Finite Group Databasebeta

← #10944084

Subgroups — #10944084

Order 768.

1506 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5523C211yesmaximal, characteristic, complemented
2C3×#2619C211yesmaximal, characteristic, complemented
3C2×#5075C211yesmaximal, characteristic, complemented
4C2×#5072C211yesmaximal, characteristic, complemented
5C2×#5037C212yesmaximal, complemented
6#71677C212yesmaximal, complemented
7#71583C211yesmaximal, characteristic, complemented
8#69660C212yesmaximal, complemented
9#69657C212yesmaximal, complemented
10#69559C211yesmaximal, characteristic, complemented
11#69556C211yesmaximal, characteristic, complemented
12#32913—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×C3×#117C2211yescharacteristic
17C22×#214C2212yes—
18C22×#213C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20#5523—24no—
21#5523—24no—
22C2×#892—21no—
23C2×D3×C4⋊C4—21no—
24C2×#67⋊C4—21no—
25C2×#67⋊C4—21no—
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#394—21no—
31C3×#394—21no—
32C3×#392—21no—
33C3×#392—21no—
34C3×#390C2212yes—
35C3×#382C2211yescharacteristic
36C2×#832C2212yes—
37C2×#832C2212yes—
38C2×#830C2212yes—
39#5075—24no—
40#5072—24no—
41#5037—28no—
42C2×#810C2212yes—
43C2×#809C2212yes—
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4529C2212yes—
47#4529C2212yes—
48#4524C2212yes—
49C2×#520—31no—
50C2×#460—31no—
51C2×#459—31no—
52C2×#449—32no—
53#2619D311yescharacteristic, complemented
54#1880—32no—
55#1879—32no—
56#1874—32no—
57#1805—31no—
58#1804—31no—
59#1798—31no—
60C22×C3×D4—22no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C22×#67—21no—
66C23×Dic3—21no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235D414yescomplemented
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×C4×D3—21no—
77C22×Dic6C2311yescharacteristic
78C3×#127—22no—
79C3×#127—22no—
80C3×#121—24no—
81C3×#121—24no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#214—24no—
95C2×#214—24no—
96C2×#214—24no—
97C2×#213—24no—
98C2×#213—24no—
99C2×#213—24no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#892—28no—
114D3×C4⋊C4—28no—
115#67⋊C4—28no—
116#67⋊C4—28no—
117C4×D12—22no—
118C4×D12—22no—
119C4×D12—22no—
120C4×D12D414yescomplemented
121C4×Dic6—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6D414yescomplemented
125C3×#95—21no—
126C3×#95—21no—
127#832—24no—
128#832—24no—
129#830—24no—
130#810—24no—
131#809—24no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×SD16—32no—
136C22×D8—32no—
137C2×#127C2×D311yescharacteristic
138C2×#121C2×D312yes—
139#520—64no—
140#520—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—61no—
148C2×#117C2×D311yescharacteristic
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#460—64no—
152#459—64no—
153#449—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×#103—32no—
157C2×#102—32no—
158C2×#102—32no—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#394—21no—
162#394—21no—
163#392—21no—
164#392—21no—
165#390C2×D312yes—
166#382C2×D311yescharacteristic
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—22no—
171C24×C3C2×D411yescharacteristic
172C23×D3—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—44no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—82no—
183C2×C3×D4—82no—
184C2×C3×D4—82no—
185C2×C3×D4—82no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C2×#67—24no—
200C2×#67—24no—
201C2×#67—24no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—24no—
206C22×Dic3—82no—
207#235—28no—
208#235—28no—
209#235—28no—
210#235—28no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12C2×D412yes—
215C2×D12C2×D412yes—
216C2×D12#4112yes—
217C2×C4×D3—24no—
218C2×C4×D3—24no—
219C2×C4×D3—24no—
220C2×C4×D3—24no—
221C2×C4×D3—24no—
222C2×C4×D3—82no—
223C2×C4×D3—82no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C2×D412yes—
228C2×Dic6C2×D412yes—
229C2×Dic6#4112yes—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4—82no—
238C3×C22⋊C4—82no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42C2×D412yes—
242C3×C42C2×D412yes—
243#214—48no—
244#213—48no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—24no—
247C4⋊Dic3—24no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3C2×D412yes—
251C4⋊Dic3C2×D412yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254C4×Dic3—24no—
255C4×Dic3—24no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×#41—34no—
263C2×#41—62no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C22×Q8—31no—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4—21no—
272C22×D4—31no—
273C22×D4—61no—
274C22×D4C22×D312yes—
275C23×C4—61no—
276C23×C4—61no—
277C23×C4—61no—
278C23×C4—61no—
279C23×C4C22×D311yescharacteristic
280C2×SD16—64no—
281C2×SD16—64no—
282C2×SD16—64no—
283C2×D8—64no—
284C2×D8—64no—
285C2×D8—64no—
286C22×C8—32no—
287C22×C8—32no—
288#127—22no—
289#127—22no—
290#121—24no—
291#121—24no—
292C4×Q8—34no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×D4—34no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—68no—
300C4×D4—68no—
301C4×D4—62no—
302#117—22no—
303#117—22no—
304#117—68no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—68no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4#6711yescharacteristic
314C2×C4⋊C4#6711yescharacteristic
315C2×C4⋊C4C22×D312yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4#6711yescharacteristic
322C2×C22⋊C4#6711yescharacteristic
323C2×C42—61no—
324C2×C42—61no—
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327C4⋊C8—62no—
328C4⋊C8—62no—
329#103—64no—
330#102—64no—
331#102—64no—
332#102—64no—
333C22⋊C8—64no—
334#95—21no—
335#95—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3—42no—
346C23×C3—42no—
347C23×C3#12112yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—22no—
351C22×D3—22no—
352C22×D3—42no—
353C22×D3—42no—
354C3×D4—42no—
355C3×D4—42no—
356C3×D4—42no—
357C3×D4—42no—
358C3×D4—84no—
359C3×D4—84no—
360C3×D4—88no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4#12012yes—
377C2×C3×C4#12012yes—
378C2×C3×C4#12112yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385#67—216no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—84no—
394D12—22no—
395D12—22no—
396D12—22no—
397D12#12114yes—
398C4×D3—216no—
399C4×D3—84no—
400C4×D3—84no—
401Dic6—22no—
402Dic6—22no—
403Dic6—22no—
404Dic6#12114yes—
405C3⋊C8—42no—
406C3⋊C8—42no—
407C24—22no—
408C24—61no—
409C24C2×#6711yescharacteristic
410#41—68no—
411#41—68no—
412#41—68no—
413#41—68no—
414C2×Q8—32no—
415C2×Q8—32no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—32no—
427C2×D4—32no—
428C2×D4—32no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4—61no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—64no—
455C22×C4—64no—
456C22×C4—242no—
457C22×C4—242no—
458C22×C4—242no—
459C22×C4D3×D412yes—
460C22×C4#23712yes—
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6711yescharacteristic
464C22×C4C2×#6711yescharacteristic
465C22×C4C2×#6711yescharacteristic
466C22×C4C23×D311yescharacteristic
467SD16—128no—
468D8—128no—
469C2×C8—62no—
470C2×C8—62no—
471C2×C8—62no—
472C2×C8—62no—
473C2×C8—62no—
474C2×C8—62no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—32no—
480C4⋊C4—32no—
481C4⋊C4—64no—
482C4⋊C4—64no—
483C4⋊C4—64no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—44no—
489C22⋊C4—64no—
490C22⋊C4—64no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—21no—
494C42—21no—
495C42—64no—
496C42—64no—
497C42D3×D412yes—
498C42D3×D412yes—
499C22×C3—22no—
500C22×C3—24no—
501C22×C3—42no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—82no—
505C22×C3—82no—
506C22×C3—82no—
507C22×C3—82no—
508C22×C3—82no—
509C22×C3—82no—
510C22×C3—82no—
511C22×C3C2×#12011yescharacteristic
512C22×C3C2×#12111yescharacteristic
513C22×C3#52511yescharacteristic
514C22×C3#57612yes—
515C22×C3#57712yes—
516C2×D3—24no—
517C2×D3—44no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—44no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4—82no—
527C3×C4C2×#12111yescharacteristic
528C3×C4C2×#12111yescharacteristic
529C3×C4#52511yescharacteristic
530C3×C4#52511yescharacteristic
531Dic3—24no—
532Dic3—42no—
533Dic3—42no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—62no—
546C23—62no—
547C23—62no—
548C23—122no—
549C23—122no—
550C23#93812yes—
551C23C22×#6711yescharacteristic
552Q8—34no—
553Q8—62no—
554Q8—62no—
555Q8—62no—
556D4—34no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—616no—
562D4—62no—
563D4—62no—
564D4—62no—
565D4—84no—
566D4—84no—
567D4—88no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—616no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—244no—
592C2×C4—244no—
593C2×C4—244no—
594C2×C4#93712yes—
595C2×C4#93712yes—
596C2×C4#93812yes—
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×D3×D411yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×#23711yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C8—122no—
604C8—122no—
605C2×C3—42no—
606C2×C3—162no—
607C2×C3#284711yescharacteristic
608C2×C3#300811yescharacteristic
609C2×C3#301011yescharacteristic
610C2×C3#304512yes—
611C2×C3#311912yes—
612D3—44no—
613C22—22no—
614C22—24no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—64no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—122no—
627C22—124no—
628C22—122no—
629C22—124no—
630C22C2×#93711yescharacteristic
631C22C2×#93811yescharacteristic
632C22#582811yescharacteristic
633C22#583512yes—
634C22#583612yes—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—64no—
641C4—82no—
642C4—122no—
643C4—122no—
644C4D3×#12111yescharacteristic
645C4D3×#12111yescharacteristic
646C4#560611yescharacteristic
647C4#560611yescharacteristic
648C3#3291311yescharacteristic, complemented, Sylow
649C2—42no—
650C2—124no—
651C2—162no—
652C2#8272311yescharacteristic
653C2#8338311yescharacteristic
654C2#8338512yes—
655C2#8339011yescharacteristic
656C2#8343412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2619C2
socleC23×C3C22×D4
hypercenter#2619D3
hypocenterC3#32913
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#5523C2semidirect
2. C3×#2619C2semidirect
3. C2×#5075C2semidirect
4. C2×#5072C2semidirect
5. C2×#5037C2semidirect
6. #71677C2semidirect
7. #71583C2semidirect
8. #69660C2semidirect
9. #69657C2semidirect
10. #69559C2semidirect
11. #69556C2semidirect
53. #2619D3semidirect
72. C2×#235D4semidirect
120. C4×D12D4semidirect
124. C4×Dic6D4semidirect
648. C3#32913semidirect

Maximal subgroups

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

subgroupindex
C2×#55232
C3×#26192
C2×#50752
C2×#50722
C2×#50372
#716772
#715832
#696602
#696572
#695592
#695562
#329133