Finite Group Databasebeta

← #10943993

Subgroups — #10943993

Order 768.

1608 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#235C211yesmaximal, characteristic, complemented
2C3×#2856C211yesmaximal, characteristic, complemented
3C2×#5075C211yesmaximal, characteristic, complemented
4C2×#5072C211yesmaximal, characteristic, complemented
5C2×#5037C212yesmaximal, complemented
6#71678C212yesmaximal, complemented
7#71530C211yesmaximal, characteristic, complemented
8#69660C212yesmaximal, complemented
9#69657C212yesmaximal, complemented
10#69588C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic, complemented
12#32904—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#121—21no—
16C2×C3×#121—21no—
17C2×C3×#121C2212yes—
18C22×C3×C42C2211yescharacteristic
19C22×#214C2212yes—
20C22×#213C2212yes—
21C2×C4×#67—21no—
22C2×C4×#67—21no—
23C2×#926C2211yescharacteristic
24C4×#235—24no—
25C4×#235—24no—
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×#872—21no—
29C2×C42×D3—21no—
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C3×#390—21no—
33C3×#390—21no—
34C3×#390C2212yes—
35C3×#384C2211yescharacteristic
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×C4×#41—31no—
50C2×#460—31no—
51C2×#459—31no—
52C2×#449—32no—
53#2856D311yescharacteristic, complemented
54#1880—32no—
55#1879—32no—
56#1875—32no—
57#1803—31no—
58#1802—31no—
59#1797—31no—
60C22×C3×D4—22no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2312yes—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235D414yescomplemented
74C22×D12C2311yescharacteristic
75C22×C4×D3—21no—
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C3×#127—22no—
80C3×#127—22no—
81C3×#121—24no—
82C3×#121—24no—
83C3×#121—44no—
84C3×#121—44no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C42—22no—
92C2×C3×C42—22no—
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D411yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#214—24no—
99C2×#214—24no—
100C2×#214—24no—
101C2×#213—24no—
102C2×#213—24no—
103C2×#213—24no—
104C4×#67—28no—
105C4×#67—28no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108#926—22no—
109#926—22no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2312yes—
118C22×C3⋊C8C2312yes—
119C4×D12—22no—
120C4×D12—22no—
121C4×D12—22no—
122C4×D12D414yescomplemented
123#872—28no—
124C42×D3—28no—
125C4×Dic6—22no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C4×Dic6D414yescomplemented
129#832—24no—
130#832—24no—
131#830—24no—
132#810—24no—
133#809—24no—
134C4⋊(C3⋊C8)—22no—
135C4⋊(C3⋊C8)—22no—
136C22×#41—31no—
137C22×SD16—32no—
138C22×D8—32no—
139C2×#127C2×D311yescharacteristic
140C2×#121—21no—
141C2×#121—21no—
142C2×#121C2×D312yes—
143C4×#41—64no—
144C4×#41—64no—
145C2×C4×Q8—31no—
146C2×C4×Q8—31no—
147C2×C4×D4—31no—
148C2×C4×D4—31no—
149C2×C4×D4—61no—
150C2×C4×D4—61no—
151C2×#117—31no—
152C2×#117—61no—
153C22×C42—61no—
154C22×C42C2×D311yescharacteristic
155#460—64no—
156#459—64no—
157#449—68no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102—32no—
164C2×C22⋊C8—32no—
165#390—21no—
166#390—21no—
167#390C2×D312yes—
168#384C2×D311yescharacteristic
169#340—32no—
170#340—32no—
171#336—32no—
172C24×C3—22no—
173C24×C3C2×D411yescharacteristic
174C23×D3—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—41no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4#4112yes—
208C22×C3×C4C2411yescharacteristic
209C2×#67—24no—
210C2×#67—24no—
211C2×#67—24no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215#235—28no—
216#235—28no—
217#235—28no—
218#235—28no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C2×D412yes—
223C2×D12C2×D412yes—
224C2×D12#4112yes—
225C2×C4×D3—24no—
226C2×C4×D3—24no—
227C2×C4×D3—24no—
228C2×C4×D3—24no—
229C2×C4×D3—24no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×D412yes—
238C2×Dic6C2×D412yes—
239C2×Dic6#4112yes—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—42no—
244C3×C22⋊C4—44no—
245C3×C22⋊C4—44no—
246C3×C22⋊C4—44no—
247C3×C42—22no—
248C3×C42—22no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42C2×D412yes—
252C3×C42C2×D412yes—
253#217—24no—
254#214—48no—
255#213—48no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D412yes—
261C4⋊Dic3C2×D412yes—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264C4×Dic3—24no—
265C4×Dic3—24no—
266C4×Dic3—24no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×#41—34no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—62no—
279C22×Q8—31no—
280C22×D4—22no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4—31no—
284C22×D4—61no—
285C22×D4C22×D312yes—
286C23×C4—61no—
287C23×C4—61no—
288C23×C4—61no—
289C23×C4#6711yescharacteristic
290C23×C4#6711yescharacteristic
291C23×C4C22×D311yescharacteristic
292C2×SD16—64no—
293C2×SD16—64no—
294C2×SD16—64no—
295C2×D8—64no—
296C2×D8—64no—
297C2×D8—64no—
298C22×C8—32no—
299C22×C8—32no—
300#127—22no—
301#127—22no—
302#121—24no—
303#121—24no—
304#121—44no—
305#121—44no—
306C4×Q8—34no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×D4—34no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—68no—
314C4×D4—68no—
315C4×D4—62no—
316#117—68no—
317#117—62no—
318#117—62no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D312yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—68no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42#6711yescharacteristic
339C2×C42#6711yescharacteristic
340C2×C42C22×D311yescharacteristic
341C2×C42C22×D311yescharacteristic
342C4⋊C8—62no—
343C4⋊C8—62no—
344#103—64no—
345#102—64no—
346#102—64no—
347#102—64no—
348C22⋊C8—64no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—42no—
356C23×C3—42no—
357C23×C3—42no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3#12112yes—
363C23×C3C22×D411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—84no—
376C3×D4—88no—
377C3×D4—84no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—44no—
392C2×C3×C4—44no—
393C2×C3×C4—42no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4#12012yes—
397C2×C3×C4#12012yes—
398C2×C3×C4#12112yes—
399C2×C3×C4#13512yes—
400C2×C3×C4AGL(1,ℤ/8)12yes—
401C2×C3×C4AGL(1,ℤ/8)12yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408#67—216no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418D12—22no—
419D12—22no—
420D12—22no—
421D12#12114yescomplemented
422C4×D3—216no—
423C4×D3—48no—
424C4×D3—48no—
425Dic6—22no—
426Dic6—22no—
427Dic6—22no—
428Dic6#12114yescomplemented
429C3⋊C8—42no—
430C3⋊C8—42no—
431C24—22no—
432C24—61no—
433C24C2×#6711yescharacteristic
434#41—68no—
435#41—68no—
436#41—68no—
437#41—68no—
438C2×Q8—32no—
439C2×Q8—32no—
440C2×Q8—32no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—32no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—64no—
463C2×D4—64no—
464C2×D4—64no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—64no—
482C22×C4—61no—
483C22×C4—64no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—64no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4D3×D412yes—
492C22×C4#23712yes—
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6711yescharacteristic
496C22×C4C2×#6711yescharacteristic
497C22×C4C2×#6711yescharacteristic
498C22×C4C23×D311yescharacteristic
499SD16—128no—
500D8—128no—
501C2×C8—62no—
502C2×C8—62no—
503C2×C8—62no—
504C2×C8—62no—
505C2×C8—62no—
506C2×C8—62no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—32no—
510C4⋊C4—32no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C22⋊C4—64no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—64no—
528C42—64no—
529C42—64no—
530C42D3×D412yes—
531C42D3×D412yes—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—82no—
540C22×C3—82no—
541C22×C3—82no—
542C22×C3—82no—
543C22×C3—82no—
544C22×C3—82no—
545C22×C3—82no—
546C22×C3C2×#12011yescharacteristic
547C22×C3C2×#12111yescharacteristic
548C22×C3#52511yescharacteristic
549C22×C3C2×#13511yescharacteristic
550C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
551C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
552C22×C3#57511yescharacteristic
553C2×D3—24no—
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—42no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—22no—
563C3×C4—41no—
564C3×C4—82no—
565C3×C4#44914yescomplemented
566C3×C4C2×#12111yescharacteristic
567C3×C4C2×#12111yescharacteristic
568C3×C4#52511yescharacteristic
569C3×C4#52511yescharacteristic
570Dic3—24no—
571Dic3—42no—
572Dic3—42no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—62no—
587C23—62no—
588C23—62no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23#93812yes—
594C23C22×#6711yescharacteristic
595Q8—34no—
596Q8—62no—
597Q8—62no—
598Q8—62no—
599D4—34no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—616no—
605D4—62no—
606D4—62no—
607D4—62no—
608D4—84no—
609D4—88no—
610D4—84no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—616no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—128no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—128no—
641C2×C4#93212yes—
642C2×C4#93212yes—
643C2×C4#93712yes—
644C2×C4#93712yes—
645C2×C4#93812yes—
646C2×C4#95012yes—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×D3×D411yescharacteristic
650C2×C4C2×#23711yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C8—122no—
654C8—122no—
655C2×C3—42no—
656C2×C3—162no—
657C2×C3#284711yescharacteristic
658C2×C3C2×#44911yescharacteristic
659C2×C3#300611yescharacteristic
660C2×C3#303511yescharacteristic
661C2×C3#303811yescharacteristic
662C2×C3#310611yescharacteristic
663C2×C3#311511yescharacteristic
664D3—44no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—64no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—122no—
681C22—122no—
682C22—122no—
683C22—122no—
684C22—122no—
685C22—122no—
686C22C2×#93211yescharacteristic
687C22C2×#93211yescharacteristic
688C22C2×#93711yescharacteristic
689C22C2×#93811yescharacteristic
690C22C2×#95011yescharacteristic
691C22#582211yescharacteristic
692C22#582811yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—41no—
697C4—64no—
698C4—82no—
699C4—122no—
700C4—122no—
701C4#503714yescomplemented
702C4D3×#12111yescharacteristic
703C4D3×#12111yescharacteristic
704C4#560611yescharacteristic
705C4#560611yescharacteristic
706C3#3290411yescharacteristic, complemented, Sylow
707C2—42no—
708C2—124no—
709C2—162no—
710C2C2×#503711yescharacteristic
711C2#8262411yescharacteristic
712C2#8272311yescharacteristic
713C2#8316511yescharacteristic
714C2#8329611yescharacteristic
715C2#8329911yescharacteristic
716C2#8330111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2856C2
socleC23×C3C22×D4
hypercenter#2856D3
hypocenterC3#32904
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×C4×#235C2semidirect
2. C3×#2856C2semidirect
3. C2×#5075C2semidirect
4. C2×#5072C2semidirect
5. C2×#5037C2semidirect
6. #71678C2semidirect
7. #71530C2semidirect
8. #69660C2semidirect
9. #69657C2semidirect
10. #69588C2semidirect
11. #69584C2semidirect
53. #2856D3semidirect
73. C2×#235D4semidirect
122. C4×D12D4semidirect
128. C4×Dic6D4semidirect
421. D12#121semidirect
428. Dic6#121semidirect
565. C3×C4#449semidirect
701. C4#5037semidirect
706. C3#32904semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
C3×#28562
C2×#50752
C2×#50722
C2×#50372
#716782
#715302
#696602
#696572
#695882
#695842
#329043