Finite Group Databasebeta

← #11885768

Subgroups — #11885768

Order 1152.

1321 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5717C211yesmaximal, characteristic, complemented
2C2×D3×#213C212yesmaximal, complemented
3D122C211yesmaximal, characteristic, complemented
4#10592772C211yesmaximal, characteristic, complemented
5C3×#5072C211yesmaximal, characteristic, complemented
6#10591515C212yesmaximal, complemented
7#10591449C212yesmaximal, complemented
8#10591448C212yesmaximal, complemented
9#10590349C211yesmaximal, characteristic, complemented
10C4×#3713C211yesmaximal, characteristic, complemented
11#10590329C211yesmaximal, characteristic, complemented
12D4×#213—31nomaximal
13#80350—31nomaximal
14C2×C3×D3×D4—21no—
15C2×C3×D3×D4C2212yescomplemented
16C2×D3×D12—21no—
17C2×D3×D12C2212yescomplemented
18C32×#127C2211yescharacteristic, complemented
19C2×#3753C2212yescomplemented
20C2×#3753C2212yescomplemented
21#64083C2211yescharacteristic, complemented
22C3×#938—21no—
23C3×#938C2212yescomplemented
24C3×#937—22no—
25C2×C3×#213C2212yescomplemented
26C3×C4×D12C2211yescharacteristic
27C3×C4×D12C2211yescharacteristic, complemented
28C3×C4×D12C2211yescharacteristic, complemented
29D3×#213—28no—
30#63923—22no—
31#63909—21no—
32#63909C2212yescomplemented
33C2×#3714C2212yescomplemented
34C2×#3713C2211yescharacteristic
35C2×D3×C3⋊C8C2212yescomplemented
36C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
37C3×#832C2212yescomplemented
38C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
39#63576C2211yescharacteristic, complemented
40#63564C2212yescomplemented
41#63562C2211yescharacteristic, complemented
42#63562C2211yescharacteristic, complemented
43(C2×D3)⋊(C3⋊C8)C2212yescomplemented
44C3×D42—31no—
45C22×#213—32no—
46C2×D3×D8—32no—
47#5717D311yescharacteristic, complemented
48D4×D12—31no—
49D4×D12—31no—
50C3×#459—31no—
51#5217—32no—
52#5156—32no—
53#5073—31no—
54#5073—31no—
55#5073—61no—
56#5072D311yescharacteristic, complemented
57#5038—32no—
58#5036—32no—
59#5015—31no—
60C4×#213—31no—
61C4×#213—31no—
62D4×C3⋊C8—31no—
63#4843—31no—
64#4834—31no—
65#4775—32no—
66#4773—32no—
67C23×C3×D3—22no—
68C22×D32—22no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4C2312yes—
72C2×C4⋊(C3⋊D3)—21no—
73C2×C4⋊(C3⋊D3)—21no—
74C2×C4⋊(C3⋊D3)C2312yes—
75C2×C3×#67—22no—
76C2×C3×#67—22no—
77C3×D3×D4—24no—
78C3×D3×D4—42no—
79C3×D3×D4—44no—
80C2×C3×D12D411yescharacteristic
81C2×C3×D12C2311yescharacteristic
82C2×C3×D12C2311yescharacteristic
83C2×C3×C4×D3—21no—
84C2×C3×C4×D3—21no—
85C2×C3×C4×D3C2312yes—
86C2×#626—22no—
87C2×#626—22no—
88D3×D12—24no—
89D3×D12—42no—
90D3×D12—44no—
91C32×C42C2311yescharacteristic
92#3753—24no—
93#3753—42no—
94C2×C3⋊(C3⋊C8)C2311yescharacteristic
95C2×C3⋊(C3⋊C8)C2311yescharacteristic
96C3×#217—22no—
97C3×#213—24no—
98C3×(C2×D3)⋊C4—21no—
99C3×(C2×D3)⋊C4—21no—
100C3×(C2×D3)⋊C4D412yes—
101C3×C4⋊Dic3D411yescharacteristic
102C3×C4⋊Dic3C2311yescharacteristic
103C3×C4⋊Dic3C2311yescharacteristic
104C2×C3×C3⋊C8C2312yes—
105#3722—22no—
106#3714—24no—
107#3713D414yescomplemented
108D3×C3⋊C8—24no—
109D4×D8—91noSylow
110C22×C3×D4—32no—
111C22×C3×D4—61no—
112C2×D3×D4—21no—
113C2×D3×D4—32no—
114C2×D3×D4—61no—
115C2×D3×D4C2×D312yescomplemented
116C22×D12—32no—
117C22×D12—61no—
118C2×C3×D8—32no—
119C3×#127—21no—
120C3×#127C2×D311yescharacteristic, complemented
121C3×#127C2×D311yescharacteristic, complemented
122C3×#121—32no—
123C3×#121—61no—
124C3×#120—62no—
125C3×C4×D4—31no—
126C3×C4×D4—31no—
127C3×C4×D4—31no—
128#938—21no—
129#938C2×D312yescomplemented
130#937—22no—
131C2×#213—31no—
132C2×#213—31no—
133C2×#213—32no—
134C2×#213—32no—
135C2×#213—32no—
136C2×#213—32no—
137C2×#213—32no—
138C2×#213—62no—
139C2×#213—62no—
140C2×#213—68no—
141C2×#213C2×D312yescomplemented
142#930—32no—
143#930—61no—
144C22×C3⋊C8—32no—
145D3×D8—68no—
146C2×D24—32no—
147C2×D3×C8—32no—
148#895—32no—
149#895—61no—
150#882—62no—
151#882—62no—
152#874—31no—
153#874—31no—
154#874—61no—
155C4×D12C2×D311yescharacteristic
156C4×D12C2×D311yescharacteristic, complemented
157C4×D12C2×D311yescharacteristic, complemented
158C3×C4⋊C8—31no—
159C3×#102—32no—
160#832—31no—
161#832—31no—
162#832C2×D312yescomplemented
163#830—32no—
164#821—32no—
165(C2×D3)⋊C8—32no—
166#809—31no—
167#809—31no—
168#809—32no—
169#807—31no—
170#807—31no—
171C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
172C4×C3⋊C8—31no—
173C4×C3⋊C8—31no—
174C4×C3⋊C8—61no—
175C23×C32—22no—
176C22×C3⋊D3—22no—
177C22×C3×D3—21no—
178C22×C3×D3—21no—
179C22×C3×D3—44no—
180C22×C3×D3—44no—
181C22×C3×D3—82no—
182C22×C3×D3C2×D412yes—
183C2×D32—44no—
184C2×D32—44no—
185C2×D32—82no—
186C32×D4—22no—
187C32×D4—22no—
188C32×D4—42no—
189C32×D4—81no—
190C2×C32×C4C2×D411yescharacteristic
191C2×C32×C4C2×D411yescharacteristic
192C2×C32×C4C2411yescharacteristic
193C4⋊(C3⋊D3)—22no—
194C4⋊(C3⋊D3)—22no—
195C4⋊(C3⋊D3)—42no—
196C4⋊(C3⋊D3)—81no—
197C3×#67—44no—
198C3×#67—44no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3C2×D412yes—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12D814yescomplemented
205C3×D12C2×D412yescomplemented
206C3×D12C2×D412yescomplemented
207C3×C4×D3—22no—
208C3×C4×D3—22no—
209C3×C4×D3—42no—
210C3×C4×D3—42no—
211#626—44no—
212#626—44no—
213C3⋊(C3⋊C8)—21no—
214C3⋊(C3⋊C8)C2×D412yescomplemented
215C3×C3⋊C8—22no—
216C22×D8—92no—
217D42—31no—
218D42—91no—
219#493—91no—
220#466—92no—
221#459—31no—
222#459—91no—
223#447—92no—
224#447—92no—
225C4×D8—91no—
226C8×D4—91no—
227C24×C3—62no—
228C23×D3—22no—
229C23×D3—62no—
230C23×D3—62no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—22no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—41no—
240C2×C3×D4—41no—
241C2×C3×D4—62no—
242C2×C3×D4—64no—
243C2×C3×D4—62no—
244C2×C3×D4—122no—
245C2×C3×D4—124no—
246C2×C3×D4C22×D312yes—
247C2×C3×D4C22×D312yes—
248C22×C3×C4—32no—
249C22×C3×C4—61no—
250C22×C3×C4—61no—
251C2×#67—22no—
252C2×#67—22no—
253C2×#67—62no—
254C2×#67—62no—
255D3×D4—24no—
256D3×D4—42no—
257D3×D4—44no—
258D3×D4—64no—
259D3×D4—122no—
260D3×D4—124no—
261C2×D12—32no—
262C2×D12—32no—
263C2×D12—62no—
264C2×D12—64no—
265C2×D12—62no—
266C2×D12—62no—
267C2×D12—61no—
268C2×D12—61no—
269C2×D12—61no—
270C2×D12—61no—
271C2×D12—121no—
272C2×D12—121no—
273C2×D12—122no—
274C2×D12—124no—
275C2×D12#6711yescharacteristic
276C2×D12C22×D311yescharacteristic
277C2×D12C22×D311yescharacteristic
278C2×C4×D3—21no—
279C2×C4×D3—21no—
280C2×C4×D3C22×D312yes—
281C3×D8—64no—
282C2×C3×C8—32no—
283C3×C4⋊C4—31no—
284C3×C4⋊C4—31no—
285C3×C4⋊C4—31no—
286C3×C22⋊C4—32no—
287C3×C22⋊C4—62no—
288C3×C22⋊C4—61no—
289C3×C22⋊C4—61no—
290C3×C42—21no—
291C3×C42C22×D311yescharacteristic
292C3×C42C22×D311yescharacteristic
293#217—22no—
294#213—24no—
295#213—34no—
296#213—34no—
297#213—64no—
298#213—64no—
299#213—64no—
300#213—124no—
301#213—122no—
302#213—122no—
303#213—124no—
304(C2×D3)⋊C4—21no—
305(C2×D3)⋊C4—21no—
306(C2×D3)⋊C4—62no—
307(C2×D3)⋊C4—62no—
308(C2×D3)⋊C4#6712yes—
309C4⋊Dic3#6711yescharacteristic
310C4⋊Dic3C22×D311yescharacteristic
311C4⋊Dic3C22×D311yescharacteristic
312C2×C3⋊C8—31no—
313C2×C3⋊C8—31no—
314C2×C3⋊C8—31no—
315C2×C3⋊C8—31no—
316C2×C3⋊C8—61no—
317C2×C3⋊C8—61no—
318C2×C3⋊C8—64no—
319C2×C3⋊C8C22×D312yes—
320D24—64no—
321D3×C8—64no—
322C22×C32—42no—
323C22×C32—42no—
324C22×C32—42no—
325C22×C32C22×D411yescharacteristic
326C2×C3⋊D3—42no—
327C2×C3⋊D3—42no—
328C2×C3⋊D3—42no—
329C2×C3×D3—22no—
330C2×C3×D3—22no—
331C2×C3×D3—22no—
332C2×C3×D3—22no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C2×C3×D3—41no—
336C2×C3×D3—41no—
337C2×C3×D3—88no—
338C2×C3×D3C2×D814yescomplemented
339D32—88no—
340C32×C4—21no—
341C32×C4C2×D812yescomplemented
342C32×C4C22×D411yescharacteristic
343C32×C4H(5,𝔽2)11yescharacteristic
344C3×Dic3—22no—
345C3×Dic3—41no—
346C3×Dic3—41no—
347C22×D4—32no—
348C22×D4—61no—
349C22×D4—92no—
350C22×D4—181no—
351C2×D8—32no—
352C2×D8—92no—
353C2×D8—91no—
354C2×D8—92no—
355C2×D8—92no—
356C2×D8—188no—
357C22×C8—92no—
358#127—91no—
359#127D3211yescharacteristic, complemented
360#121—32no—
361#121—61no—
362#121—92no—
363#121—181no—
364#120—62no—
365#120—182no—
366C4×D4—31no—
367C4×D4—31no—
368C4×D4—31no—
369#107—91no—
370C4⋊C8—31no—
371#102—32no—
372#102—92no—
373#102—91no—
374#102—91no—
375C22⋊C8—92no—
376C4×C8—91no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3—32no—
380C23×C3—42no—
381C23×C3—61no—
382C23×C3—61no—
383C23×C3—124no—
384C23×C3—124no—
385C23×C3—242no—
386C22×D3—21no—
387C22×D3—21no—
388C22×D3—44no—
389C22×D3—44no—
390C22×D3—62no—
391C22×D3—62no—
392C22×D3—82no—
393C22×D3—122no—
394C22×D3—124no—
395C22×D3—124no—
396C22×D3—124no—
397C22×D3—124no—
398C22×D3—242no—
399C22×D3—242no—
400C22×D3C2×#6712yes—
401C3×D4—22no—
402C3×D4—22no—
403C3×D4—22no—
404C3×D4—22no—
405C3×D4—34no—
406C3×D4—32no—
407C3×D4—32no—
408C3×D4—42no—
409C3×D4—42no—
410C3×D4—42no—
411C3×D4—42no—
412C3×D4—61no—
413C3×D4—61no—
414C3×D4—82no—
415C3×D4—81no—
416C3×D4—81no—
417C3×D4—82no—
418C3×D4—124no—
419C3×D4—124no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—32no—
424C2×C3×C4—62no—
425C2×C3×C4—62no—
426C2×C3×C4—61no—
427C2×C3×C4—61no—
428C2×C3×C4—122no—
429C2×C3×C4—122no—
430C2×C3×C4D3×D411yescharacteristic
431C2×C3×C4D3×D411yescharacteristic
432C2×C3×C4C2×#6711yescharacteristic
433C2×C3×C4C2×#6711yescharacteristic
434C2×C3×C4C23×D311yescharacteristic
435C2×C3×C4C23×D311yescharacteristic
436#67—44no—
437#67—44no—
438#67—124no—
439#67—124no—
440C2×Dic3—21no—
441C2×Dic3—21no—
442C2×Dic3C2×#6712yes—
443D12—21no—
444D12—21no—
445D12—62no—
446D12—62no—
447D12—62no—
448D12—62no—
449D12—122no—
450D12—122no—
451D12—124no—
452D12—124no—
453D12—122no—
454D12—122no—
455D12—242no—
456D12—241no—
457D12—241no—
458D12—242no—
459D12#21314yescomplemented
460D12D3×D412yescomplemented
461D12D3×D412yescomplemented
462C4×D3—22no—
463C4×D3—22no—
464C4×D3—42no—
465C4×D3—42no—
466C3×C8—62no—
467C3⋊C8—22no—
468C3⋊C8—32no—
469C3⋊C8—32no—
470C3⋊C8—62no—
471C3⋊C8—62no—
472C3⋊C8—61no—
473C3⋊C8—61no—
474C2×C32—82no—
475C2×C32D4211yescharacteristic
476C2×C32C22×D811yescharacteristic
477C2×C32#57611yescharacteristic
478C3⋊D3—82no—
479C3×D3—24no—
480C3×D3—42no—
481C3×D3—42no—
482C24—62no—
483C24—182no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—31no—
487C2×D4—31no—
488C2×D4—31no—
489C2×D4—62no—
490C2×D4—64no—
491C2×D4—62no—
492C2×D4—92no—
493C2×D4—122no—
494C2×D4—124no—
495C2×D4—182no—
496C2×D4—184no—
497C2×D4—182no—
498C2×D4—181no—
499C2×D4—181no—
500C2×D4—362no—
501C2×D4—364no—
502C2×D4C2×D3212yes—
503C22×C4—32no—
504C22×C4—61no—
505C22×C4—61no—
506D8—64no—
507D8—94no—
508D8—184no—
509D8—184no—
510D8—362no—
511C2×C8—32no—
512C2×C8—91no—
513C2×C8—91no—
514C2×C8—184no—
515C4⋊C4—31no—
516C4⋊C4—31no—
517C4⋊C4—31no—
518C22⋊C4—32no—
519C22⋊C4—62no—
520C22⋊C4—61no—
521C22⋊C4—61no—
522C22⋊C4—182no—
523C42C2×D3211yescharacteristic
524C22×C3—21no—
525C22×C3—34no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—62no—
533C22×C3—62no—
534C22×C3—62no—
535C22×C3—62no—
536C22×C3—82no—
537C22×C3—82no—
538C22×C3—82no—
539C22×C3—121no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—121no—
543C22×C3—248no—
544C22×C3C2×D3×D411yescharacteristic
545C22×C3C22×#6711yescharacteristic
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—22no—
549C2×D3—22no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—88no—
555C2×D3—122no—
556C2×D3—122no—
557C2×D3—122no—
558C2×D3—122no—
559C2×D3—122no—
560C2×D3—122no—
561C2×D3—242no—
562C2×D3—242no—
563C2×D3—242no—
564C2×D3—248no—
565C2×D3—248no—
566C2×D3C2×#21314yescomplemented
567C3×C4—22no—
568C3×C4—21no—
569C3×C4—21no—
570C3×C4—21no—
571C3×C4—21no—
572C3×C4—22no—
573C3×C4—62no—
574C3×C4—121no—
575C3×C4—121no—
576C3×C4D3×D812yescomplemented
577C3×C4C2×#21312yescomplemented
578C3×C4C2×D3×D411yescharacteristic
579C3×C4C2×D3×D411yescharacteristic
580C3×C4#100411yescharacteristic
581C3×C4#100911yescharacteristic
582Dic3—22no—
583Dic3—41no—
584Dic3—41no—
585C32D4×D811yescharacteristic, complemented, Sylow
586C23—22no—
587C23—32no—
588C23—61no—
589C23—61no—
590C23—124no—
591C23—124no—
592C23—182no—
593C23—242no—
594C23—364no—
595C23—364no—
596C23—722no—
597D4—22no—
598D4—22no—
599D4—32no—
600D4—32no—
601D4—34no—
602D4—42no—
603D4—61no—
604D4—61no—
605D4—81no—
606D4—124no—
607D4—124no—
608D4—182no—
609D4—182no—
610D4—364no—
611D4—364no—
612D4—362no—
613D4—721no—
614C2×C4—32no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4D3×#6711yescharacteristic
622C2×C4D3×#6711yescharacteristic
623C2×C4C22×D3211yescharacteristic
624C8—62no—
625C8—92no—
626C8—181no—
627C2×C3—21no—
628C2×C3—21no—
629C2×C3—21no—
630C2×C3—64no—
631C2×C3—82no—
632C2×C3—82no—
633C2×C3—122no—
634C2×C3—122no—
635C2×C3—162no—
636C2×C3#560911yescharacteristic
637C2×C3C2×D3×D811yescharacteristic
638C2×C3#577411yescharacteristic
639C2×C3C22×#21311yescharacteristic
640C2×C3D4×#6711yescharacteristic
641C2×C3#583511yescharacteristic
642D3—24no—
643D3—42no—
644D3—42no—
645D3—242no—
646D3—242no—
647D3—482no—
648C22—34no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—62no—
653C22—62no—
654C22—62no—
655C22—62no—
656C22—121no—
657C22—121no—
658C22—121no—
659C22—121no—
660C22—248no—
661C22—362no—
662C22—362no—
663C22—362no—
664C22—728no—
665C22C2×D3×#6711yescharacteristic
666C4—21no—
667C4—62no—
668C4—121no—
669C4—121no—
670C4D3×#21312yescomplemented
671C4D32×D411yescharacteristic
672C4#6431411yescharacteristic
673C3—21no—
674C3#8035011yescharacteristic, complemented
675C3D4×#21311yescharacteristic, complemented
676C2—64no—
677C2—82no—
678C2—122no—
679C2—122no—
680C2—722no—
681C2C2×D3×#21311yescharacteristic
682C2#1059509711yescharacteristic
683C2#1059528811yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#127C22
socleC22×C32C22×D4
hypercenter#127D32
hypocenterC32D4×D8
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×#5717C2semidirect
2. C2×D3×#213C2semidirect
3. D122C2semidirect
4. #10592772C2semidirect
5. C3×#5072C2semidirect
6. #10591515C2semidirect
7. #10591449C2semidirect
8. #10591448C2semidirect
9. #10590349C2semidirect
10. C4×#3713C2semidirect
11. #10590329C2semidirect
15. C2×C3×D3×D4C22semidirect
17. C2×D3×D12C22semidirect
18. C32×#127C22semidirect
19. C2×#3753C22semidirect
20. C2×#3753C22semidirect
21. #64083C22semidirect
23. C3×#938C22semidirect
25. C2×C3×#213C22semidirect
27. C3×C4×D12C22semidirect
28. C3×C4×D12C22semidirect
32. #63909C22semidirect
33. C2×#3714C22semidirect
35. C2×D3×C3⋊C8C22semidirect
36. C4×C3⋊(C3⋊C8)C22semidirect
37. C3×#832C22semidirect
38. C3×C4⋊(C3⋊C8)C22semidirect
39. #63576C22semidirect
40. #63564C22semidirect
41. #63562C22semidirect
42. #63562C22semidirect
43. (C2×D3)⋊(C3⋊C8)C22semidirect
47. #5717D3semidirect
56. #5072D3semidirect
107. #3713D4semidirect
115. C2×D3×D4C2×D3semidirect
120. C3×#127C2×D3semidirect
121. C3×#127C2×D3semidirect
129. #938C2×D3semidirect
141. C2×#213C2×D3semidirect
156. C4×D12C2×D3semidirect
157. C4×D12C2×D3semidirect
162. #832C2×D3semidirect
171. C4⋊(C3⋊C8)C2×D3semidirect
204. C3×D12D8semidirect
205. C3×D12C2×D4semidirect
206. C3×D12C2×D4semidirect
214. C3⋊(C3⋊C8)C2×D4semidirect
338. C2×C3×D3C2×D8semidirect
341. C32×C4C2×D8semidirect
359. #127D32semidirect
459. D12#213semidirect
460. D12D3×D4semidirect
461. D12D3×D4semidirect
566. C2×D3C2×#213semidirect
576. C3×C4D3×D8semidirect
577. C3×C4C2×#213semidirect
585. C32D4×D8semidirect
670. C4D3×#213semidirect
674. C3#80350semidirect
675. C3D4×#213semidirect

Maximal subgroups

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

subgroupindex
C3×#57172
C2×D3×#2132
D1222
#105927722
C3×#50722
#105915152
#105914492
#105914482
#105903492
C4×#37132
#105903292
D4×#2133
#803503