Finite Group Databasebeta

← #10943994

Subgroups — #10943994

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×#239C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×#5172C212yesmaximal, complemented
4#83192C211yesmaximal, characteristic, complemented
5C2×#5072C211yesmaximal, characteristic, complemented
6#72048C212yesmaximal, complemented
7#71626C211yesmaximal, characteristic, complemented
8#70479C211yesmaximal, characteristic, complemented
9#70475C211yesmaximal, characteristic, complemented
10#70208C212yesmaximal, complemented
11#70201C212yesmaximal, complemented
12#32904—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×Q8×Dic3C2211yescharacteristic
18C2×#938—21no—
19C2×#938—21no—
20C2×#938C2212yes—
21C22×#213C2212yes—
22C4×#239—24no—
23C4×#239—24no—
24C2×#893—21no—
25C2×#893C2211yescharacteristic
26C2×#892—21no—
27C2×#892C2211yescharacteristic
28C2×C4×D12—21no—
29C2×C4×D12C2211yescharacteristic
30C2×C42×D3—21no—
31C2×C42×D3C2211yescharacteristic
32C3×#460—24no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×#835C2212yes—
36#5221—21no—
37#5221—21no—
38#5221C2212yes—
39C2×#832C2212yes—
40#5172—28no—
41C2×#821C2212yes—
42C2×(C2×D3)⋊C8C2212yes—
43#5072—24no—
44#4935C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×C4×#41—31no—
50C2×#460D311yescharacteristic, complemented
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#1797—31no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2312yes—
64C22×#67—22no—
65C2×#239—22no—
66C2×#239—22no—
67C2×#239—22no—
68C2×#239—22no—
69C2×#239—22no—
70C2×#239D414yescomplemented
71C22×D12—21no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×C4×D3D411yescharacteristic
77C22×C4×D3D411yescharacteristic
78C22×C4×D3C2311yescharacteristic
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C22×C3×C8C2312yes—
83C3×#127—22no—
84C3×#127—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8D414yescomplemented
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—22no—
92C2×#217—22no—
93C2×#217—22no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96#938—24no—
97#938—24no—
98#938—44no—
99#938—44no—
100C2×#213—24no—
101C2×#213—24no—
102C2×#213—24no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2312yes—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2312yes—
117#893—28no—
118#893—22no—
119#893—22no—
120#892—28no—
121#892—22no—
122#892—22no—
123C4×D12—28no—
124C4×D12—22no—
125C4×D12D414yescomplemented
126C42×D3—28no—
127C42×D3—22no—
128C42×D3—22no—
129C3×C4⋊C8—22no—
130C3×#102—24no—
131#835—24no—
132#832—24no—
133#821—24no—
134(C2×D3)⋊C8—24no—
135C4⋊(C3⋊C8)—22no—
136C22×#41—31no—
137C22×SD16C2×D312yes—
138C22×D8—32no—
139C2×#127C2×D311yescharacteristic
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
143C4×#41—64no—
144C4×#41—64no—
145C2×C4×Q8—31no—
146C2×C4×Q8C2×D311yescharacteristic
147C2×C4×D4—31no—
148C2×C4×D4—31no—
149C2×C4×D4—61no—
150C2×C4×D4—61no—
151C2×#117—31no—
152C2×#117—61no—
153C22×C42—31no—
154C22×C42—61no—
155#460—24no—
156#459—64no—
157#449—68no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102C2×D312yes—
164C2×C22⋊C8—32no—
165#390—32no—
166#390—61no—
167#390—61no—
168#384—31no—
169#340—32no—
170#340—32no—
171#336—32no—
172C24×C3—22no—
173C23×D3—21no—
174C23×D3C2×D411yescharacteristic
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8#4112yes—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C2×#67—44no—
199C2×#67—44no—
200C2×#67—44no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206#239—28no—
207#239—28no—
208#239—28no—
209#239—28no—
210C2×D12—24no—
211C2×D12—24no—
212C2×D12—24no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12C2×D412yes—
217C2×D12C2×D412yes—
218C2×D12#4112yes—
219C2×C4×D3—24no—
220C2×C4×D3—24no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—24no—
226C2×C4×D3—24no—
227C2×C4×D3—22no—
228C2×C4×D3—24no—
229C2×C4×D3—22no—
230C2×C4×D3—22no—
231C2×C4×D3—22no—
232C2×C4×D3—22no—
233C2×C4×D3—41no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3C2×D412yes—
239C2×C4×D3#4112yes—
240C3×SD16—48no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C2×C3×C8—22no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D412yes—
247C3×C42—24no—
248C3×C42C2×D412yes—
249C3×C42C2×D412yes—
250#217—44no—
251#217—44no—
252#217—44no—
253#213—48no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—22no—
260C4⋊Dic3—22no—
261C4⋊Dic3—42no—
262C4⋊Dic3—42no—
263C4⋊Dic3C2×D412yes—
264C4×Dic3—24no—
265C4×Dic3—24no—
266C4×Dic3—22no—
267C4×Dic3—22no—
268C4×Dic3—21no—
269C4×Dic3—21no—
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×Q8C22×D311yescharacteristic
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—31no—
283C22×D4—61no—
284C22×D4—62no—
285C22×D4C22×D312yes—
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—61no—
290C23×C4—61no—
291C23×C4—61no—
292C2×SD16—24no—
293C2×SD16—24no—
294C2×SD16—24no—
295C2×D8—64no—
296C2×D8—64no—
297C2×D8—64no—
298C22×C8—32no—
299C22×C8C22×D312yes—
300#127—22no—
301#127—22no—
302#121—64no—
303#121—64no—
304#121—124no—
305#121—124no—
306C4×Q8—22no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8#6714yescomplemented
310C4×D4—34no—
311C4×D4—68no—
312C4×D4—62no—
313C4×D4—68no—
314C4×D4—62no—
315C4×D4—62no—
316#117—62no—
317#117—68no—
318#117—62no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
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—21no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—68no—
337C2×C42—61no—
338C2×C42—61no—
339C2×C42—62no—
340C2×C42—62no—
341C2×C42C22×D311yescharacteristic
342C4⋊C8—22no—
343C4⋊C8—62no—
344#103—64no—
345#102—24no—
346#102—64no—
347#102—64no—
348C22⋊C8—64no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3—42no—
356C23×C3C22×D411yescharacteristic
357C22×D3—21no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—21no—
361C22×D3—22no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3#12112yes—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8#12114yescomplemented
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—84no—
380C3×D4—84no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12012yes—
390C2×C3×C4#12012yes—
391C2×C3×C4#13512yes—
392C2×C3×C4AGL(1,ℤ/8)12yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399#67—88no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412C2×Dic3—42no—
413C2×Dic3#12112yes—
414C2×Dic3AGL(1,ℤ/8)12yes—
415D12—216no—
416D12—22no—
417D12—22no—
418D12—22no—
419D12#12114yescomplemented
420C4×D3—216no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—44no—
426C4×D3—48no—
427C4×D3—44no—
428C4×D3—48no—
429C3×C8—42no—
430C3⋊C8—42no—
431C24—22no—
432C24—31no—
433C24—61no—
434#41—68no—
435#41—68no—
436#41—68no—
437#41—68no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8D3×D412yes—
442C2×Q8#23712yes—
443C2×Q8C2×#6712yes—
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—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—64no—
460C2×D4—64no—
461C2×D4—64no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—124no—
466C2×D4—124no—
467C2×D4—124no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—31no—
475C22×C4—64no—
476C22×C4—62no—
477C22×C4—64no—
478C22×C4—64no—
479C22×C4—64no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—64no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—121no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4D3×D411yescharacteristic
497C22×C4D3×D411yescharacteristic
498C22×C4C23×D311yescharacteristic
499SD16—48no—
500D8—128no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—62no—
505C2×C8—62no—
506C2×C8—62no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—32no—
510C4⋊C4—64no—
511C4⋊C4—61no—
512C4⋊C4—62no—
513C4⋊C4—62no—
514C4⋊C4—122no—
515C4⋊C4—122no—
516C4⋊C4C2×#6712yes—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C22⋊C4—124no—
522C22⋊C4—124no—
523C42—24no—
524C42—64no—
525C42—64no—
526C42—62no—
527C42—62no—
528C42—61no—
529C42—61no—
530C42D3×D412yes—
531C42C2×#6712yes—
532C22×C3—82no—
533C22×C3—82no—
534C22×C3—82no—
535C22×C3—82no—
536C22×C3—82no—
537C22×C3—82no—
538C22×C3—82no—
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—22no—
547C2×D3—24no—
548C2×D3—22no—
549C2×D3—22no—
550C2×D3—41no—
551C2×D3—42no—
552C2×D3—41no—
553C2×D3—42no—
554C2×D3—41no—
555C2×D3—42no—
556C2×D3—41no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C3×C4—24no—
561C3×C4—22no—
562C3×C4—42no—
563C3×C4#44914yescomplemented
564C3×C4C2×#12111yescharacteristic
565C3×C4C2×#12111yescharacteristic
566C3×C4#52511yescharacteristic
567C3×C4#52511yescharacteristic
568Dic3—22no—
569Dic3—22no—
570Dic3—41no—
571Dic3—42no—
572Dic3—82no—
573C23—32no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—62no—
586C23—61no—
587C23—62no—
588C23—62no—
589C23—61no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23C2×D3×D411yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—22no—
598Q8#93814yescomplemented
599D4—34no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—616no—
605D4—62no—
606D4—62no—
607D4—62no—
608D4—84no—
609D4—84no—
610D4—248no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—32no—
618C2×C4—32no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—616no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—62no—
633C2×C4—61no—
634C2×C4—124no—
635C2×C4—128no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—124no—
639C2×C4—128no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4#91412yes—
644C2×C4#91612yes—
645C2×C4#93712yes—
646C2×C4#93712yes—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4C22×#6711yescharacteristic
653C8—42no—
654C8—122no—
655C2×C3—162no—
656C2×C3#284711yescharacteristic
657C2×C3C2×#44911yescharacteristic
658C2×C3#300611yescharacteristic
659C2×C3#303511yescharacteristic
660C2×C3#303811yescharacteristic
661C2×C3#310611yescharacteristic
662C2×C3#311511yescharacteristic
663D3—42no—
664D3—44no—
665C22—62no—
666C22—64no—
667C22—62no—
668C22—62no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—122no—
677C22—121no—
678C22—122no—
679C22—121no—
680C22—122no—
681C22—121no—
682C22—122no—
683C22—121no—
684C22—122no—
685C22—122no—
686C22D3×#12111yescharacteristic
687C22#560611yescharacteristic
688C22C2×#91411yescharacteristic
689C22C2×#91611yescharacteristic
690C22D3×AGL(1,ℤ/8)11yescharacteristic
691C22#577311yescharacteristic
692C22C2×#93711yescharacteristic
693C4—24no—
694C4—22no—
695C4—42no—
696C4—62no—
697C4—62no—
698C4—121no—
699C4—122no—
700C4—242no—
701C4#517214yescomplemented
702C4D3×#12111yescharacteristic
703C4#560611yescharacteristic
704C4C2×#93811yescharacteristic
705C4#582811yescharacteristic
706C3#3290411yescharacteristic, complemented, Sylow
707C2—122no—
708C2—124no—
709C2—162no—
710C2#8036111yescharacteristic
711C2#8036311yescharacteristic
712C2#8038211yescharacteristic
713C2#8038411yescharacteristic
714C2#8272311yescharacteristic
715C2C2×#517211yescharacteristic
716C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
hypercenterC2×#460D3
hypocenterC3#32904
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×C4×#239C2semidirect
2. C2×C3×#460C2semidirect
3. C2×#5172C2semidirect
4. #83192C2semidirect
5. C2×#5072C2semidirect
6. #72048C2semidirect
7. #71626C2semidirect
8. #70479C2semidirect
9. #70475C2semidirect
10. #70208C2semidirect
11. #70201C2semidirect
50. C2×#460D3semidirect
70. C2×#239D4semidirect
86. C3×C4×Q8D4semidirect
125. C4×D12D4semidirect
309. C4×Q8#67semidirect
374. C3×Q8#121semidirect
419. D12#121semidirect
563. C3×C4#449semidirect
598. Q8#938semidirect
701. C4#5172semidirect
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×#2392
C2×C3×#4602
C2×#51722
#831922
C2×#50722
#720482
#716262
#704792
#704752
#702082
#702012
#329043