Finite Group Databasebeta

← #10831854

Subgroups — #10831854

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85766C211yesmaximal, characteristic, complemented
2C3×C4×#120C211yesmaximal, characteristic
3#84031C212yesmaximal, complemented
4#71669C212yesmaximal
5#71619C211yesmaximal, characteristic, complemented
6#69597C212yesmaximal, complemented
7#69594C212yesmaximal, complemented
8#69497C212yesmaximal, complemented
9#69457C212yesmaximal, complemented
10#24350—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C2×C3×#120C2211yescharacteristic
13C2×C3×C4×D4—21no—
14C2×C3×C4×D4C2211yescharacteristic
15#5798C2212yes—
16C2×#924—21no—
17C2×#924—22no—
18C2×#924C2212yes—
19#5543—22no—
20#5539—24no—
21D4×Dic6—22no—
22#5500—21no—
23#5483—21no—
24#5483C2212yes—
25#5470—21no—
26C3×#386—21no—
27C3×#386C2211yescharacteristic
28C3×#385—21no—
29C3×#385C2211yescharacteristic
30C3×#379C2211yescharacteristic
31C3×C4×C22⋊C4—21no—
32C3×C4×C22⋊C4C2211yescharacteristic
33#5247C2214yes—
34C2×#834C2212yes—
35#5212—22no—
36#5212C2212yes—
37#5210—24no—
38#5210C2214yes—
39#4527C2214yes—
40#4527C2214yes—
41#4526C2212yes—
42#3501—31no—
43#2848—32no—
44C4×#120D311yescharacteristic
45#2062—32no—
46#2029—32no—
47#1996—32no—
48#1889—32no—
49#1867—32no—
50#1799—31no—
51C25×C3C2311yescharacteristic
52C22×C3×D4—21no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—22no—
55C23×C3×C4—21no—
56C23×C3×C4—22no—
57C23×C3×C4—21no—
58C23×C3×C4—44no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6D412yescomplemented
64C3×#120—22no—
65C3×#120Q814yescomplemented
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4—44no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4D411yescharacteristic
76C2×C3×C22⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4C2311yescharacteristic
79C2×C3×C42—21no—
80C2×C3×C42D411yescharacteristic
81C2×#217—22no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217C2314yes—
85C2×#217C2314yes—
86D4×Dic3—24no—
87D4×Dic3—24no—
88#926—22no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3D412yescomplemented
92#924—22no—
93#924—22no—
94#924—22no—
95#924—22no—
96#924—22no—
97#924—22no—
98#924—22no—
99#924—22no—
100#924—48no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4D414yescomplemented
104#890—24no—
105#888—24no—
106#878—24no—
107#878—24no—
108#878—24no—
109#878—24no—
110#870—22no—
111#869—21no—
112#869—21no—
113C4×Dic6—22no—
114C4×Dic6—22no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118#834—24no—
119C3⋊#95—22no—
120C3⋊#95—24no—
121C24×C4C2×D311yescharacteristic
122#554—64no—
123#554—62no—
124D4×Q8—62no—
125#544—61no—
126#543—32no—
127#543—61no—
128#533—61no—
129C2×C22⋊Q8—32no—
130C2×C22⋊Q8—61no—
131C2×C22⋊Q8—62no—
132C2×#120C2×D311yescharacteristic
133#518—32no—
134C2×C4×D4—21no—
135C2×C4×D4C2×D311yescharacteristic
136C2×#99—32no—
137#386—21no—
138#386C2×D311yescharacteristic
139#385—21no—
140#385—34no—
141#385—64no—
142#385C2×D311yescharacteristic
143#380—32no—
144#380—62no—
145#379—34no—
146#379C2×D311yescharacteristic
147C4×C22⋊C4—21no—
148C4×C22⋊C4C2×D311yescharacteristic
149#342—32no—
150#342—34no—
151#342—34no—
152C24×C3—22no—
153C24×C3—21no—
154C24×C3—21no—
155C24×C3—22no—
156C24×C3—42no—
157C24×C3—42no—
158C24×C3C2×D411yescharacteristic
159C24×C3C2×Q812yescomplemented
160C24×C3C2411yescharacteristic
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—24no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4C2×Q812yes—
167C2×C3×D4#4112yes—
168C22×C3×C4—28no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—44no—
179C22×C3×C4—42no—
180C22×C3×C4—44no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4—44no—
184C22×C3×C4—42no—
185C22×C3×C4—41no—
186C22×C3×C4—42no—
187C22×C3×C4—41no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—24no—
198C22×Dic3C2×D414yes—
199C22×Dic3C2×D414yes—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—44no—
205C2×Dic6—82no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—42no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4C2×Q812yescomplemented
217C3×C42—21no—
218C3×C42—21no—
219C3×C42—42no—
220#217—24no—
221#217—24no—
222#217—24no—
223#217—24no—
224#217—24no—
225#217—48no—
226#217—48no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3—44no—
234Dic3⋊C4—24no—
235Dic3⋊C4—24no—
236Dic3⋊C4—24no—
237Dic3⋊C4—24no—
238Dic3⋊C4—48no—
239C4×Dic3—24no—
240C25C22×D311yescharacteristic
241C22×Q8—32no—
242C22×Q8—61no—
243C22×D4—21no—
244C22×D4C22×D311yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—44no—
250C23×C4#6711yescharacteristic
251C23×C4C22×D311yescharacteristic
252C23×C4C22×D311yescharacteristic
253C4⋊Q8—61no—
254C4⋊Q8—61no—
255C4⋊Q8—64no—
256#125—62no—
257#125—64no—
258C22⋊Q8—64no—
259C22⋊Q8—64no—
260C22⋊Q8—64no—
261C22⋊Q8—64no—
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264C22⋊Q8—62no—
265C22⋊Q8—62no—
266C22⋊Q8—62no—
267C22⋊Q8—62no—
268C22⋊Q8—62no—
269C22⋊Q8—62no—
270C22⋊Q8—128no—
271#120—22no—
272#120Dic614yescomplemented
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—44no—
278C4×D4—64no—
279C4×D4—64no—
280#117—62no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—34no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4#6711yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—34no—
294C2×C22⋊C4—34no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4#6711yescharacteristic
299C2×C22⋊C4C22×D311yescharacteristic
300C2×C22⋊C4C22×D311yescharacteristic
301C2×C22⋊C4C22×D311yescharacteristic
302C2×C42—21no—
303C2×C42#6711yescharacteristic
304#99—64no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—64no—
309#95—62no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—24no—
317C23×C3—21no—
318C23×C3—41no—
319C23×C3—44no—
320C23×C3—41no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3—41no—
325C23×C3—44no—
326C23×C3—48no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—42no—
332C23×C3C22⋊Q812yescomplemented
333C23×C3C22×D412yes—
334C23×C3C22×D411yescharacteristic
335C23×C3C22×Q811yescharacteristic
336C23×C3C2×#4111yescharacteristic
337C23×C32−1+412yes—
338C3×D4—24no—
339C3×D4—24no—
340C3×D4—48no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—48no—
353C2×C3×C4—44no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—81no—
367C2×C3×C4#12011yescharacteristic
368C2×C3×C4#12012yes—
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4C22⋊Q812yescomplemented
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—44no—
374C2×Dic3—44no—
375C2×Dic3—44no—
376C2×Dic3—44no—
377C2×Dic3—44no—
378C2×Dic3—82no—
379C2×Dic3—82no—
380Dic6—44no—
381Dic6—82no—
382Dic6—82no—
383Dic6—82no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24C2×Dic612yescomplemented
391C24C2×#6711yescharacteristic
392C24C23×D311yescharacteristic
393C2×Q8—62no—
394C2×Q8—62no—
395C2×Q8—62no—
396C2×Q8—62no—
397C2×Q8—124no—
398C2×Q8—242no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C2×Dic612yes—
405C2×D4#23512yes—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—34no—
415C22×C4—34no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—64no—
431C22×C4D3×D412yes—
432C22×C4D3×D412yes—
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—42no—
440C4⋊C4—62no—
441C4⋊C4—64no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—64no—
445C4⋊C4—62no—
446C4⋊C4—64no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C4⋊C4—64no—
450C4⋊C4—128no—
451C4⋊C4—124no—
452C22⋊C4—21no—
453C22⋊C4—22no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—64no—
464C22⋊C4—64no—
465C22⋊C4—128no—
466C22⋊C4—128no—
467C22⋊C4C2×Dic612yescomplemented
468C42—21no—
469C42—21no—
470C42—42no—
471C42—64no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—24no—
477C22×C3—28no—
478C22×C3—22no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—44no—
484C22×C3—44no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—44no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—41no—
493C22×C3—42no—
494C22×C3—81no—
495C22×C3C2×#12011yescharacteristic
496C22×C3C2×C22⋊Q811yescharacteristic
497C22×C3#53611yescharacteristic
498C22×C3#54812yes—
499C22×C3D4×Q812yes—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—81no—
508C3×C4B(4,𝔽2)12yes—
509Dic3—44no—
510Dic3—44no—
511Dic3—84no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—21no—
519C23—21no—
520C23—41no—
521C23—44no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—44no—
528C23—48no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23#92412yescomplemented
535C23C22×Dic611yescharacteristic
536C23C2×#23511yescharacteristic
537C23C2×D3×D412yes—
538C23#101012yes—
539C23C22×#6711yescharacteristic
540Q8—124no—
541Q8—242no—
542Q8—242no—
543Q8—242no—
544D4—24no—
545D4—24no—
546D4—48no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—81no—
575C2×C4—124no—
576C2×C4—124no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4—124no—
580C2×C4—242no—
581C2×C4—242no—
582C2×C4#92412yescomplemented
583C2×C4#93711yescharacteristic
584C2×C4#93712yes—
585C2×C4#93711yescharacteristic
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—44no—
591C2×C3—42no—
592C2×C3#240911yescharacteristic
593C2×C3C2×B(4,𝔽2)11yescharacteristic
594C2×C3#302511yescharacteristic
595C22—21no—
596C22—22no—
597C22—21no—
598C22—22no—
599C22—24no—
600C22—28no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—81no—
618C22D4×Dic612yes—
619C22#555212yes—
620C22C2×#92411yescharacteristic
621C22C2×#93711yescharacteristic
622C22#582511yescharacteristic
623C4—21no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—124no—
632C4—124no—
633C4—244no—
634C4#503212yes—
635C3#2435011yescharacteristic, complemented, Sylow
636C2—21no—
637C2—21no—
638C2—22no—
639C2—22no—
640C2—44no—
641C2—42no—
642C2#8234911yescharacteristic
643C2C2×#503211yescharacteristic
644C2#8293611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×C4×#120C2
socleC22×C3C2×#120
prime radicalC2×C3×#120C22
hypercenterC4×#120D3
hypocenterC3#24350
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
1. #85766C2semidirect
3. #84031C2semidirect
5. #71619C2semidirect
6. #69597C2semidirect
7. #69594C2semidirect
8. #69497C2semidirect
9. #69457C2semidirect
63. C22×Dic6D4semidirect
65. C3×#120Q8semidirect
91. C2×C4⋊Dic3D4semidirect
103. C2×Dic3⋊C4D4semidirect
159. C24×C3C2×Q8semidirect
216. C3×C22⋊C4C2×Q8semidirect
272. #120Dic6semidirect
332. C23×C3C22⋊Q8semidirect
370. C2×C3×C4C22⋊Q8semidirect
390. C24C2×Dic6semidirect
467. C22⋊C4C2×Dic6semidirect
534. C23#924semidirect
582. C2×C4#924semidirect
635. C3#24350semidirect

Maximal subgroups

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

subgroupindex
#857662
C3×C4×#1202
#840312
#716692
#716192
#695972
#695942
#694972
#694572
#243503