Finite Group Databasebeta

← #10830943

Subgroups — #10830943

Order 768.

1776 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#213C211yesmaximal, characteristic, complemented
2#84053C211yesmaximal, characteristic, complemented
3C2×#4782C212yesmaximal, complemented
4C2×#4730C211yesmaximal, characteristic, complemented
5#74016C211yesmaximal, characteristic, complemented
6C3×#1786C211yesmaximal, characteristic, complemented
7#70212C212yesmaximal, complemented
8#70204C212yesmaximal, complemented
9#70191C212yesmaximal, complemented
10#69959C211yesmaximal, characteristic, complemented
11#69511C211yesmaximal, characteristic, complemented
12#24272—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C2×#953—21no—
16C2×#938C2212yescomplemented
17C22×#213—28no—
18C22×#213—24no—
19C22×#213C2212yes—
20C22×#213C2212yes—
21C2×C4×#67C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C22×ΓL(1,𝔽25)C2211yescharacteristic, complemented
24C2×#895C2212yescomplemented
25C2×#882—21no—
26C2×C3×#102C2212yescomplemented
27C2×C3×#102C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5247—21no—
31#5223—21no—
32#5222—21no—
33#5221C2212yescomplemented
34#5212C2211yescharacteristic
35C2×#821C2212yescomplemented
36C2×#821C2212yescomplemented
37C2×(C2×D3)⋊C8C2211yescharacteristic, complemented
38C2×#815C2211yescharacteristic, complemented
39C2×Dic3⋊C8C2211yescharacteristic, complemented
40#4988C2212yescomplemented
41C2×#808C2212yescomplemented
42#4955—21no—
43#4782—28no—
44#4730—24no—
45#4672—21no—
46#4671—21no—
47#4529C2212yescomplemented
48C23×D8—31no—
49C2×#469—32no—
50C2×#435—31no—
51#2408—31no—
52#2408—31no—
53#2054—32no—
54#1999—32no—
55#1972—32no—
56#1807—31no—
57#1786—31no—
58#1786D311yescharacteristic, complemented
59C25×C3—21no—
60C24×D3—21no—
61C22×C3×D4—24no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4—42no—
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C22×#67—22no—
69C22×#67D411yescharacteristic
70C22×D12—22no—
71C22×D12—24no—
72C22×D12—22no—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12—42no—
76C22×D12C2312yes—
77C22×C4×D3C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82#953—44no—
83C2×#217—21no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217D411yescharacteristic
88#938—24no—
89#938—42no—
90C2×#213—232no—
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—24no—
94C2×#213—24no—
95C2×#213—24no—
96C2×#213D418yescomplemented
97C4×#67—22no—
98C4×#67—22no—
99C2×(C2×D3)⋊C4—22no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×C4⋊Dic3C2312yes—
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8—24no—
107C22×C3⋊C8D414yescomplemented
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×ΓL(1,𝔽25)—22no—
111C2×ΓL(1,𝔽25)—24no—
112#895—24no—
113#895—42no—
114#882—44no—
115C3×#102—24no—
116C3×#102—42no—
117C3×C22⋊C8—22no—
118C3×#95—21no—
119C3⋊#95—21no—
120#821—24no—
121#821—42no—
122(C2×D3)⋊C8—22no—
123#815—22no—
124Dic3⋊C8—22no—
125#808—24no—
126C23×D4—31no—
127C23×D4C2×D311yescharacteristic
128C22×D8—32no—
129C22×D8—32no—
130C22×D8—64no—
131C22×D8—68no—
132C22×M16—31no—
133C23×C8—31no—
134C2×#121—32no—
135C2×#121—32no—
136C2×#120—61no—
137C2×#120—61no—
138C2×C4×D4—31no—
139#469—68no—
140#435—64no—
141C2×#106—32no—
142C2×C4⋊C8—31no—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102C2×D312yescomplemented
146C2×#102C2×D312yescomplemented
147C2×C22⋊C8—31no—
148C2×C22⋊C8C2×D311yescharacteristic
149C2×#97—31no—
150#394—61no—
151#394—61no—
152#392—61no—
153#392—61no—
154#390—32no—
155#390—32no—
156#380—31no—
157#380C2×D311yescharacteristic
158#379—61no—
159#379—61no—
160#340—32no—
161C24×C3—22no—
162C24×C3—44no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3—42no—
166C24×C3—82no—
167C24×C3C2×D411yescharacteristic
168C23×D3—22no—
169C23×D3—44no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—42no—
173C23×D3—82no—
174C23×D3C2×D411yescharacteristic
175C2×C3×D4—28no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—48no—
182C2×C3×D4—44no—
183C2×C3×D4—44no—
184C2×C3×D4—42no—
185C2×C3×D4C2×D414yes—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—22no—
194C2×#67—22no—
195C2×#67—22no—
196C2×#67—44no—
197C2×#67—44no—
198C2×#67—82no—
199C22×Dic3—22no—
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C2×D12—28no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—48no—
210C2×D12—44no—
211C2×D12—44no—
212C2×D12—42no—
213C2×D12—44no—
214C2×D12—44no—
215C2×D12—82no—
216C2×D12C2×D414yes—
217C2×C4×D3—21no—
218C2×C4×D3—22no—
219C2×C4×D3—21no—
220C2×C4×D3—22no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—42no—
228#217—22no—
229#217—44no—
230#217—44no—
231#217—42no—
232#217—82no—
233#213—216no—
234#213—216no—
235(C2×D3)⋊C4—22no—
236(C2×D3)⋊C4—44no—
237(C2×D3)⋊C4—44no—
238(C2×D3)⋊C4—82no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241Dic3⋊C4—22no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—28no—
246C2×C3⋊C8C2×D412yes—
247C2×C3⋊C8C2×D414yes—
248C2×C3⋊C8C2×D412yes—
249ΓL(1,𝔽25)—44no—
250C25—21no—
251C25—61no—
252C22×D4—24no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—21no—
256C22×D4—31no—
257C22×D4—32no—
258C22×D4—42no—
259C22×D4—64no—
260C22×D4—62no—
261C22×D4—62no—
262C22×D4—62no—
263C22×D4—62no—
264C22×D4—61no—
265C22×D4—122no—
266C22×D4C22×D312yes—
267C23×C4—31no—
268C23×C4C22×D311yescharacteristic
269C2×D8—38no—
270C2×D8—64no—
271C2×D8—64no—
272C2×D8—64no—
273C2×D8—64no—
274C2×D8—632no—
275C2×D8—64no—
276C2×M16—62no—
277C2×M16—64no—
278C22×C8—31no—
279C22×C8—34no—
280C22×C8—31no—
281C22×C8—64no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284#121—64no—
285#121—64no—
286#121—122no—
287#121—122no—
288#120—124no—
289#120—124no—
290C4×D4—62no—
291C4×D4—62no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—32no—
294C2×C4⋊C4C22×D312yes—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—31no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—62no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—62no—
304C2×C22⋊C4—62no—
305C2×C42—31no—
306#106—64no—
307C4⋊C8—62no—
308#102—24no—
309#102—42no—
310#102—64no—
311#102—122no—
312C22⋊C8—22no—
313C22⋊C8—62no—
314#97—62no—
315#95—21no—
316#95—61no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—24no—
320C23×C3—48no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—88no—
328C23×C3—88no—
329C23×C3—84no—
330C23×C3—84no—
331C23×C3#12112yes—
332C23×C3AGL(1,ℤ/8)12yes—
333C23×C3AGL(1,ℤ/8)12yes—
334C23×C3C22×D411yescharacteristic
335C22×D3—21no—
336C22×D3—24no—
337C22×D3—22no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—22no—
341C22×D3—48no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—88no—
349C22×D3—88no—
350C22×D3—84no—
351C22×D3—84no—
352C3×D4—24no—
353C3×D4—24no—
354C3×D4—48no—
355C3×D4—42no—
356C3×D4—42no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4#12112yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D412yes—
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)12yes—
369#67—44no—
370#67—88no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381D12—24no—
382D12—24no—
383D12—48no—
384D12—42no—
385D12—42no—
386D12—88no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—42no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—22no—
393C3⋊C8#12014yescomplemented
394C24—22no—
395C24—31no—
396C24—44no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—62no—
401C24—82no—
402C24—124no—
403C24—122no—
404C24—122no—
405C24—122no—
406C24—242no—
407C24D3×D411yescharacteristic
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—34no—
415C2×D4—48no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—68no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—128no—
429C2×D4—124no—
430C2×D4—124no—
431C2×D4—124no—
432C2×D4—124no—
433C2×D4—124no—
434C2×D4—124no—
435C2×D4—122no—
436C2×D4—242no—
437C2×D4—242no—
438C2×D4D3×D414yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4#23512yes—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D412yes—
453C22×C4C23×D311yescharacteristic
454D8—616no—
455D8—616no—
456M16—124no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—32no—
462C2×C8—34no—
463C2×C8—32no—
464C2×C8—68no—
465C2×C8—62no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C22⋊C4—42no—
472C22⋊C4—62no—
473C22⋊C4—62no—
474C22⋊C4—124no—
475C22⋊C4—124no—
476C22⋊C4—124no—
477C22⋊C4—124no—
478C22⋊C4—122no—
479C22⋊C4—242no—
480C22⋊C4—242no—
481C42—61no—
482C42—61no—
483C22×C3—22no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—22no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—816no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3#46914yescomplemented
496C22×C3C2×#12111yescharacteristic
497C22×C3D4212yes—
498C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57612yes—
501C2×D3—41no—
502C2×D3—44no—
503C2×D3—42no—
504C2×D3—44no—
505C2×D3—42no—
506C2×D3—44no—
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—816no—
510C2×D3—82no—
511C2×D3—82no—
512C2×D3—82no—
513C2×D3—82no—
514C3×C4—22no—
515C3×C4—82no—
516C3×C4C2×#12011yescharacteristic
517C3×C4#53511yescharacteristic
518C3×C4#54612yes—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—82no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—48no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23—62no—
537C23—64no—
538C23—62no—
539C23—88no—
540C23—88no—
541C23—84no—
542C23—84no—
543C23—128no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23—248no—
551C23—248no—
552C23—244no—
553C23—244no—
554C23#88412yes—
555C23#91112yes—
556C23#91412yes—
557C23C2×D3×D411yescharacteristic
558D4—24no—
559D4—24no—
560D4—48no—
561D4—42no—
562D4—42no—
563D4—64no—
564D4—64no—
565D4—128no—
566D4—124no—
567D4—122no—
568D4—122no—
569D4—248no—
570D4—248no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—122no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4#88412yes—
592C2×C4C2×#23511yescharacteristic
593C2×C4C2×D3×D412yes—
594C2×C4C2×D3×D411yescharacteristic
595C2×C4#100412yes—
596C8—34no—
597C8—41no—
598C8—41no—
599C8—62no—
600C2×C3—24no—
601C2×C3—84no—
602C2×C3#240811yescharacteristic
603C2×C3C2×#46911yescharacteristic
604C2×C3#305811yescharacteristic
605C2×C3#326412yes—
606C2×C3#326412yes—
607D3—81no—
608D3—84no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—816no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—122no—
622C22—124no—
623C22—121no—
624C22—121no—
625C22—124no—
626C22—122no—
627C22—124no—
628C22—121no—
629C22—2416no—
630C22—242no—
631C22—242no—
632C22—242no—
633C22—242no—
634C22#478214yescomplemented
635C22C2×#88411yescharacteristic
636C22#559012yes—
637C22C2×#91111yescharacteristic
638C22C2×#91411yescharacteristic
639C22#577412yes—
640C4—22no—
641C4—82no—
642C4—121no—
643C4—121no—
644C4—121no—
645C4—242no—
646C4#549012yes—
647C4D3×#12011yescharacteristic
648C4#559211yescharacteristic
649C3#2427211yescharacteristic, complemented, Sylow
650C2—24no—
651C2—84no—
652C2—241no—
653C2—244no—
654C2#7382411yescharacteristic
655C2C2×#478211yescharacteristic
656C2#7893211yescharacteristic
657C2#8002212yes—
658C2#8016612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
hypercenter#1786D3
hypocenterC3#24272
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. C23×#213C2semidirect
2. #84053C2semidirect
3. C2×#4782C2semidirect
4. C2×#4730C2semidirect
5. #74016C2semidirect
6. C3×#1786C2semidirect
7. #70212C2semidirect
8. #70204C2semidirect
9. #70191C2semidirect
10. #69959C2semidirect
11. #69511C2semidirect
16. C2×#938C22semidirect
23. C22×ΓL(1,𝔽25)C22semidirect
24. C2×#895C22semidirect
26. C2×C3×#102C22semidirect
27. C2×C3×#102C22semidirect
33. #5221C22semidirect
35. C2×#821C22semidirect
36. C2×#821C22semidirect
37. C2×(C2×D3)⋊C8C22semidirect
38. C2×#815C22semidirect
39. C2×Dic3⋊C8C22semidirect
40. #4988C22semidirect
41. C2×#808C22semidirect
47. #4529C22semidirect
58. #1786D3semidirect
96. C2×#213D4semidirect
107. C22×C3⋊C8D4semidirect
145. C2×#102C2×D3semidirect
146. C2×#102C2×D3semidirect
393. C3⋊C8#120semidirect
495. C22×C3#469semidirect
634. C22#4782semidirect
649. C3#24272semidirect

Maximal subgroups

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

subgroupindex
C23×#2132
#840532
C2×#47822
C2×#47302
#740162
C3×#17862
#702122
#702042
#701912
#699592
#695112
#242723