Finite Group Databasebeta

← #10816025

Subgroups — #10816025

Order 768.

1008 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2898C211yesmaximal, characteristic, complemented
2C2×#5245C211yesmaximal, characteristic
3#84111C211yesmaximal, characteristic, complemented
4#82564C211yesmaximal, characteristic, complemented
5#78820C211yesmaximal, characteristic, complemented
6#72018C211yesmaximal, characteristic, complemented
7#72008C211yesmaximal, characteristic, complemented
8#71665C211yesmaximal, characteristic, complemented
9#70343C211yesmaximal, characteristic, complemented
10#70338C211yesmaximal, characteristic, complemented
11#70303C211yesmaximal, characteristic, complemented
12#70298C211yesmaximal, characteristic, complemented
13#70289C211yesmaximal, characteristic, complemented
14#70263C211yesmaximal, characteristic, complemented
15#70226C211yesmaximal, characteristic, complemented
16#22895—31nomaximal, Sylow
17C2×C3×2−1+4C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19C3×#520C2211yescharacteristic
20#5827C2211yescharacteristic
21C2×#949C2211yescharacteristic
22C2×#949C2211yescharacteristic
23C2×#948C2211yescharacteristic
24C2×#947C2211yescharacteristic
25#5521C2211yescharacteristic
26C3×#419—21no—
27C3×#419C2211yescharacteristic
28C3×#417—21no—
29C3×#417C2211yescharacteristic
30C3×#416—21no—
31C3×#416C2211yescharacteristic
32C2×C3×#103—21no—
33C2×C3×#103C2211yescharacteristic
34#5245C418yescomplemented
35C2×#837C2211yescharacteristic
36C2×#837C2211yescharacteristic
37#5233—21no—
38#5233C2211yescharacteristic
39#5232—21no—
40#5232C2211yescharacteristic
41C2×#836C2211yescharacteristic
42#5225—21no—
43#5225C2211yescharacteristic
44C2×#835—21no—
45C2×#835C2211yescharacteristic
46C2×#823C2211yescharacteristic
47C2×#821C2211yescharacteristic
48#5101C2211yescharacteristic
49C2×C8×Dic3C2211yescharacteristic
50#5026—21no—
51C4×#215—21no—
52#5014—21no—
53C4×#213—21no—
54#4997C2211yescharacteristic
55#4995C2211yescharacteristic
56#4977C2211yescharacteristic
57C2×#809C2211yescharacteristic
58#4964C2211yescharacteristic
59#4817—21no—
60#4788—21no—
61#4785—21no—
62#4756—21no—
63#4558C2211yescharacteristic
64#4555C2211yescharacteristic
65#4554C2211yescharacteristic
66#4552C2211yescharacteristic
67#4532C2211yescharacteristic
68#4530C2211yescharacteristic
69C2×#454—31no—
70#2946—31no—
71#2946—31no—
72#2898—31no—
73#2898D311yescharacteristic, complemented
74#1958—31no—
75#1930—31no—
76#1930—31no—
77#1907—31no—
78#1907—31no—
79#1885—31no—
80#1860—31no—
81#1860—31no—
82#1764—31no—
83#1764—31no—
84C3×2−1+4—22no—
85C3×2−1+4—22no—
86C3×2−1+4—41no—
87C3×2−1+4C2×C414yescomplemented
88C2×C3×#41—21no—
89C2×C3×#41—21no—
90C2×C3×#41—41no—
91C2×C3×#41C2311yescharacteristic
92C2×C3×#41C2311yescharacteristic
93C22×C3×Q8—21no—
94C22×C3×Q8—21no—
95C22×C3×Q8C2311yescharacteristic
96C22×D12C2311yescharacteristic
97C3×#131—22no—
98C3×#131—41no—
99C2×C3×M16—21no—
100C2×C3×M16C2311yescharacteristic
101C22×C3×C8—21no—
102C22×C3×C8C2311yescharacteristic
103C3×C4×Q8—21no—
104C3×C4×D4—21no—
105C3×C4×D4—41no—
106C3×#117—21no—
107C3×#117C2311yescharacteristic
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4C2311yescharacteristic
110#949—22no—
111#949—22no—
112#949C2×C414yescomplemented
113#949C2×C414yescomplemented
114#948—22no—
115#948—41no—
116#947—41no—
117#947C2×C414yescomplemented
118Q8×Dic3—21no—
119C2×#215—21no—
120C2×#215—21no—
121D4×Dic3—21no—
122D4×Dic3—41no—
123C2×#213—21no—
124C2×#213—21no—
125C2×(C2×D3)⋊C4—41no—
126#926—21no—
127#926C2311yescharacteristic
128C2×C4⋊Dic3—21no—
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4×Dic3C2311yescharacteristic
131C2×#208—21no—
132C2×#208C2311yescharacteristic
133C2×#208C2311yescharacteristic
134C2×#208C2311yescharacteristic
135C22×C3⋊C8—21no—
136C22×C3⋊C8C2311yescharacteristic
137#893—21no—
138#893—21no—
139D3×C22⋊C4—41no—
140C4×D12—21no—
141C4×D12—21no—
142C3×#103—21no—
143C3×#103—21no—
144C3×#103—21no—
145C3×#103—21no—
146C3×#103—21no—
147C3×#103D411yescharacteristic
148C3×#103D411yescharacteristic
149C3×#102—21no—
150C3×#102D411yescharacteristic
151C3×#102D411yescharacteristic
152#837C2×C414yescomplemented
153#837C2×C414yescomplemented
154#836C2×C414yescomplemented
155#835—21no—
156#835—21no—
157#835—21no—
158#835—21no—
159#835—21no—
160#835D411yescharacteristic
161#835D411yescharacteristic
162#832—21no—
163#832D411yescharacteristic
164#832D411yescharacteristic
165C3⋊#95—41no—
166#823—41no—
167#821—21no—
168#821—21no—
169#815—21no—
170C8×Dic3—21no—
171C8×Dic3—41no—
172#809—21no—
173#809—21no—
174#808—21no—
175#808—21no—
176#807—21no—
177#807—21no—
178#803—21no—
179C4×C3⋊C8—21no—
180C2×2−1+4C2×D311yescharacteristic
181C2×AGL(1,ℤ/8)—31no—
182C2×AGL(1,ℤ/8)—31no—
183C2×#131—31no—
184C2×#131C2×D311yescharacteristic
185C2×#124—31no—
186#520—31no—
187#520C2×D311yescharacteristic
188#518—31no—
189#454—38no—
190#442—61no—
191#442—61no—
192#440—61no—
193#440—61no—
194C4×SD16—61no—
195C4×SD16—61no—
196C4×D8—61no—
197C4×D8—61no—
198#428—31no—
199#428—31no—
200C2×#104—31no—
201C2×#104—31no—
202#419—21no—
203#419—31no—
204#419—61no—
205#419C2×D311yescharacteristic
206#417—21no—
207#417—31no—
208#417—61no—
209#417C2×D311yescharacteristic
210#416—21no—
211#416—31no—
212#416—61no—
213#416C2×D311yescharacteristic
214C2×#103—21no—
215C2×#103—31no—
216C2×#103—61no—
217C2×#103C2×D311yescharacteristic
218C2×#102—31no—
219C2×#102—31no—
220C2×#101—31no—
221C2×#100—31no—
222#405—31no—
223#405—31no—
224C2×C4×C8—31no—
225#382—31no—
226#343—31no—
227#343—31no—
228#341—31no—
229#341—31no—
230#339—31no—
231#339—31no—
232C23×D3—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41—21no—
236C3×#41—21no—
237C3×#41—24no—
238C3×#41—24no—
239C3×#41—44no—
240C3×#41—42no—
241C3×#41—42no—
242C3×#41—44no—
243C3×#41C22×C412yes—
244C3×#41C22×C412yes—
245C3×#41#4112yes—
246C3×#41#4112yes—
247C2×C3×Q8—21no—
248C2×C3×Q8—24no—
249C2×C3×Q8—24no—
250C2×C3×Q8—21no—
251C2×C3×Q8—21no—
252C2×C3×Q8—21no—
253C2×C3×Q8—21no—
254C2×C3×Q8—41no—
255C2×C3×Q8—42no—
256C2×C3×Q8—42no—
257C2×C3×Q8C22×C412yes—
258C2×C3×Q8C2×D411yescharacteristic
259C2×C3×Q8C2×D411yescharacteristic
260C2×C3×D4—21no—
261C2×C3×D4—21no—
262C2×C3×D4—41no—
263C2×C3×D4C2×D411yescharacteristic
264C2×C3×D4C2×D411yescharacteristic
265C22×C3×C4—21no—
266C22×C3×C4—21no—
267C22×C3×C4—21no—
268C22×C3×C4—21no—
269C22×C3×C4—41no—
270C22×C3×C4—41no—
271C22×C3×C4—41no—
272C22×C3×C4C2411yescharacteristic
273C22×Dic3—21no—
274C22×Dic3—21no—
275C22×Dic3—41no—
276C2×D12—21no—
277C2×D12—21no—
278C2×D12—21no—
279C2×D12—21no—
280C2×D12—21no—
281C2×D12C22×C412yes—
282C2×C4×D3—41no—
283C2×C4×D3—41no—
284C3×M16—21no—
285C3×M16—21no—
286C3×M16—42no—
287C2×C3×C8—21no—
288C2×C3×C8—21no—
289C2×C3×C8—21no—
290C2×C3×C8—42no—
291C2×C3×C8C2×D411yescharacteristic
292C2×C3×C8C2×D411yescharacteristic
293C3×C4⋊C4—21no—
294C3×C4⋊C4—21no—
295C3×C4⋊C4—41no—
296C3×C4⋊C4C2×D411yescharacteristic
297C3×C4⋊C4C2×D411yescharacteristic
298C3×C22⋊C4—21no—
299C3×C22⋊C4—41no—
300C3×C42—21no—
301C3×C42—41no—
302#217—21no—
303#217—41no—
304#215—24no—
305#215—24no—
306#213—24no—
307#213—24no—
308(C2×D3)⋊C4—44no—
309(C2×D3)⋊C4—41no—
310(C2×D3)⋊C4—41no—
311C4⋊Dic3—21no—
312C4⋊Dic3—21no—
313C4⋊Dic3—41no—
314C4⋊Dic3C2×D411yescharacteristic
315C4⋊Dic3C2×D411yescharacteristic
316C4×Dic3—21no—
317C4×Dic3—21no—
318C4×Dic3—41no—
319C4×Dic3C22×C412yes—
320#208—21no—
321#208—21no—
322#208—21no—
323#208—21no—
324#208—42no—
325#208C22×C412yes—
326#208C22×C412yes—
327C2×C3⋊C8—21no—
328C2×C3⋊C8—21no—
329C2×C3⋊C8—21no—
330C2×C3⋊C8—21no—
331C2×C3⋊C8—21no—
332C2×C3⋊C8—42no—
333C2×C3⋊C8C2×D411yescharacteristic
334C2×C3⋊C8C2×D411yescharacteristic
3352−1+4—22no—
3362−1+4—22no—
3372−1+4—41no—
3382−1+4C4×D314yescomplemented
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—41no—
342C2×#41C22×D311yescharacteristic
343C2×#41C22×D311yescharacteristic
344C22×Q8—21no—
345C22×Q8—21no—
346C22×Q8C22×D311yescharacteristic
347C22×D4—31no—
348AGL(1,ℤ/8)—34no—
349AGL(1,ℤ/8)—34no—
350AGL(1,ℤ/8)—62no—
351AGL(1,ℤ/8)—62no—
352C2×SD16—61no—
353C2×SD16—61no—
354C2×D8—61no—
355C2×D8—61no—
356#131—22no—
357#131—41no—
358#131—62no—
359#131—121no—
360C2×M16—21no—
361C2×M16—31no—
362C2×M16—31no—
363C2×M16—31no—
364C2×M16—61no—
365C2×M16C22×D311yescharacteristic
366C22×C8—21no—
367C22×C8—31no—
368C22×C8—61no—
369C22×C8C22×D311yescharacteristic
370#124—34no—
371#124—121no—
372C4×Q8—21no—
373C4×Q8—61no—
374C4×D4—21no—
375C4×D4—41no—
376C4×D4—61no—
377C4×D4—61no—
378C4×D4—61no—
379C4×D4—61no—
380C4×D4—61no—
381C4×D4—121no—
382#117—21no—
383#117—31no—
384#117—61no—
385#117C22×D311yescharacteristic
386C2×C4⋊C4—21no—
387C2×C4⋊C4—31no—
388C2×C4⋊C4—61no—
389C2×C4⋊C4C22×D311yescharacteristic
390C2×C22⋊C4—121no—
391C2×C22⋊C4—121no—
392C2×C42—31no—
393#107—61no—
394#107—61no—
395#106—61no—
396#106—61no—
397#104—34no—
398#104—34no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—21no—
403#103—21no—
404#103—31no—
405#103—31no—
406#103—61no—
407#103—61no—
408#103—61no—
409#103—61no—
410#103—61no—
411#103#6711yescharacteristic
412#103#6711yescharacteristic
413#102—21no—
414#102—31no—
415#102—31no—
416#102—61no—
417#102—61no—
418#102—61no—
419#102—61no—
420#102—61no—
421#102#6711yescharacteristic
422#102#6711yescharacteristic
423#101—34no—
424#100—121no—
425#97—61no—
426#97—61no—
427C4×C8—61no—
428C4×C8—61no—
429C4×C8—121no—
430#95—121no—
431C23×C3—21no—
432C23×C3—21no—
433C23×C3H(5,𝔽2)11yescharacteristic
434C22×D3—22no—
435C22×D3—41no—
436C22×D3—41no—
437C22×D3—81no—
438C22×D3—81no—
439C3×Q8—21no—
440C3×Q8—21no—
441C3×Q8—22no—
442C3×Q8—22no—
443C3×Q8—44no—
444C3×Q8—41no—
445C3×Q8—41no—
446C3×Q8C4×D412yes—
447C3×Q8C4×D412yes—
448C3×D4—22no—
449C3×D4—22no—
450C3×D4—22no—
451C3×D4—22no—
452C3×D4—21no—
453C3×D4—21no—
454C3×D4—44no—
455C3×D4C4×D412yes—
456C3×D4C4×D412yes—
457C2×C3×C4—21no—
458C2×C3×C4—21no—
459C2×C3×C4—22no—
460C2×C3×C4—22no—
461C2×C3×C4—22no—
462C2×C3×C4—22no—
463C2×C3×C4—22no—
464C2×C3×C4—21no—
465C2×C3×C4—21no—
466C2×C3×C4—44no—
467C2×C3×C4—44no—
468C2×C3×C4—41no—
469C2×C3×C4—41no—
470C2×C3×C4—41no—
471C2×C3×C4—81no—
472C2×C3×C4—161no—
473C2×C3×C4C23×C411yescharacteristic
474C2×C3×C4C22×D411yescharacteristic
475C2×C3×C4C22×D411yescharacteristic
476C2×C3×C4C2×#4111yescharacteristic
477C2×C3×C4C2×#4111yescharacteristic
478C2×C3×C4H(5,𝔽2)11yescharacteristic
479C2×Dic3—22no—
480C2×Dic3—21no—
481C2×Dic3—21no—
482C2×Dic3—41no—
483C2×Dic3—41no—
484C2×Dic3—41no—
485C2×Dic3—81no—
486C2×Dic3—161no—
487D12—22no—
488D12—22no—
489D12—41no—
490D12—41no—
491C4×D3—161no—
492C4×D3—161no—
493C3×C8—21no—
494C3×C8—21no—
495C3×C8—41no—
496C3⋊C8—22no—
497C3⋊C8—22no—
498C3⋊C8—21no—
499C3⋊C8—21no—
500C3⋊C8—41no—
501C24—61no—
502#41—21no—
503#41—21no—
504#41—21no—
505#41—21no—
506#41—24no—
507#41—24no—
508#41—44no—
509#41—42no—
510#41—42no—
511#41—44no—
512#41C2×C4×D312yes—
513#41C2×C4×D312yes—
514#41#23512yes—
515#41#23712yes—
516C2×Q8—21no—
517C2×Q8—24no—
518C2×Q8—24no—
519C2×Q8—21no—
520C2×Q8—21no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—41no—
524C2×Q8—42no—
525C2×Q8—42no—
526C2×Q8C2×C4×D312yes—
527C2×Q8D3×D411yescharacteristic
528C2×Q8C2×#6711yescharacteristic
529C2×D4—21no—
530C2×D4—21no—
531C2×D4—32no—
532C2×D4—41no—
533C2×D4—61no—
534C2×D4—61no—
535C2×D4—61no—
536C2×D4—61no—
537C2×D4—61no—
538C2×D4D3×D411yescharacteristic
539C2×D4C2×#6711yescharacteristic
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—41no—
545C22×C4—41no—
546C22×C4—41no—
547C22×C4—61no—
548C22×C4—61no—
549C22×C4—121no—
550C22×C4—121no—
551C22×C4—121no—
552C22×C4C23×D311yescharacteristic
553SD16—64no—
554SD16—64no—
555D8—64no—
556D8—64no—
557M16—21no—
558M16—21no—
559M16—32no—
560M16—32no—
561M16—42no—
562M16—61no—
563M16—61no—
564M16—61no—
565M16—61no—
566M16—122no—
567C2×C8—21no—
568C2×C8—21no—
569C2×C8—21no—
570C2×C8—31no—
571C2×C8—31no—
572C2×C8—42no—
573C2×C8—61no—
574C2×C8—61no—
575C2×C8—61no—
576C2×C8—61no—
577C2×C8—61no—
578C2×C8—122no—
579C2×C8C2×#6711yescharacteristic
580C2×C8C2×#6711yescharacteristic
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—31no—
584C4⋊C4—31no—
585C4⋊C4—41no—
586C4⋊C4—61no—
587C4⋊C4—61no—
588C4⋊C4—121no—
589C4⋊C4C2×#6711yescharacteristic
590C4⋊C4C2×#6711yescharacteristic
591C22⋊C4—21no—
592C22⋊C4—41no—
593C22⋊C4—61no—
594C22⋊C4—124no—
595C22⋊C4—121no—
596C22⋊C4—121no—
597C22⋊C4—121no—
598C42—21no—
599C42—32no—
600C42—41no—
601C42—61no—
602C42—61no—
603C42—121no—
604C22×C3—21no—
605C22×C3—21no—
606C22×C3—22no—
607C22×C3—22no—
608C22×C3—41no—
609C22×C3—41no—
610C22×C3—41no—
611C22×C3—41no—
612C22×C3#51811yescharacteristic
613C22×C3D4211yescharacteristic
614C22×C3#55111yescharacteristic
615C2×D3—42no—
616C2×D3—81no—
617C2×D3—81no—
618C2×D3—82no—
619C2×D3—161no—
620C3×C4—22no—
621C3×C4—22no—
622C3×C4—42no—
623C3×C4—41no—
624C3×C4—41no—
625C3×C4—81no—
626C3×C4C2×C4×D411yescharacteristic
627C3×C4C2×C4×D411yescharacteristic
628C3×C4#54611yescharacteristic
629C3×C4#54611yescharacteristic
630Dic3—42no—
631Dic3—41no—
632Dic3—41no—
633Dic3—81no—
634C23—21no—
635C23—21no—
636C23—62no—
637C23—121no—
638C23—121no—
639C23—241no—
640C23—241no—
641C23#100411yescharacteristic
642Q8—21no—
643Q8—21no—
644Q8—22no—
645Q8—22no—
646Q8—44no—
647Q8—41no—
648Q8—41no—
649Q8#67⋊C412yes—
650Q8C4×#6712yes—
651D4—22no—
652D4—22no—
653D4—22no—
654D4—22no—
655D4—21no—
656D4—21no—
657D4—44no—
658D4—62no—
659D4—62no—
660D4—121no—
661D4—121no—
662D4#67⋊C412yes—
663D4C4×#6712yes—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—62no—
679C2×C4—61no—
680C2×C4—61no—
681C2×C4—81no—
682C2×C4—121no—
683C2×C4—121no—
684C2×C4—121no—
685C2×C4—161no—
686C2×C4—241no—
687C2×C4—481no—
688C2×C4—481no—
689C2×C4—481no—
690C2×C4C22×C4×D311yescharacteristic
691C2×C4C2×#23511yescharacteristic
692C2×C4C2×D3×D411yescharacteristic
693C2×C4C2×#23711yescharacteristic
694C2×C4#100411yescharacteristic
695C2×C4C22×#6711yescharacteristic
696C8—21no—
697C8—21no—
698C8—41no—
699C8—62no—
700C8—61no—
701C8—61no—
702C8—62no—
703C8—121no—
704C2×C3—21no—
705C2×C3—21no—
706C2×C3—42no—
707C2×C3—42no—
708C2×C3#235811yescharacteristic
709C2×C3#329011yescharacteristic
710C2×C3#329011yescharacteristic
711D3—82no—
712C22—21no—
713C22—21no—
714C22—22no—
715C22—22no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—122no—
721C22—242no—
722C22—241no—
723C22—241no—
724C22—481no—
725C22#548611yescharacteristic
726C22#549511yescharacteristic
727C22D4×#6711yescharacteristic
728C4—22no—
729C4—22no—
730C4—41no—
731C4—41no—
732C4—42no—
733C4—81no—
734C4—122no—
735C4—121no—
736C4—121no—
737C4—241no—
738C4C2×#67⋊C411yescharacteristic
739C4#549011yescharacteristic
740C4C2×C4×#6711yescharacteristic
741C4#580611yescharacteristic
742C3#2289511yescharacteristic, complemented, Sylow
743C2—21no—
744C2—21no—
745C2—42no—
746C2—42no—
747C2—242no—
748C2#8209911yescharacteristic
749C2#8401211yescharacteristic
750C2#8402011yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2898C2
socleC22×C3D42
prime radicalC2×#5245C2
cosocleC22×C3×C4C24
hypercenter#2898D3
hypocenterC3#22895
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2898

Complements

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

NH ≅ G/Ntype
1. C3×#2898C2semidirect
3. #84111C2semidirect
4. #82564C2semidirect
5. #78820C2semidirect
6. #72018C2semidirect
7. #72008C2semidirect
8. #71665C2semidirect
9. #70343C2semidirect
10. #70338C2semidirect
11. #70303C2semidirect
12. #70298C2semidirect
13. #70289C2semidirect
14. #70263C2semidirect
15. #70226C2semidirect
34. #5245C4semidirect
73. #2898D3semidirect
87. C3×2−1+4C2×C4semidirect
112. #949C2×C4semidirect
113. #949C2×C4semidirect
117. #947C2×C4semidirect
152. #837C2×C4semidirect
153. #837C2×C4semidirect
154. #836C2×C4semidirect
338. 2−1+4C4×D3semidirect
742. C3#22895semidirect

Maximal subgroups

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

subgroupindex
C3×#28982
C2×#52452
#841112
#825642
#788202
#720182
#720082
#716652
#703432
#703382
#703032
#702982
#702892
#702632
#702262
#228953