Finite Group Databasebeta

← #11784873

Subgroups — #11784873

Order 1152.

1321 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64307C211yesmaximal, characteristic
2#10592832C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic, complemented
4C2×C3×#823C211yesmaximal, characteristic, complemented
5#10591318C211yesmaximal, characteristic, complemented
6C2×#63570C211yesmaximal, characteristic, complemented
7#10591088C211yesmaximal, characteristic, complemented
8#70384—31nomaximal
9#70086—31nomaximal
10C2×#3828C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C2×D3×#67—22no—
14#64307—22no—
15#64307—22no—
16#64307—22no—
17#64307C414yescomplemented
18C2×#3802—21no—
19C2×D3×D12—21no—
20C2×#3796—21no—
21C2×#3795—21no—
22C2×C32×M16C2211yescharacteristic, complemented
23C2×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
24C2×C3×C4×Dic3C2211yescharacteristic, complemented
25C2×C3×#208C2211yescharacteristic, complemented
26C2×(C2×D3)⋊Dic3—22no—
27#63648—21no—
28#63643—21no—
29C3×#825—22no—
30C3×#823—22no—
31#63570—22no—
32#63566—21no—
33#63563—21no—
34C2×#1009—31no—
35C2×#1004—31no—
36C2×C3×#104—31no—
37C2×C3×#100—31no—
38#5232—31no—
39#5221—31no—
40C2×#825—31no—
41C2×#825D311yescharacteristic, complemented
42#5133—31no—
43#5133—31no—
44#5133—31no—
45#5133—61no—
46C2×#823D311yescharacteristic, complemented
47#4939—31no—
48C2×#805—31no—
49C23×C3×D3D412yes—
50C22×D32—22no—
51C22×C32×C4C2311yescharacteristic
52C2×#638—21no—
53C22×C3⋊Dic3—21no—
54#3828—21no—
55#3828—21no—
56#3828C2×C412yes—
57#3828D412yes—
58C2×C4⋊(C3⋊D3)D411yescharacteristic
59C2×C4×C3⋊D3—21no—
60C2×#634D411yescharacteristic
61C2×C3×#67—21no—
62C22×C3×Dic3—21no—
63C3×#235—21no—
64C3×#235—21no—
65C3×#235C2×C412yes—
66C3×#235D412yes—
67C2×C3×D12—21no—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12C2×C412yes—
72C2×C3×D12D412yes—
73C2×C3×D12D411yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×Dic6D411yescharacteristic
76D3×#67—28no—
77D3×#67—28no—
78C2×#626—22no—
79C2×#625—22no—
80C2×D3×Dic3—22no—
81#3802—24no—
82#3802—24no—
83D3×D12—24no—
84D3×D12—24no—
85#3796—24no—
86#3796—24no—
87#3795—24no—
88#3795—24no—
89C32×M16—21no—
90C32×M16—21no—
91C2×C32×C8—21no—
92C4⋊(C3⋊Dic3)—21no—
93C4⋊(C3⋊Dic3)—21no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3—21no—
96C3×#208—21no—
97C3×#208—21no—
98C2×C3×C3⋊C8—21no—
99(C2×D3)⋊Dic3—44no—
100#1792—91noSylow
101C2×C3×#41—31no—
102C22×C3×D4—31no—
103C22×#67—62no—
104#1009—34no—
105#1009—62no—
106#1009—62no—
107#1009—62no—
108C2×#239—61no—
109#1004—34no—
110#1004—62no—
111#1004—62no—
112#1004—62no—
113C2×#237—61no—
114C2×#237—61no—
115C2×D3×D4—61no—
116C2×D3×D4—61no—
117C2×D3×D4—62no—
118C2×D3×D4—61no—
119C2×#235—31no—
120C2×#235—31no—
121C2×#235—61no—
122C2×#235—61no—
123C2×#235C2×D311yescharacteristic
124C22×D12—61no—
125C22×D12C2×D311yescharacteristic
126C2×C3×M16—21no—
127C2×C3×M16—31no—
128C2×C3×M16C2×D311yescharacteristic, complemented
129C2×C3×M16C2×D311yescharacteristic, complemented
130C2×C3×C42—31no—
131C2×#217—62no—
132C2×(C2×D3)⋊C4—62no—
133C2×C4⋊Dic3—31no—
134C2×C4⋊Dic3—31no—
135C2×C4⋊Dic3—61no—
136C2×C4×Dic3C2×D311yescharacteristic, complemented
137C2×#208C2×D311yescharacteristic, complemented
138C3×#104—62no—
139C3×#100—62no—
140#835—61no—
141#832—61no—
142#825—22no—
143#825—62no—
144#823—22no—
145#821—61no—
146#821—61no—
147#821—62no—
148#821—61no—
149#816—61no—
150#816—61no—
151#816—62no—
152#816—61no—
153#805—62no—
154C23×C32C2×D411yescharacteristic
155C22×C3⋊D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—22no—
158C22×C3×D3—22no—
159C22×C3×D3—42no—
160C22×C3×D3—42no—
161C22×C3×D3C22⋊C414yes—
162C2×D32—28no—
163C2×D32—82no—
164C2×D32—82no—
165C2×C32×C4C22×C411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171#638—22no—
172#638—22no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—41no—
175C2×C3⋊Dic3—42no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)C22⋊C412yescomplemented
178C4×C3⋊D3—22no—
179C4×C3⋊D3—22no—
180#634—21no—
181#634C22⋊C412yescomplemented
182C3×#67—22no—
183C3×#67—22no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—41no—
188C3×D12—22no—
189C3×D12—22no—
190C3×D12—22no—
191C3×D12—22no—
192C3×D12—21no—
193C3×D12C22⋊C412yescomplemented
194C3×C4×D3—22no—
195C3×C4×D3—22no—
196C3×Dic6—21no—
197C3×Dic6C22⋊C412yescomplemented
198#626—28no—
199#625—28no—
200D3×Dic3—28no—
201C32×C8—41no—
202C3×C3⋊C8—41no—
203C2×H(5,𝔽2)—91no—
204C2×#104—31no—
205C2×#104—91no—
206#417—91no—
207#417—91no—
208C2×#100—31no—
209#390—91no—
210C24×C3—32no—
211C23×D3—62no—
212C23×D3—62no—
213C23×D3D1212yes—
214C3×#41—32no—
215C3×#41—32no—
216C3×#41—61no—
217C3×#41—61no—
218C2×C3×Q8—31no—
219C2×C3×D4—32no—
220C2×C3×D4—31no—
221C2×C3×D4—32no—
222C2×C3×D4—61no—
223C2×C3×D4—61no—
224C2×C3×D4—61no—
225C2×C3×D4—61no—
226C2×C3×D4—61no—
227C22×C3×C4—21no—
228C22×C3×C4—61no—
229C22×C3×C4—61no—
230C22×C3×C4C22×D311yescharacteristic
231C22×C3×C4C22×D311yescharacteristic
232C2×#67—21no—
233C2×#67—62no—
234C2×#67—68no—
235C2×#67—68no—
236C2×#67—62no—
237C2×#67—62no—
238C2×#67—61no—
239C2×#67—61no—
240C2×#67—121no—
241C22×Dic3—21no—
242C22×Dic3—62no—
243C22×Dic3—61no—
244C22×Dic3—61no—
245C22×Dic3—121no—
246#239—64no—
247#239—64no—
248#237—64no—
249#237—64no—
250#237—64no—
251#237—64no—
252D3×D4—64no—
253D3×D4—64no—
254D3×D4—64no—
255D3×D4—64no—
256D3×D4—64no—
257D3×D4—68no—
258D3×D4—68no—
259D3×D4—64no—
260#235—21no—
261#235—21no—
262#235—32no—
263#235—32no—
264#235—32no—
265#235—32no—
266#235—62no—
267#235—62no—
268#235—64no—
269#235—64no—
270#235—61no—
271#235—61no—
272#235—61no—
273#235—61no—
274#235—121no—
275#235—121no—
276#235C4×D312yes—
277#235#6712yes—
278C2×D12—21no—
279C2×D12—21no—
280C2×D12—21no—
281C2×D12—21no—
282C2×D12—31no—
283C2×D12—31no—
284C2×D12—61no—
285C2×D12—64no—
286C2×D12—64no—
287C2×D12—62no—
288C2×D12C4×D312yes—
289C2×D12D1211yescharacteristic
290C2×D12#6712yes—
291C2×C4×D3—21no—
292C2×C4×D3—62no—
293C2×C4×D3—61no—
294C2×C4×D3—61no—
295C2×C4×D3—121no—
296C2×Dic6—31no—
297C2×Dic6—31no—
298C2×Dic6—61no—
299C2×Dic6D1211yescharacteristic
300C3×M16—21no—
301C3×M16—21no—
302C3×M16—21no—
303C3×M16—21no—
304C3×M16—41no—
305C3×M16—41no—
306C3×M16—61no—
307C3×M16—61no—
308C2×C3×C8—22no—
309C2×C3×C8—21no—
310C2×C3×C8—21no—
311C2×C3×C8—61no—
312C3×C42—61no—
313C3×C42—61no—
314#217—124no—
315(C2×D3)⋊C4—124no—
316C4⋊Dic3—62no—
317C4⋊Dic3—61no—
318C4⋊Dic3—61no—
319C4⋊Dic3—61no—
320C4⋊Dic3—61no—
321C4⋊Dic3—121no—
322C4×Dic3—21no—
323C4×Dic3—21no—
324#208—21no—
325#208—21no—
326C2×C3⋊C8—21no—
327C22×C32—21no—
328C22×C32—21no—
329C22×C32C2×C22⋊C411yescharacteristic
330C22×C32#12011yescharacteristic
331C22×C32#12011yescharacteristic
332C2×C3⋊D3—22no—
333C2×C3⋊D3—41no—
334C2×C3⋊D3—41no—
335C2×C3×D3—24no—
336C2×C3×D3—22no—
337C2×C3×D3—41no—
338C2×C3×D3—42no—
339C2×C3×D3—41no—
340C2×C3×D3—42no—
341C2×C3×D3—44no—
342C2×C3×D3—82no—
343D32—88no—
344C32×C4C2×C22⋊C411yescharacteristic
345C32×C4C2×C22⋊C411yescharacteristic
346C32×C4#12011yescharacteristic
347C32×C4#12011yescharacteristic
348C3⋊Dic3—22no—
349C3⋊Dic3—81no—
350C3×Dic3—22no—
351C3×Dic3—42no—
352H(5,𝔽2)—94no—
353H(5,𝔽2)—182no—
354H(5,𝔽2)—182no—
355H(5,𝔽2)—182no—
356C2×#41—31no—
357C2×#41—91no—
358C2×#41—181no—
359C2×#41—181no—
360C22×D4—31no—
361C22×D4—181no—
362C22×D4—181no—
363C22×D4—182no—
364C2×M16—31no—
365C2×M16D3211yescharacteristic, complemented
366C2×C4⋊C4—91no—
367C2×C22⋊C4—182no—
368C2×C42—31no—
369#104—62no—
370#104—182no—
371#103—181no—
372#103—181no—
373#102—181no—
374#102—181no—
375#100—62no—
376C23×C3—21no—
377C23×C3—34no—
378C23×C3—62no—
379C23×C3—61no—
380C23×C3—62no—
381C23×C3—122no—
382C23×C3—122no—
383C23×C3C2×D1211yescharacteristic
384C23×C3D3×D411yescharacteristic
385C22×D3—21no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—42no—
389C22×D3—42no—
390C22×D3—68no—
391C22×D3—68no—
392C22×D3—61no—
393C22×D3—61no—
394C22×D3—121no—
395C22×D3—242no—
396C22×D3—242no—
397C22×D3—242no—
398C22×D3—242no—
399C22×D3(C2×D3)⋊C414yes—
400C3×Q8—32no—
401C3×Q8—61no—
402C3×D4—32no—
403C3×D4—62no—
404C3×D4—62no—
405C3×D4—62no—
406C3×D4—62no—
407C3×D4—62no—
408C3×D4—61no—
409C3×D4—62no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—61no—
417C2×C3×C4—62no—
418C2×C3×C4—62no—
419C2×C3×C4—62no—
420C2×C3×C4—121no—
421C2×C3×C4—121no—
422C2×C3×C4C2×C4×D311yescharacteristic
423C2×C3×C4C2×C4×D311yescharacteristic
424C2×C3×C4C2×D1211yescharacteristic
425C2×C3×C4D3×D411yescharacteristic
426C2×C3×C4D3×D411yescharacteristic
427C2×C3×C4D3×D411yescharacteristic
428C2×C3×C4D3×D411yescharacteristic
429C2×C3×C4D3×D411yescharacteristic
430C2×C3×C4D3×D411yescharacteristic
431C2×C3×C4D3×D411yescharacteristic
432C2×C3×C4C2×#6711yescharacteristic
433C2×C3×C4C2×#6711yescharacteristic
434#67—22no—
435#67—22no—
436#67—68no—
437#67—68no—
438#67—68no—
439#67—62no—
440#67—62no—
441#67—62no—
442#67—62no—
443#67—122no—
444#67—122no—
445C2×Dic3—21no—
446C2×Dic3—22no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—68no—
450C2×Dic3—61no—
451C2×Dic3—61no—
452C2×Dic3—121no—
453C2×Dic3—122no—
454C2×Dic3—122no—
455C2×Dic3—122no—
456C2×Dic3—121no—
457C2×Dic3—121no—
458C2×Dic3—242no—
459D12—22no—
460D12—22no—
461D12—21no—
462D12—22no—
463D12—22no—
464D12—32no—
465D12—32no—
466D12—62no—
467D12—68no—
468D12—61no—
469D12—61no—
470D12—121no—
471D12(C2×D3)⋊C412yescomplemented
472C4×D3—22no—
473C4×D3—22no—
474C4×D3—68no—
475C4×D3—62no—
476C4×D3—62no—
477C4×D3—62no—
478C4×D3—62no—
479C4×D3—122no—
480C4×D3—122no—
481Dic6—21no—
482Dic6—32no—
483Dic6—32no—
484Dic6—62no—
485Dic6—61no—
486Dic6—61no—
487Dic6—121no—
488Dic6(C2×D3)⋊C412yescomplemented
489C3×C8—42no—
490C3×C8—41no—
491C3×C8—41no—
492C3×C8—121no—
493C3⋊C8—41no—
494C2×C32—21no—
495C2×C32—21no—
496C2×C32#37911yescharacteristic
497C2×C32#45311yescharacteristic
498C2×C32#45511yescharacteristic
499C3⋊D3—42no—
500C3×D3—42no—
501C3×D3—44no—
502C24—32no—
503C24—182no—
504#41—32no—
505#41—32no—
506#41—61no—
507#41—61no—
508#41—92no—
509#41—92no—
510#41—184no—
511#41—184no—
512#41—184no—
513#41—184no—
514#41—181no—
515#41—181no—
516C2×Q8—31no—
517C2×Q8—91no—
518C2×D4—32no—
519C2×D4—31no—
520C2×D4—32no—
521C2×D4—61no—
522C2×D4—61no—
523C2×D4—61no—
524C2×D4—61no—
525C2×D4—61no—
526C2×D4—91no—
527C2×D4—184no—
528C2×D4—184no—
529C2×D4—182no—
530C2×D4—184no—
531C2×D4—184no—
532C2×D4—188no—
533C2×D4—188no—
534C2×D4—182no—
535C2×D4—181no—
536C22×C4—61no—
537C22×C4—61no—
538C22×C4—182no—
539C22×C4—181no—
540C22×C4—181no—
541C22×C4C2×D3211yescharacteristic
542M16—21no—
543M16—21no—
544M16—61no—
545M16—61no—
546C2×C8—21no—
547C2×C8—61no—
548C4⋊C4—181no—
549C4⋊C4—181no—
550C22⋊C4—364no—
551C42—61no—
552C42—61no—
553C22×C3—21no—
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—21no—
560C22×C3—41no—
561C22×C3—41no—
562C22×C3—64no—
563C22×C3—62no—
564C22×C3—124no—
565C22×C3—121no—
566C22×C3—121no—
567C22×C3—122no—
568C22×C3—122no—
569C22×C3—242no—
570C22×C3D3×C22⋊C411yescharacteristic
571C22×C3#88211yescharacteristic
572C22×C3#88211yescharacteristic
573C22×C3#88211yescharacteristic
574C22×C3C2×(C2×D3)⋊C411yescharacteristic
575C22×C3#93711yescharacteristic
576C2×D3—22no—
577C2×D3—24no—
578C2×D3—42no—
579C2×D3—42no—
580C2×D3—41no—
581C2×D3—41no—
582C2×D3—44no—
583C2×D3—62no—
584C2×D3—62no—
585C2×D3—82no—
586C2×D3—122no—
587C2×D3—121no—
588C2×D3—121no—
589C2×D3—121no—
590C2×D3—121no—
591C2×D3—241no—
592C2×D3—241no—
593C2×D3—248no—
594C2×D3—248no—
595C3×C4—21no—
596C3×C4—21no—
597C3×C4—21no—
598C3×C4—21no—
599C3×C4—62no—
600C3×C4—122no—
601C3×C4D3×C22⋊C411yescharacteristic
602C3×C4D3×C22⋊C411yescharacteristic
603C3×C4D3×C22⋊C411yescharacteristic
604C3×C4#88211yescharacteristic
605C3×C4C2×(C2×D3)⋊C411yescharacteristic
606C3×C4#93711yescharacteristic
607C3×C4#93711yescharacteristic
608C3×C4#93711yescharacteristic
609Dic3—22no—
610Dic3—42no—
611Dic3—62no—
612Dic3—62no—
613Dic3—122no—
614Dic3—242no—
615Dic3—241no—
616Dic3—241no—
617C32#179211yescharacteristic, complemented, Sylow
618C23—34no—
619C23—61no—
620C23—62no—
621C23—62no—
622C23—122no—
623C23—122no—
624C23—188no—
625C23—181no—
626C23—722no—
627C23—722no—
628C23D3×D1211yescharacteristic
629Q8—32no—
630Q8—61no—
631Q8—92no—
632Q8—181no—
633D4—32no—
634D4—62no—
635D4—62no—
636D4—62no—
637D4—62no—
638D4—62no—
639D4—61no—
640D4—62no—
641D4—92no—
642D4—188no—
643D4—188no—
644D4—182no—
645D4—182no—
646D4—181no—
647C2×C4—61no—
648C2×C4—62no—
649C2×C4—62no—
650C2×C4—62no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—188no—
654C2×C4—181no—
655C2×C4—182no—
656C2×C4—182no—
657C2×C4—362no—
658C2×C4—361no—
659C2×C4C4×D3211yescharacteristic
660C2×C4D3×D1211yescharacteristic
661C2×C4#380211yescharacteristic
662C2×C4D3×#6711yescharacteristic
663C2×C4D3×#6711yescharacteristic
664C2×C4#381111yescharacteristic
665C8—41no—
666C8—121no—
667C2×C3—21no—
668C2×C3—21no—
669C2×C3—21no—
670C2×C3—21no—
671C2×C3—21no—
672C2×C3—21no—
673C2×C3—21no—
674C2×C3—41no—
675C2×C3—41no—
676C2×C3—122no—
677C2×C3—124no—
678C2×C3#474611yescharacteristic
679C2×C3#474711yescharacteristic
680C2×C3#482211yescharacteristic
681C2×C3#482511yescharacteristic
682C2×C3#495511yescharacteristic
683C2×C3#495511yescharacteristic
684D3—44no—
685D3—42no—
686D3—122no—
687D3—122no—
688D3—242no—
689C22—21no—
690C22—21no—
691C22—62no—
692C22—64no—
693C22—124no—
694C22—122no—
695C22—121no—
696C22—122no—
697C22—121no—
698C22—182no—
699C22—242no—
700C22—361no—
701C22—361no—
702C22—728no—
703C22D3×(C2×D3)⋊C411yescharacteristic
704C22#6392211yescharacteristic
705C22#6392311yescharacteristic
706C4—62no—
707C4—122no—
708C4—182no—
709C4—721no—
710C4D3×(C2×D3)⋊C411yescharacteristic
711C4#6392111yescharacteristic
712C4#6392211yescharacteristic
713C4#6398311yescharacteristic
714C3—21no—
715C3#7008611yescharacteristic, complemented
716C3#7038411yescharacteristic, complemented
717C2—21no—
718C2—21no—
719C2—124no—
720C2—122no—
721C2—362no—
722C2#1059101511yescharacteristic
723C2#1059101611yescharacteristic
724C2#1059139911yescharacteristic
subgroupG/H
centerC22#63923
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#120
prime radicalC2×#64307C2
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1792
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10592832C2semidirect
3. C2×C3×#825C2semidirect
4. C2×C3×#823C2semidirect
5. #10591318C2semidirect
6. C2×#63570C2semidirect
7. #10591088C2semidirect
17. #64307C4semidirect
22. C2×C32×M16C22semidirect
23. C2×C4⋊(C3⋊Dic3)C22semidirect
24. C2×C3×C4×Dic3C22semidirect
25. C2×C3×#208C22semidirect
41. C2×#825D3semidirect
46. C2×#823D3semidirect
128. C2×C3×M16C2×D3semidirect
129. C2×C3×M16C2×D3semidirect
136. C2×C4×Dic3C2×D3semidirect
137. C2×#208C2×D3semidirect
177. C4⋊(C3⋊D3)C22⋊C4semidirect
181. #634C22⋊C4semidirect
193. C3×D12C22⋊C4semidirect
197. C3×Dic6C22⋊C4semidirect
365. C2×M16D32semidirect
471. D12(C2×D3)⋊C4semidirect
488. Dic6(C2×D3)⋊C4semidirect
617. C32#1792semidirect
715. C3#70086semidirect
716. C3#70384semidirect

Maximal subgroups

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

subgroupindex
C2×#643072
#105928322
C2×C3×#8252
C2×C3×#8232
#105913182
C2×#635702
#105910882
#703843
#700863