Finite Group Databasebeta

← #10827739

Subgroups — #10827739

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3455C211yesmaximal, characteristic, complemented
2#83322C211yesmaximal, characteristic, complemented
3#82936C211yesmaximal, characteristic, complemented
4#82536C211yesmaximal, characteristic, complemented
5#71842C212yesmaximal, complemented
6#71817C211yesmaximal, characteristic, complemented
7#71715C212yesmaximal, complemented
8#71714C211yesmaximal, characteristic, complemented
9#71645C212yesmaximal, complemented
10#71644C211yesmaximal, characteristic, complemented
11#71591C212yesmaximal, complemented
12#23995—31nomaximal, Sylow
13#5964C2211yescharacteristic
14C3×#542—21no—
15C3×#542—21no—
16C3×#541—21no—
17C3×#540—21no—
18C3×#539—22no—
19C3×#538—22no—
20C3×#533C2211yescharacteristic, complemented
21C3×#532C2211yescharacteristic
22C3×#529—22no—
23C3×#529—22no—
24C3×#525C2211yescharacteristic
25C3×#525C2211yescharacteristic, complemented
26C3×C4×#41C2211yescharacteristic
27C3×C4×#41C2211yescharacteristic, complemented
28C2×C3×#117C2211yescharacteristic, complemented
29#5822C2212yes—
30#5819C2211yescharacteristic
31#5791C2212yes—
32#5240C2211yescharacteristic
33#5237C2211yescharacteristic
34#5236C2212yescomplemented
35#5207C2212yescomplemented
36#5095—21no—
37#5077—21no—
38#5062—21no—
39#5061—21no—
40#5053—22no—
41#5032—22no—
42#5006C2212yes—
43#5005—22no—
44#5005C2212yes—
45C2×#805—22no—
46C2×#805C2212yes—
47C2×#805C2212yes—
48#4533C2212yescomplemented
49#4531C2212yescomplemented
50#4523C2212yescomplemented
51#4523C2212yescomplemented
52#3455D311yescharacteristic, complemented
53#3025—31no—
54#3018—31no—
55#2901—31no—
56#2031—32no—
57#2010—32no—
58#1906—32no—
59#1897—32no—
60#1891—31no—
61#1884—31no—
62#1827—31no—
63C2×C3×#41C2311yescharacteristic
64C2×C3×#41C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66#1010—22no—
67#1009—22no—
68D3×#41—42no—
69#1007—21no—
70#1004—21no—
71#1004—22no—
72C2×#235—22no—
73C2×#235C2312yes—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8D411yescharacteristic, complemented
76C3×#127D411yescharacteristic, complemented
77C3×#126—22no—
78C3×#126—22no—
79C3×#126—22no—
80C3×#126—22no—
81C3×#125—22no—
82C3×#125—22no—
83C3×#125—21no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124D411yescharacteristic, complemented
87C3×#124D411yescharacteristic, complemented
88C3×#123—22no—
89C3×#123—22no—
90C3×#123—22no—
91C3×#123—22no—
92C3×#123—21no—
93C3×#123D412yes—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—22no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—21no—
98C3×C22⋊Q8—22no—
99C3×#121—22no—
100C3×#121—22no—
101C3×#121—21no—
102C3×#121—21no—
103C3×#121—22no—
104C3×#120—21no—
105C3×#120D412yes—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—22no—
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—22no—
113C3×C4×D4—22no—
114C3×C4×D4—22no—
115C3×C4×D4—22no—
116C3×#117—21no—
117C3×#117—21no—
118C3×#117—21no—
119C3×#117—21no—
120C3×#117—21no—
121C3×#117D412yescomplemented
122C3×#117C2311yescharacteristic
123C3×#117C2311yescharacteristic
124C2×C3×C4⋊C4—21no—
125C2×C3×C22⋊C4—21no—
126C2×C3×C42—21no—
127C2×C3×C42D412yescomplemented
128C2×C3×C42C2311yescharacteristic
129C2×C3×C42C2311yescharacteristic
130#953—22no—
131#951—22no—
132#950—42no—
133#949—22no—
134#948—42no—
135#942—22no—
136#932—22no—
137#930—22no—
138#929—22no—
139C4×#67—42no—
140#926C2312yes—
141#924—22no—
142C2×#208—22no—
143C2×#208C2312yes—
144C2×#208C2312yes—
145#837—42no—
146#836—21no—
147#834—22no—
148#833—21no—
149#830—42no—
150#806—24no—
151#805—22no—
152#805—24no—
153#805—22no—
154#805—22no—
155#805—22no—
156#585—31no—
157#575—32no—
158#568—31no—
159#542—21no—
160#542—21no—
161#541—21no—
162#540—21no—
163#539—22no—
164#538—22no—
165#533C2×D311yescharacteristic, complemented
166#532C2×D311yescharacteristic
167#529—22no—
168#529—22no—
169#525—32no—
170#525C2×D311yescharacteristic
171#525C2×D311yescharacteristic, complemented
172C4×#41C2×D311yescharacteristic
173C4×#41C2×D311yescharacteristic, complemented
174C2×#117C2×D311yescharacteristic, complemented
175#458—62no—
176B(4,𝔽2)—62no—
177#456—61no—
178#455—61no—
179#454—61no—
180#453—61no—
181#421—32no—
182#421—32no—
183#421—62no—
184C2×#104—32no—
185C2×#104—32no—
186C2×#104—62no—
187#413—31no—
188#410—31no—
189#410—32no—
190#407—32no—
191#344—32no—
192#339—32no—
193#339—32no—
194H(3,ℤ/4)—32no—
195C24×C3C2×D411yescharacteristic
196C3×#41—22no—
197C3×#41—22no—
198C3×#41—22no—
199C3×#41—22no—
200C2×C3×Q8—22no—
201C2×C3×Q8C2×D411yescharacteristic
202C2×C3×Q8C2×D411yescharacteristic
203C2×C3×D4—21no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—22no—
207C2×C3×D4C2×D412yes—
208C2×C3×D4C2×D411yescharacteristic
209C2×C3×D4C2×D411yescharacteristic
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2411yescharacteristic
222C2×#67—22no—
223C2×#67—42no—
224C2×#67—42no—
225#239—42no—
226D3×Q8—42no—
227#237—42no—
228#237—44no—
229D3×D4—42no—
230D3×D4—44no—
231#235—24no—
232#235—22no—
233#235—22no—
234#235—22no—
235#235—22no—
236#235—42no—
237#235—42no—
238C2×D12—22no—
239C2×D12#4112yes—
240C2×C4×D3—22no—
241C2×C4×D3—42no—
242C2×Dic6—22no—
243C2×Dic6#4112yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C22⋊C4—21no—
259C3×C22⋊C4—21no—
260C3×C22⋊C4—21no—
261C3×C22⋊C4—21no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—22no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4C2×D412yes—
272C3×C42—22no—
273C3×C42—22no—
274C3×C42—21no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42C2×D411yescharacteristic
278C3×C42C2×D411yescharacteristic
279C3×C42C2×D412yes—
280C3×C42C2×D411yescharacteristic
281C3×C42C2×D411yescharacteristic
282#217—22no—
283#217—42no—
284#216—42no—
285#215—42no—
286#214—42no—
287#213—42no—
288(C2×D3)⋊C4—42no—
289C4⋊Dic3—22no—
290Dic3⋊C4—42no—
291C4×Dic3—22no—
292#208—22no—
293#208—22no—
294#208—22no—
295#208—22no—
296#208—42no—
297C2×C3⋊C8—22no—
298C2×C3⋊C8—22no—
299C2×C3⋊C8—42no—
3002−1+4—62no—
3012−1+4—61no—
302H(5,𝔽2)—62no—
303H(5,𝔽2)—62no—
304H(5,𝔽2)—61no—
305C2×#41—32no—
306C2×#41—62no—
307C2×#41—122no—
308C2×#41C22×D311yescharacteristic
309C2×#41C22×D311yescharacteristic
310C23×C4C22×D311yescharacteristic
311#137—62no—
312#137—62no—
313AGL(1,ℤ/8)—62no—
314AGL(1,ℤ/8)—62no—
315#135—122no—
316#131—122no—
317C2×M16—32no—
318C2×M16—32no—
319C2×M16—62no—
320C4⋊Q8—21no—
321C4⋊Q8#6711yescharacteristic, complemented
322#127#6711yescharacteristic, complemented
323#126—22no—
324#126—22no—
325#126—22no—
326#126—22no—
327#125—22no—
328#125—22no—
329#125—21no—
330#124—22no—
331#124—22no—
332#124#6711yescharacteristic, complemented
333#124#6711yescharacteristic, complemented
334#123—22no—
335#123—22no—
336#123—22no—
337#123—22no—
338#123—21no—
339#123—62no—
340#123#6712yes—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—22no—
346C22⋊Q8—62no—
347#121—22no—
348#121—22no—
349#121—21no—
350#121—21no—
351#121—22no—
352#121—62no—
353#120—21no—
354#120—62no—
355#120#6712yes—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—22no—
359C4×D4—22no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—122no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—21no—
372#117—32no—
373#117#6712yescomplemented
374#117C22×D311yescharacteristic
375#117C22×D311yescharacteristic
376C2×C4⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C42—21no—
379C2×C42#6712yescomplemented
380C2×C42C22×D311yescharacteristic
381C2×C42C22×D311yescharacteristic
382#104—62no—
383#104—62no—
384#104—62no—
385#104—62no—
386#104—64no—
387#104—122no—
388#101—61no—
389#100—61no—
390#99—62no—
391#99—64no—
392C22⋊C8—122no—
393C23×C3—21no—
394C23×C3—21no—
395C23×C3—22no—
396C23×C3—81no—
397C23×C3#12012yes—
398C23×C3C22×D411yescharacteristic
399C22×D3—22no—
400C22×D3—42no—
401C3×Q8—22no—
402C3×Q8—22no—
403C3×Q8—82no—
404C3×D4—24no—
405C3×D4—22no—
406C3×D4—22no—
407C3×D4—42no—
408C3×D4—82no—
409C3×D4—82no—
410C3×D4—82no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—22no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—22no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—42no—
427C2×C3×C4—81no—
428C2×C3×C4—82no—
429C2×C3×C4—82no—
430C2×C3×C4—81no—
431C2×C3×C4#12012yes—
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C2×#4112yes—
435C2×C3×C4H(5,𝔽2)12yes—
436#67—44no—
437#67—42no—
438#67—42no—
439#67—84no—
440C2×Dic3—22no—
441C2×Dic3—42no—
442C2×Dic3—42no—
443D12—22no—
444D12—22no—
445D12—42no—
446C4×D3—44no—
447C4×D3—42no—
448C4×D3—42no—
449Dic6—22no—
450Dic6—22no—
451Dic6—42no—
452C3⋊C8—42no—
453C3⋊C8—42no—
454C24C2×#6711yescharacteristic
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—64no—
460#41—62no—
461#41—62no—
462#41—62no—
463#41—62no—
464#41—122no—
465#41—124no—
466#41—122no—
467#41—122no—
468#41—122no—
469C2×Q8—22no—
470C2×Q8—32no—
471C2×Q8—62no—
472C2×Q8—122no—
473C2×Q8C2×#6711yescharacteristic
474C2×Q8C2×#6711yescharacteristic
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—32no—
480C2×D4—62no—
481C2×D4—62no—
482C2×D4—122no—
483C2×D4—124no—
484C2×D4—122no—
485C2×D4—122no—
486C2×D4C2×#6712yes—
487C2×D4C2×#6711yescharacteristic
488C2×D4C2×#6711yescharacteristic
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—62no—
499C22×C4—122no—
500C22×C4C2×#6711yescharacteristic
501C22×C4C2×#6712yes—
502C22×C4C23×D311yescharacteristic
503Q16—122no—
504SD16—122no—
505SD16—122no—
506D8—122no—
507M16—62no—
508M16—62no—
509M16—62no—
510M16—62no—
511M16—122no—
512C2×C8—62no—
513C2×C8—62no—
514C2×C8—122no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—62no—
530C4⋊C4—122no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—62no—
545C22⋊C4—122no—
546C22⋊C4—122no—
547C22⋊C4C2×#6712yes—
548C42—22no—
549C42—22no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—62no—
554C42C2×#6711yescharacteristic
555C42C2×#6711yescharacteristic
556C42C2×#6712yes—
557C42C2×#6711yescharacteristic
558C42C2×#6711yescharacteristic
559C22×C3—22no—
560C22×C3—41no—
561C22×C3—42no—
562C22×C3—42no—
563C22×C3—42no—
564C22×C3—41no—
565C22×C3—41no—
566C22×C3—42no—
567C22×C3—41no—
568C22×C3—41no—
569C22×C3C2×#12011yescharacteristic
570C22×C3#54612yes—
571C2×D3—42no—
572C2×D3—82no—
573C2×D3—162no—
574C3×C4—22no—
575C3×C4—41no—
576C3×C4—41no—
577C3×C4—41no—
578C3×C4—42no—
579C3×C4—42no—
580C3×C4—41no—
581C3×C4—41no—
582C3×C4—41no—
583C3×C4#52511yescharacteristic
584C3×C4#53511yescharacteristic
585C3×C4#54612yes—
586Dic3—42no—
587Dic3—82no—
588C23—22no—
589C23—21no—
590C23—21no—
591C23—62no—
592C23—81no—
593C23—122no—
594C23#95312yes—
595C23C22×#6711yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—62no—
599Q8—62no—
600Q8—82no—
601Q8—122no—
602D4—24no—
603D4—22no—
604D4—22no—
605D4—42no—
606D4—62no—
607D4—62no—
608D4—82no—
609D4—82no—
610D4—82no—
611D4—122no—
612D4—124no—
613D4—122no—
614D4—122no—
615D4—244no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—62no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—124no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4#95312yes—
643C2×C4D3×#4112yes—
644C2×C4#100912yes—
645C2×C4C22×#6711yescharacteristic
646C2×C4C22×#6711yescharacteristic
647C8—122no—
648C8—122no—
649C2×C3—22no—
650C2×C3—21no—
651C2×C3—42no—
652C2×C3—81no—
653C2×C3—81no—
654C2×C3#240411yescharacteristic
655D3—82no—
656C22—22no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—41no—
665C22—41no—
666C22—122no—
667C22—242no—
668C22—482no—
669C22#582912yes—
670C22C2×#95311yescharacteristic
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—42no—
676C4—42no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—122no—
681C4—242no—
682C4#582811yescharacteristic
683C4#582912yes—
684C4#583011yescharacteristic
685C3#2399511yescharacteristic, complemented, Sylow
686C2—21no—
687C2—22no—
688C2—42no—
689C2—81no—
690C2—81no—
691C2—242no—
692C2#8409611yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3455C2
socleC2×C3#2404
hypercenter#3455D3
hypocenterC3#23995
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. C3×#3455C2semidirect
2. #83322C2semidirect
3. #82936C2semidirect
4. #82536C2semidirect
5. #71842C2semidirect
6. #71817C2semidirect
7. #71715C2semidirect
8. #71714C2semidirect
9. #71645C2semidirect
10. #71644C2semidirect
11. #71591C2semidirect
20. C3×#533C22semidirect
25. C3×#525C22semidirect
27. C3×C4×#41C22semidirect
28. C2×C3×#117C22semidirect
34. #5236C22semidirect
35. #5207C22semidirect
48. #4533C22semidirect
49. #4531C22semidirect
50. #4523C22semidirect
51. #4523C22semidirect
52. #3455D3semidirect
75. C3×C4⋊Q8D4semidirect
76. C3×#127D4semidirect
86. C3×#124D4semidirect
87. C3×#124D4semidirect
121. C3×#117D4semidirect
127. C2×C3×C42D4semidirect
165. #533C2×D3semidirect
171. #525C2×D3semidirect
173. C4×#41C2×D3semidirect
174. C2×#117C2×D3semidirect
321. C4⋊Q8#67semidirect
322. #127#67semidirect
332. #124#67semidirect
333. #124#67semidirect
373. #117#67semidirect
379. C2×C42#67semidirect
685. C3#23995semidirect

Maximal subgroups

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

subgroupindex
C3×#34552
#833222
#829362
#825362
#718422
#718172
#717152
#717142
#716452
#716442
#715912
#239953