Finite Group Databasebeta

← #10848732

Subgroups — #10848732

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic, complemented
3C2×#5074C211yesmaximal, characteristic, complemented
4#82077C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#71765C211yesmaximal, characteristic, complemented
7#71685C212yesmaximal, complemented
8#71684C212yesmaximal, complemented
9#71613C212yesmaximal, complemented
10#71612C212yesmaximal, complemented
11#71529C211yesmaximal, characteristic, complemented
12#25602—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#532—24no—
15C3×#532—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C22×C3×C42C2211yescharacteristic
24C22×#214C2212yes—
25C22×#213C2212yes—
26C2×#930C2211yescharacteristic
27C2×#924C2211yescharacteristic
28C2×#875C2211yescharacteristic
29C2×#832C2212yes—
30C2×#832C2212yes—
31#5213—21no—
32#5213—21no—
33C2×#830C2212yes—
34#5089—24no—
35#5074—24no—
36C2×#810C2212yes—
37C2×#809C2212yes—
38#4940—21no—
39#4940—21no—
40#4939C2211yescharacteristic
41#4936C2211yescharacteristic
42#4929C2211yescharacteristic
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#532D311yescharacteristic, complemented
49C2×#496—31no—
50C2×#489—31no—
51#2850—31no—
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1961—31no—
56#1879—32no—
57#1877—32no—
58#1749—31no—
59C2×C3×#41—28no—
60C2×C3×#41—44no—
61C2×C3×#41—44no—
62C22×C3×Q8—21no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4C2312yes—
67C23×C3×C4—22no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C22×#67—21no—
72C22×D12C2311yescharacteristic
73C22×Dic6C2311yescharacteristic
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8D414yescomplemented
76C3×#127—22no—
77C3×#127D414yescomplemented
78C3×#124—28no—
79C3×#121—28no—
80C3×#121—24no—
81C3×#121—24no—
82C3×C4×D4—28no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42—22no—
90C2×C3×C42—22no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#214—24no—
97C2×#214—24no—
98C2×#214—42no—
99C2×#213—24no—
100C2×#213—24no—
101C2×#213—42no—
102#930—42no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—42no—
111C2×Dic3⋊C4—21no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114#875—22no—
115#875—22no—
116#832—42no—
117#832—42no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122#830—24no—
123#810—24no—
124#809—24no—
125#803—22no—
126#803—22no—
127C22×#41—21no—
128C22×SD16—32no—
129C22×D8—32no—
130#532—24no—
131#532—24no—
132C2×C4⋊Q8C2×D311yescharacteristic
133C2×#127C2×D311yescharacteristic
134C2×#124—21no—
135C2×#124—31no—
136C2×C22⋊Q8—31no—
137C2×#121—21no—
138C2×#121—31no—
139C2×#121C2×D312yes—
140C2×C4×D4—21no—
141C2×C4×D4C2×D312yes—
142C22×C42C2×D311yescharacteristic
143#496—64no—
144#489—64no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×C22⋊C8—32no—
150C2×#97—31no—
151C2×#97—31no—
152#391—31no—
153#390—31no—
154#388—61no—
155#388—61no—
156#386—61no—
157#386—61no—
158#382—31no—
159#340—32no—
160#340—32no—
161#336—32no—
162C24×C3—22no—
163C24×C3C2×D411yescharacteristic
164C23×D3—21no—
165C3×#41—432no—
166C2×C3×Q8—24no—
167C2×C3×Q8—42no—
168C2×C3×Q8—42no—
169C2×C3×D4—28no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4—42no—
180C2×C3×D4—42no—
181C2×C3×D4—44no—
182C2×C3×D4—44no—
183C2×C3×D4C2×D414yes—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—28no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—41no—
195C22×C3×C4—41no—
196C22×C3×C4—44no—
197C22×C3×C4—44no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C2×#67—42no—
205C2×#67—42no—
206C2×#67—42no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—22no—
213C2×D12—22no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4C2×D414yes—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—28no—
223C3×C42—22no—
224C3×C42—22no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229#217—42no—
230#214—48no—
231#213—48no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239Dic3⋊C4—42no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8#4114yescomplemented
246C2×#41—28no—
247C2×#41—44no—
248C2×#41—44no—
249C22×Q8—21no—
250C22×Q8—31no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—31no—
255C22×D4—61no—
256C22×D4C22×D312yes—
257C23×C4—22no—
258C23×C4#6711yescharacteristic
259C23×C4#6711yescharacteristic
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C2×D8—64no—
265C2×D8—64no—
266C2×D8—122no—
267C22×C8—32no—
268C22×C8—32no—
269C4⋊Q8—22no—
270C4⋊Q8#6714yescomplemented
271#127—22no—
272#127#6714yescomplemented
273#124—28no—
274#124—62no—
275#124—62no—
276C22⋊Q8—122no—
277#121—28no—
278#121—24no—
279#121—24no—
280#121—122no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D312yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42#6711yescharacteristic
300C2×C42#6711yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#103—64no—
304#102—64no—
305#102—122no—
306#102—122no—
307C22⋊C8—64no—
308#97—62no—
309#97—62no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—24no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3AGL(1,ℤ/8)12yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C3×Q8—48no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—48no—
340C3×D4—48no—
341C3×D4—416no—
342C3×D4—42no—
343C3×D4—42no—
344C2×C3×C4—28no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—24no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—44no—
359C2×C3×C4—416no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4#12012yes—
365C2×C3×C4#12012yes—
366C2×C3×C4AGL(1,ℤ/8)12yes—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4112yes—
370C2×C3×C4H(5,𝔽2)12yes—
371#67—84no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381D12—42no—
382D12—42no—
383Dic6—42no—
384Dic6—42no—
385C3⋊C8—24no—
386C3⋊C8—42no—
387C24—22no—
388C24—61no—
389C24C2×#6711yescharacteristic
390#41—432no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—62no—
395C2×Q8—62no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—122no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4C2×#6714yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C22×C4C23×D311yescharacteristic
443SD16—128no—
444D8—128no—
445C2×C8—34no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8—62no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—122no—
456C4⋊C4C2×#6714yes—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42C2×#6712yes—
470C42C2×#6712yes—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3C2×#12011yescharacteristic
486C22×C3#53211yescharacteristic
487C22×C3#53511yescharacteristic
488C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
490C22×C3#57511yescharacteristic
491C22×C3#57511yescharacteristic
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—44no—
504C3×C4—41no—
505C3×C4#54612yes—
506C3×C4#55012yes—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—24no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23#93212yes—
531C23C22×#6711yescharacteristic
532Q8—48no—
533Q8—122no—
534Q8—122no—
535D4—24no—
536D4—24no—
537D4—48no—
538D4—48no—
539D4—416no—
540D4—42no—
541D4—42no—
542D4—122no—
543D4—122no—
544D4—244no—
545C2×C4—28no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—416no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4#93212yes—
575C2×C4#95312yes—
576C2×C4#95312yes—
577C2×C4D3×#4112yes—
578C2×C4#100912yes—
579C2×C4C22×#6711yescharacteristic
580C2×C4C22×#6711yescharacteristic
581C8—64no—
582C8—122no—
583C2×C3—42no—
584C2×C3—84no—
585C2×C3#243111yescharacteristic
586C2×C3#300311yescharacteristic
587C2×C3#300511yescharacteristic
588C2×C3#317111yescharacteristic
589C2×C3#318011yescharacteristic
590C2×C3#321311yescharacteristic
591C2×C3#321411yescharacteristic
592D3—161no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22C2×#93211yescharacteristic
615C22C2×#93211yescharacteristic
616C22#582211yescharacteristic
617C22#582211yescharacteristic
618C22#583011yescharacteristic
619C22#583211yescharacteristic
620C22C2×#95311yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—44no—
625C4—41no—
626C4—41no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#582912yes—
631C4#583412yes—
632C3#2560211yescharacteristic, complemented, Sylow
633C2—42no—
634C2—84no—
635C2—481no—
636C2#8329211yescharacteristic
637C2#8329311yescharacteristic
638C2#8330611yescharacteristic
639C2#8330811yescharacteristic
640C2#8403811yescharacteristic
641C2#8408111yescharacteristic
642C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#25602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×#5089C2semidirect
3. C2×#5074C2semidirect
4. #82077C2semidirect
5. #71788C2semidirect
6. #71765C2semidirect
7. #71685C2semidirect
8. #71684C2semidirect
9. #71613C2semidirect
10. #71612C2semidirect
11. #71529C2semidirect
48. C2×#532D3semidirect
75. C3×C4⋊Q8D4semidirect
77. C3×#127D4semidirect
245. C2×C3⋊C8#41semidirect
270. C4⋊Q8#67semidirect
272. #127#67semidirect
632. C3#25602semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50892
C2×#50742
#820772
#717882
#717652
#716852
#716842
#716132
#716122
#715292
#256023