Finite Group Databasebeta

← #11868045

Subgroups — #11868045

Order 1152.

957 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5748C211yesmaximal, characteristic, complemented
2#10594786C211yesmaximal, characteristic, complemented
3C2×#63822C211yesmaximal, characteristic, complemented
4C2×#63815C211yesmaximal, characteristic, complemented
5C2×#63649C211yesmaximal, characteristic, complemented
6C3×#5132C211yesmaximal, characteristic, complemented
7#10591076C211yesmaximal, characteristic, complemented
8#10590982C212yesmaximal, complemented
9#10590981C212yesmaximal, complemented
10#10590980C212yesmaximal, complemented
11#10590979C212yesmaximal, complemented
12#84001—31nomaximal
13#78719—31nomaximal
14C2×C3×#235C2211yescharacteristic
15C2×C3×#235C2211yescharacteristic
16C2×C3×#235C2211yescharacteristic
17C2×C32×M16C2211yescharacteristic
18C2×#3748C2211yescharacteristic
19C22×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C3×#909—21no—
22C3×#909C2212yes—
23C3×#908—21no—
24C3×#908C2212yes—
25C3×#904—22no—
26C2×#3718C2211yescharacteristic, complemented
27#63827—21no—
28#63827C2212yes—
29C2×#3715C2211yescharacteristic, complemented
30C2×#3715C2211yescharacteristic, complemented
31#63823—21no—
32#63823C2212yes—
33#63822—22no—
34#63822—21no—
35#63822—21no—
36#63822—21no—
37#63822—21no—
38C2×#3713C2211yescharacteristic, complemented
39C2×#3711—21no—
40#63815—21no—
41#63815—21no—
42#63815C2212yes—
43#63815C2212yes—
44C2×D3×C3⋊C8—21no—
45#63649C2212yescomplemented
46#63649C2212yescomplemented
47C3×#824C2212yescomplemented
48C3×#823C2212yescomplemented
49#63560C2212yescomplemented
50#63558C2212yescomplemented
51C3×#575—31no—
52#5822—31no—
53#5822—31no—
54C2×#950—31no—
55C2×#950—31no—
56C2×#948—31no—
57#5748D311yescharacteristic, complemented
58C2×#907—31no—
59C3×#413—31no—
60#5237—31no—
61#5204—32no—
62#5203—32no—
63#5200—32no—
64#5199—32no—
65#5132—31no—
66#5132D311yescharacteristic, complemented
67C2×#822—31no—
68C2×#822—31no—
69C2×#822—61no—
70#4755—32no—
71#4754—32no—
72#4749—32no—
73#4748—32no—
74C22×C32×C4C2311yescharacteristic
75C2×C3×#67—21no—
76C2×C3×#67—21no—
77C2×C3×#67—21no—
78C3×#235—21no—
79C3×#235—21no—
80C3×#235—21no—
81C3×#235—21no—
82C3×#235—21no—
83C3×#235—21no—
84C3×#235—21no—
85C3×#235—21no—
86C3×#235—21no—
87C3×#235—21no—
88C3×#235D412yes—
89C3×#235C2312yes—
90C2×C3×D12D411yescharacteristic
91C2×C3×D12C2311yescharacteristic
92C2×C3×D12C2311yescharacteristic
93C2×C3×C4×D3—21no—
94C2×C3×C4×D3—21no—
95C2×C3×C4×D3—21no—
96C2×C3×Dic6D411yescharacteristic
97C2×C3×Dic6C2311yescharacteristic
98C2×C3×Dic6C2311yescharacteristic
99C32×M16—21no—
100C32×M16C2312yes—
101C2×C32×C8—21no—
102#3748—21no—
103#3748C2312yes—
104C2×C3⋊(C3⋊C8)—21no—
105C2×C3⋊(C3⋊C8)D411yescharacteristic
106C2×C3⋊(C3⋊C8)D411yescharacteristic
107C2×C3⋊(C3⋊C8)D411yescharacteristic
108C2×C3⋊(C3⋊C8)D411yescharacteristic
109C2×C3⋊(C3⋊C8)C2311yescharacteristic
110C2×C3⋊(C3⋊C8)C2311yescharacteristic
111C3×#208—21no—
112C3×#208C2312yes—
113C2×C3×C3⋊C8—21no—
114C3×Dic12—22no—
115C3×D24—22no—
116C3×#204—22no—
117C3×#204—22no—
118#3718—22no—
119#3718—21no—
120#3718—21no—
121#3715—22no—
122#3715—22no—
123#3715—21no—
124#3715—21no—
125#3715—21no—
126#3715—21no—
127#3713—22no—
128#3713—21no—
129#3713—21no—
130#3711—22no—
131#3711—22no—
132D3×C3⋊C8—22no—
133D3×C3⋊C8—22no—
134#3067—91noSylow
135C2×C3×#41—31no—
136C2×C3×#41—31no—
137C2×C3×#41—31no—
138C2×#235C2×D311yescharacteristic
139C2×#235C2×D311yescharacteristic
140C2×#235C2×D311yescharacteristic
141C3×#137—32no—
142C3×#137—61no—
143C3×AGL(1,ℤ/8)—32no—
144C3×AGL(1,ℤ/8)—61no—
145C3×#135—62no—
146C2×C3×M16—21no—
147C2×C3×M16—31no—
148C2×C3×M16C2×D311yescharacteristic
149C2×C3×M16C2×D311yescharacteristic
150#951—32no—
151#951—61no—
152#950—61no—
153#950—61no—
154#950—61no—
155#950—61no—
156#950—61no—
157#950—61no—
158#950—61no—
159#950—61no—
160#950—62no—
161#950—62no—
162#949—32no—
163#949—61no—
164#948—32no—
165#948—32no—
166#948—61no—
167#948—61no—
168C2×#216—31no—
169C2×#216—31no—
170#942—32no—
171#942—61no—
172C2×#215—31no—
173C2×#215—31no—
174C2×#214—31no—
175C2×#214—31no—
176#932—32no—
177#932—61no—
178C2×#213—31no—
179C2×#213—31no—
180C2×#208—31no—
181C2×#208—31no—
182C2×#208—61no—
183C2×#208—61no—
184C2×#208C2×D311yescharacteristic
185C22×C3⋊C8—31no—
186C22×C3⋊C8—31no—
187C22×C3⋊C8—61no—
188C22×C3⋊C8—61no—
189#909—21no—
190#909C2×D312yes—
191#908—21no—
192#908C2×D312yes—
193#907—32no—
194#907—32no—
195#907—61no—
196#907—61no—
197#904—22no—
198C2×ΓL(1,𝔽25)—61no—
199C2×D3×C8—61no—
200C3×#101—32no—
201C3×#100—32no—
202#836—32no—
203#833—32no—
204#824—32no—
205#824C2×D312yescomplemented
206#823—32no—
207#823C2×D312yescomplemented
208#822—32no—
209#822—32no—
210#822—32no—
211#822—32no—
212#822—62no—
213#822—62no—
214C23×C32#4111yescharacteristic
215C22×C3×D3—21no—
216C22×C3×D3—21no—
217C22×C3×D3—21no—
218C2×C32×C4C2×D411yescharacteristic
219C2×C32×C4C2×D411yescharacteristic
220C2×C32×C4C2×D411yescharacteristic
221C2×C32×C4C2×D411yescharacteristic
222C2×C32×C4#4111yescharacteristic
223C2×C32×C4C2411yescharacteristic
224C3×#67—22no—
225C3×#67—22no—
226C3×#67—41no—
227C3×#67—41no—
228C3×#67—41no—
229C3×#67—41no—
230C2×C3×Dic3—21no—
231C2×C3×Dic3—21no—
232C2×C3×Dic3—21no—
233C3×D12—21no—
234C3×D12—21no—
235C3×D12—21no—
236C3×D12—21no—
237C3×D12—21no—
238C3×D12C2×D412yes—
239C3×C4×D3—22no—
240C3×C4×D3—22no—
241C3×C4×D3—41no—
242C3×C4×D3—41no—
243C3×C4×D3—41no—
244C3×C4×D3—41no—
245C3×Dic6—21no—
246C3×Dic6—21no—
247C3×Dic6—21no—
248C3×Dic6—21no—
249C3×Dic6—21no—
250C3×Dic6C2×D412yes—
251C32×C8—22no—
252C3⋊(C3⋊C8)—22no—
253C3⋊(C3⋊C8)C2×D411yescharacteristic
254C3⋊(C3⋊C8)C2×D411yescharacteristic
255C3⋊(C3⋊C8)C2×D411yescharacteristic
256C3⋊(C3⋊C8)C2×D411yescharacteristic
257C3×C3⋊C8—22no—
258#575—31no—
259#575—91no—
260C2×#135—91no—
261C2×#131—91no—
262#473—92no—
263#472—92no—
264#471—92no—
265#471—92no—
266C2×#108—91no—
267#413—31no—
268#413—91no—
269C3×#41—32no—
270C3×#41—32no—
271C3×#41—61no—
272C3×#41—61no—
273C3×#41—61no—
274C3×#41—61no—
275C3×#41—61no—
276C3×#41—61no—
277C3×#41—61no—
278C3×#41—61no—
279C3×#41—61no—
280C3×#41—61no—
281C2×C3×Q8—31no—
282C2×C3×Q8—31no—
283C2×C3×Q8—31no—
284C2×C3×D4—31no—
285C2×C3×D4—31no—
286C2×C3×D4—31no—
287C2×C3×D4—61no—
288C2×C3×D4—61no—
289C2×C3×D4—61no—
290C22×C3×C4—21no—
291C22×C3×C4—61no—
292C22×C3×C4—61no—
293C22×C3×C4—61no—
294C22×C3×C4C22×D311yescharacteristic
295C22×C3×C4C22×D311yescharacteristic
296C2×#67—21no—
297C2×#67—21no—
298C2×#67—21no—
299#235—21no—
300#235—21no—
301#235—21no—
302#235—21no—
303#235—21no—
304#235—21no—
305#235—21no—
306#235—21no—
307#235—21no—
308#235—21no—
309#235#6712yes—
310#235C22×D312yes—
311C2×D12#6711yescharacteristic
312C2×D12C22×D311yescharacteristic
313C2×D12C22×D311yescharacteristic
314C2×C4×D3—21no—
315C2×C4×D3—21no—
316C2×C4×D3—21no—
317C2×Dic6#6711yescharacteristic
318C2×Dic6C22×D311yescharacteristic
319C2×Dic6C22×D311yescharacteristic
320C3×Q16—62no—
321C3×SD16—62no—
322C3×SD16—62no—
323C3×D8—62no—
324C3×M16—21no—
325C3×M16—21no—
326C3×M16—22no—
327C3×M16—32no—
328C3×M16—41no—
329C3×M16—61no—
330C3×M16C22×D312yes—
331C3×M16C22×D312yes—
332C2×C3×C8—21no—
333C2×C3×C8—21no—
334C2×C3×C8—41no—
335C2×C3×C8—61no—
336#216—61no—
337#216—61no—
338#216—61no—
339#216—61no—
340#216—62no—
341#216—62no—
342#215—61no—
343#215—61no—
344#215—61no—
345#215—61no—
346#215—62no—
347#215—62no—
348#214—61no—
349#214—61no—
350#214—61no—
351#214—61no—
352#214—62no—
353#214—62no—
354#213—61no—
355#213—61no—
356#213—61no—
357#213—61no—
358#213—62no—
359#213—62no—
360#208—21no—
361#208—32no—
362#208—32no—
363#208—62no—
364#208—62no—
365#208—62no—
366#208—61no—
367#208—61no—
368#208—121no—
369#208C22×D312yes—
370C2×C3⋊C8—21no—
371C2×C3⋊C8—31no—
372C2×C3⋊C8—31no—
373C2×C3⋊C8—31no—
374C2×C3⋊C8—31no—
375C2×C3⋊C8—31no—
376C2×C3⋊C8—31no—
377C2×C3⋊C8—31no—
378C2×C3⋊C8—31no—
379C2×C3⋊C8—31no—
380C2×C3⋊C8—31no—
381C2×C3⋊C8—31no—
382C2×C3⋊C8—31no—
383C2×C3⋊C8—61no—
384C2×C3⋊C8—61no—
385C2×C3⋊C8—61no—
386C2×C3⋊C8—61no—
387C2×C3⋊C8—61no—
388C2×C3⋊C8—61no—
389C2×C3⋊C8—62no—
390C2×C3⋊C8—62no—
391C2×C3⋊C8—61no—
392C2×C3⋊C8—61no—
393C2×C3⋊C8—121no—
394Dic12—22no—
395D24—22no—
396#204—22no—
397#204—22no—
398ΓL(1,𝔽25)—62no—
399ΓL(1,𝔽25)—62no—
400D3×C8—62no—
401D3×C8—62no—
402C22×C32—41no—
403C22×C32#12111yescharacteristic
404C22×C32#12111yescharacteristic
405C22×C32C2×#4111yescharacteristic
406C2×C3×D3—22no—
407C2×C3×D3—42no—
408C2×C3×D3—41no—
409C2×C3×D3—41no—
410C2×C3×D3—81no—
411C2×C3×D3—81no—
412C32×C4#12111yescharacteristic
413C32×C4#12111yescharacteristic
414C32×C4C22×D411yescharacteristic
415C32×C4C22×D411yescharacteristic
416C3×Dic3—22no—
417C3×Dic3—41no—
418C3×Dic3—41no—
419C2×#41—31no—
420C2×#41—31no—
421C2×#41—31no—
422#137—32no—
423#137—61no—
424#137—92no—
425#137—181no—
426AGL(1,ℤ/8)—32no—
427AGL(1,ℤ/8)—61no—
428AGL(1,ℤ/8)—92no—
429AGL(1,ℤ/8)—181no—
430#135—62no—
431#135—181no—
432#135—181no—
433#135—181no—
434#135—181no—
435#135—182no—
436C2×Q16—91no—
437C2×SD16—91no—
438C2×SD16—91no—
439C2×D8—91no—
440#131—92no—
441#131—92no—
442#131—181no—
443#131—181no—
444C2×M16—31no—
445C2×M16—91no—
446C2×M16—181no—
447C2×M16D3211yescharacteristic
448C22×C8—91no—
449C22×C8—181no—
450#108—92no—
451#108—92no—
452#101—32no—
453#101—92no—
454#100—32no—
455#100—92no—
456C23×C3—21no—
457C23×C3—61no—
458C23×C3—61no—
459C23×C3—61no—
460C23×C3#23511yescharacteristic
461C23×C3#23711yescharacteristic
462C22×D3—21no—
463C22×D3—21no—
464C22×D3—21no—
465C3×Q8—32no—
466C3×Q8—61no—
467C3×Q8—61no—
468C3×Q8—61no—
469C3×Q8—61no—
470C3×Q8—61no—
471C3×D4—32no—
472C3×D4—62no—
473C3×D4—62no—
474C3×D4—61no—
475C3×D4—61no—
476C3×D4—61no—
477C3×D4—61no—
478C3×D4—61no—
479C3×D4—121no—
480C3×D4—121no—
481C3×D4—121no—
482C3×D4—121no—
483C2×C3×C4—21no—
484C2×C3×C4—21no—
485C2×C3×C4—21no—
486C2×C3×C4—21no—
487C2×C3×C4—21no—
488C2×C3×C4—21no—
489C2×C3×C4—61no—
490C2×C3×C4—62no—
491C2×C3×C4—62no—
492C2×C3×C4—61no—
493C2×C3×C4—61no—
494C2×C3×C4—121no—
495C2×C3×C4—121no—
496C2×C3×C4—121no—
497C2×C3×C4—121no—
498C2×C3×C4#23511yescharacteristic
499C2×C3×C4D3×D411yescharacteristic
500C2×C3×C4D3×D411yescharacteristic
501C2×C3×C4D3×D411yescharacteristic
502C2×C3×C4D3×D411yescharacteristic
503C2×C3×C4D3×D411yescharacteristic
504C2×C3×C4D3×D411yescharacteristic
505C2×C3×C4#23711yescharacteristic
506C2×C3×C4C2×#6711yescharacteristic
507C2×C3×C4C2×#6711yescharacteristic
508C2×C3×C4C23×D311yescharacteristic
509C2×C3×C4C23×D311yescharacteristic
510#67—22no—
511#67—22no—
512#67—41no—
513#67—41no—
514#67—41no—
515#67—41no—
516C2×Dic3—21no—
517C2×Dic3—21no—
518C2×Dic3—21no—
519D12—21no—
520D12—21no—
521D12—21no—
522D12—21no—
523D12—21no—
524D12C2×#6712yes—
525C4×D3—22no—
526C4×D3—22no—
527C4×D3—41no—
528C4×D3—41no—
529C4×D3—41no—
530C4×D3—41no—
531Dic6—21no—
532Dic6—21no—
533Dic6—21no—
534Dic6—21no—
535Dic6—21no—
536Dic6C2×#6712yes—
537C3×C8—22no—
538C3×C8—22no—
539C3×C8—42no—
540C3×C8—62no—
541C3⋊C8—22no—
542C3⋊C8—31no—
543C3⋊C8—31no—
544C3⋊C8—31no—
545C3⋊C8—31no—
546C3⋊C8—31no—
547C3⋊C8—31no—
548C3⋊C8—31no—
549C3⋊C8—31no—
550C3⋊C8—61no—
551C3⋊C8—61no—
552C3⋊C8—61no—
553C3⋊C8—61no—
554C3⋊C8—62no—
555C3⋊C8—62no—
556C3⋊C8—122no—
557C2×C32—21no—
558C2×C32—21no—
559C2×C32—21no—
560C2×C32C2×#12111yescharacteristic
561C3×D3—42no—
562C3×D3—81no—
563C3×D3—81no—
564#41—32no—
565#41—32no—
566#41—61no—
567#41—61no—
568#41—61no—
569#41—61no—
570#41—61no—
571#41—61no—
572#41—61no—
573#41—61no—
574#41—61no—
575#41—61no—
576C2×Q8—31no—
577C2×Q8—31no—
578C2×Q8—31no—
579C2×D4—31no—
580C2×D4—31no—
581C2×D4—31no—
582C2×D4—61no—
583C2×D4—61no—
584C2×D4—61no—
585C22×C4—61no—
586C22×C4—61no—
587C22×C4—61no—
588C22×C4C2×D3211yescharacteristic
589Q16—62no—
590Q16—181no—
591Q16—181no—
592Q16—182no—
593SD16—62no—
594SD16—62no—
595SD16—181no—
596SD16—181no—
597SD16—181no—
598SD16—181no—
599SD16—182no—
600SD16—182no—
601D8—62no—
602D8—181no—
603D8—181no—
604D8—182no—
605M16—21no—
606M16—32no—
607M16—61no—
608M16—92no—
609M16—182no—
610M16—182no—
611M16—181no—
612M16C2×D3212yes—
613C2×C8—21no—
614C2×C8—61no—
615C2×C8—91no—
616C2×C8—91no—
617C2×C8—91no—
618C2×C8—91no—
619C2×C8—91no—
620C2×C8—91no—
621C2×C8—182no—
622C2×C8—182no—
623C2×C8—181no—
624C22×C3—21no—
625C22×C3—21no—
626C22×C3—21no—
627C22×C3—41no—
628C22×C3—41no—
629C22×C3—42no—
630C22×C3—62no—
631C22×C3—122no—
632C22×C3—121no—
633C22×C3—121no—
634C22×C3—241no—
635C22×C3—241no—
636C22×C3#88411yescharacteristic
637C22×C3#88411yescharacteristic
638C22×C3#93911yescharacteristic
639C22×C3#93911yescharacteristic
640C22×C3C2×#23511yescharacteristic
641C22×C3C2×#23711yescharacteristic
642C2×D3—22no—
643C2×D3—42no—
644C2×D3—41no—
645C2×D3—41no—
646C2×D3—81no—
647C2×D3—81no—
648C3×C4—21no—
649C3×C4—21no—
650C3×C4—21no—
651C3×C4—21no—
652C3×C4—62no—
653C3×C4—121no—
654C3×C4—121no—
655C3×C4#88411yescharacteristic
656C3×C4#88411yescharacteristic
657C3×C4#93911yescharacteristic
658C3×C4#93911yescharacteristic
659C3×C4C2×D3×D411yescharacteristic
660C3×C4C2×D3×D411yescharacteristic
661C3×C4C2×D3×D411yescharacteristic
662C3×C4C22×#6711yescharacteristic
663Dic3—22no—
664Dic3—41no—
665Dic3—41no—
666C32#306711yescharacteristic, complemented, Sylow
667C23—61no—
668C23—61no—
669C23—61no—
670C23#379511yescharacteristic
671Q8—32no—
672Q8—61no—
673Q8—61no—
674Q8—61no—
675Q8—61no—
676Q8—61no—
677D4—32no—
678D4—62no—
679D4—62no—
680D4—61no—
681D4—61no—
682D4—61no—
683D4—61no—
684D4—61no—
685D4—121no—
686D4—121no—
687D4—121no—
688D4—121no—
689C2×C4—61no—
690C2×C4—62no—
691C2×C4—62no—
692C2×C4—61no—
693C2×C4—61no—
694C2×C4—121no—
695C2×C4—121no—
696C2×C4—121no—
697C2×C4—121no—
698C2×C4#379511yescharacteristic
699C2×C4#380211yescharacteristic
700C2×C4#380211yescharacteristic
701C2×C4D3×#6711yescharacteristic
702C2×C4D3×#6711yescharacteristic
703C2×C4C22×D3211yescharacteristic
704C8—22no—
705C8—62no—
706C8—91no—
707C8—91no—
708C8—91no—
709C8—91no—
710C8—182no—
711C2×C3—22no—
712C2×C3—21no—
713C2×C3—21no—
714C2×C3—21no—
715C2×C3—21no—
716C2×C3—21no—
717C2×C3—21no—
718C2×C3—21no—
719C2×C3—41no—
720C2×C3—41no—
721C2×C3—122no—
722C2×C3—241no—
723C2×C3—241no—
724C2×C3C2×#88411yescharacteristic
725C2×C3C2×#93911yescharacteristic
726D3—42no—
727D3—81no—
728D3—81no—
729C22—41no—
730C22—62no—
731C22—122no—
732C22—121no—
733C22—121no—
734C22—241no—
735C22—241no—
736C22#6388511yescharacteristic
737C22#6388511yescharacteristic
738C22C2×#379511yescharacteristic
739C4—62no—
740C4—121no—
741C4—121no—
742C4#6388511yescharacteristic
743C4#6388511yescharacteristic
744C4C2×#380211yescharacteristic
745C4C2×D3×#6711yescharacteristic
746C3—21no—
747C3#7871911yescharacteristic, complemented
748C3#8400111yescharacteristic, complemented
749C2—21no—
750C2—21no—
751C2—21no—
752C2—122no—
753C2—241no—
754C2—241no—
755C2C2×#6388511yescharacteristic
subgroupG/H
centerC4#63885
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC2×C32C2×#121
hypercenterC2×M16D32
hypocenterC32#3067
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C3×#5748C2semidirect
2. #10594786C2semidirect
3. C2×#63822C2semidirect
4. C2×#63815C2semidirect
5. C2×#63649C2semidirect
6. C3×#5132C2semidirect
7. #10591076C2semidirect
8. #10590982C2semidirect
9. #10590981C2semidirect
10. #10590980C2semidirect
11. #10590979C2semidirect
26. C2×#3718C22semidirect
29. C2×#3715C22semidirect
30. C2×#3715C22semidirect
38. C2×#3713C22semidirect
45. #63649C22semidirect
46. #63649C22semidirect
47. C3×#824C22semidirect
48. C3×#823C22semidirect
49. #63560C22semidirect
50. #63558C22semidirect
57. #5748D3semidirect
66. #5132D3semidirect
205. #824C2×D3semidirect
207. #823C2×D3semidirect
666. C32#3067semidirect
747. C3#78719semidirect
748. C3#84001semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
#105947862
C2×#638222
C2×#638152
C2×#636492
C3×#51322
#105910762
#105909822
#105909812
#105909802
#105909792
#840013
#787193