Finite Group Databasebeta

← #10848739

Subgroups — #10848739

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×#5477C211yesmaximal, characteristic, complemented
2C3×#2850C211yesmaximal, characteristic, complemented
3C2×#5058C211yesmaximal, characteristic, complemented
4C2×#5055C211yesmaximal, characteristic, complemented
5#71699C212yesmaximal, complemented
6#71684C212yesmaximal, complemented
7#71529C211yesmaximal, characteristic, complemented
8#69687C212yesmaximal, complemented
9#69657C212yesmaximal, complemented
10#69555C211yesmaximal, characteristic, complemented
11#69553C211yesmaximal, characteristic, complemented
12#25602—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×#124C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#121C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#215C2212yes—
19C22×#213C2212yes—
20C2×#930—21no—
21C2×#930C2212yes—
22#5477—24no—
23#5477—24no—
24C2×#875—21no—
25C2×#874C2211yescharacteristic
26C2×C4×D12—21no—
27C2×C4×D12C2212yes—
28C2×#869C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#388—21no—
32C3×#388—21no—
33C3×#386—21no—
34C3×#386—21no—
35C3×#382C2211yescharacteristic
36C2×#835C2212yes—
37C2×#832C2212yes—
38C2×#830C2212yes—
39#5058—24no—
40#5055—24no—
41C2×#809C2212yes—
42C2×#809C2212yes—
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#532—31no—
49C2×#496—31no—
50C2×#489—31no—
51#2850D311yescharacteristic, complemented
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1961—31no—
56#1879—32no—
57#1877—32no—
58#1749—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×#67—22no—
66C2×#235—28no—
67C2×#235—44no—
68C2×#235—44no—
69C22×D12—21no—
70C22×D12—21no—
71C22×D12C2312yes—
72C22×C4×D3—22no—
73C22×Dic6—21no—
74C3×#124—22no—
75C3×#124—22no—
76C3×C22⋊Q8—42no—
77C3×#121—42no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#215—24no—
94C2×#215—24no—
95C2×#215—42no—
96C2×#213—24no—
97C2×#213—24no—
98C2×#213—42no—
99#930—28no—
100#930—24no—
101#930—24no—
102C2×(C2×D3)⋊C4—22no—
103C2×(C2×D3)⋊C4—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2312yes—
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2312yes—
108#875—28no—
109#874—22no—
110#874D414yescomplemented
111C4×D12—28no—
112C4×D12—24no—
113C4×D12—24no—
114#869—22no—
115#869D414yescomplemented
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120#835—24no—
121#832—24no—
122#830—24no—
123#809—42no—
124#809—42no—
125#803—22no—
126#803—22no—
127C22×#41—61no—
128C22×SD16—32no—
129C22×D8—32no—
130#532—64no—
131#532—64no—
132C2×C4⋊Q8—31no—
133C2×#127—31no—
134C2×#124—61no—
135C2×#124C2×D311yescharacteristic
136C2×C22⋊Q8C2×D311yescharacteristic
137C2×#121—32no—
138C2×#121—61no—
139C2×#121C2×D311yescharacteristic
140C2×C4×D4—32no—
141C2×C4×D4—61no—
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#391C2×D311yescharacteristic
153#390C2×D311yescharacteristic
154#388—21no—
155#388—21no—
156#386—21no—
157#386—21no—
158#382C2×D311yescharacteristic
159#340—32no—
160#340—32no—
161#336—32no—
162C24×C3—21no—
163C24×C3C2×D411yescharacteristic
164C23×D3—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—41no—
188C22×C3×C4—41no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C2×#67—28no—
196C2×#67—44no—
197C2×#67—44no—
198C22×Dic3—22no—
199#235—432no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—24no—
205C2×D12—24no—
206C2×D12—22no—
207C2×D12—42no—
208C2×D12—42no—
209C2×D12—42no—
210C2×D12—42no—
211C2×D12C2×D414yes—
212C2×C4×D3—28no—
213C2×C4×D3—44no—
214C2×C4×D3—44no—
215C2×Dic6—24no—
216C2×Dic6—42no—
217C2×Dic6—42no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—42no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42C2×D412yes—
232C3×C42C2×D412yes—
233#215—48no—
234#213—48no—
235(C2×D3)⋊C4—28no—
236(C2×D3)⋊C4—28no—
237C4⋊Dic3—24no—
238C4⋊Dic3—22no—
239C4⋊Dic3C2×D414yes—
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—68no—
247C2×#41—124no—
248C2×#41—124no—
249C22×Q8—61no—
250C22×Q8C22×D311yescharacteristic
251C22×D4—21no—
252C22×D4—32no—
253C22×D4—62no—
254C22×D4—61no—
255C22×D4—61no—
256C22×D4C22×D311yescharacteristic
257C23×C4—62no—
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—34no—
270C4⋊Q8—62no—
271#127—34no—
272#127—62no—
273#124—22no—
274#124—22no—
275#124—68no—
276C22⋊Q8—42no—
277#121—42no—
278#121—64no—
279#121—64no—
280#121—68no—
281C4×D4—64no—
282C4×D4—64no—
283C4×D4—68no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
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—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3AGL(1,ℤ/8)12yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—24no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—84no—
341C2×C3×C4—28no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
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—21no—
354C2×C3×C4—44no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4#12012yes—
367C2×C3×C4#12012yes—
368C2×C3×C4AGL(1,ℤ/8)12yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4112yes—
372C2×C3×C4H(5,𝔽2)12yes—
373#67—416no—
374C2×Dic3—24no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377D12—24no—
378D12—24no—
379D12—48no—
380D12—48no—
381D12—42no—
382D12—42no—
383C4×D3—416no—
384Dic6—48no—
385C3⋊C8—24no—
386C3⋊C8—42no—
387C24—21no—
388C24—62no—
389C24C2×#6711yescharacteristic
390#41—1232no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—64no—
396C2×Q8—122no—
397C2×Q8—122no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—34no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—68no—
407C2×D4—64no—
408C2×D4—64no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—122no—
415C2×D4—122no—
416C2×D4—122no—
417C2×D4—122no—
418C2×D4—124no—
419C2×D4—124no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—68no—
434C22×C4—62no—
435C22×C4—124no—
436C22×C4—124no—
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—22no—
452C4⋊C4—22no—
453C4⋊C4—34no—
454C4⋊C4—42no—
455C4⋊C4—64no—
456C4⋊C4—62no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—68no—
464C22⋊C4—68no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42D3×D412yes—
470C42D3×D412yes—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
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—44no—
493C2×D3—44no—
494C2×D3—44no—
495C2×D3—82no—
496C2×D3—82no—
497C2×D3—82no—
498C2×D3—82no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4#54612yes—
508C3×C4#55012yes—
509Dic3—44no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—64no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23#94912yes—
531C23C22×#6711yescharacteristic
532Q8—42no—
533Q8—42no—
534Q8—128no—
535D4—42no—
536D4—42no—
537D4—64no—
538D4—64no—
539D4—84no—
540D4—128no—
541D4—128no—
542D4—1216no—
543D4—122no—
544D4—122no—
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—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—64no—
571C2×C4—1216no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4#93712yes—
575C2×C4#93712yes—
576C2×C4#94912yes—
577C2×C4C2×D3×D411yescharacteristic
578C2×C4C2×D3×D411yescharacteristic
579C2×C4#100412yes—
580C2×C4D3×#4112yes—
581C8—64no—
582C8—122no—
583C2×C3—42no—
584C2×C3—161no—
585C2×C3#243111yescharacteristic
586C2×C3#300311yescharacteristic
587C2×C3#300511yescharacteristic
588C2×C3#317111yescharacteristic
589C2×C3#318011yescharacteristic
590C2×C3#321311yescharacteristic
591C2×C3#321411yescharacteristic
592D3—84no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—124no—
608C22—124no—
609C22—124no—
610C22—242no—
611C22—242no—
612C22—242no—
613C22—242no—
614C22C2×#93711yescharacteristic
615C22#580511yescharacteristic
616C22C2×#94911yescharacteristic
617C22C2×#94911yescharacteristic
618C22#582211yescharacteristic
619C22#582211yescharacteristic
620C22#583211yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—124no—
630C4#559912yes—
631C4#561212yes—
632C3#2560211yescharacteristic, complemented, Sylow
633C2—42no—
634C2—161no—
635C2—244no—
636C2#8265211yescharacteristic
637C2#8292711yescharacteristic
638C2#8292811yescharacteristic
639C2#8294711yescharacteristic
640C2#8313511yescharacteristic
641C2#8313711yescharacteristic
642C2#8330711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2850C2
socleC23×C3C22×D4
hypercenter#2850D3
hypocenterC3#25602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C3×#2850C2semidirect
3. C2×#5058C2semidirect
4. C2×#5055C2semidirect
5. #71699C2semidirect
6. #71684C2semidirect
7. #71529C2semidirect
8. #69687C2semidirect
9. #69657C2semidirect
10. #69555C2semidirect
11. #69553C2semidirect
51. #2850D3semidirect
110. #874D4semidirect
115. #869D4semidirect
245. C2×C3⋊C8#41semidirect
632. C3#25602semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C3×#28502
C2×#50582
C2×#50552
#716992
#716842
#715292
#696872
#696572
#695552
#695532
#256023