Finite Group Databasebeta

← #10729012

Subgroups — #10729012

Order 768.

1626 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#388C211yesmaximal, characteristic
2C2×#4976C212yesmaximal, complemented
3C2×#4949C211yesmaximal, characteristic
4C2×#4945C212yesmaximal
5C2×#4928C211yesmaximal, characteristic, complemented
6C2×#4932C211yesmaximal, characteristic, complemented
7#71547C212yesmaximal
8#71546C211yesmaximal, characteristic
9#69423C211yesmaximal, characteristic, complemented
10#69421C212yesmaximal, complemented
11#69420C211yesmaximal, characteristic, complemented
12#14910—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2212yes—
26C3×#388—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37#4976—28no—
38#4949—24no—
39#4945—28no—
40#4932—24no—
41#4928—24no—
42C2×#399—31no—
43C2×#389—31no—
44C2×#389—32no—
45C2×#388—32no—
46C2×#388D311yescharacteristic
47C2×#376—31no—
48#1445—32no—
49#1445—31no—
50#1444—31no—
51#1439—31no—
52#1439—32no—
53C25×C3C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42D414yes—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×C4×Dic3—24no—
98C2×C4×Dic3—24no—
99C2×C4×Dic3D418yes—
100C3×#95—22no—
101C3×#95—22no—
102C3⋊#95—24no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95D414yes—
110C22×C4⋊C4—32no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C22⋊C4—32no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4C2×D311yescharacteristic
119C22×C22⋊C4C2×D311yescharacteristic
120C22×C22⋊C4C2×D311yescharacteristic
121C22×C42—32no—
122C22×C42C2×D311yescharacteristic
123#399—64no—
124#389—68no—
125#389—64no—
126#388—24no—
127#388—68no—
128#376—64no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D311yescharacteristic
139C24×C3—22no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—21no—
145C24×C3—21no—
146C24×C3—21no—
147C24×C3—22no—
148C24×C3—21no—
149C24×C3#4112yes—
150C24×C3C2411yescharacteristic
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4#4112yes—
182C22×C3×C4#4112yes—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3C2×D414yes—
207C22×Dic3C2×D414yes—
208C22×Dic3C2×D414yes—
209C22×Dic3#4114yes—
210C22×Dic3#4114yes—
211C3×C4⋊C4—44no—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—44no—
215C3×C42—28no—
216#217—216no—
217#217—48no—
218C4⋊Dic3—28no—
219C4⋊Dic3—44no—
220Dic3⋊C4—216no—
221Dic3⋊C4—48no—
222Dic3⋊C4—48no—
223C4×Dic3—216no—
224C25C22×D311yescharacteristic
225C23×C4—32no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—32no—
229C23×C4C22×D311yescharacteristic
230C23×C4C22×D311yescharacteristic
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—62no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—38no—
268C2×C42—64no—
269C2×C42—64no—
270C2×C42#6714yes—
271#95—22no—
272#95—22no—
273#95—34no—
274#95—64no—
275#95—64no—
276#95—64no—
277#95—64no—
278#95—64no—
279#95—62no—
280#95—64no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—41no—
304C23×C3C22×D411yescharacteristic
305C23×C3C22×D412yes—
306C23×C3C22×D411yescharacteristic
307C23×C3C2×#4111yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4112yes—
311C23×C3C2×#4112yes—
312C23×C32−1+412yes—
313C23×C32−1+411yescharacteristic
314C23×C32−1+411yescharacteristic
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4#12114yes—
346C2×C3×C4#12714yes—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3#12118yes—
375C24—21no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—21no—
384C24—21no—
385C24#23512yes—
386C24C23×D311yescharacteristic
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—34no—
413C22×C4—34no—
414C22×C4—34no—
415C22×C4—34no—
416C22×C4—34no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4#23512yes—
441C22×C4D3×D412yes—
442C22×C4D3×D414yes—
443C22×C4#23712yes—
444C22×C4C2×#6712yes—
445C22×C4C2×#6712yes—
446C22×C4C2×#6712yes—
447C4⋊C4—44no—
448C4⋊C4—616no—
449C4⋊C4—68no—
450C4⋊C4—128no—
451C4⋊C4—124no—
452C4⋊C4—128no—
453C22⋊C4—28no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—616no—
457C22⋊C4—128no—
458C42—28no—
459C42—616no—
460C22×C3—42no—
461C22×C3—44no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—44no—
465C22×C3—81no—
466C22×C3—82no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—82no—
470C22×C3—81no—
471C22×C3C2×#12111yescharacteristic
472C22×C3C2×#12112yes—
473C22×C3#52511yescharacteristic
474C22×C3#52512yes—
475C22×C3#52512yes—
476C22×C3#52511yescharacteristic
477C22×C3C2×#12711yescharacteristic
478C22×C3#53211yescharacteristic
479C22×C3#53212yes—
480C22×C3#53612yes—
481C22×C3#53612yes—
482C22×C3#53611yescharacteristic
483C22×C3#53611yescharacteristic
484C22×C3#54712yes—
485C22×C3#54711yescharacteristic
486C22×C3#54811yescharacteristic
487C22×C3#55212yes—
488C22×C3#55212yes—
489C22×C3#55212yes—
490C22×C3#55212yes—
491C22×C3#55211yescharacteristic
492C22×C3#55612yes—
493C22×C3#55611yescharacteristic
494C3×C4—24no—
495C3×C4—24no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4—81no—
500Dic3—28no—
501Dic3—44no—
502Dic3—44no—
503Dic3—82no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23C2×#23511yescharacteristic
528C23C2×#23511yescharacteristic
529C23C2×D3×D411yescharacteristic
530C23C2×D3×D412yes—
531C23C2×#23711yescharacteristic
532C23D3×#4112yes—
533C23D3×#4112yes—
534C23#101012yes—
535C23#101011yescharacteristic
536C23#101011yescharacteristic
537C23C22×#6711yescharacteristic
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—38no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4#93914yes—
597C2×C4#94014yes—
598C2×C3—82no—
599C2×C3#239711yescharacteristic
600C2×C3#241312yes—
601C2×C3#241311yescharacteristic
602C2×C3#241511yescharacteristic
603C2×C3#241512yes—
604C2×C3#260412yes—
605C2×C3#260411yescharacteristic
606C2×C3#261111yescharacteristic
607C2×C3#261112yes—
608C2×C3#265811yescharacteristic
609C2×C3#285611yescharacteristic
610C22—42no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—82no—
619C22—81no—
620C22—81no—
621C22#552912yes—
622C22#553112yes—
623C22#554111yescharacteristic
624C22#554211yescharacteristic
625C22#554712yes—
626C22#554712yes—
627C22#555111yescharacteristic
628C22#555211yescharacteristic
629C22#559612yes—
630C22D3×#12112yes—
631C22#565012yes—
632C22#565512yes—
633C22#567512yes—
634C22#567512yes—
635C22#570212yes—
636C22#570812yes—
637C22#579111yescharacteristic
638C22C2×#93911yescharacteristic
639C22C2×#94011yescharacteristic
640C22#582511yescharacteristic
641C22#582511yescharacteristic
642C22#582811yescharacteristic
643C22#583211yescharacteristic
644C4—24no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—68no—
649C4—81no—
650C4—81no—
651C4—124no—
652C4—124no—
653C4—242no—
654C3#1491011yescharacteristic, complemented, Sylow
655C2—82no—
656C2#7549612yes—
657C2#7564812yes—
658C2#7565112yes—
659C2#7584312yes—
660C2#8226111yescharacteristic
661C2#8234711yescharacteristic
662C2#8235111yescharacteristic
663C2#8270011yescharacteristic
664C2#8284311yescharacteristic
665C2#8298011yescharacteristic
666C2#8308811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#388C2
prime radicalC25×C3C23
hypercenterC2×#388D3
hypocenterC3#14910
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#4976C2semidirect
5. C2×#4928C2semidirect
6. C2×#4932C2semidirect
9. #69423C2semidirect
10. #69421C2semidirect
11. #69420C2semidirect
654. C3#14910semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3882
C2×#49762
C2×#49492
C2×#49452
C2×#49282
C2×#49322
#715472
#715462
#694232
#694212
#694202
#149103