Finite Group Databasebeta

← #11878867

Subgroups — #11878867

Order 1152.

1040 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#874C211yesmaximal, characteristic
2#10593873C211yesmaximal, characteristic, complemented
3#10592547C211yesmaximal, characteristic, complemented
4C3×#4848C211yesmaximal, characteristic, complemented
5C3×#4833C211yesmaximal, characteristic, complemented
6#10590946C212yesmaximal, complemented
7#10590942C212yesmaximal, complemented
8#10590878C212yesmaximal, complemented
9#10590877C212yesmaximal, complemented
10#10590377C211yesmaximal, characteristic, complemented
11#10590367C211yesmaximal, characteristic, complemented
12#80387—31nomaximal
13#79305—31nomaximal
14C2×D3×D12—21no—
15C2×D3×D12—21no—
16C2×D3×D12C2212yes—
17C2×#3744C2212yescomplemented
18#64083C2211yescharacteristic
19C4×#634C2211yescharacteristic, complemented
20C2×C3×#204C2212yescomplemented
21C3×#874C2211yescharacteristic
22C3×C42×D3C2211yescharacteristic
23C3×C4×Dic6C2211yescharacteristic, complemented
24#63918C2211yescharacteristic, complemented
25C4×#627—21no—
26#63911—21no—
27#63911—21no—
28#63911C2212yes—
29#63867—21no—
30#63866C2211yescharacteristic, complemented
31#63857C2211yescharacteristic
32#63856C2211yescharacteristic
33#63855—21no—
34#63851—21no—
35C2×#3717C2212yescomplemented
36C2×#3716C2212yescomplemented
37C32×C4⋊C8C2211yescharacteristic, complemented
38#63648C2212yescomplemented
39C3×#821C2212yescomplemented
40C3×(C2×D3)⋊C8C2212yes—
41C3×C8×Dic3C2212yescomplemented
42C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
43#63564C2212yescomplemented
44#63563C2212yescomplemented
45D3×#127—31no—
46#5578—31no—
47C2×#874—31no—
48#5470—31no—
49C3×#460—31no—
50C3×#431—31no—
51#5175—32no—
52#5169—32no—
53#5075—31no—
54#5027—31no—
55#4848—31no—
56#4848—31no—
57#4848—31no—
58#4848—31no—
59#4848—61no—
60#4848D311yescharacteristic, complemented
61#4833D311yescharacteristic, complemented
62#4776—32no—
63#4760—32no—
64#4731—32no—
65#4731—32no—
66#4694—32no—
67#4693—32no—
68C22×D32—22no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)—21no—
71C2×C4⋊(C3⋊D3)C2312yes—
72C2×#634C2311yescharacteristic
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12C2312yes—
76C2×C3×C4×D3D411yescharacteristic
77C2×C3×C4×D3D411yescharacteristic
78C2×C3×C4×D3C2311yescharacteristic
79C2×C3×Dic6C2311yescharacteristic
80C2×#627—21no—
81C2×#626—22no—
82C2×#626—22no—
83C2×#626—22no—
84D3×D12—42no—
85D3×D12—42no—
86D3×D12—44no—
87D3×D12—44no—
88C2×C32×C8C2312yes—
89C32×C42C2311yescharacteristic
90C4⋊(C3⋊Dic3)C2311yescharacteristic
91#3750—21no—
92C4×C3⋊Dic3—21no—
93#3744—42no—
94C3×C4⋊Dic3C2311yescharacteristic
95C3×Dic3⋊C4—21no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3D411yescharacteristic
98C3×C4×Dic3D411yescharacteristic
99C3×C4×Dic3C2311yescharacteristic
100C2×C3×C3⋊C8C2312yes—
101C3×#204—42no—
102#3724—21no—
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—21no—
105Dic3⋊Dic3—21no—
106Dic3⋊Dic3—21no—
107(C2×D3)⋊Dic3—21no—
108(C2×D3)⋊Dic3—21no—
109#3717—42no—
110#3716—42no—
111#3208—91noSylow
112C2×D3×D4—32no—
113C2×D3×D4—61no—
114C2×D3×D4—61no—
115C22×D12—32no—
116C22×D12—61no—
117C22×D12—61no—
118C2×C3×SD16—32no—
119C3×#127—31no—
120C3×C4×Q8—31no—
121C2×C3×C42—31no—
122#946—31no—
123#944—31no—
124C2×#215—32no—
125#940—32no—
126#940—61no—
127#940—61no—
128C2×#214—32no—
129#926—61no—
130#924—31no—
131#924—31no—
132#924—61no—
133C2×#204—32no—
134C2×#204—32no—
135C2×#204—32no—
136C2×#204—32no—
137C2×#204—62no—
138C2×#204C2×D312yescomplemented
139#897—31no—
140#896—61no—
141#889—61no—
142#888—31no—
143#874—32no—
144#874—31no—
145#874—31no—
146#874—61no—
147#874—61no—
148#874—61no—
149#874C2×D311yescharacteristic
150#872—61no—
151C42×D3C2×D311yescharacteristic
152#870—61no—
153#869—31no—
154#869—31no—
155C4×Dic6—31no—
156C4×Dic6—31no—
157C4×Dic6—61no—
158C4×Dic6—61no—
159C4×Dic6—61no—
160C4×Dic6C2×D311yescharacteristic, complemented
161C3×C4⋊C8—21no—
162C3×C4⋊C8—31no—
163C3×C4⋊C8C2×D311yescharacteristic, complemented
164C3×C4⋊C8C2×D311yescharacteristic, complemented
165C3×#102—32no—
166C3×C22⋊C8—32no—
167C3×C4×C8—32no—
168#832—32no—
169#821—32no—
170#821—32no—
171#821—32no—
172#821—32no—
173#821—62no—
174#821C2×D312yescomplemented
175(C2×D3)⋊C8C2×D312yes—
176C8×Dic3C2×D312yescomplemented
177#809—32no—
178C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
179C22×C3⋊D3—22no—
180C22×C3×D3—22no—
181C22×C3×D3C2×D411yescharacteristic
182C2×D32—44no—
183C2×D32—44no—
184C2×D32—44no—
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2411yescharacteristic
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190C4⋊(C3⋊D3)—22no—
191C4⋊(C3⋊D3)—22no—
192C4⋊(C3⋊D3)—42no—
193C4⋊(C3⋊D3)—42no—
194#634—21no—
195#634—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3C2×D411yescharacteristic
199C2×C3×Dic3C2×D411yescharacteristic
200C2×C3×Dic3C2×D411yescharacteristic
201C3×D12—22no—
202C3×D12—22no—
203C3×D12—42no—
204C3×D12—42no—
205C3×C4×D3—21no—
206C3×C4×D3—21no—
207C3×C4×D3—22no—
208C3×C4×D3SD1614yescomplemented
209C3×Dic6—21no—
210C3×Dic6—21no—
211#627—42no—
212#626—44no—
213#626—44no—
214#626—44no—
215C32×C8—22no—
216C3×C3⋊C8—22no—
217#533—91no—
218C2×#127—91no—
219#492—92no—
220#486—92no—
221#460—31no—
222#460—91no—
223#460—91no—
224#460—91no—
225#450—92no—
226#450—92no—
227#431—31no—
228C23×D3—62no—
229C23×D3—62no—
230C2×C3×Q8—31no—
231C2×C3×D4—32no—
232C2×C3×D4—61no—
233C2×C3×D4—61no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C2×#67—62no—
238C2×#67—62no—
239C2×#67—62no—
240D3×D4—122no—
241D3×D4—122no—
242D3×D4—124no—
243D3×D4—124no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12—32no—
247C2×D12—32no—
248C2×D12—62no—
249C2×D12—62no—
250C2×D12—62no—
251C2×D12—62no—
252C2×D12—61no—
253C2×D12—61no—
254C2×D12—61no—
255C2×D12—61no—
256C2×D12—121no—
257C2×D12—121no—
258C2×D12—122no—
259C2×D12—122no—
260C2×D12—124no—
261C2×D12—124no—
262C2×D12C22×D312yes—
263C2×C4×D3D1211yescharacteristic
264C2×C4×D3D1211yescharacteristic
265C2×C4×D3C22×D311yescharacteristic
266C2×Dic6—31no—
267C2×Dic6—31no—
268C2×Dic6—61no—
269C2×Dic6—61no—
270C2×Dic6—61no—
271C2×Dic6C22×D311yescharacteristic
272C3×SD16—122no—
273C2×C3×C8—22no—
274C2×C3×C8—32no—
275C2×C3×C8C22×D312yes—
276C2×C3×C8C22×D312yes—
277C3×C4⋊C4—31no—
278C3×C4⋊C4—61no—
279C3×C42—21no—
280C3×C42—31no—
281C3×C42—31no—
282C3×C42—31no—
283C3×C42—61no—
284C3×C42C22×D311yescharacteristic
285C3×C42C22×D311yescharacteristic
286#217—61no—
287#217—61no—
288#215—122no—
289#214—122no—
290(C2×D3)⋊C4—61no—
291(C2×D3)⋊C4—61no—
292C4⋊Dic3—31no—
293C4⋊Dic3—31no—
294C4⋊Dic3—61no—
295C4⋊Dic3—61no—
296C4⋊Dic3—61no—
297C4⋊Dic3—61no—
298C4⋊Dic3—61no—
299C4⋊Dic3C22×D311yescharacteristic
300Dic3⋊C4—21no—
301Dic3⋊C4—61no—
302Dic3⋊C4—61no—
303Dic3⋊C4—61no—
304Dic3⋊C4—61no—
305Dic3⋊C4—61no—
306Dic3⋊C4—61no—
307Dic3⋊C4—61no—
308Dic3⋊C4—61no—
309Dic3⋊C4—121no—
310C4×Dic3—21no—
311C4×Dic3—61no—
312C4×Dic3—61no—
313C4×Dic3—121no—
314C4×Dic3D1211yescharacteristic
315C4×Dic3D1211yescharacteristic
316C4×Dic3C22×D311yescharacteristic
317C2×C3⋊C8C22×D312yes—
318#204—42no—
319#204—122no—
320#204—122no—
321#204—124no—
322#204—122no—
323#204—122no—
324C22×C32C22×D411yescharacteristic
325C2×C3⋊D3—42no—
326C2×C3⋊D3—42no—
327C2×C3⋊D3—42no—
328C2×C3×D3—21no—
329C2×C3×D3—21no—
330C2×C3×D3—42no—
331C2×C3×D3—42no—
332C2×C3×D3—42no—
333C2×C3×D3C2×SD1612yes—
334D32—88no—
335C32×C4—21no—
336C32×C4C2×SD1612yes—
337C32×C4H(5,𝔽2)11yescharacteristic
338C32×C4H(5,𝔽2)11yescharacteristic
339C3⋊Dic3—41no—
340C3⋊Dic3—41no—
341C3×Dic3—21no—
342C3×Dic3—41no—
343C3×Dic3—41no—
344C3×Dic3C2×SD1612yes—
345C3×Dic3AGL(1,ℤ/8)12yes—
346C22×D4—92no—
347C22×D4—181no—
348C22×D4—181no—
349C2×SD16—32no—
350C2×SD16—92no—
351C2×SD16—92no—
352C2×SD16—92no—
353C4⋊Q8—91no—
354C4⋊Q8—91no—
355#127—31no—
356#127—92no—
357#127—91no—
358#127—181no—
359#127—181no—
360#125—181no—
361C22⋊Q8—91no—
362C22⋊Q8—91no—
363C22⋊Q8—181no—
364C4×Q8—31no—
365C4×Q8—91no—
366C4×Q8—181no—
367#117—181no—
368C2×C42—31no—
369C4⋊C8—31no—
370C4⋊C8D3211yescharacteristic, complemented
371#102—32no—
372#102—92no—
373#102—92no—
374#102—92no—
375C22⋊C8—32no—
376C4×C8—32no—
377C23×C3—31no—
378C23×C3—62no—
379C22×D3—22no—
380C22×D3—62no—
381C22×D3—62no—
382C22×D3—122no—
383C22×D3—124no—
384C22×D3—124no—
385C22×D3—124no—
386C22×D3—124no—
387C22×D3—124no—
388C22×D3—124no—
389C22×D3C2×D1211yescharacteristic
390C3×Q8—61no—
391C3×Q8—61no—
392C3×D4—62no—
393C3×D4—62no—
394C3×D4—122no—
395C3×D4—122no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—34no—
402C2×C3×C4—31no—
403C2×C3×C4—62no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4C2×D1211yescharacteristic
409C2×C3×C4C2×D1211yescharacteristic
410C2×C3×C4D3×D411yescharacteristic
411C2×C3×C4D3×D411yescharacteristic
412C2×C3×C4C23×D311yescharacteristic
413C2×C3×C4C23×D311yescharacteristic
414#67—124no—
415#67—124no—
416#67—124no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—61no—
420C2×Dic3—61no—
421C2×Dic3—61no—
422C2×Dic3—61no—
423C2×Dic3—121no—
424C2×Dic3—121no—
425C2×Dic3C2×D1211yescharacteristic
426C2×Dic3C2×D1211yescharacteristic
427C2×Dic3C2×D1211yescharacteristic
428D12—22no—
429D12—22no—
430D12—42no—
431D12—42no—
432D12—62no—
433D12—62no—
434D12—62no—
435D12—62no—
436D12—122no—
437D12—122no—
438D12—122no—
439D12—122no—
440D12—124no—
441D12—124no—
442D12—124no—
443D12—122no—
444D12—122no—
445D12—242no—
446D12—242no—
447C4×D3—21no—
448C4×D3—21no—
449C4×D3—22no—
450C4×D3#20414yescomplemented
451Dic6—21no—
452Dic6—21no—
453Dic6—62no—
454Dic6—61no—
455Dic6—61no—
456Dic6—61no—
457Dic6—61no—
458Dic6—121no—
459Dic6—122no—
460Dic6—122no—
461C3×C8—22no—
462C3×C8—22no—
463C3×C8—42no—
464C3×C8—62no—
465C3⋊C8—22no—
466C2×C32#53511yescharacteristic
467C2×C32C22×SD1611yescharacteristic
468C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
469C3⋊D3—82no—
470C3×D3—22no—
471C3×D3—82no—
472C24—182no—
473C2×Q8—31no—
474C2×Q8—91no—
475C2×Q8—181no—
476C2×D4—32no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—92no—
480C2×D4—182no—
481C2×D4—182no—
482C2×D4—182no—
483C2×D4—181no—
484C2×D4—181no—
485C2×D4—364no—
486C2×D4—362no—
487C2×D4—362no—
488C2×D4—364no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492SD16—122no—
493SD16—362no—
494SD16—362no—
495SD16—362no—
496C2×C8—32no—
497C2×C8C2×D3212yes—
498C4⋊C4—31no—
499C4⋊C4—61no—
500C4⋊C4—91no—
501C4⋊C4—181no—
502C4⋊C4—181no—
503C4⋊C4—181no—
504C4⋊C4—181no—
505C4⋊C4—181no—
506C4⋊C4—181no—
507C22⋊C4—181no—
508C22⋊C4—181no—
509C42—31no—
510C42—31no—
511C42—31no—
512C42—61no—
513C42—181no—
514C42C2×D3211yescharacteristic
515C22×C3—21no—
516C22×C3—32no—
517C22×C3—61no—
518C22×C3—61no—
519C22×C3—122no—
520C22×C3—122no—
521C22×C3—122no—
522C22×C3C22×D1211yescharacteristic
523C22×C3C2×D3×D411yescharacteristic
524C2×D3—21no—
525C2×D3—21no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—122no—
530C2×D3—122no—
531C2×D3—122no—
532C2×D3—122no—
533C2×D3—122no—
534C2×D3—122no—
535C2×D3—242no—
536C2×D3—242no—
537C2×D3—242no—
538C2×D3—248no—
539C2×D3—248no—
540C2×D3C2×#20412yes—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—22no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—32no—
547C3×C4—32no—
548C3×C4—41no—
549C3×C4—61no—
550C3×C4—121no—
551C3×C4—121no—
552C3×C4C2×#20412yes—
553C3×C4D3×SD1612yes—
554C3×C4#100411yescharacteristic
555C3×C4#100911yescharacteristic
556C3×C4#100911yescharacteristic
557C3×C4#100911yescharacteristic
558Dic3—21no—
559Dic3—41no—
560Dic3—41no—
561Dic3—122no—
562Dic3—121no—
563Dic3—121no—
564Dic3—121no—
565Dic3—121no—
566Dic3—241no—
567Dic3C2×#20412yes—
568Dic3#90812yes—
569C32#320811yescharacteristic, complemented, Sylow
570C23—31no—
571C23—62no—
572C23—182no—
573C23—364no—
574C23—364no—
575C23—364no—
576Q8—61no—
577Q8—61no—
578Q8—181no—
579Q8—181no—
580Q8—362no—
581D4—62no—
582D4—62no—
583D4—122no—
584D4—122no—
585D4—182no—
586D4—182no—
587D4—362no—
588D4—364no—
589D4—364no—
590D4—364no—
591D4—362no—
592C2×C4—31no—
593C2×C4—31no—
594C2×C4—34no—
595C2×C4—31no—
596C2×C4—62no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—181no—
602C2×C4—181no—
603C2×C4D3×D1211yescharacteristic
604C2×C4D3×D1211yescharacteristic
605C2×C4C22×D3211yescharacteristic
606C8—22no—
607C8—62no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—62no—
612C2×C3—242no—
613C2×C3#552511yescharacteristic
614C2×C3#560711yescharacteristic
615C2×C3C22×#20411yescharacteristic
616C2×C3C2×#90811yescharacteristic
617C2×C3C2×D3×SD1611yescharacteristic
618C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
619D3—22no—
620D3—82no—
621D3—242no—
622D3—242no—
623D3—482no—
624C22—32no—
625C22—61no—
626C22—61no—
627C22—122no—
628C22—122no—
629C22—122no—
630C22—362no—
631C22—362no—
632C22—362no—
633C22—728no—
634C22C2×D3×D1211yescharacteristic
635C4—21no—
636C4—32no—
637C4—32no—
638C4—61no—
639C4—121no—
640C4—121no—
641C4—361no—
642C4—361no—
643C4D3×#20412yes—
644C4#6431411yescharacteristic
645C4#6432011yescharacteristic
646C3—21no—
647C3#7930511yescharacteristic, complemented
648C3#8038711yescharacteristic, complemented
649C2—62no—
650C2—242no—
651C2—722no—
652C2#1059462811yescharacteristic
653C2C2×D3×#20411yescharacteristic
654C2D3×#90811yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊C8C22
socleC22×C32C22×D4
prime radicalD3×#874C2
hypercenterC4⋊C8D32
hypocenterC32#3208
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10593873C2semidirect
3. #10592547C2semidirect
4. C3×#4848C2semidirect
5. C3×#4833C2semidirect
6. #10590946C2semidirect
7. #10590942C2semidirect
8. #10590878C2semidirect
9. #10590877C2semidirect
10. #10590377C2semidirect
11. #10590367C2semidirect
17. C2×#3744C22semidirect
19. C4×#634C22semidirect
20. C2×C3×#204C22semidirect
23. C3×C4×Dic6C22semidirect
24. #63918C22semidirect
30. #63866C22semidirect
35. C2×#3717C22semidirect
36. C2×#3716C22semidirect
37. C32×C4⋊C8C22semidirect
38. #63648C22semidirect
39. C3×#821C22semidirect
41. C3×C8×Dic3C22semidirect
42. C3×C4⋊(C3⋊C8)C22semidirect
43. #63564C22semidirect
44. #63563C22semidirect
60. #4848D3semidirect
61. #4833D3semidirect
138. C2×#204C2×D3semidirect
160. C4×Dic6C2×D3semidirect
163. C3×C4⋊C8C2×D3semidirect
164. C3×C4⋊C8C2×D3semidirect
174. #821C2×D3semidirect
176. C8×Dic3C2×D3semidirect
178. C4⋊(C3⋊C8)C2×D3semidirect
208. C3×C4×D3SD16semidirect
370. C4⋊C8D32semidirect
450. C4×D3#204semidirect
569. C32#3208semidirect
647. C3#79305semidirect
648. C3#80387semidirect

Maximal subgroups

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

subgroupindex
D3×#8742
#105938732
#105925472
C3×#48482
C3×#48332
#105909462
#105909422
#105908782
#105908772
#105903772
#105903672
#803873
#793053