Finite Group Databasebeta

← #10944001

Subgroups — #10944001

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
2#82081C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×#4843C211yesmaximal, characteristic, complemented
5C2×#4734C212yesmaximal, complemented
6C3×#1797C211yesmaximal, characteristic, complemented
7#70485C211yesmaximal, characteristic, complemented
8#70479C211yesmaximal, characteristic, complemented
9#69933C212yesmaximal, complemented
10#68686C212yesmaximal, complemented
11#68673C212yesmaximal, complemented
12#32904—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930—21no—
17C2×#930C2212yes—
18C2×C4×#67—21no—
19C2×C4×#67—21no—
20C2×#926C2211yescharacteristic
21C22×D24C2212yes—
22C22×#204C2212yes—
23C2×#874C2211yescharacteristic
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
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×C22⋊C8C2212yes—
35C2×#821C2212yes—
36C2×#821C2212yes—
37C2×#821C2212yes—
38C2×#816C2212yes—
39#4939—21no—
40#4939—21no—
41#4939C2212yes—
42#4935C2211yescharacteristic
43#4848—24no—
44#4843—24no—
45#4734—28no—
46C3×#336C2212yes—
47#4551C2212yes—
48#4551C2212yes—
49C2×C4×#41—31no—
50C2×#460—31no—
51C2×#459—31no—
52C2×#449—32no—
53#2856—31no—
54#1880—32no—
55#1879—32no—
56#1875—32no—
57#1803—31no—
58#1802—31no—
59#1797D311yescharacteristic, complemented
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235D414yescomplemented
71C22×D12—21no—
72C22×D12—21no—
73C22×D12C2312yes—
74C22×D12C2311yescharacteristic
75C22×C4×D3—21no—
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2312yes—
80C22×C3×C8C2312yes—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88#930—24no—
89#930—24no—
90#930—44no—
91#930—44no—
92C4×#67—28no—
93C4×#67—28no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×D24—24no—
112C2×D24—24no—
113C2×D24—24no—
114C2×#204—24no—
115C2×#204—24no—
116C2×#204—24no—
117#874—22no—
118#874—22no—
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
129C3×C4⋊C8—22no—
130C3×C4⋊C8—22no—
131C3×C22⋊C8—24no—
132#821—24no—
133#821—24no—
134#821—24no—
135#816—24no—
136C22×#41—31no—
137C22×SD16—32no—
138C22×D8—32no—
139C2×#127—31no—
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
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⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102—32no—
164C2×C22⋊C8C2×D312yes—
165#390—32no—
166#390—61no—
167#390—61no—
168#384—31no—
169#340—32no—
170#340—32no—
171#336C2×D312yes—
172C24×C3C2×D411yescharacteristic
173C23×D3—21no—
174C23×D3—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—41no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—24no—
194C2×#67—24no—
195C2×#67—24no—
196C2×#67—44no—
197C2×#67—44no—
198C2×#67—44no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203#235—28no—
204#235—28no—
205#235—28no—
206#235—28no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—42no—
217C2×D12—42no—
218C2×D12—42no—
219C2×D12—42no—
220C2×D12—42no—
221C2×D12—42no—
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—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C2×C3×C8—22no—
244C2×C3×C8—22no—
245C2×C3×C8—22no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42C2×D412yes—
251C3×C42C2×D412yes—
252#217—24no—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—44no—
256(C2×D3)⋊C4—44no—
257(C2×D3)⋊C4—44no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—22no—
261C4⋊Dic3—22no—
262C4⋊Dic3—42no—
263C4⋊Dic3—42no—
264C4⋊Dic3C2×D412yes—
265C4⋊Dic3C2×D412yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268C4×Dic3—24no—
269C4×Dic3—24no—
270C4×Dic3—24no—
271D24—48no—
272#204—48no—
273C2×#41—34no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—62no—
279C22×Q8—31no—
280C22×D4—31no—
281C22×D4—32no—
282C22×D4—61no—
283C22×D4—62no—
284C22×D4—61no—
285C22×D4—61no—
286C23×C4—61no—
287C23×C4—61no—
288C23×C4—61no—
289C23×C4D1211yescharacteristic
290C23×C4D1211yescharacteristic
291C23×C4C22×D311yescharacteristic
292C2×SD16—64no—
293C2×SD16—64no—
294C2×SD16—64no—
295C2×D8—64no—
296C2×D8—64no—
297C2×D8—64no—
298C22×C8C22×D312yes—
299C22×C8C22×D312yes—
300#127—62no—
301#127—62no—
302#121—64no—
303#121—64no—
304#121—124no—
305#121—124no—
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—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—62no—
330C2×C22⋊C4—62no—
331C2×C22⋊C4—62no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—68no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42D1211yescharacteristic
339C2×C42D1211yescharacteristic
340C2×C42C22×D311yescharacteristic
341C2×C42C22×D311yescharacteristic
342C4⋊C8—22no—
343C4⋊C8—22no—
344#103—64no—
345#102—64no—
346#102—64no—
347#102—64no—
348C22⋊C8—24no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3#12112yes—
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—42no—
365C22×D3—42no—
366C22×D3—42no—
367C22×D3—42no—
368C22×D3—42no—
369C22×D3—42no—
370C22×D3—42no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—44no—
385C2×C3×C4—44no—
386C2×C3×C4#12012yes—
387C2×C3×C4#12012yes—
388C2×C3×C4#12112yes—
389C2×C3×C4#13512yes—
390C2×C3×C4AGL(1,ℤ/8)12yes—
391C2×C3×C4AGL(1,ℤ/8)12yes—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398#67—216no—
399#67—88no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—42no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412D12—22no—
413D12—22no—
414D12—22no—
415D12—42no—
416D12—42no—
417D12—42no—
418D12—42no—
419D12—84no—
420D12—84no—
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—61no—
432C24—62no—
433C24C2×D1211yescharacteristic
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—32no—
445C2×D4—32no—
446C2×D4—32no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—64no—
450C2×D4—61no—
451C2×D4—62no—
452C2×D4—62no—
453C2×D4—62no—
454C2×D4—61no—
455C2×D4—62no—
456C2×D4—61no—
457C2×D4—62no—
458C2×D4—62no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4—124no—
462C2×D4—124no—
463C2×D4—124no—
464C2×D4—122no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—122no—
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—41no—
478C22×C4—64no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—61no—
482C22×C4—64no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—64no—
486C22×C4—62no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4C2×D1211yescharacteristic
492C22×C4C2×D1211yescharacteristic
493C22×C4C2×D1211yescharacteristic
494C22×C4C2×D1211yescharacteristic
495C22×C4C2×D1211yescharacteristic
496C22×C4D3×D412yes—
497C22×C4#23912yes—
498C22×C4C23×D311yescharacteristic
499SD16—128no—
500D8—128no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C4⋊C4—32no—
508C4⋊C4—32no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—62no—
514C4⋊C4—62no—
515C4⋊C4—122no—
516C4⋊C4—122no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C22⋊C4—124no—
522C22⋊C4—124no—
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×C3C2×#12011yescharacteristic
540C22×C3C2×#12111yescharacteristic
541C22×C3#52511yescharacteristic
542C22×C3C2×#13511yescharacteristic
543C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
544C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
545C22×C3#57511yescharacteristic
546C2×D3—24no—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—82no—
554C2×D3—82no—
555C2×D3—82no—
556C2×D3—82no—
557C2×D3—82no—
558C2×D3—82no—
559C2×D3—82no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—22no—
563C3×C4—41no—
564C3×C4#44914yescomplemented
565C3×C4C2×#12111yescharacteristic
566C3×C4C2×#12111yescharacteristic
567C3×C4#52511yescharacteristic
568C3×C4#52511yescharacteristic
569Dic3—24no—
570Dic3—42no—
571Dic3—42no—
572Dic3—82no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—122no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23#89512yes—
594C23C22×D1211yescharacteristic
595Q8—34no—
596Q8—62no—
597Q8—62no—
598Q8—62no—
599D4—34no—
600D4—616no—
601D4—62no—
602D4—62no—
603D4—62no—
604D4—122no—
605D4—122no—
606D4—122no—
607D4—122no—
608D4—244no—
609D4—248no—
610D4—244no—
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—616no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—128no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—128no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4#88212yes—
642C2×C4#88212yes—
643C2×C4#89512yes—
644C2×C4#90412yes—
645C2×C4#90812yes—
646C2×C4#90812yes—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×D3×D411yescharacteristic
650C2×C4C2×#23911yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C8—42no—
654C8—42no—
655C2×C3—42no—
656C2×C3#284711yescharacteristic
657C2×C3C2×#44911yescharacteristic
658C2×C3#300611yescharacteristic
659C2×C3#303511yescharacteristic
660C2×C3#303811yescharacteristic
661C2×C3#310611yescharacteristic
662C2×C3#311511yescharacteristic
663D3—44no—
664D3—162no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—64no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—122no—
677C22—122no—
678C22—122no—
679C22—242no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22—242no—
684C22—242no—
685C22—242no—
686C22C2×#88211yescharacteristic
687C22C2×#89511yescharacteristic
688C22#552411yescharacteristic
689C22C2×#90411yescharacteristic
690C22C2×#90811yescharacteristic
691C22C2×#90811yescharacteristic
692C22#574811yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—41no—
697C4—64no—
698C4—122no—
699C4—122no—
700C4—242no—
701C4#473414yescomplemented
702C4D3×#12111yescharacteristic
703C4D3×#12111yescharacteristic
704C4#562711yescharacteristic
705C4#562711yescharacteristic
706C3#3290411yescharacteristic, complemented, Sylow
707C2—42no—
708C2—124no—
709C2—482no—
710C2#7404811yescharacteristic
711C2C2×#473411yescharacteristic
712C2#7863911yescharacteristic
713C2#7902111yescharacteristic
714C2#7905611yescharacteristic
715C2#7930111yescharacteristic
716C2#7934711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
hypercenter#1797D3
hypocenterC3#32904
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×C4×#235C2semidirect
2. #82081C2semidirect
3. C2×#4848C2semidirect
4. C2×#4843C2semidirect
5. C2×#4734C2semidirect
6. C3×#1797C2semidirect
7. #70485C2semidirect
8. #70479C2semidirect
9. #69933C2semidirect
10. #68686C2semidirect
11. #68673C2semidirect
59. #1797D3semidirect
70. C2×#235D4semidirect
122. C4×D12D4semidirect
128. C4×Dic6D4semidirect
421. D12#121semidirect
428. Dic6#121semidirect
564. C3×C4#449semidirect
701. C4#4734semidirect
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
#820812
C2×#48482
C2×#48432
C2×#47342
C3×#17972
#704852
#704792
#699332
#686862
#686732
#329043