Finite Group Databasebeta

← #11662149

Subgroups — #11662149

Order 768.

2890 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87268C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic, complemented
3C22×#908C211yesmaximal, characteristic, complemented
4C2×#5521C211yesmaximal, characteristic, complemented
5C3×#2896C211yesmaximal, characteristic, complemented
6#83750C211yesmaximal, characteristic, complemented
7#82213C211yesmaximal, characteristic, complemented
8#78976C214yesmaximal, complemented
9#78944C214yesmaximal, complemented
10#78910C218yesmaximal, complemented
11#78863C214yesmaximal, complemented
12#78862C214yesmaximal, complemented
13#60423—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C23×D12C2211yescharacteristic
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19#5833—22no—
20#5832C2214yescomplemented
21#5830C2212yescomplemented
22#5828—22no—
23#5825C2212yescomplemented
24C2×#949—28no—
25C2×#949C2214yes—
26C2×#949C2214yescomplemented
27#5816C2212yescomplemented
28C22×#215C2212yescomplemented
29#5805C2212yescomplemented
30#5804—22no—
31C22×#213C2212yescomplemented
32#5791—22no—
33C2×#926C2211yescharacteristic
34C22×#208C2211yescharacteristic
35C2×#908—28no—
36C2×#908C2214yes—
37C2×#908C2214yescomplemented
38C22×D24C2212yescomplemented
39C22×#204C2212yescomplemented
40#5521—22no—
41#5521—22no—
42#5521C2214yescomplemented
43#5521C2214yescomplemented
44C2×#893C2214yescomplemented
45C2×D3×C22⋊C4—22no—
46C2×C4×D12—22no—
47C3×#419C2212yescomplemented
48C3×#418C2212yescomplemented
49C3×#417C2214yescomplemented
50C3×#416C2214yescomplemented
51#5119C2214yescomplemented
52#5118C2214yescomplemented
53#5102C2214yescomplemented
54#5004C2212yescomplemented
55#5002C2214yescomplemented
56#5001C2212yescomplemented
57#4966C2214yescomplemented
58#4819C22116yescomplemented
59#4818C22116yescomplemented
60#4787C22116yescomplemented
61#4786C22116yescomplemented
62C22×AGL(1,ℤ/8)—31no—
63C22×AGL(1,ℤ/8)—31no—
64#3449—31no—
65C2×#518—31no—
66#3042—34no—
67#3042—34no—
68#3041—34no—
69#3041—34no—
70#3036—38no—
71#2905—31no—
72#2896—31no—
73#2896D311yescharacteristic, complemented
74C24×D3—21no—
75C2×C3×#41—21no—
76C2×C3×#41—21no—
77C2×C3×#41—24no—
78C2×C3×#41—24no—
79C2×C3×#41C2312yes—
80C2×C3×#41C2312yes—
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4—21no—
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4—21no—
85C23×C3×C4C2311yescharacteristic
86C22×#67—21no—
87C2×#235—21no—
88C2×#235—21no—
89C2×#235—24no—
90C2×#235—24no—
91C2×#235C2312yes—
92C2×#235C2312yes—
93C22×D12—21no—
94C22×D12—21no—
95C22×D12—21no—
96C22×D12—24no—
97C22×D12D418yescomplemented
98C22×D12C2311yescharacteristic
99C22×D12C2312yes—
100C22×D12C2312yes—
101C22×D12C2312yes—
102C22×C4×D3—21no—
103C22×C4×D3—24no—
104C22×Dic6C2311yescharacteristic
105C2×C3×M16—24no—
106C2×C3×M16C2312yes—
107C2×C3×M16C2312yes—
108C22×C3×C8C2312yes—
109C3×#117—21no—
110C3×#117—21no—
111C3×#117C2312yes—
112C3×#117C2312yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C22⋊C4—21no—
115C2×C3×C42—21no—
116#953—24no—
117C2×#217D412yes—
118#949—216no—
119#949—216no—
120#947C2314yescomplemented
121#946—24no—
122#944C2314yescomplemented
123C2×#215—28no—
124C2×#215C2318yes—
125#940C2314yescomplemented
126#939—28no—
127#938—24no—
128#937—24no—
129#936C2314yescomplemented
130#935—24no—
131C2×#213—28no—
132C2×#213C2318yes—
133#930—24no—
134#929—24no—
135C4×#67—28no—
136C2×(C2×D3)⋊C4—24no—
137#926D412yes—
138#926D412yes—
139#926C2312yes—
140#926C2312yes—
141C2×C4⋊Dic3D412yes—
142#924—24no—
143C2×C4×Dic3C2312yes—
144C2×#208—24no—
145C2×#208C2312yes—
146C2×#208C2312yes—
147C22×C3⋊C8C2312yes—
148#908—216no—
149#908—216no—
150C2×D24—28no—
151C2×D24C2318yes—
152C2×#204—28no—
153C2×#204C2318yes—
154#893—28no—
155#893C23116yes—
156D3×C22⋊C4—28no—
157D3×C22⋊C4—28no—
158C4×D12—28no—
159C4×D12—28no—
160C3×#103C2318yescomplemented
161C3×#102C2318yescomplemented
162Dic3⋊C8C23116yescomplemented
163#810C2318yescomplemented
164#809C2318yescomplemented
165C22×#41—31no—
166C22×#41C2×D311yescharacteristic
167C23×D4—31no—
168C2×AGL(1,ℤ/8)—34no—
169C2×AGL(1,ℤ/8)—34no—
170C2×AGL(1,ℤ/8)—34no—
171C2×AGL(1,ℤ/8)—34no—
172C2×AGL(1,ℤ/8)—68no—
173C2×AGL(1,ℤ/8)—68no—
174C22×SD16—32no—
175C22×SD16—32no—
176C22×D8—32no—
177C22×D8—32no—
178C22×M16—31no—
179C22×M16C2×D311yescharacteristic
180#537—62no—
181#536—32no—
182#536—32no—
183#535—32no—
184#535—32no—
185#534—62no—
186#532—34no—
187#525—62no—
188#525—62no—
189#518—34no—
190#518—34no—
191#518—62no—
192#518—62no—
193C2×C4×D4—34no—
194C2×C4×D4—62no—
195C2×#117—31no—
196C2×#117C2×D311yescharacteristic
197C22×C22⋊C4—62no—
198#463—316no—
199#463—316no—
200#461—316no—
201#459—316no—
202#424—34no—
203#424—34no—
204#423—34no—
205#419—32no—
206#419C2×D312yescomplemented
207#418—32no—
208#418C2×D312yescomplemented
209#417—34no—
210#417C2×D314yescomplemented
211#416—34no—
212#416C2×D314yescomplemented
213C24×C3—21no—
214C24×C3C2×D411yescharacteristic
215C23×D3—22no—
216C23×D3—28no—
217C23×D3—22no—
218C23×D3—22no—
219C23×D3—21no—
220C23×D3—84no—
221C3×#41—24no—
222C3×#41—24no—
223C3×#41—24no—
224C3×#41—24no—
225C3×#41—44no—
226C3×#41—44no—
227C2×C3×Q8—21no—
228C2×C3×Q8—22no—
229C2×C3×Q8—22no—
230C2×C3×Q8C2412yes—
231C2×C3×D4—22no—
232C2×C3×D4—22no—
233C2×C3×D4—22no—
234C2×C3×D4—21no—
235C2×C3×D4—22no—
236C2×C3×D4—44no—
237C2×C3×D4C2412yes—
238C22×C3×C4—22no—
239C22×C3×C4—21no—
240C22×C3×C4—22no—
241C22×C3×C4—22no—
242C22×C3×C4—44no—
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4C2×D411yescharacteristic
246C22×C3×C4C2×D414yes—
247C22×C3×C4#4114yes—
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2411yescharacteristic
250C22×C3×C4C2411yescharacteristic
251C2×#67—22no—
252C2×#67—22no—
253C2×#67—28no—
254C2×#67—44no—
255C22×Dic3—21no—
256C22×Dic3C2×D414yes—
257#235—24no—
258#235—24no—
259#235—24no—
260#235—24no—
261#235—44no—
262#235—44no—
263C2×D12—28no—
264C2×D12—28no—
265C2×D12—28no—
266C2×D12—22no—
267C2×D12—22no—
268C2×D12—22no—
269C2×D12—24no—
270C2×D12—24no—
271C2×D12—24no—
272C2×D12—22no—
273C2×D12—22no—
274C2×D12—21no—
275C2×D12C2×D418yes—
276C2×D12C2×D418yes—
277C2×D12C2×D418yes—
278C2×D12C2412yes—
279C2×D12C2414yes—
280C2×C4×D3—22no—
281C2×C4×D3—22no—
282C2×C4×D3—216no—
283C2×C4×D3—44no—
284C2×C4×D3—88no—
285C2×Dic6—22no—
286C2×Dic6—22no—
287C2×Dic6—21no—
288C2×Dic6C2412yes—
289C3×M16—24no—
290C3×M16—24no—
291C2×C3×C8—24no—
292C2×C3×C8C2414yes—
293C3×C4⋊C4—22no—
294C3×C4⋊C4C2414yes—
295C3×C22⋊C4—22no—
296C3×C22⋊C4—22no—
297C3×C42—22no—
298C3×C42—22no—
299#217—28no—
300#217C2×D414yes—
301#217C2×D414yes—
302#215—216no—
303#213—216no—
304(C2×D3)⋊C4—216no—
305(C2×D3)⋊C4—28no—
306C4⋊Dic3C2×D414yes—
307C4⋊Dic3C2×D414yes—
308Dic3⋊C4—28no—
309C4×Dic3C2×D414yes—
310C4×Dic3C2414yes—
311#208—24no—
312#208—24no—
313C2×C3⋊C8—24no—
314C2×C3⋊C8C2414yes—
315D24—216no—
316#204—216no—
317C25—61no—
318C2×#41—21no—
319C2×#41—21no—
320C2×#41—24no—
321C2×#41—24no—
322C2×#41—32no—
323C2×#41—32no—
324C2×#41—61no—
325C2×#41—61no—
326C2×#41—64no—
327C2×#41—64no—
328C2×#41C22×D312yes—
329C2×#41C22×D312yes—
330C22×Q8—31no—
331C22×Q8C22×D311yescharacteristic
332C22×D4—21no—
333C22×D4—38no—
334C22×D4—32no—
335C22×D4—32no—
336C22×D4—32no—
337C22×D4—31no—
338C22×D4—61no—
339C22×D4—64no—
340C22×D4—61no—
341C22×D4—61no—
342C22×D4—61no—
343C22×D4C22×D311yescharacteristic
344C23×C4—21no—
345C23×C4—64no—
346C23×C4—61no—
347C23×C4C22×D311yescharacteristic
348AGL(1,ℤ/8)—616no—
349AGL(1,ℤ/8)—616no—
350AGL(1,ℤ/8)—616no—
351AGL(1,ℤ/8)—616no—
352C2×SD16—38no—
353C2×SD16—38no—
354C2×SD16—68no—
355C2×SD16—68no—
356C2×D8—38no—
357C2×D8—38no—
358C2×D8—68no—
359C2×D8—68no—
360C2×M16—24no—
361C2×M16—32no—
362C2×M16—32no—
363C2×M16—64no—
364C2×M16C22×D312yes—
365C2×M16C22×D312yes—
366C22×C8—32no—
367C22×C8C22×D312yes—
368C4⋊Q8—34no—
369#127—34no—
370#124—34no—
371#124—34no—
372#123—64no—
373#123—64no—
374C22⋊Q8—64no—
375C22⋊Q8—64no—
376#121—64no—
377#121—64no—
378#121—68no—
379#120—64no—
380#120—64no—
381C4×D4—316no—
382C4×D4—68no—
383C4×D4—68no—
384C4×D4—68no—
385C4×D4—68no—
386#117—21no—
387#117—21no—
388#117—32no—
389#117—32no—
390#117—32no—
391#117—32no—
392#117C22×D312yes—
393#117C22×D312yes—
394C2×C4⋊C4—32no—
395C2×C4⋊C4C22×D312yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—32no—
398C2×C22⋊C4—68no—
399C2×C22⋊C4—68no—
400C2×C22⋊C4—64no—
401C2×C42—21no—
402C2×C42—32no—
403C4⋊C8—316no—
404#103—38no—
405#103C22×D318yescomplemented
406#102—38no—
407#102C22×D318yescomplemented
408C23×C3—22no—
409C23×C3—42no—
410C23×C3—42no—
411C23×C3—82no—
412C23×C3#12114yes—
413C23×C3C22×D411yescharacteristic
414C23×C3C22×D411yescharacteristic
415C23×C3C22×D411yescharacteristic
416C22×D3—28no—
417C22×D3—24no—
418C22×D3—28no—
419C22×D3—28no—
420C22×D3—22no—
421C22×D3—48no—
422C22×D3—42no—
423C22×D3—82no—
424C22×D3—84no—
425C22×D3—84no—
426C22×D3—816no—
427C3×Q8—22no—
428C3×Q8—22no—
429C3×Q8—42no—
430C3×D4—22no—
431C3×D4—22no—
432C3×D4—44no—
433C3×D4—44no—
434C3×D4—42no—
435C2×C3×C4—22no—
436C2×C3×C4—24no—
437C2×C3×C4—44no—
438C2×C3×C4—42no—
439C2×C3×C4—44no—
440C2×C3×C4—44no—
441C2×C3×C4#12114yes—
442C2×C3×C4#12114yes—
443C2×C3×C4#12114yes—
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C22×D411yescharacteristic
446C2×C3×C4C22×D411yescharacteristic
447C2×C3×C4C22×D412yes—
448C2×C3×C4C22×D412yes—
449C2×C3×C4C22×D412yes—
450C2×C3×C4C2×#4112yes—
451C2×C3×C4C2×#4112yes—
452C2×C3×C4C2×#4112yes—
453C2×C3×C4C2511yescharacteristic
454#67—44no—
455#67—44no—
456#67—88no—
457C2×Dic3—22no—
458C2×Dic3—44no—
459C2×Dic3—42no—
460C2×Dic3C22×D418yes—
461D12—28no—
462D12—28no—
463D12—28no—
464D12—22no—
465D12—24no—
466D12—22no—
467D12—42no—
468D12C22×D418yes—
469C4×D3—44no—
470C4×D3—44no—
471C4×D3—816no—
472Dic6—22no—
473Dic6—22no—
474Dic6—42no—
475C3×C8—24no—
476C3⋊C8—24no—
477C24—21no—
478C24—62no—
479C24—68no—
480C24—62no—
481C24—62no—
482C24—61no—
483C24—244no—
484C24D3×D411yescharacteristic
485#41—24no—
486#41—24no—
487#41—24no—
488#41—24no—
489#41—44no—
490#41—44no—
491#41—64no—
492#41—64no—
493#41—64no—
494#41—64no—
495#41—124no—
496#41—124no—
497C2×Q8—21no—
498C2×Q8—22no—
499C2×Q8—22no—
500C2×Q8—32no—
501C2×Q8—61no—
502C2×Q8—62no—
503C2×Q8—62no—
504C2×Q8C23×D312yes—
505C2×D4—22no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4—21no—
510C2×D4—38no—
511C2×D4—38no—
512C2×D4—38no—
513C2×D4—34no—
514C2×D4—32no—
515C2×D4—44no—
516C2×D4—68no—
517C2×D4—62no—
518C2×D4—62no—
519C2×D4—68no—
520C2×D4—68no—
521C2×D4—68no—
522C2×D4—64no—
523C2×D4—64no—
524C2×D4—64no—
525C2×D4—62no—
526C2×D4—62no—
527C2×D4—62no—
528C2×D4—62no—
529C2×D4—62no—
530C2×D4—61no—
531C2×D4—124no—
532C2×D4C23×D312yes—
533C22×C4—22no—
534C22×C4—21no—
535C22×C4—22no—
536C22×C4—22no—
537C22×C4—34no—
538C22×C4—44no—
539C22×C4—616no—
540C22×C4—61no—
541C22×C4—62no—
542C22×C4—62no—
543C22×C4—124no—
544C22×C4—248no—
545C22×C4#23514yes—
546C22×C4D3×D411yescharacteristic
547C22×C4D3×D411yescharacteristic
548C22×C4D3×D411yescharacteristic
549C22×C4D3×D414yes—
550C22×C4C23×D311yescharacteristic
551C22×C4C23×D311yescharacteristic
552C22×C4C23×D311yescharacteristic
553SD16—616no—
554SD16—616no—
555D8—616no—
556D8—616no—
557M16—24no—
558M16—24no—
559M16—64no—
560M16—64no—
561C2×C8—24no—
562C2×C8—34no—
563C2×C8—64no—
564C2×C8C23×D314yes—
565C4⋊C4—22no—
566C4⋊C4—34no—
567C4⋊C4—34no—
568C4⋊C4—68no—
569C4⋊C4C23×D314yes—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—34no—
573C22⋊C4—34no—
574C22⋊C4—616no—
575C22⋊C4—68no—
576C22⋊C4—68no—
577C42—22no—
578C42—22no—
579C42—34no—
580C42—34no—
581C22×C3—22no—
582C22×C3—22no—
583C22×C3—22no—
584C22×C3—44no—
585C22×C3—44no—
586C22×C3—42no—
587C22×C3—84no—
588C22×C3C2×#12112yes—
589C22×C3C2×#12112yes—
590C22×C3C2×#12112yes—
591C22×C3C23×D411yescharacteristic
592C2×D3—28no—
593C2×D3—416no—
594C2×D3—48no—
595C2×D3—44no—
596C2×D3—42no—
597C2×D3—88no—
598C2×D3—88no—
599C2×D3—84no—
600C3×C4—44no—
601C3×C4—42no—
602C3×C4C2×#12112yes—
603C3×C4C2×#12112yes—
604C3×C4C2×#12112yes—
605C3×C4C23×D411yescharacteristic
606C3×C4C22×#4111yescharacteristic
607Dic3—28no—
608Dic3—42no—
609C23—22no—
610C23—42no—
611C23—42no—
612C23—64no—
613C23—68no—
614C23—68no—
615C23—68no—
616C23—62no—
617C23—82no—
618C23—128no—
619C23—122no—
620C23—244no—
621C23—244no—
622C23—242no—
623C23—2416no—
624C23#88414yes—
625C23C2×D3×D411yescharacteristic
626C23C2×D3×D411yescharacteristic
627C23C2×D3×D411yescharacteristic
628Q8—22no—
629Q8—22no—
630Q8—42no—
631Q8—62no—
632Q8—62no—
633Q8—122no—
634D4—22no—
635D4—22no—
636D4—38no—
637D4—44no—
638D4—44no—
639D4—42no—
640D4—64no—
641D4—62no—
642D4—62no—
643D4—68no—
644D4—68no—
645D4—68no—
646D4—124no—
647D4—124no—
648D4—122no—
649D4—248no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—38no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—62no—
658C2×C4—124no—
659C2×C4—122no—
660C2×C4—124no—
661C2×C4—124no—
662C2×C4—2416no—
663C2×C4#88414yes—
664C2×C4#88414yes—
665C2×C4#88414yes—
666C2×C4C2×#23512yes—
667C2×C4C2×#23512yes—
668C2×C4C2×#23512yes—
669C2×C4C2×D3×D412yes—
670C2×C4C2×D3×D412yes—
671C2×C4C2×D3×D412yes—
672C2×C4C2×D3×D411yescharacteristic
673C2×C4C2×D3×D411yescharacteristic
674C2×C4C2×D3×D411yescharacteristic
675C2×C4C24×D311yescharacteristic
676C8—24no—
677C8—64no—
678C2×C3—22no—
679C2×C3—22no—
680C2×C3—22no—
681C2×C3—82no—
682C2×C3C22×#12111yescharacteristic
683C2×C3#358512yes—
684D3—48no—
685D3—82no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—44no—
690C22—44no—
691C22—42no—
692C22—68no—
693C22—84no—
694C22—1216no—
695C22—128no—
696C22—124no—
697C22—122no—
698C22—248no—
699C22—248no—
700C22—244no—
701C22C2×#88412yes—
702C22C2×#88412yes—
703C22C2×#88412yes—
704C22C22×D3×D411yescharacteristic
705C4—44no—
706C4—42no—
707C4—68no—
708C4—122no—
709C4C2×#88412yes—
710C4C2×#88412yes—
711C4C2×#88412yes—
712C4C22×#23511yescharacteristic
713C4C22×D3×D411yescharacteristic
714C3#6042311yescharacteristic, complemented, Sylow
715C2—22no—
716C2—22no—
717C2—22no—
718C2—82no—
719C2—128no—
720C2—242no—
721C2C22×#88411yescharacteristic
722C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2896C2
socleC22×C3C23×D4
hypercenter#2896D3
hypocenterC3#60423
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
1. #87268C2semidirect
2. C22×#949C2semidirect
3. C22×#908C2semidirect
4. C2×#5521C2semidirect
5. C3×#2896C2semidirect
6. #83750C2semidirect
7. #82213C2semidirect
8. #78976C2semidirect
9. #78944C2semidirect
10. #78910C2semidirect
11. #78863C2semidirect
12. #78862C2semidirect
20. #5832C22semidirect
21. #5830C22semidirect
23. #5825C22semidirect
26. C2×#949C22semidirect
27. #5816C22semidirect
28. C22×#215C22semidirect
29. #5805C22semidirect
31. C22×#213C22semidirect
37. C2×#908C22semidirect
38. C22×D24C22semidirect
39. C22×#204C22semidirect
42. #5521C22semidirect
43. #5521C22semidirect
44. C2×#893C22semidirect
47. C3×#419C22semidirect
48. C3×#418C22semidirect
49. C3×#417C22semidirect
50. C3×#416C22semidirect
51. #5119C22semidirect
52. #5118C22semidirect
53. #5102C22semidirect
54. #5004C22semidirect
55. #5002C22semidirect
56. #5001C22semidirect
57. #4966C22semidirect
58. #4819C22semidirect
59. #4818C22semidirect
60. #4787C22semidirect
61. #4786C22semidirect
73. #2896D3semidirect
97. C22×D12D4semidirect
120. #947C23semidirect
122. #944C23semidirect
125. #940C23semidirect
129. #936C23semidirect
160. C3×#103C23semidirect
161. C3×#102C23semidirect
162. Dic3⋊C8C23semidirect
163. #810C23semidirect
164. #809C23semidirect
206. #419C2×D3semidirect
208. #418C2×D3semidirect
210. #417C2×D3semidirect
212. #416C2×D3semidirect
405. #103C22×D3semidirect
407. #102C22×D3semidirect
714. C3#60423semidirect

Maximal subgroups

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

subgroupindex
#872682
C22×#9492
C22×#9082
C2×#55212
C3×#28962
#837502
#822132
#789762
#789442
#789102
#788632
#788622
#604233