Finite Group Databasebeta

← #11901213

Subgroups — #11901213

Order 1152.

3285 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3804C212yesmaximal, complemented
2#10595964C211yesmaximal, characteristic, complemented
3C3×#5493C211yesmaximal, characteristic, complemented
4C2×C3×#877C211yesmaximal, characteristic, complemented
5#10595519C212yesmaximal, complemented
6#10594911C214yesmaximal, complemented
7#10594910C214yesmaximal, complemented
8#10594081C214yesmaximal, complemented
9#10594080C214yesmaximal, complemented
10#10594079C214yesmaximal, complemented
11#10594078C214yesmaximal, complemented
12#86294—31nomaximal
13#85735—31nomaximal
14C22×#638C2212yes—
15C22×C3×#67C2212yes—
16C23×C3×Dic3C2211yescharacteristic
17C22×#627C2212yescomplemented
18C22×#626C2212yescomplemented
19C22×#624C2212yescomplemented
20C2×#3804—216no—
21C2×#3804C2214yes—
22C2×#3804C2214yes—
23C2×#3804C2214yescomplemented
24C2×#3804C2214yescomplemented
25C22×D3×Dic3C2212yescomplemented
26C2×C32×C22⋊C4C2211yescharacteristic
27#64139C2214yescomplemented
28#64095C2214yescomplemented
29#64091C2214yescomplemented
30C2×C3×#217C2211yescharacteristic
31C3×#930C2214yescomplemented
32C2×C3×Dic3⋊C4C2212yescomplemented
33C2×C3×C4×Dic3C2212yescomplemented
34C3×#886C2214yescomplemented
35C3×#882C2214yescomplemented
36C3×#877C2214yescomplemented
37C3×#877C2214yescomplemented
38#63981C2214yescomplemented
39#63978C2218yescomplemented
40#63977C2214yescomplemented
41#63966C2214yescomplemented
42#63962C2214yescomplemented
43#63918C2214yescomplemented
44#63911C2214yescomplemented
45#63910C2218yescomplemented
46#63895C2218yescomplemented
47#63894C2214yescomplemented
48#63893C2218yescomplemented
49Dic3×D12C2214yescomplemented
50#63886C2218yescomplemented
51#63880C2218yescomplemented
52#63852C2218yescomplemented
53C22×#237—32no—
54C22×#235—32no—
55C3×#534—31no—
56C2×C3×#117—31no—
57#5804—32no—
58#5656—34no—
59#5648—34no—
60#5597—34no—
61#5596—34no—
62#5592—34no—
63#5586—34no—
64#5526—34no—
65#5525—34no—
66#5524—34no—
67#5524—34no—
68#5510—34no—
69#5493—32no—
70#5493—31no—
71#5493—31no—
72#5493—61no—
73#5493D311yescharacteristic, complemented
74C2×#877D311yescharacteristic, complemented
75#5477—34no—
76C24×C32C2311yescharacteristic
77C23×C3⋊D3C2312yes—
78C23×C3×D3C2312yes—
79C22×C32×C4C2312yes—
80C2×#638—28no—
81C2×#638C2314yes—
82C2×#638C2314yes—
83C22×C3⋊Dic3C2312yes—
84C2×C4⋊(C3⋊D3)C2314yes—
85C2×C3×#67—28no—
86C2×C3×#67C2314yes—
87C2×C3×#67C2314yes—
88C22×C3×Dic3D418yescomplemented
89C22×C3×Dic3C2312yes—
90C22×C3×Dic3C2312yes—
91C22×C3×Dic3C2312yes—
92C22×C3×Dic3C2312yes—
93C22×C3×Dic3C2311yescharacteristic
94C22×C3×Dic3C2311yescharacteristic
95C2×C3×D12C2314yes—
96C2×#627—28no—
97C2×#627C2314yes—
98C2×#627C2314yes—
99C2×#626—28no—
100C2×#626C2314yes—
101C2×#626C2314yes—
102C2×#626C2318yes—
103C2×#624—28no—
104C2×#624C2314yes—
105C2×#624C2314yes—
106#3804—216no—
107#3804—216no—
108#3804—216no—
109#3804—216no—
110C2×D3×Dic3—28no—
111C2×D3×Dic3C2314yes—
112C2×D3×Dic3C2314yes—
113C32×C22⋊C4C2314yes—
114#3752C2318yes—
115C4⋊(C3⋊Dic3)C2314yes—
116C3×#217C2314yes—
117C3×(C2×D3)⋊C4C2318yes—
118C3×C4⋊Dic3C2314yes—
119C3×Dic3⋊C4C2318yes—
120C3×C4×Dic3C2314yes—
121C3×C4×Dic3C2314yes—
122Dic3⋊Dic3C2318yes—
123#3722C2318yes—
124(C2×D3)⋊Dic3C2318yes—
125#3449—91noSylow
126C22×C3×D4—32no—
127C23×C3×C4—31no—
128C22×#67—32no—
129C22×#67—32no—
130C22×#67—32no—
131C22×#67—62no—
132C22×#67C2×D312yes—
133C23×Dic3—32no—
134C23×Dic3C2×D311yescharacteristic
135C2×#237—34no—
136C2×#237—34no—
137C2×#237—34no—
138C2×#237—34no—
139C2×#237—616no—
140C2×#235—34no—
141C2×#235—34no—
142C2×#235—34no—
143C2×#235—34no—
144C2×#235—616no—
145C22×D12—32no—
146C22×C4×D3—32no—
147C22×C4×D3—32no—
148C22×C4×D3—32no—
149C22×Dic6—32no—
150C22×Dic6—32no—
151C3×#123—34no—
152C3×#121—34no—
153C3×#120—34no—
154C3×#117—34no—
155C3×#117—34no—
156C2×C3×C4⋊C4—32no—
157C2×C3×C22⋊C4—21no—
158C2×C3×C22⋊C4—31no—
159C2×C3×C22⋊C4C2×D311yescharacteristic
160C2×C3×C22⋊C4C2×D311yescharacteristic
161C2×C3×C42—32no—
162#953—34no—
163C2×#217C2×D311yescharacteristic
164#940—34no—
165#939—38no—
166#939—38no—
167#937—34no—
168#935—34no—
169D4×Dic3—34no—
170#930—34no—
171#930—34no—
172#930—38no—
173#930—64no—
174#930C2×D314yescomplemented
175#929—34no—
176C2×Dic3⋊C4C2×D312yescomplemented
177C2×C4×Dic3C2×D312yescomplemented
178#897—38no—
179#897—38no—
180#896—38no—
181#895—38no—
182#895—38no—
183#893—34no—
184#888—34no—
185#886—34no—
186#886—34no—
187#886—34no—
188#886—34no—
189#886—64no—
190#886C2×D314yescomplemented
191#885—38no—
192#884—38no—
193#882—34no—
194#882—34no—
195#882—34no—
196#882—34no—
197#882—64no—
198#882C2×D314yescomplemented
199#67⋊C4—38no—
200#878—38no—
201#877C2×D314yescomplemented
202#877C2×D314yescomplemented
203#875—38no—
204#874—34no—
205C4×D12—34no—
206C4×D12—38no—
207C4×D12—34no—
208#869—34no—
209C23×C32—42no—
210C23×C32C2×D414yes—
211C23×C32C2412yes—
212C23×C32C2411yescharacteristic
213C22×C3⋊D3—24no—
214C22×C3⋊D3—44no—
215C22×C3⋊D3C2414yes—
216C22×C3×D3—24no—
217C22×C3×D3—44no—
218C22×C3×D3C2414yes—
219C2×C32×C4—42no—
220C2×C32×C4C2414yes—
221#638—28no—
222#638—28no—
223C2×C3⋊Dic3—24no—
224C2×C3⋊Dic3C2414yes—
225C4⋊(C3⋊D3)—44no—
226C3×#67—28no—
227C3×#67—28no—
228C2×C3×Dic3—24no—
229C2×C3×Dic3—42no—
230C2×C3×Dic3C2×D418yes—
231C2×C3×Dic3C2×D414yes—
232C2×C3×Dic3C2×D414yes—
233C2×C3×Dic3C2×D418yes—
234C2×C3×Dic3C2414yes—
235C2×C3×Dic3C2414yes—
236C2×C3×Dic3C2411yescharacteristic
237C2×C3×Dic3C2411yescharacteristic
238C2×C3×Dic3C2412yes—
239C3×D12—44no—
240#627—28no—
241#627—28no—
242#626—28no—
243#626—28no—
244#626—48no—
245#624—28no—
246#624—28no—
247D3×Dic3—28no—
248D3×Dic3—28no—
249C22×#41—92no—
250#537—94no—
251#536—94no—
252#535—94no—
253#534—31no—
254#534—92no—
255#534—91no—
256#532—94no—
257#525—94no—
258#525—94no—
259C2×#117—31no—
260C24×C3—21no—
261C24×C3—32no—
262C24×C3C22×D311yescharacteristic
263C24×C3C22×D311yescharacteristic
264C23×D3—32no—
265C23×D3—32no—
266C23×D3—62no—
267C23×D3C22×D312yes—
268C2×C3×D4—34no—
269C2×C3×D4—34no—
270C2×C3×D4—34no—
271C2×C3×D4—68no—
272C22×C3×C4—22no—
273C22×C3×C4—38no—
274C22×C3×C4—31no—
275C22×C3×C4—31no—
276C22×C3×C4—32no—
277C22×C3×C4—32no—
278C22×C3×C4—32no—
279C22×C3×C4—32no—
280C22×C3×C4C22×D312yes—
281C22×C3×C4C22×D312yes—
282C2×#67—28no—
283C2×#67—34no—
284C2×#67—34no—
285C2×#67—34no—
286C2×#67—34no—
287C2×#67—34no—
288C2×#67—34no—
289C2×#67—38no—
290C2×#67—64no—
291C2×#67—64no—
292C2×#67—68no—
293C2×#67—68no—
294C2×#67—68no—
295C2×#67—128no—
296C2×#67C22×D314yes—
297C2×#67C22×D314yes—
298C22×Dic3—32no—
299C22×Dic3—32no—
300C22×Dic3—34no—
301C22×Dic3—34no—
302C22×Dic3—62no—
303C22×Dic3—68no—
304C22×Dic3D1218yescomplemented
305C22×Dic3C22×D312yes—
306C22×Dic3C22×D312yes—
307C22×Dic3C22×D312yes—
308C22×Dic3C22×D312yes—
309C22×Dic3C22×D311yescharacteristic
310C22×Dic3C22×D311yescharacteristic
311#237—616no—
312#237—616no—
313#237—616no—
314#237—616no—
315#235—616no—
316#235—616no—
317#235—616no—
318#235—616no—
319C2×D12—34no—
320C2×D12—34no—
321C2×D12—38no—
322C2×D12—34no—
323C2×D12—34no—
324C2×D12—64no—
325C2×D12—68no—
326C2×D12C22×D314yes—
327C2×C4×D3—34no—
328C2×C4×D3—34no—
329C2×C4×D3—34no—
330C2×C4×D3—34no—
331C2×C4×D3—34no—
332C2×C4×D3—34no—
333C2×C4×D3—68no—
334C2×C4×D3—68no—
335C2×C4×D3—68no—
336C2×Dic6—34no—
337C2×Dic6—34no—
338C2×Dic6—34no—
339C2×Dic6—34no—
340C2×Dic6—68no—
341C2×Dic6—68no—
342C3×C4⋊C4—34no—
343C3×C4⋊C4—38no—
344C3×C22⋊C4—24no—
345C3×C22⋊C4—34no—
346C3×C22⋊C4—38no—
347C3×C22⋊C4C22×D314yes—
348C3×C22⋊C4C22×D314yes—
349C3×C42—34no—
350C3×C42—34no—
351#217—38no—
352#217C22×D314yes—
353(C2×D3)⋊C4—38no—
354(C2×D3)⋊C4—38no—
355(C2×D3)⋊C4—38no—
356(C2×D3)⋊C4—38no—
357(C2×D3)⋊C4—38no—
358(C2×D3)⋊C4—68no—
359(C2×D3)⋊C4C22×D318yes—
360C4⋊Dic3—34no—
361C4⋊Dic3—34no—
362C4⋊Dic3—38no—
363C4⋊Dic3—64no—
364C4⋊Dic3C22×D314yes—
365Dic3⋊C4—38no—
366Dic3⋊C4C22×D318yes—
367C4×Dic3C22×D314yes—
368C4×Dic3C22×D314yes—
369C22×C32—22no—
370C22×C32—24no—
371C22×C32—44no—
372C22×C32C22×D414yes—
373C22×C32C22×D412yes—
374C22×C32C2511yescharacteristic
375C2×C3⋊D3—24no—
376C2×C3⋊D3—28no—
377C2×C3⋊D3—48no—
378C2×C3×D3—24no—
379C2×C3×D3—28no—
380C2×C3×D3—48no—
381C32×C4—24no—
382C3⋊Dic3—24no—
383C3×Dic3—24no—
384C3×Dic3—24no—
385C3×Dic3C22×D412yes—
386C3×Dic3C22×D414yes—
387C3×Dic3C22×D412yes—
388C2×#41—94no—
389C2×#41—94no—
390C2×#41—94no—
391C2×#41—94no—
392C2×#41—1816no—
393C22×Q8—92no—
394C22×D4—32no—
395C22×D4—92no—
396C22×D4—92no—
397C23×C4—31no—
398C23×C4—92no—
399C23×C4—92no—
400C4⋊Q8—94no—
401#127—94no—
402#124—98no—
403#123—34no—
404#123—94no—
405#123—94no—
406#123—94no—
407C22⋊Q8—98no—
408C22⋊Q8—98no—
409#121—34no—
410#121—98no—
411#121—98no—
412#121—98no—
413#121—94no—
414#120—34no—
415#120—94no—
416#120—94no—
417#120—94no—
418C4×D4—98no—
419C4×D4—94no—
420C4×D4—94no—
421#117—34no—
422#117—34no—
423C2×C4⋊C4—32no—
424C2×C22⋊C4—31no—
425C2×C22⋊C4D3211yescharacteristic
426C2×C42—32no—
427C23×C3—22no—
428C23×C3—21no—
429C23×C3—24no—
430C23×C3—34no—
431C23×C3—42no—
432C23×C3—42no—
433C23×C3—64no—
434C23×C3—82no—
435C23×C3—124no—
436C23×C3C2×D1214yes—
437C23×C3D3×D414yes—
438C23×C3C23×D312yes—
439C23×C3C23×D311yescharacteristic
440C23×C3C23×D312yes—
441C23×C3C23×D311yescharacteristic
442C22×D3—24no—
443C22×D3—34no—
444C22×D3—34no—
445C22×D3—44no—
446C22×D3—64no—
447C22×D3—64no—
448C22×D3—64no—
449C22×D3—124no—
450C22×D3—124no—
451C22×D3—124no—
452C22×D3—244no—
453C22×D3C23×D314yes—
454C3×D4—68no—
455C3×D4—68no—
456C3×D4—124no—
457C2×C3×C4—24no—
458C2×C3×C4—38no—
459C2×C3×C4—38no—
460C2×C3×C4—34no—
461C2×C3×C4—34no—
462C2×C3×C4—31no—
463C2×C3×C4—31no—
464C2×C3×C4—32no—
465C2×C3×C4—34no—
466C2×C3×C4—34no—
467C2×C3×C4—42no—
468C2×C3×C4—42no—
469C2×C3×C4—64no—
470C2×C3×C4—82no—
471C2×C3×C4—122no—
472C2×C3×C4C23×D314yes—
473C2×C3×C4C23×D314yes—
474#67—28no—
475#67—28no—
476#67—68no—
477#67—68no—
478#67—68no—
479#67—68no—
480#67—68no—
481#67—68no—
482#67—128no—
483#67—128no—
484#67—128no—
485C2×Dic3—24no—
486C2×Dic3—34no—
487C2×Dic3—34no—
488C2×Dic3—42no—
489C2×Dic3—64no—
490C2×Dic3—64no—
491C2×Dic3—64no—
492C2×Dic3—68no—
493C2×Dic3—68no—
494C2×Dic3—124no—
495C2×Dic3C2×D1218yes—
496C2×Dic3C2×D1218yes—
497C2×Dic3C2×D1214yes—
498C2×Dic3C2×D1214yes—
499C2×Dic3C23×D314yes—
500C2×Dic3C23×D314yes—
501C2×Dic3C23×D311yescharacteristic
502C2×Dic3C23×D311yescharacteristic
503C2×Dic3C23×D312yes—
504D12—44no—
505D12—68no—
506D12—68no—
507D12—124no—
508D12—124no—
509D12—128no—
510D12—244no—
511C4×D3—68no—
512C4×D3—68no—
513C4×D3—68no—
514C4×D3—68no—
515C4×D3—68no—
516C4×D3—68no—
517Dic6—68no—
518Dic6—68no—
519Dic6—68no—
520Dic6—68no—
521C2×C32—24no—
522C2×C32—22no—
523C2×C32C23×D411yescharacteristic
524C2×C32#58512yes—
525C3⋊D3—44no—
526C3×D3—44no—
527C24—32no—
528C24—92no—
529C24C2×D3211yescharacteristic
530#41—1816no—
531#41—1816no—
532#41—1816no—
533#41—1816no—
534C2×Q8—94no—
535C2×Q8—94no—
536C2×Q8—188no—
537C2×D4—34no—
538C2×D4—34no—
539C2×D4—34no—
540C2×D4—68no—
541C2×D4—94no—
542C2×D4—94no—
543C2×D4—98no—
544C2×D4—94no—
545C2×D4—94no—
546C2×D4—94no—
547C2×D4—188no—
548C2×D4—188no—
549C22×C4—31no—
550C22×C4—31no—
551C22×C4—32no—
552C22×C4—32no—
553C22×C4—38no—
554C22×C4—32no—
555C22×C4—32no—
556C22×C4—94no—
557C22×C4—94no—
558C22×C4—94no—
559C22×C4—94no—
560C22×C4—92no—
561C22×C4—188no—
562C22×C4—188no—
563C22×C4C2×D3212yes—
564C4⋊C4—38no—
565C4⋊C4—34no—
566C4⋊C4—98no—
567C4⋊C4—94no—
568C22⋊C4—34no—
569C22⋊C4—38no—
570C22⋊C4—98no—
571C22⋊C4—98no—
572C22⋊C4—98no—
573C22⋊C4C2×D3214yes—
574C42—34no—
575C42—34no—
576C22×C3—22no—
577C22×C3—24no—
578C22×C3—22no—
579C22×C3—24no—
580C22×C3—24no—
581C22×C3—22no—
582C22×C3—21no—
583C22×C3—42no—
584C22×C3—44no—
585C22×C3—44no—
586C22×C3—44no—
587C22×C3—64no—
588C22×C3—68no—
589C22×C3—84no—
590C22×C3—128no—
591C22×C3C22×D1214yes—
592C22×C3C22×D1212yes—
593C22×C3C2×D3×D414yes—
594C22×C3C2×D3×D412yes—
595C22×C3C24×D311yescharacteristic
596C22×C3C24×D311yescharacteristic
597C2×D3—24no—
598C2×D3—28no—
599C2×D3—48no—
600C2×D3—64no—
601C2×D3—64no—
602C2×D3—68no—
603C2×D3—68no—
604C2×D3—128no—
605C2×D3—124no—
606C2×D3—128no—
607C2×D3—128no—
608C2×D3—248no—
609C3×C4—24no—
610C3×C4—24no—
611C3×C4—32no—
612C3×C4—34no—
613C3×C4—32no—
614C3×C4—44no—
615C3×C4—64no—
616C3×C4—64no—
617Dic3—24no—
618Dic3—24no—
619Dic3—64no—
620Dic3—64no—
621Dic3—124no—
622Dic3C22×D1212yes—
623Dic3C22×D1214yes—
624Dic3C22×D1212yes—
625C32#344911yescharacteristic, complemented, Sylow
626C23—34no—
627C23—42no—
628C23—64no—
629C23—94no—
630C23—124no—
631C23—184no—
632C23—364no—
633C23D3×D1214yes—
634C23C22×D3212yes—
635C23C22×D3211yescharacteristic
636Q8—188no—
637Q8—188no—
638D4—68no—
639D4—68no—
640D4—124no—
641D4—188no—
642D4—188no—
643D4—188no—
644D4—188no—
645D4—364no—
646D4—368no—
647C2×C4—38no—
648C2×C4—38no—
649C2×C4—34no—
650C2×C4—34no—
651C2×C4—31no—
652C2×C4—31no—
653C2×C4—32no—
654C2×C4—34no—
655C2×C4—34no—
656C2×C4—42no—
657C2×C4—64no—
658C2×C4—94no—
659C2×C4—122no—
660C2×C4—188no—
661C2×C4—188no—
662C2×C4—188no—
663C2×C4—188no—
664C2×C4—184no—
665C2×C4C22×D3214yes—
666C2×C3—24no—
667C2×C3—22no—
668C2×C3—24no—
669C2×C3—22no—
670C2×C3—21no—
671C2×C3—22no—
672C2×C3—44no—
673C2×C3—42no—
674C2×C3—124no—
675C2×C3C23×D1211yescharacteristic
676C2×C3C22×D3×D411yescharacteristic
677C2×C3#596412yes—
678C2×C3#596612yes—
679D3—44no—
680D3—124no—
681D3—124no—
682D3—244no—
683C22—22no—
684C22—24no—
685C22—44no—
686C22—64no—
687C22—68no—
688C22—128no—
689C22—184no—
690C22—188no—
691C22—368no—
692C22C2×D3×D1214yes—
693C22C2×D3×D1212yes—
694C22C23×D3211yescharacteristic
695C4—24no—
696C4—32no—
697C4—34no—
698C4—32no—
699C4—64no—
700C4—64no—
701C4—184no—
702C3—21no—
703C3#8573511yescharacteristic, complemented
704C3#8629411yescharacteristic, complemented
705C2—24no—
706C2—22no—
707C2—124no—
708C2—364no—
709C2C22×D3×D1211yescharacteristic
710C2#1059667812yes—
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC2×C32×C22⋊C4C22
hypercenterC2×C22⋊C4D32
hypocenterC32#3449
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#3804C2semidirect
2. #10595964C2semidirect
3. C3×#5493C2semidirect
4. C2×C3×#877C2semidirect
5. #10595519C2semidirect
6. #10594911C2semidirect
7. #10594910C2semidirect
8. #10594081C2semidirect
9. #10594080C2semidirect
10. #10594079C2semidirect
11. #10594078C2semidirect
17. C22×#627C22semidirect
18. C22×#626C22semidirect
19. C22×#624C22semidirect
23. C2×#3804C22semidirect
24. C2×#3804C22semidirect
25. C22×D3×Dic3C22semidirect
27. #64139C22semidirect
28. #64095C22semidirect
29. #64091C22semidirect
31. C3×#930C22semidirect
32. C2×C3×Dic3⋊C4C22semidirect
33. C2×C3×C4×Dic3C22semidirect
34. C3×#886C22semidirect
35. C3×#882C22semidirect
36. C3×#877C22semidirect
37. C3×#877C22semidirect
38. #63981C22semidirect
39. #63978C22semidirect
40. #63977C22semidirect
41. #63966C22semidirect
42. #63962C22semidirect
43. #63918C22semidirect
44. #63911C22semidirect
45. #63910C22semidirect
46. #63895C22semidirect
47. #63894C22semidirect
48. #63893C22semidirect
49. Dic3×D12C22semidirect
50. #63886C22semidirect
51. #63880C22semidirect
52. #63852C22semidirect
73. #5493D3semidirect
74. C2×#877D3semidirect
88. C22×C3×Dic3D4semidirect
174. #930C2×D3semidirect
176. C2×Dic3⋊C4C2×D3semidirect
177. C2×C4×Dic3C2×D3semidirect
190. #886C2×D3semidirect
198. #882C2×D3semidirect
201. #877C2×D3semidirect
202. #877C2×D3semidirect
304. C22×Dic3D12semidirect
625. C32#3449semidirect
703. C3#85735semidirect
704. C3#86294semidirect

Maximal subgroups

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

subgroupindex
C22×#38042
#105959642
C3×#54932
C2×C3×#8772
#105955192
#105949112
#105949102
#105940812
#105940802
#105940792
#105940782
#862943
#857353