Finite Group Databasebeta

← #11878854

Subgroups — #11878854

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
1C3×D3×#127C211yesmaximal, characteristic
2#10593873C211yesmaximal, characteristic, complemented
3#10592774C211yesmaximal, characteristic, complemented
4C3×#5075C211yesmaximal, characteristic, complemented
5#10591490C212yesmaximal, complemented
6#10591487C212yesmaximal, complemented
7#10591459C212yesmaximal, complemented
8#10591458C212yesmaximal, complemented
9#10590377C211yesmaximal, characteristic, complemented
10#10590364C211yesmaximal, characteristic, complemented
11#10590248C211yesmaximal, characteristic, complemented
12#83169—31nomaximal
13#80387—31nomaximal
14C2×C3×D3×D4—21no—
15C2×C3×D3×D4—21no—
16C2×C3×D3×D4C2212yes—
17C32×#127C2211yescharacteristic
18C2×#3754C2212yescomplemented
19C4×#634C2211yescharacteristic, complemented
20C3×#940—21no—
21C3×#940—21no—
22C3×#940C2212yes—
23C2×C3×#214C2212yescomplemented
24C3×#874C2211yescharacteristic
25C3×C42×D3C2211yescharacteristic
26C3×C4×Dic6C2211yescharacteristic, complemented
27#63918C2211yescharacteristic, complemented
28C4×#627—21no—
29#63867—21no—
30#63866C2211yescharacteristic, complemented
31#63857C2211yescharacteristic
32#63856C2211yescharacteristic
33#63855—21no—
34#63851—21no—
35C2×#3716C2212yescomplemented
36C2×#3715C2212yescomplemented
37#63659C2212yescomplemented
38C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
39C3×#832C2212yescomplemented
40C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
41#63563C2212yescomplemented
42#63562C2212yescomplemented
43(C2×D3)⋊(C3⋊C8)C2212yes—
44Dic3×C3⋊C8C2212yescomplemented
45C2×C3×#127—31no—
46D3×#127D311yescharacteristic
47#5578—31no—
48#5470—31no—
49C3×#460—31no—
50#5218—32no—
51#5218—32no—
52#5175—32no—
53#5169—32no—
54#5076—32no—
55#5075—31no—
56#5075—31no—
57#5075—31no—
58#5075—31no—
59#5075—61no—
60#5075D311yescharacteristic, complemented
61#5070—32no—
62#5027—31no—
63#4925—31no—
64#4848—31no—
65#4833—31no—
66#4776—32no—
67#4760—32no—
68C23×C3×D3—22no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4C2312yes—
72C2×#634C2311yescharacteristic
73C2×C3×#67—22no—
74C2×C3×#67—22no—
75C2×C3×#67—22no—
76C3×D3×D4—42no—
77C3×D3×D4—42no—
78C3×D3×D4—44no—
79C3×D3×D4—44no—
80C2×C3×D12—21no—
81C2×C3×D12—21no—
82C2×C3×D12C2312yes—
83C2×C3×C4×D3D411yescharacteristic
84C2×C3×C4×D3D411yescharacteristic
85C2×C3×C4×D3C2311yescharacteristic
86C2×C3×Dic6C2311yescharacteristic
87C2×#627—21no—
88C32×C42C2311yescharacteristic
89#3754—42no—
90C4⋊(C3⋊Dic3)C2311yescharacteristic
91#3750—21no—
92C4×C3⋊Dic3—21no—
93C2×C3⋊(C3⋊C8)C2312yes—
94C3×#214—42no—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×Dic3⋊C4—21no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3D411yescharacteristic
99C3×C4×Dic3D411yescharacteristic
100C3×C4×Dic3C2311yescharacteristic
101C2×C3×C3⋊C8C2312yes—
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#3716—42no—
110#3715—42no—
111#3208—91noSylow
112C22×C3×D4—32no—
113C22×C3×D4—61no—
114C22×C3×D4—61no—
115C2×D3×D4—21no—
116C2×D3×D4—21no—
117C2×D3×D4C2×D312yes—
118C2×C3×SD16—32no—
119C3×#127—21no—
120C3×#127—32no—
121C3×#127—31no—
122C3×#127—61no—
123C3×#127—61no—
124C3×#127C2×D311yescharacteristic
125C3×#127C2×D311yescharacteristic
126C3×C4×Q8—31no—
127C2×C3×C42—31no—
128#946—31no—
129#944—31no—
130C2×#215—32no—
131#940—21no—
132#940—21no—
133#940C2×D312yes—
134C2×#214—32no—
135C2×#214—32no—
136C2×#214—32no—
137C2×#214—32no—
138C2×#214—62no—
139C2×#214C2×D312yescomplemented
140#926—61no—
141#924—31no—
142#924—31no—
143#924—61no—
144C2×#204—32no—
145#897—31no—
146#896—61no—
147#889—61no—
148#888—31no—
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—31no—
162C3×#102—32no—
163#832—32no—
164#832—32no—
165#832—32no—
166#832—32no—
167#832—62no—
168#832C2×D312yescomplemented
169#830—32no—
170#821—32no—
171(C2×D3)⋊C8—32no—
172C8×Dic3—32no—
173#809—32no—
174C4⋊(C3⋊C8)—31no—
175C4⋊(C3⋊C8)—31no—
176C4⋊(C3⋊C8)—61no—
177C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
178C4×C3⋊C8—32no—
179C23×C32—22no—
180C22×C3×D3—22no—
181C22×C3×D3—44no—
182C22×C3×D3—44no—
183C22×C3×D3—44no—
184C22×C3×D3C2×D411yescharacteristic
185C32×D4—22no—
186C32×D4—22no—
187C32×D4—42no—
188C32×D4—42no—
189C2×C32×C4C2×D411yescharacteristic
190C2×C32×C4C2×D411yescharacteristic
191C2×C32×C4C2411yescharacteristic
192C2×C3⋊Dic3—21no—
193C2×C3⋊Dic3—21no—
194#634—21no—
195#634—21no—
196C3×#67—44no—
197C3×#67—44no—
198C3×#67—44no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3C2×D411yescharacteristic
202C2×C3×Dic3C2×D411yescharacteristic
203C2×C3×Dic3C2×D411yescharacteristic
204C3×D12—22no—
205C3×D12—22no—
206C3×D12—42no—
207C3×D12—42no—
208C3×C4×D3—21no—
209C3×C4×D3—21no—
210C3×C4×D3—22no—
211C3×C4×D3SD1614yescomplemented
212C3×Dic6—21no—
213C3×Dic6—21no—
214#627—42no—
215C3⋊(C3⋊C8)—22no—
216C3×C3⋊C8—22no—
217#533—91no—
218C2×#127—31no—
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—91no—
228C24×C3—62no—
229C23×D3—22no—
230C2×C3×Q8—31no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—22no—
236C2×C3×D4—32no—
237C2×C3×D4—41no—
238C2×C3×D4—41no—
239C2×C3×D4—62no—
240C2×C3×D4—62no—
241C2×C3×D4—62no—
242C2×C3×D4—61no—
243C2×C3×D4—61no—
244C2×C3×D4—124no—
245C2×C3×D4—122no—
246C2×C3×D4—122no—
247C2×C3×D4—124no—
248C2×C3×D4C22×D312yes—
249C2×C3×D4C22×D312yes—
250C22×C3×C4—31no—
251C22×C3×C4—31no—
252C22×C3×C4—31no—
253C2×#67—22no—
254C2×#67—22no—
255C2×#67—22no—
256D3×D4—42no—
257D3×D4—42no—
258D3×D4—44no—
259D3×D4—44no—
260C2×D12—21no—
261C2×D12—21no—
262C2×D12C22×D312yes—
263C2×C4×D3#6711yescharacteristic
264C2×C4×D3#6711yescharacteristic
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—32no—
274C3×C4⋊C4—31no—
275C3×C4⋊C4—61no—
276C3×C42—21no—
277C3×C42—31no—
278C3×C42—31no—
279C3×C42—31no—
280C3×C42—61no—
281C3×C42C22×D311yescharacteristic
282C3×C42C22×D311yescharacteristic
283#217—61no—
284#217—61no—
285#215—122no—
286#214—42no—
287#214—122no—
288#214—122no—
289#214—122no—
290#214—122no—
291#214—124no—
292(C2×D3)⋊C4—61no—
293(C2×D3)⋊C4—61no—
294C4⋊Dic3—31no—
295C4⋊Dic3—31no—
296C4⋊Dic3—61no—
297C4⋊Dic3—61no—
298C4⋊Dic3—61no—
299C4⋊Dic3—61no—
300C4⋊Dic3—61no—
301C4⋊Dic3C22×D311yescharacteristic
302Dic3⋊C4—21no—
303Dic3⋊C4—61no—
304Dic3⋊C4—61no—
305Dic3⋊C4—61no—
306Dic3⋊C4—61no—
307Dic3⋊C4—61no—
308Dic3⋊C4—61no—
309Dic3⋊C4—61no—
310Dic3⋊C4—61no—
311Dic3⋊C4—121no—
312C4×Dic3—21no—
313C4×Dic3—61no—
314C4×Dic3—61no—
315C4×Dic3—121no—
316C4×Dic3#6711yescharacteristic
317C4×Dic3#6711yescharacteristic
318C4×Dic3C22×D311yescharacteristic
319C2×C3⋊C8—32no—
320C2×C3⋊C8—32no—
321C2×C3⋊C8—62no—
322C2×C3⋊C8C22×D312yes—
323#204—122no—
324C22×C32—42no—
325C22×C32—42no—
326C22×C32—42no—
327C22×C32C22×D411yescharacteristic
328C2×C3×D3—21no—
329C2×C3×D3—21no—
330C2×C3×D3—42no—
331C2×C3×D3—42no—
332C2×C3×D3—42no—
333C2×C3×D3—88no—
334C2×C3×D3C2×SD1612yes—
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—32no—
347C22×D4—61no—
348C22×D4—61no—
349C2×SD16—32no—
350C2×SD16—92no—
351C2×SD16—92no—
352C2×SD16—92no—
353C4⋊Q8—91no—
354C4⋊Q8—91no—
355#127—32no—
356#127—31no—
357#127—61no—
358#127—61no—
359#127D3211yescharacteristic
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⋊C8—91no—
371#102—32no—
372#102—92no—
373#102—92no—
374#102—92no—
375C22⋊C8—92no—
376C4×C8—92no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3—31no—
380C23×C3—42no—
381C23×C3—62no—
382C23×C3—124no—
383C23×C3—124no—
384C23×C3—124no—
385C22×D3—22no—
386C22×D3—44no—
387C22×D3—44no—
388C22×D3—44no—
389C22×D3C2×#6711yescharacteristic
390C3×Q8—61no—
391C3×Q8—61no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—42no—
401C3×D4—42no—
402C3×D4—62no—
403C3×D4—62no—
404C3×D4—82no—
405C3×D4—82no—
406C3×D4—122no—
407C3×D4—124no—
408C3×D4—124no—
409C3×D4—124no—
410C3×D4—122no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—31no—
415C2×C3×C4—31no—
416C2×C3×C4—34no—
417C2×C3×C4—31no—
418C2×C3×C4—62no—
419C2×C3×C4—61no—
420C2×C3×C4—61no—
421C2×C3×C4—61no—
422C2×C3×C4—61no—
423C2×C3×C4D3×D411yescharacteristic
424C2×C3×C4D3×D411yescharacteristic
425C2×C3×C4C2×#6711yescharacteristic
426C2×C3×C4C2×#6711yescharacteristic
427C2×C3×C4C23×D311yescharacteristic
428C2×C3×C4C23×D311yescharacteristic
429#67—44no—
430#67—44no—
431#67—44no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—61no—
435C2×Dic3—61no—
436C2×Dic3—61no—
437C2×Dic3—61no—
438C2×Dic3—121no—
439C2×Dic3—121no—
440C2×Dic3C2×#6711yescharacteristic
441C2×Dic3C2×#6711yescharacteristic
442C2×Dic3C2×#6711yescharacteristic
443D12—22no—
444D12—22no—
445D12—42no—
446D12—42no—
447C4×D3—21no—
448C4×D3—21no—
449C4×D3—22no—
450C4×D3#21414yescomplemented
451Dic6—21no—
452Dic6—21no—
453Dic6—62no—
454Dic6—61no—
455Dic6—61no—
456Dic6—61no—
457Dic6—61no—
458Dic6—121no—
459Dic6—122no—
460Dic6—122no—
461C3×C8—62no—
462C3⋊C8—22no—
463C3⋊C8—62no—
464C3⋊C8—62no—
465C3⋊C8—122no—
466C2×C32—82no—
467C2×C32#53511yescharacteristic
468C2×C32C22×SD1611yescharacteristic
469C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
470C3×D3—22no—
471C3×D3—82no—
472C24—62no—
473C2×Q8—31no—
474C2×Q8—91no—
475C2×Q8—181no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—32no—
479C2×D4—62no—
480C2×D4—62no—
481C2×D4—62no—
482C2×D4—61no—
483C2×D4—61no—
484C2×D4—124no—
485C2×D4—122no—
486C2×D4—122no—
487C2×D4—124no—
488C2×D4C2×D3212yes—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492SD16—122no—
493SD16—362no—
494SD16—362no—
495SD16—362no—
496C2×C8—32no—
497C2×C8—92no—
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—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—61no—
524C22×C3—61no—
525C22×C3—82no—
526C22×C3—82no—
527C22×C3—82no—
528C22×C3—122no—
529C22×C3—122no—
530C22×C3—122no—
531C22×C3—248no—
532C22×C3C2×D3×D411yescharacteristic
533C22×C3C22×#6711yescharacteristic
534C2×D3—21no—
535C2×D3—21no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—88no—
540C2×D3C2×#21412yes—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—22no—
546C3×C4—32no—
547C3×C4—32no—
548C3×C4—41no—
549C3×C4—61no—
550C3×C4—121no—
551C3×C4—121no—
552C3×C4D3×SD1612yes—
553C3×C4C2×#21412yes—
554C3×C4#100411yescharacteristic
555C3×C4#100411yescharacteristic
556C3×C4#100411yescharacteristic
557C3×C4#100911yescharacteristic
558Dic3—21no—
559Dic3—41no—
560Dic3—41no—
561Dic3—122no—
562Dic3—121no—
563Dic3—121no—
564Dic3—121no—
565Dic3—121no—
566Dic3—241no—
567Dic3#93212yes—
568Dic3C2×#21412yes—
569C32#320811yescharacteristic, complemented, Sylow
570C23—22no—
571C23—31no—
572C23—62no—
573C23—124no—
574C23—124no—
575C23—124no—
576Q8—61no—
577Q8—61no—
578Q8—181no—
579Q8—181no—
580Q8—362no—
581D4—22no—
582D4—22no—
583D4—42no—
584D4—42no—
585D4—62no—
586D4—62no—
587D4—122no—
588D4—124no—
589D4—124no—
590D4—124no—
591D4—122no—
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×#6711yescharacteristic
604C2×C4D3×#6711yescharacteristic
605C2×C4C22×D3211yescharacteristic
606C8—62no—
607C8—182no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—62no—
612C2×C3—82no—
613C2×C3—82no—
614C2×C3—162no—
615C2×C3—242no—
616C2×C3#560711yescharacteristic
617C2×C3C2×D3×SD1611yescharacteristic
618C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
619C2×C3C2×#93211yescharacteristic
620C2×C3C22×#21411yescharacteristic
621C2×C3#580511yescharacteristic
622D3—22no—
623D3—82no—
624C22—32no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—61no—
629C22—61no—
630C22—122no—
631C22—122no—
632C22—122no—
633C22—248no—
634C22C2×D3×#6711yescharacteristic
635C4—21no—
636C4—32no—
637C4—32no—
638C4—61no—
639C4—121no—
640C4—121no—
641C4—361no—
642C4—361no—
643C4D3×#21412yes—
644C4#6431311yescharacteristic
645C4#6431411yescharacteristic
646C3—21no—
647C3#8038711yescharacteristic, complemented
648C3#8316911yescharacteristic, complemented
649C2—62no—
650C2—82no—
651C2—242no—
652C2D3×#93211yescharacteristic
653C2C2×D3×#21411yescharacteristic
654C2#1059523111yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#127C22
socleC22×C32C22×D4
prime radicalC3×D3×#127C2
hypercenter#127D32
hypocenterC32#3208
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
2. #10593873C2semidirect
3. #10592774C2semidirect
4. C3×#5075C2semidirect
5. #10591490C2semidirect
6. #10591487C2semidirect
7. #10591459C2semidirect
8. #10591458C2semidirect
9. #10590377C2semidirect
10. #10590364C2semidirect
11. #10590248C2semidirect
18. C2×#3754C22semidirect
19. C4×#634C22semidirect
23. C2×C3×#214C22semidirect
26. C3×C4×Dic6C22semidirect
27. #63918C22semidirect
30. #63866C22semidirect
35. C2×#3716C22semidirect
36. C2×#3715C22semidirect
37. #63659C22semidirect
38. C4⋊(C3⋊(C3⋊C8))C22semidirect
39. C3×#832C22semidirect
40. C3×C4⋊(C3⋊C8)C22semidirect
41. #63563C22semidirect
42. #63562C22semidirect
44. Dic3×C3⋊C8C22semidirect
60. #5075D3semidirect
139. C2×#214C2×D3semidirect
160. C4×Dic6C2×D3semidirect
168. #832C2×D3semidirect
177. C4⋊(C3⋊C8)C2×D3semidirect
211. C3×C4×D3SD16semidirect
450. C4×D3#214semidirect
569. C32#3208semidirect
647. C3#80387semidirect
648. C3#83169semidirect

Maximal subgroups

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

subgroupindex
C3×D3×#1272
#105938732
#105927742
C3×#50752
#105914902
#105914872
#105914592
#105914582
#105903772
#105903642
#105902482
#831693
#803873