811yescharacteristic
514C3×C4#93911yescharacteristic
515C3×C4#95311yescharacteristic
516C3×C4C3×#12011yescharacteristic
517C3×C4C3×#12011yescharacteristic
518C3×C4C3×#12111yescharacteristic
519C3×C4C3×#12111yescharacteristic
520Dic3—24no—
521Dic3—81no—
522C32#201411yescharacteristic, complemented, Sylow
523C23—22no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—62no—
529C23—122no—
530C23—122no—
531C23—121no—
532C23—121no—
533C23C3×D3×D411yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
542D4—62no—
543D4—62no—
544D4—1216no—
545D4—121no—
546D4—121no—
547C2×C4—22no—
548C2×C4—34no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—122no—
554C2×C4—121no—
555C2×C4C3×D3×D411yescharacteristic
556C2×C4C3×D3×D411yescharacteristic
557C2×C4C3×#23711yescharacteristic
558C2×C4C2×C3×#6711yescharacteristic
559C2×C4C2×C3×#6711yescharacteristic
560C2×C4C2×C3×#6711yescharacteristic
561C8—22no—
562C8—62no—
563C2×C3—22no—
564C2×C3—22no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—84no—
569C2×C3—82no—
570C2×C3—82no—
571C2×C3—242no—
572C2×C3#516312yes—
573C2×C3#516912yes—
574C2×C3#517511yescharacteristic
575C2×C3#517611yescharacteristic
576C2×C3#522311yescharacteristic
577C2×C3C3×#39211yescharacteristic
578C2×C3C3×#45012yes—
579C2×C3C3×#46012yes—
580C2×C3C3×#49211yescharacteristic
581C2×C3C3×#49611yescharacteristic
582D3—82no—
583C22—44no—
584C22—42no—
585C22—82no—
586C22—81no—
587C22—81no—
588C22—124no—
589C22—122no—
590C22—242no—
591C22—241no—
592C22—241no—
593C22C3×D3×SD1612yes—
594C22C3×#91412yes—
595C22C3×#93711yescharacteristic
596C22C3×#93911yescharacteristic
597C22C3×#94011yescharacteristic
598C4—42no—
599C4—64no—
600C4—241no—
601C4C3×#93711yescharacteristic
602C4C3×#93811yescharacteristic
603C4C3×#93911yescharacteristic
604C4C3×#95311yescharacteristic
605C3—21no—
606C3#7205611yescharacteristic, direct product factor
607C3C3×#201411yescharacteristic, complemented
608C2—82no—
609C2—242no—
610C2C3×#516312yes—
611C2C3×#516912yes—
612C2C3×#517511yescharacteristic
613C2C3×#517611yescharacteristic
614C2C3×#522311yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC2×C3×#940C2
hypercenterC22×C3×SD16D3
hypocenterC3C3×#2014
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C32×SD16C2semidirect
2. C22×C3×#215C2semidirect
4. C3×#5229C2semidirect
5. C2×C3×#832C2semidirect
6. C2×C3×#821C2semidirect
7. C3×#4550C2semidirect
9. #72056C3direct (N × H)
31. C22×C3×SD16D3semidirect
32. C22×C3×SD16C2×C3semidirect
34. C22×#215C2×C3semidirect
39. #5229C2×C3semidirect
40. C2×#832C2×C3semidirect
41. C2×#821C2×C3semidirect
43. #4550C2×C3semidirect
51. C2×C32×Q8D4semidirect
74. C2×C32×C8D4semidirect
82. C2×C3×C3⋊C8D4semidirect
161. C2×C3×Dic3SD16semidirect
169. C22×SD16C3×D3semidirect
196. C2×C3×Q8#67semidirect
197. C2×C3×Q8C3×D4semidirect
258. C2×C3×C8#67semidirect
259. C2×C3×C8C3×D4semidirect
271. C2×C3⋊C8C3×D4semidirect
407. C2×Dic3C3×SD16semidirect
431. C2×Q8C3×#67semidirect
458. C2×C8C3×#67semidirect
522. C32#2014semidirect
606. C3#72056direct (N × H)
607. C3C3×#2014semidirect

Maximal subgroups

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

subgroupindex
C22×C32×SD162
C22×C3×#2152
C2×C3×#9402
C3×#52292
C2×C3×#8322
C2×C3×#8212
C3×#45502
C3×#20143
#720563