Finite Group Databasebeta

← #10967687

Subgroups — #10967687

Order 768.

2962 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic, complemented
2C2×C4×#237C211yesmaximal, characteristic, complemented
3C3×#2563C211yesmaximal, characteristic, complemented
4#83243C211yesmaximal, characteristic, complemented
5#82378C211yesmaximal, characteristic, complemented
6#76808C214yesmaximal, complemented
7#76777C214yesmaximal, complemented
8#76313C212yesmaximal, complemented
9#76293C212yesmaximal, complemented
10#74166C212yesmaximal, complemented
11#74149C212yesmaximal, complemented
12#74019C214yesmaximal, complemented
13#73865C214yesmaximal, complemented
14#73712C211yesmaximal, characteristic, complemented
15C4×#892C211yesmaximal, characteristic, complemented
16#34977—31nomaximal, Sylow
17C22×#239C2212yes—
18C22×#237C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#947C2212yes—
22C2×#940C2212yes—
23C2×D4×Dic3C2211yescharacteristic
24C2×#930C2214yes—
25C2×C4×#67C2212yes—
26C22×C4×Dic3C2212yes—
27#5725—28no—
28C4×#237—28no—
29C2×#895C2212yes—
30C2×#895C2212yes—
31C2×#893C2212yes—
32C2×#893C2212yes—
33C2×#892C2212yes—
34C2×#892C2211yescharacteristic
35C2×#887C2211yescharacteristic
36C2×#67⋊C4C2212yes—
37C2×#67⋊C4C2218yes—
38C2×#877C2212yes—
39C2×#875C2212yes—
40C2×#874C2211yescharacteristic
41C2×C42×D3C2211yescharacteristic
42C2×C4×Dic6C2211yescharacteristic
43C3×#393C2214yes—
44C3×#390C2212yes—
45C3×#382C2212yes—
46C3×#380C2214yes—
47C3×C4×C4⋊C4C2211yescharacteristic
48#5230C2214yes—
49#4988C2214yes—
50#4988C2212yes—
51#4975C2212yes—
52Dic3×C4⋊C4C2212yes—
53#4973C2212yes—
54#4972C2214yes—
55#4960C2214yes—
56#4960C2218yes—
57#4958C2218yes—
58#4957C2218yes—
59#4939C2212yes—
60C4×(C2×D3)⋊C4C2212yes—
61C4×C4⋊Dic3C2211yescharacteristic
62C42×Dic3C2212yes—
63#4672C2214yes—
64#4672C2218yes—
65#4669C2218yes—
66#4669C2218yes—
67#4666C2218yes—
68#4666C2212yes—
69#4648C2212yes—
70C2×#532—31no—
71C2×C4×#41—31no—
72#2610—34no—
73#2567—31no—
74#2563D311yescharacteristic, complemented
75#2548—31no—
76#2541—31no—
77#2500—32no—
78#2431—32no—
79#2414—34no—
80#2388—34no—
81C4×#127—32no—
82C4×#124—32no—
83C4×#121—34no—
84C4×#117—31no—
85C22×C3×Q8C2312yes—
86C22×C3×D4C2311yescharacteristic
87C23×C3×C4C2312yes—
88C22×#67C2312yes—
89C22×#67C2318yes—
90C23×Dic3C2312yes—
91C23×Dic3C2312yes—
92C2×#239—28no—
93C2×#239—48no—
94C2×#237—24no—
95C2×#237—24no—
96C2×#237—24no—
97C22×D12C2312yes—
98C22×D12C2312yes—
99C22×D12C2312yes—
100C22×D12C2314yes—
101C22×C4×D3C2314yes—
102C22×C4×D3C2311yescharacteristic
103C22×C4×D3C2311yescharacteristic
104C22×C4×D3C2311yescharacteristic
105C22×Dic6C2311yescharacteristic
106C3×C4⋊Q8—24no—
107C3×C4×D4—24no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2318yes—
114C2×C3×C42C2312yes—
115C2×C3×C42C2311yescharacteristic
116C2×#217C2312yes—
117#947—28no—
118#940—28no—
119D4×Dic3—24no—
120#930—216no—
121C4×#67—28no—
122C2×(C2×D3)⋊C4C2318yes—
123C2×(C2×D3)⋊C4C2318yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2318yes—
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2312yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2312yes—
133C2×Dic3⋊C4C2318yes—
134C2×C4×Dic3—28no—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2314yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2311yescharacteristic
144#895—28no—
145#895—28no—
146#893—28no—
147#893—28no—
148#892—24no—
149#892—28no—
150#887—24no—
151#67⋊C4—28no—
152#67⋊C4—416no—
153#877—28no—
154#875—28no—
155#874—24no—
156C42×D3D418yescomplemented
157C4×Dic6—24no—
158C3×#95C2312yes—
159C3×#95C2318yes—
160C3⋊#95C2312yes—
161C3⋊#95C2314yes—
162C3⋊#95C2318yes—
163C22×#41—32no—
164C22×#41—31no—
165#532—68no—
166C2×C4⋊Q8C2×D311yescharacteristic
167C2×#127—32no—
168C2×#127—31no—
169C2×#124—32no—
170C2×#124—32no—
171C2×#121—34no—
172C2×#121—32no—
173C2×#121—32no—
174C4×#41—68no—
175C2×C4×Q8—31no—
176C2×C4×Q8—31no—
177C2×C4×D4—31no—
178C2×C4×D4—32no—
179C2×C4×D4—32no—
180C2×C4×D4—32no—
181C2×C4×D4—32no—
182C2×C4×D4—38no—
183C2×C4×D4C2×D311yescharacteristic
184C2×#117—31no—
185C2×#117—32no—
186C2×#117—32no—
187C22×C42—31no—
188C22×C42—32no—
189#395—34no—
190#394—34no—
191#394—38no—
192#393C2×D314yes—
193#392—34no—
194#392—38no—
195#391—32no—
196#390—34no—
197#390—32no—
198#390—38no—
199#390—32no—
200#390C2×D312yes—
201#389—34no—
202#388—38no—
203#388—38no—
204#386—38no—
205#384—32no—
206#382C2×D312yes—
207#380C2×D314yes—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—31no—
210C4×C4⋊C4C2×D311yescharacteristic
211C4×C22⋊C4—32no—
212C4×C22⋊C4—32no—
213C4×C22⋊C4—38no—
214#376—32no—
215C43—32no—
216C24×C3C2412yes—
217C23×D3C2414yes—
218C23×D3C2411yescharacteristic
219C2×C3×Q8—24no—
220C2×C3×Q8—44no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4#4114yescomplemented
224C22×C3×C4—24no—
225C22×C3×C4—28no—
226C22×C3×C4C2414yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C2×#67—24no—
233C2×#67—24no—
234C2×#67—216no—
235C2×#67—24no—
236C2×#67—216no—
237C2×#67—48no—
238C22×Dic3—24no—
239C22×Dic3—24no—
240C22×Dic3—24no—
241C22×Dic3—24no—
242C22×Dic3—28no—
243C22×Dic3C2412yes—
244C22×Dic3C2411yescharacteristic
245C22×Dic3C2411yescharacteristic
246C22×Dic3C2414yes—
247C22×Dic3C2412yes—
248C22×Dic3C2411yescharacteristic
249#239—432no—
250#237—232no—
251C2×D12—28no—
252C2×D12—24no—
253C2×D12—24no—
254C2×D12—24no—
255C2×D12—24no—
256C2×D12—28no—
257C2×D12—24no—
258C2×D12—44no—
259C2×D12—48no—
260C2×C4×D3—24no—
261C2×C4×D3—22no—
262C2×C4×D3—22no—
263C2×C4×D3—24no—
264C2×C4×D3—28no—
265C2×C4×D3—216no—
266C2×C4×D3C2×D414yes—
267C2×C4×D3C2×D414yes—
268C2×C4×D3C2×D414yes—
269C2×Dic6—22no—
270C2×Dic6—22no—
271C2×Dic6#4114yescomplemented
272C3×C4⋊C4—22no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—24no—
276C3×C22⋊C4—24no—
277C3×C22⋊C4—48no—
278C3×C42—24no—
279C3×C42C2×D414yes—
280#217—24no—
281(C2×D3)⋊C4—24no—
282(C2×D3)⋊C4—28no—
283(C2×D3)⋊C4—28no—
284(C2×D3)⋊C4—216no—
285(C2×D3)⋊C4—28no—
286(C2×D3)⋊C4—216no—
287(C2×D3)⋊C4—48no—
288C4⋊Dic3—22no—
289C4⋊Dic3#4118yescomplemented
290Dic3⋊C4—24no—
291Dic3⋊C4—24no—
292Dic3⋊C4—48no—
293C4×Dic3—24no—
294C4×Dic3—24no—
295C4×Dic3—24no—
296C4×Dic3—24no—
297C4×Dic3—28no—
298C4×Dic3—28no—
299C4×Dic3C2×D414yes—
300C4×Dic3C2×D414yes—
301C4×Dic3C2×D414yes—
302C2×#41—64no—
303C2×#41—68no—
304C2×#41—64no—
305C2×#41—64no—
306C2×#41—128no—
307C22×Q8—31no—
308C22×Q8C22×D312yes—
309C22×D4—32no—
310C22×D4—38no—
311C22×D4—32no—
312C22×D4—32no—
313C22×D4—34no—
314C22×D4—32no—
315C22×D4C22×D311yescharacteristic
316C23×C4—31no—
317C23×C4—31no—
318C23×C4—31no—
319C23×C4—32no—
320C23×C4—32no—
321C23×C4—34no—
322C23×C4C22×D312yes—
323C4⋊Q8—24no—
324#127—68no—
325#127—64no—
326#124—68no—
327#124—68no—
328#121—68no—
329#121—68no—
330#121—616no—
331C4×Q8—64no—
332C4×Q8—64no—
333C4×D4—24no—
334C4×D4—68no—
335C4×D4—68no—
336C4×D4—64no—
337C4×D4—68no—
338C4×D4—68no—
339C4×D4—1216no—
340#117—68no—
341#117—68no—
342#117—64no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—38no—
348C2×C4⋊C4C22×D311yescharacteristic
349C2×C4⋊C4C22×D312yes—
350C2×C4⋊C4C22×D312yes—
351C2×C4⋊C4C22×D312yes—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—34no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4—34no—
357C2×C22⋊C4—38no—
358C2×C22⋊C4—38no—
359C2×C22⋊C4—38no—
360C2×C22⋊C4C22×D312yes—
361C2×C22⋊C4C22×D318yes—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—38no—
366C2×C42—32no—
367C2×C42—34no—
368C2×C42—34no—
369C2×C42—32no—
370C2×C42—32no—
371C2×C42—32no—
372C2×C42—68no—
373C2×C42C22×D312yes—
374C2×C42C22×D311yescharacteristic
375#95—34no—
376#95—32no—
377#95—38no—
378#95C22×D312yes—
379#95C22×D318yes—
380C23×C3—22no—
381C23×C3—22no—
382C23×C3—24no—
383C23×C3—24no—
384C23×C3C2×#4114yes—
385C23×C3C2511yescharacteristic
386C22×D3—22no—
387C22×D3—21no—
388C22×D3—21no—
389C22×D3—22no—
390C22×D3—24no—
391C22×D3—24no—
392C22×D3—28no—
393C22×D3—24no—
394C22×D3—28no—
395C22×D3C22×D412yes—
396C3×Q8—48no—
397C3×D4—28no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4—24no—
403C2×C3×C4—24no—
404C2×C3×C4—24no—
405C2×C3×C4—28no—
406C2×C3×C4—48no—
407C2×C3×C4C22×D412yes—
408C2×C3×C4C22×D412yes—
409C2×C3×C4C22×D412yes—
410C2×C3×C4C2×#4112yes—
411C2×C3×C4C2×#4112yes—
412C2×C3×C4C2×#4114yes—
413#67—216no—
414#67—432no—
415C2×Dic3—216no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—24no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—24no—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—48no—
426C2×Dic3C22×D412yes—
427C2×Dic3C22×D412yes—
428C2×Dic3C22×D412yes—
429C2×Dic3C2×#4118yes—
430C2×Dic3C2×#4114yes—
431C2×Dic3H(5,𝔽2)14yes—
432D12—48no—
433D12—416no—
434D12—48no—
435D12—48no—
436C4×D3—28no—
437C4×D3—28no—
438C4×D3—28no—
439C4×D3—416no—
440Dic6—28no—
441C24—31no—
442C24—34no—
443C24C23×D312yes—
444#41—632no—
445#41—1232no—
446C2×Q8—24no—
447C2×Q8—34no—
448C2×Q8—44no—
449C2×Q8—62no—
450C2×Q8—62no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—68no—
454C2×D4—64no—
455C2×D4—616no—
456C2×D4—64no—
457C2×D4—64no—
458C2×D4—616no—
459C2×D4—64no—
460C2×D4—64no—
461C2×D4—64no—
462C2×D4—64no—
463C2×D4—68no—
464C2×D4—64no—
465C2×D4—124no—
466C2×D4—128no—
467C2×D4—128no—
468C2×D4#23914yescomplemented
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—34no—
474C22×C4—32no—
475C22×C4—31no—
476C22×C4—34no—
477C22×C4—34no—
478C22×C4—34no—
479C22×C4—32no—
480C22×C4—64no—
481C22×C4—68no—
482C22×C4—616no—
483C22×C4—64no—
484C22×C4—64no—
485C22×C4—64no—
486C22×C4—68no—
487C22×C4—64no—
488C22×C4—62no—
489C22×C4—64no—
490C22×C4—62no—
491C22×C4C23×D314yes—
492C22×C4C23×D314yes—
493C22×C4C23×D312yes—
494C22×C4C23×D311yescharacteristic
495C22×C4C23×D311yescharacteristic
496C22×C4C23×D311yescharacteristic
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—38no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—62no—
505C4⋊C4—128no—
506C22⋊C4—24no—
507C22⋊C4—48no—
508C22⋊C4—68no—
509C22⋊C4—68no—
510C22⋊C4—68no—
511C22⋊C4—616no—
512C22⋊C4—616no—
513C22⋊C4—64no—
514C22⋊C4—64no—
515C22⋊C4—128no—
516C42—24no—
517C42—34no—
518C42—34no—
519C42—34no—
520C42—64no—
521C42—64no—
522C42—68no—
523C42—68no—
524C42—64no—
525C42—64no—
526C42D3×D414yes—
527C22×C3—24no—
528C22×C3—28no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3C23×D411yescharacteristic
533C22×C3C22×#4111yescharacteristic
534C22×C3C22×#4111yescharacteristic
535C22×C3C22×#4112yes—
536C22×C3C2×H(5,𝔽2)12yes—
537C2×D3—22no—
538C2×D3—22no—
539C2×D3—22no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—44no—
543C2×D3—44no—
544C2×D3—44no—
545C2×D3—48no—
546C2×D3—48no—
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—44no—
550C3×C4—44no—
551C3×C4#52514yes—
552C3×C4#53214yes—
553Dic3—24no—
554Dic3—22no—
555Dic3—22no—
556Dic3—28no—
557Dic3—42no—
558Dic3#53514yes—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—24no—
563C23—32no—
564C23—61no—
565C23—62no—
566C23—61no—
567C23—62no—
568C23—64no—
569C23—64no—
570C23—64no—
571C23—68no—
572C23—68no—
573C23C2×#23914yes—
574C23C24×D311yescharacteristic
575Q8—48no—
576Q8—68no—
577D4—28no—
578D4—616no—
579D4—128no—
580D4—1216no—
581D4—1232no—
582D4—128no—
583D4—128no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—32no—
593C2×C4—34no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—38no—
597C2×C4—34no—
598C2×C4—48no—
599C2×C4—68no—
600C2×C4—616no—
601C2×C4—62no—
602C2×C4—64no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—64no—
608C2×C4—68no—
609C2×C4—68no—
610C2×C4—62no—
611C2×C4—64no—
612C2×C4—1216no—
613C2×C4—128no—
614C2×C4C2×D3×D412yes—
615C2×C4C2×D3×D412yes—
616C2×C4C2×D3×D412yes—
617C2×C4C2×#23912yes—
618C2×C4D3×#4114yes—
619C2×C4D3×#4112yes—
620C2×C3—44no—
621C2×C3C2×#52511yescharacteristic
622C2×C3C2×#53211yescharacteristic
623C2×C3C2×#53511yescharacteristic
624C2×C3#348112yes—
625C2×C3#348312yes—
626D3—42no—
627D3—84no—
628C22—28no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—62no—
634C22—62no—
635C22—62no—
636C22—122no—
637C22—122no—
638C22—124no—
639C22—124no—
640C22—128no—
641C22—128no—
642C22—124no—
643C22C22×D3×D411yescharacteristic
644C22C22×#23911yescharacteristic
645C22C2×D3×#4112yes—
646C22C2×D3×#4111yescharacteristic
647C22D3×H(5,𝔽2)12yes—
648C4—22no—
649C4—34no—
650C4—42no—
651C4—44no—
652C4—44no—
653C4—62no—
654C4—68no—
655C4—62no—
656C4—64no—
657C4—122no—
658C4#562714yes—
659C4#566814yes—
660C3#3497711yescharacteristic, complemented, Sylow
661C2—44no—
662C2—122no—
663C2—244no—
664C2C2×#562711yescharacteristic
665C2C2×#566811yescharacteristic
666C2D3×#53511yescharacteristic
667C2#8692412yes—
668C2#8695912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2563C2
hypercenter#2563D3
hypocenterC3#34977
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#5725C2semidirect
2. C2×C4×#237C2semidirect
3. C3×#2563C2semidirect
4. #83243C2semidirect
5. #82378C2semidirect
6. #76808C2semidirect
7. #76777C2semidirect
8. #76313C2semidirect
9. #76293C2semidirect
10. #74166C2semidirect
11. #74149C2semidirect
12. #74019C2semidirect
13. #73865C2semidirect
14. #73712C2semidirect
15. C4×#892C2semidirect
74. #2563D3semidirect
156. C42×D3D4semidirect
223. C2×C3×D4#41semidirect
271. C2×Dic6#41semidirect
289. C4⋊Dic3#41semidirect
468. C2×D4#239semidirect
660. C3#34977semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×C4×#2372
C3×#25632
#832432
#823782
#768082
#767772
#763132
#762932
#741662
#741492
#740192
#738652
#737122
C4×#8922
#349773