Finite Group Databasebeta

← #11864367

Subgroups — #11864367

Order 1152.

1953 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64366C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C2×#63833C211yesmaximal, characteristic, complemented
4C22×#3717C211yesmaximal, characteristic, complemented
5C3×#5234C211yesmaximal, characteristic, complemented
6#10591184C214yesmaximal, complemented
7#10591183C214yesmaximal, complemented
8#10591105C211yesmaximal, characteristic, complemented
9#10591096C211yesmaximal, characteristic, complemented
10#83404—31nomaximal
11#78671—31nomaximal
12C2×C32×#41C2211yescharacteristic
13#64366—24no—
14#64366—24no—
15C2×#3832—21no—
16C2×D4×C3⋊D3—22no—
17C2×D4×C3⋊D3—21no—
18C2×#3828C2212yes—
19C2×#3828C2211yescharacteristic
20C22×C4⋊(C3⋊D3)C2212yes—
21C22×C4⋊(C3⋊D3)C2211yescharacteristic
22C22×C3×Dic6C2211yescharacteristic, complemented
23C3×#944—21no—
24C3×#936—21no—
25C3×#935—22no—
26C3×#926C2211yescharacteristic, complemented
27C3×#924C2214yescomplemented
28C2×C3×#208C2211yescharacteristic, complemented
29C22×C3×C3⋊C8C2211yescharacteristic, complemented
30C2×#3719C2212yescomplemented
31#63833—24no—
32C2×#3717—24no—
33C2×#3717C2212yescomplemented
34C2×#3717C2214yescomplemented
35C3×#830C2214yescomplemented
36#63567C2212yescomplemented
37#63564C2214yescomplemented
38#63564C2212yescomplemented
39C2×#1009—31no—
40C2×#1009—31no—
41C2×#1009—61no—
42C3×#536—31no—
43#5825D311yescharacteristic, complemented
44C2×#942—31no—
45C22×#215—31no—
46C2×#909—31no—
47C22×#204—31no—
48C3×#408—31no—
49#5234D311yescharacteristic, complemented
50#5131—31no—
51#5113—31no—
52#5047—34no—
53#5046—34no—
54#5002—31no—
55#5001—31no—
56#4733—34no—
57#4731—34no—
58C23×C3⋊D3—22no—
59C23×C3⋊D3—21no—
60C32×#41—22no—
61C32×#41—22no—
62C2×C32×Q8D411yescharacteristic
63C2×C32×D4D412yes—
64C2×C32×D4D411yescharacteristic
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C2×#638—22no—
68C2×#638—21no—
69#3832—28no—
70D4×C3⋊D3—216no—
71D4×C3⋊D3—28no—
72#3828—22no—
73#3828—24no—
74#3828—24no—
75#3828D414yes—
76C2×C4⋊(C3⋊D3)—22no—
77C2×C4⋊(C3⋊D3)—21no—
78C2×C4⋊(C3⋊D3)—22no—
79C2×C4⋊(C3⋊D3)—24no—
80C2×C4⋊(C3⋊D3)—24no—
81C2×C4⋊(C3⋊D3)D414yes—
82C2×C4⋊(C3⋊D3)C2311yescharacteristic
83C2×C4⋊(C3⋊D3)C2312yes—
84C2×C4×C3⋊D3—22no—
85C2×C4×C3⋊D3—21no—
86C2×#634C2311yescharacteristic
87C22×C3×Dic3—21no—
88C2×C3×Dic6—22no—
89C2×C3×Dic6—22no—
90C2×C3×Dic6—21no—
91C2×C3×Dic6C2312yes—
92C3×#217—24no—
93C3×#217—21no—
94C3×C4⋊Dic3C2312yes—
95C3×Dic3⋊C4—24no—
96C3×C4×Dic3—21no—
97C3×#208—22no—
98C2×C3×C3⋊C8—22no—
99C2×C3×C3⋊C8C2312yes—
100C2×C3×C3⋊C8C2312yes—
101#3719—24no—
102#3717—28no—
103#3717—24no—
104#3009—91noSylow
105C2×C3×#41—21no—
106C2×C3×#41C2×D311yescharacteristic
107C2×C3×#41C2×D311yescharacteristic
108C22×C3×Q8—31no—
109#1009—64no—
110#1009—64no—
111#1009—64no—
112#1009—64no—
113#1009—68no—
114#1009—68no—
115C2×#239—62no—
116C2×#239—61no—
117C2×#239—61no—
118C2×D3×D4—64no—
119C2×D3×D4—62no—
120C2×D3×D4—62no—
121C2×D3×D4—61no—
122C2×D3×D4—62no—
123C2×D3×D4—61no—
124C2×#235—31no—
125C2×#235—31no—
126C2×#235—32no—
127C2×#235—32no—
128C2×#235—61no—
129C2×#235—62no—
130C22×D12—31no—
131C22×D12—31no—
132C22×D12—32no—
133C22×D12—32no—
134C22×D12—61no—
135C22×D12—62no—
136C22×Dic6C2×D311yescharacteristic, complemented
137C2×C3×M16—31no—
138C22×C3×C8—31no—
139C3×C4⋊Q8—61no—
140C3×#124—61no—
141C3×#123—62no—
142C3×C22⋊Q8—34no—
143C3×#117—31no—
144#944—21no—
145C2×#216—32no—
146#942—64no—
147C2×#215—34no—
148C2×#215—32no—
149C2×#215—64no—
150#936—21no—
151#935—22no—
152#926C2×D311yescharacteristic, complemented
153#924C2×D314yescomplemented
154C2×#208C2×D311yescharacteristic, complemented
155C22×C3⋊C8C2×D311yescharacteristic, complemented
156#909—64no—
157C2×Dic12—32no—
158C2×#204—34no—
159C2×#204—32no—
160C2×#204—64no—
161C3×C22⋊C8—34no—
162#830C2×D314yescomplemented
163#821—34no—
164#821—32no—
165#816—32no—
166#810—32no—
167#809—34no—
168#809—32no—
169C23×C32C2×D412yes—
170C23×C32C2×D411yescharacteristic
171C22×C3⋊D3—22no—
172C22×C3⋊D3—21no—
173C22×C3⋊D3—28no—
174C22×C3⋊D3—24no—
175C22×C3⋊D3—82no—
176C22×C3⋊D3—84no—
177C32×Q8—22no—
178C32×D4—24no—
179C32×D4—22no—
180C2×C32×C4—22no—
181C2×C32×C4—22no—
182C2×C32×C4C2×D412yes—
183C2×C32×C4C2×D411yescharacteristic
184C2×C32×C4C2×D412yes—
185C2×C32×C4C2×D412yes—
186C2×C32×C4C2411yescharacteristic
187#638—28no—
188#638—24no—
189C2×C3⋊Dic3—21no—
190C4⋊(C3⋊D3)—28no—
191C4⋊(C3⋊D3)—24no—
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)—22no—
194C4⋊(C3⋊D3)C2×D412yes—
195C4⋊(C3⋊D3)C2×D414yes—
196C4×C3⋊D3—28no—
197C4×C3⋊D3—24no—
198#634—21no—
199#634C2×D412yes—
200C2×C3×Dic3—22no—
201C2×C3×Dic3—22no—
202C2×C3×Dic3—42no—
203C3×Dic6—22no—
204C3×Dic6—22no—
205C3×Dic6—42no—
206C3×C3⋊C8—22no—
207C3×C3⋊C8—22no—
208C2×H(5,𝔽2)—91no—
209C2×#137—91no—
210C22×SD16—91no—
211#536—31no—
212#452—94no—
213#450—94no—
214#418—91no—
215#416—91no—
216#408—31no—
217C23×D3—64no—
218C23×D3—62no—
219C23×D3—62no—
220C23×D3—61no—
221C23×D3—62no—
222C23×D3—61no—
223C3×#41—24no—
224C3×#41—24no—
225C3×#41—22no—
226C3×#41—22no—
227C3×#41—22no—
228C3×#41—22no—
229C2×C3×Q8—21no—
230C2×C3×Q8—32no—
231C2×C3×Q8—61no—
232C2×C3×Q8—62no—
233C2×C3×Q8—62no—
234C2×C3×Q8D1211yescharacteristic
235C2×C3×Q8#6711yescharacteristic
236C2×C3×D4—22no—
237C2×C3×D4—21no—
238C2×C3×D4D1212yes—
239C2×C3×D4D1211yescharacteristic
240C2×C3×D4#6712yes—
241C2×C3×D4#6711yescharacteristic
242C22×C3×C4—22no—
243C22×C3×C4—21no—
244C22×C3×C4—61no—
245C22×C3×C4C22×D312yes—
246C22×C3×C4C22×D312yes—
247C22×C3×C4C22×D311yescharacteristic
248C22×C3×C4C22×D311yescharacteristic
249C2×#67—64no—
250C2×#67—62no—
251C2×#67—62no—
252C2×#67—61no—
253C2×#67—62no—
254C2×#67—61no—
255C22×Dic3—21no—
256#239—616no—
257#239—68no—
258#239—68no—
259D3×D4—632no—
260D3×D4—616no—
261D3×D4—616no—
262D3×D4—616no—
263D3×D4—68no—
264D3×D4—68no—
265#235—34no—
266#235—34no—
267#235—64no—
268#235—64no—
269#235—64no—
270#235—64no—
271#235—64no—
272#235—68no—
273#235—68no—
274#235—64no—
275#235—62no—
276#235—62no—
277C2×D12—34no—
278C2×D12—34no—
279C2×D12—32no—
280C2×D12—31no—
281C2×D12—32no—
282C2×D12—31no—
283C2×D12—62no—
284C2×D12—61no—
285C2×D12—64no—
286C2×D12—64no—
287C2×D12—64no—
288C2×D12—64no—
289C2×D12—64no—
290C2×D12—64no—
291C2×D12—62no—
292C2×D12—68no—
293C2×D12—68no—
294C2×D12—64no—
295C2×D12—62no—
296C2×D12—61no—
297C2×D12—62no—
298C2×D12—62no—
299C2×D12—62no—
300C2×D12—61no—
301C2×C4×D3—64no—
302C2×C4×D3—62no—
303C2×C4×D3—62no—
304C2×C4×D3—61no—
305C2×C4×D3—62no—
306C2×C4×D3—61no—
307C2×Dic6—22no—
308C2×Dic6—22no—
309C2×Dic6—21no—
310C2×Dic6—31no—
311C2×Dic6—31no—
312C2×Dic6—61no—
313C2×Dic6C22×D312yes—
314C3×M16—62no—
315C2×C3×C8—32no—
316C2×C3×C8—32no—
317C2×C3×C8—62no—
318C3×C4⋊C4—32no—
319C3×C4⋊C4—64no—
320C3×C22⋊C4—61no—
321C3×C22⋊C4—64no—
322C3×C42—61no—
323#217—24no—
324#217—21no—
325#216—64no—
326#215—68no—
327#215—64no—
328C4⋊Dic3C22×D312yes—
329Dic3⋊C4—24no—
330C4×Dic3—21no—
331#208—22no—
332C2×C3⋊C8—22no—
333C2×C3⋊C8C22×D312yes—
334C2×C3⋊C8C22×D312yes—
335Dic12—64no—
336#204—68no—
337#204—64no—
338C22×C32—22no—
339C22×C32—22no—
340C22×C32—24no—
341C22×C32#12012yes—
342C22×C32C22×D411yescharacteristic
343C2×C3⋊D3—24no—
344C2×C3⋊D3—22no—
345C2×C3⋊D3—44no—
346C2×C3⋊D3—42no—
347C2×C3⋊D3—84no—
348C2×C3⋊D3—88no—
349C32×C4—22no—
350C32×C4#12012yes—
351C32×C4C22×D411yescharacteristic
352C32×C4C22×D411yescharacteristic
353C3⋊Dic3—22no—
354C3×Dic3—42no—
355C3×Dic3—42no—
356H(5,𝔽2)—184no—
357H(5,𝔽2)—184no—
358C2×#41—91no—
359C2×#41—92no—
360C2×#41—181no—
361C2×#41D3211yescharacteristic
362C22×Q8—31no—
363C22×D4—91no—
364C22×D4—92no—
365C22×D4—182no—
366C22×D4—181no—
367#137—184no—
368C2×Q16—92no—
369C2×SD16—94no—
370C2×SD16—92no—
371C2×SD16—184no—
372C2×M16—31no—
373C22×C8—31no—
374C4⋊Q8—61no—
375#124—61no—
376#123—62no—
377C22⋊Q8—34no—
378#117—31no—
379#103—92no—
380#102—92no—
381#102—94no—
382C22⋊C8—34no—
383C23×C3—22no—
384C23×C3—21no—
385C23×C3C2×D1212yes—
386C23×C3C2×D1211yescharacteristic
387C23×C3C2×#6712yes—
388C23×C3C2×#6711yescharacteristic
389C22×D3—64no—
390C22×D3—62no—
391C22×D3—62no—
392C22×D3—61no—
393C22×D3—62no—
394C22×D3—61no—
395C22×D3—616no—
396C22×D3—68no—
397C22×D3—68no—
398C22×D3—68no—
399C22×D3—64no—
400C22×D3—64no—
401C22×D3—242no—
402C22×D3—244no—
403C22×D3—242no—
404C22×D3—244no—
405C22×D3—244no—
406C22×D3—248no—
407C3×Q8—24no—
408C3×Q8—22no—
409C3×Q8—22no—
410C3×Q8—62no—
411C3×Q8—62no—
412C3×Q8—122no—
413C3×D4—28no—
414C3×D4—24no—
415C3×D4—24no—
416C3×D4—24no—
417C3×D4—22no—
418C3×D4—22no—
419C2×C3×C4—24no—
420C2×C3×C4—24no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—21no—
426C2×C3×C4—22no—
427C2×C3×C4—21no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—62no—
431C2×C3×C4—62no—
432C2×C3×C4—122no—
433C2×C3×C4C2×D1212yes—
434C2×C3×C4C2×D1211yescharacteristic
435C2×C3×C4D3×D412yes—
436C2×C3×C4D3×D412yes—
437C2×C3×C4D3×D412yes—
438C2×C3×C4D3×D412yes—
439C2×C3×C4C2×#6712yes—
440C2×C3×C4C2×#6711yescharacteristic
441C2×C3×C4C23×D311yescharacteristic
442C2×C3×C4C23×D311yescharacteristic
443#67—616no—
444#67—68no—
445#67—68no—
446#67—68no—
447#67—64no—
448#67—64no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—42no—
452C2×Dic3—62no—
453C2×Dic3—61no—
454C2×Dic3—61no—
455D12—34no—
456D12—32no—
457D12—34no—
458D12—32no—
459D12—64no—
460D12—62no—
461D12—62no—
462D12—64no—
463D12—616no—
464D12—68no—
465D12—68no—
466D12—68no—
467D12—64no—
468D12—64no—
469D12—62no—
470D12—61no—
471D12—62no—
472D12—61no—
473C4×D3—616no—
474C4×D3—68no—
475C4×D3—68no—
476C4×D3—68no—
477C4×D3—64no—
478C4×D3—64no—
479Dic6—22no—
480Dic6—22no—
481Dic6—32no—
482Dic6—32no—
483Dic6—42no—
484Dic6—62no—
485Dic6—62no—
486Dic6—61no—
487Dic6—61no—
488C3×C8—62no—
489C3×C8—62no—
490C3⋊C8—22no—
491C3⋊C8—22no—
492C2×C32—22no—
493C2×C32—42no—
494C2×C32C2×#12011yescharacteristic
495C2×C32#57712yes—
496C3⋊D3—44no—
497C3⋊D3—42no—
498C24—182no—
499C24—181no—
500#41—22no—
501#41—22no—
502#41—94no—
503#41—184no—
504#41—184no—
505#41—188no—
506#41—182no—
507C2×Q8—32no—
508C2×Q8—61no—
509C2×Q8—62no—
510C2×Q8—62no—
511C2×Q8—91no—
512C2×Q8#62611yescharacteristic
513C2×D4—92no—
514C2×D4—91no—
515C2×D4—94no—
516C2×D4—184no—
517C2×D4—184no—
518C2×D4—1816no—
519C2×D4—188no—
520C2×D4—182no—
521C2×D4—181no—
522C2×D4—182no—
523C2×D4—181no—
524C2×D4—182no—
525C2×D4#62612yes—
526C2×D4#62611yescharacteristic
527C22×C4—61no—
528C22×C4—182no—
529C22×C4—181no—
530C22×C4C2×D3212yes—
531C22×C4C2×D3211yescharacteristic
532Q16—184no—
533SD16—188no—
534SD16—184no—
535M16—62no—
536C2×C8—32no—
537C2×C8—32no—
538C2×C8—62no—
539C4⋊C4—32no—
540C4⋊C4—64no—
541C22⋊C4—61no—
542C22⋊C4—64no—
543C42—61no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—24no—
547C22×C3—22no—
548C22×C3—24no—
549C22×C3—22no—
550C22×C3—24no—
551C22×C3—21no—
552C22×C3—22no—
553C22×C3—42no—
554C22×C3—44no—
555C22×C3#88212yes—
556C22×C3#93712yes—
557C22×C3C22×D1211yescharacteristic
558C22×C3C22×#6711yescharacteristic
559C2×D3—68no—
560C2×D3—64no—
561C2×D3—64no—
562C2×D3—62no—
563C2×D3—64no—
564C2×D3—62no—
565C2×D3—128no—
566C2×D3—124no—
567C2×D3—124no—
568C2×D3—122no—
569C2×D3—124no—
570C2×D3—122no—
571C2×D3—244no—
572C2×D3—244no—
573C2×D3—248no—
574C2×D3—2416no—
575C2×D3—248no—
576C2×D3—248no—
577C3×C4—24no—
578C3×C4—22no—
579C3×C4—22no—
580C3×C4—21no—
581C3×C4—21no—
582C3×C4—22no—
583C3×C4—122no—
584C3×C4—122no—
585C3×C4#88212yes—
586C3×C4#93712yes—
587C3×C4C2×D3×D411yescharacteristic
588C3×C4C2×D3×D411yescharacteristic
589C3×C4C2×D3×D411yescharacteristic
590C3×C4C2×D3×D411yescharacteristic
591Dic3—42no—
592Dic3—42no—
593Dic3—64no—
594Dic3—62no—
595Dic3—62no—
596C32#300911yescharacteristic, complemented, Sylow
597C23—188no—
598C23—182no—
599C23—181no—
600C23—184no—
601C23—722no—
602C23—724no—
603C23C2×#62612yes—
604C23C2×#62611yescharacteristic
605Q8—22no—
606Q8—62no—
607Q8—62no—
608Q8—92no—
609Q8—122no—
610Q8—181no—
611D4—24no—
612D4—22no—
613D4—94no—
614D4—92no—
615D4—188no—
616D4—188no—
617D4—184no—
618D4—181no—
619D4—182no—
620D4—184no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—122no—
626C2×C4—188no—
627C2×C4—181no—
628C2×C4—184no—
629C2×C4C2×#62612yes—
630C2×C4C2×#62611yescharacteristic
631C2×C4#381112yes—
632C2×C4#381112yes—
633C2×C4C22×D3211yescharacteristic
634C8—62no—
635C8—62no—
636C2×C3—22no—
637C2×C3—22no—
638C2×C3—22no—
639C2×C3—21no—
640C2×C3—42no—
641C2×C3—44no—
642C2×C3—42no—
643C2×C3—42no—
644C2×C3C2×#88211yescharacteristic
645C2×C3#575112yes—
646C2×C3C2×#93711yescharacteristic
647C2×C3#583712yes—
648D3—128no—
649D3—124no—
650D3—124no—
651D3—122no—
652D3—124no—
653D3—122no—
654C22—22no—
655C22—24no—
656C22—22no—
657C22—184no—
658C22—182no—
659C22—364no—
660C22—362no—
661C22—724no—
662C22—728no—
663C22#6398112yes—
664C22C22×#62611yescharacteristic
665C4—22no—
666C4—122no—
667C4—122no—
668C4—182no—
669C4#6398112yes—
670C4C2×#381111yescharacteristic
671C4C2×#381111yescharacteristic
672C3—21no—
673C3#7867111yescharacteristic, complemented
674C3#8340411yescharacteristic, complemented
675C2—22no—
676C2—42no—
677C2—364no—
678C2—362no—
679C2#1059544812yes—
680C2C2×#6398111yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×#41C22
socleC22×C32C22×D4
prime radicalC2×#64366C2
hypercenterC2×#41D32
hypocenterC32#3009
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C3×#5825C2semidirect
3. C2×#63833C2semidirect
4. C22×#3717C2semidirect
5. C3×#5234C2semidirect
6. #10591184C2semidirect
7. #10591183C2semidirect
8. #10591105C2semidirect
9. #10591096C2semidirect
22. C22×C3×Dic6C22semidirect
26. C3×#926C22semidirect
27. C3×#924C22semidirect
28. C2×C3×#208C22semidirect
29. C22×C3×C3⋊C8C22semidirect
30. C2×#3719C22semidirect
33. C2×#3717C22semidirect
34. C2×#3717C22semidirect
35. C3×#830C22semidirect
36. #63567C22semidirect
37. #63564C22semidirect
38. #63564C22semidirect
43. #5825D3semidirect
49. #5234D3semidirect
136. C22×Dic6C2×D3semidirect
152. #926C2×D3semidirect
153. #924C2×D3semidirect
154. C2×#208C2×D3semidirect
155. C22×C3⋊C8C2×D3semidirect
162. #830C2×D3semidirect
596. C32#3009semidirect
673. C3#78671semidirect
674. C3#83404semidirect

Maximal subgroups

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

subgroupindex
C2×#643662
C3×#58252
C2×#638332
C22×#37172
C3×#52342
#105911842
#105911832
#105911052
#105910962
#834043
#786713