Finite Group Databasebeta

← #10943986

Subgroups — #10943986

Order 768.

1576 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic, complemented
3C2×#5227C211yesmaximal, characteristic, complemented
4C2×#5163C212yesmaximal, complemented
5#82700C211yesmaximal, characteristic
6#72048C212yesmaximal, complemented
7#71677C211yesmaximal, characteristic, complemented
8#70240C212yesmaximal, complemented
9#70221C212yesmaximal, complemented
10#69932C211yesmaximal, characteristic, complemented
11#69908C211yesmaximal, characteristic, complemented
12#32903—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#121C2211yescharacteristic
16C2×Q8×Dic3—28no—
17C2×Q8×Dic3—21no—
18C2×Q8×Dic3—21no—
19C2×Q8×Dic3C2212yes—
20C2×Q8×Dic3C2212yes—
21C22×#215C2212yes—
22C2×#940C2211yescharacteristic
23C2×#939—21no—
24C2×#939—21no—
25C2×#930C2211yescharacteristic
26C22×C4⋊Dic3—21no—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4×Dic3—21no—
29C22×C4×Dic3C2211yescharacteristic
30C3×#448—24no—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33#5227—24no—
34C2×#835C2212yes—
35#5221C2211yescharacteristic
36C2×#832C2212yes—
37C2×#830C2211yescharacteristic
38#5163—28no—
39C2×#821C2212yes—
40C2×Dic3⋊C8C2212yes—
41#4988C2211yescharacteristic
42#4975C2211yescharacteristic
43#4957—21no—
44#4957—21no—
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C22×C4×Q8—31no—
49C2×#460—32no—
50C2×#448—31no—
51C2×#448D311yescharacteristic, complemented
52#2856—31no—
53#1880—32no—
54#1880—32no—
55#1874—31no—
56#1874—31no—
57#1803—32no—
58#1797—31no—
59C22×C3×Q8—24no—
60C22×C3×Q8—24no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8D414yescomplemented
64C22×C3×Q8C2312yes—
65C22×C3×D4—21no—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67—21no—
71C22×#67—21no—
72C23×Dic3—21no—
73C23×Dic3D411yescharacteristic
74C23×Dic3D411yescharacteristic
75C22×D12C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C22×C3×C8C2312yes—
80C3×#121—22no—
81C3×#121—22no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×#217—21no—
85C2×#217—21no—
86Q8×Dic3—28no—
87Q8×Dic3—28no—
88Q8×Dic3—24no—
89Q8×Dic3—24no—
90Q8×Dic3—24no—
91Q8×Dic3D418yescomplemented
92C2×#215—24no—
93C2×#215—24no—
94C2×#215—24no—
95#940—24no—
96#939—44no—
97#939—44no—
98#930—22no—
99#930—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—28no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—24no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2312yes—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—28no—
113C2×C4×Dic3—22no—
114C2×C4×Dic3—22no—
115C2×C4×Dic3—24no—
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121C3×#103—24no—
122C3×C22⋊C8—22no—
123#835—24no—
124#832—24no—
125#830—22no—
126#821—24no—
127Dic3⋊C8—24no—
128C23×Q8C2×D311yescharacteristic
129C22×SD16—32no—
130C22×SD16C2×D312yes—
131C2×#127—31no—
132C2×#121—31no—
133C2×#121—61no—
134C2×#121—61no—
135C2×#121C2×D311yescharacteristic
136C2×C4×Q8—32no—
137C2×C4×Q8—32no—
138C2×C4×Q8—61no—
139C2×C4×Q8—61no—
140C2×C4×Q8—68no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—61no—
143C22×C42—31no—
144C22×C42—61no—
145#460—68no—
146#448—24no—
147#448—64no—
148C2×C4⋊C8—32no—
149C2×#103—32no—
150C2×#103C2×D312yes—
151C2×#102—32no—
152C2×#102—32no—
153C2×C22⋊C8—31no—
154C2×C22⋊C8C2×D311yescharacteristic
155#390—31no—
156#390—31no—
157#390—61no—
158#390—61no—
159#384—31no—
160#340—32no—
161#340—32no—
162#336—32no—
163C24×C3—21no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C2×C3×Q8—24no—
167C2×C3×Q8—216no—
168C2×C3×Q8—24no—
169C2×C3×Q8—216no—
170C2×C3×Q8—24no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8C2×D414yes—
175C2×C3×Q8C2×D414yes—
176C2×C3×Q8#4114yes—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4#4112yes—
194C22×C3×C4C2411yescharacteristic
195C2×#67—42no—
196C2×#67—42no—
197C2×#67—42no—
198C2×#67—42no—
199C2×#67—42no—
200C2×#67—42no—
201C2×#67—42no—
202C2×#67—42no—
203C2×#67—42no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—21no—
211C22×Dic3—41no—
212C22×Dic3—42no—
213C22×Dic3—42no—
214C22×Dic3SD1614yescomplemented
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C2×D12—22no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—22no—
223C3×SD16—48no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C22⋊C4—42no—
230#217—42no—
231#217—42no—
232#215—48no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236C4⋊Dic3—28no—
237C4⋊Dic3—24no—
238C4⋊Dic3—22no—
239C4⋊Dic3—28no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3C2×D414yes—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245C4×Dic3—28no—
246C4×Dic3—22no—
247C4×Dic3—28no—
248C4×Dic3—22no—
249C4×Dic3—22no—
250C4×Dic3C2×D414yes—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C22×Q8—24no—
255C22×Q8—24no—
256C22×Q8—21no—
257C22×Q8—22no—
258C22×Q8#6714yescomplemented
259C22×Q8C22×D312yes—
260C22×D4—21no—
261C22×D4—31no—
262C22×D4—61no—
263C22×D4—61no—
264C22×D4—61no—
265C22×D4C22×D311yescharacteristic
266C23×C4—21no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—64no—
277C22×C8—32no—
278C22×C8C22×D312yes—
279#127—64no—
280#121—22no—
281#121—22no—
282#121—62no—
283#121—62no—
284#121—124no—
285#121—124no—
286C4×Q8—38no—
287C4×Q8—64no—
288C4×Q8—64no—
289C4×Q8—68no—
290C4×Q8—68no—
291C4×Q8—64no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4—68no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—64no—
313C2×C42—62no—
314C2×C42—62no—
315C2×C42—68no—
316C4⋊C8—64no—
317#103—24no—
318#103—64no—
319#102—64no—
320#102—64no—
321C22⋊C8—22no—
322C22⋊C8—62no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3#12112yes—
335C23×C3C2×SD1612yes—
336C23×C3AGL(1,ℤ/8)12yes—
337C23×C3C22×D411yescharacteristic
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C3×Q8—216no—
346C3×Q8—24no—
347C3×Q8—24no—
348C3×Q8—24no—
349C3×Q8#12118yescomplemented
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—84no—
353C2×C3×C4—216no—
354C2×C3×C4—24no—
355C2×C3×C4—21no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—21no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4#12112yes—
365C2×C3×C4C22×D412yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4112yes—
368C2×C3×C4C2×#4111yescharacteristic
369#67—84no—
370#67—84no—
371#67—84no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—28no—
377C2×Dic3—21no—
378C2×Dic3—24no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—44no—
382C2×Dic3—48no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3#12012yes—
389C2×Dic3#12012yes—
390C2×Dic3C2×SD1612yes—
391C2×Dic3C2×SD1612yes—
392C2×Dic3#13712yes—
393D12—42no—
394D12—42no—
395C3×C8—42no—
396C3⋊C8—42no—
397C24—21no—
398C24—61no—
399C24D3×D411yescharacteristic
400C2×Q8—24no—
401C2×Q8—216no—
402C2×Q8—24no—
403C2×Q8—216no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8D3×D414yes—
409C2×Q8#23714yes—
410C2×Q8C2×#6714yes—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—62no—
419C2×D4—61no—
420C2×D4—62no—
421C2×D4—61no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4—122no—
429C2×D4—122no—
430C2×D4—122no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—34no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—64no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—121no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4D3×D411yescharacteristic
454C22×C4#23712yes—
455C22×C4C2×#6712yes—
456C22×C4C23×D311yescharacteristic
457SD16—48no—
458SD16—128no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—62no—
463C2×C8—62no—
464C2×C8—62no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—34no—
468C4⋊C4—68no—
469C4⋊C4—68no—
470C4⋊C4—64no—
471C4⋊C4—62no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C22⋊C4—42no—
477C22⋊C4—122no—
478C22⋊C4—122no—
479C22⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C42—34no—
483C42—62no—
484C42—62no—
485C42—68no—
486C42—68no—
487C42—62no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3#46014yescomplemented
502C22×C3C2×#12011yescharacteristic
503C22×C3C2×#12111yescharacteristic
504C22×C3#52511yescharacteristic
505C22×C3C22×SD1611yescharacteristic
506C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
507C22×C3C2×#13711yescharacteristic
508C22×C3#57511yescharacteristic
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C3×C4—28no—
517C3×C4—22no—
518C3×C4—81no—
519C3×C4C2×#12112yes—
520C3×C4#52512yes—
521Dic3—22no—
522Dic3—22no—
523Dic3—41no—
524Dic3—44no—
525Dic3—81no—
526Dic3#44814yescomplemented
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23D3×SD1612yes—
546C23#91412yes—
547C23#93912yes—
548C23C2×D3×D411yescharacteristic
549Q8—216no—
550Q8—24no—
551Q8—24no—
552Q8—24no—
553Q8#93918yescomplemented
554D4—42no—
555D4—42no—
556D4—84no—
557D4—122no—
558D4—122no—
559D4—244no—
560D4—244no—
561D4—244no—
562C2×C4—216no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—68no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—64no—
586C2×C4—61no—
587C2×C4—124no—
588C2×C4—128no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4#93912yes—
595C2×C4C2×D3×D412yes—
596C2×C4C2×#23712yes—
597C2×C4C2×#23711yescharacteristic
598C2×C4C22×#6711yescharacteristic
599C8—42no—
600C8—122no—
601C2×C3—24no—
602C2×C3—161no—
603C2×C3#284711yescharacteristic
604C2×C3C2×#44811yescharacteristic
605C2×C3#300611yescharacteristic
606C2×C3C2×#46011yescharacteristic
607C2×C3#303711yescharacteristic
608C2×C3#310311yescharacteristic
609C2×C3#311211yescharacteristic
610D3—161no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22#516314yescomplemented
632C22#558611yescharacteristic
633C22D3×#12011yescharacteristic
634C22C2×D3×SD1611yescharacteristic
635C22C2×#91411yescharacteristic
636C22D3×#13711yescharacteristic
637C22#577711yescharacteristic
638C22C2×#93911yescharacteristic
639C4—28no—
640C4—22no—
641C4—34no—
642C4—62no—
643C4—62no—
644C4—81no—
645C4—121no—
646C4—124no—
647C4—241no—
648C4#558612yes—
649C4C2×#93912yes—
650C3#3290311yescharacteristic, complemented, Sylow
651C2—24no—
652C2—161no—
653C2—481no—
654C2#8002811yescharacteristic
655C2#8003011yescharacteristic
656C2D3×#44811yescharacteristic
657C2#8004011yescharacteristic
658C2#8260111yescharacteristic
659C2C2×#516311yescharacteristic
660C2#8396611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#448C2
socleC23×C3C22×D4
prime radical#82700C2
hypercenterC2×#448D3
hypocenterC3#32903
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic3C2semidirect
2. C2×C3×#448C2semidirect
3. C2×#5227C2semidirect
4. C2×#5163C2semidirect
6. #72048C2semidirect
7. #71677C2semidirect
8. #70240C2semidirect
9. #70221C2semidirect
10. #69932C2semidirect
11. #69908C2semidirect
51. C2×#448D3semidirect
63. C22×C3×Q8D4semidirect
91. Q8×Dic3D4semidirect
214. C22×Dic3SD16semidirect
258. C22×Q8#67semidirect
349. C3×Q8#121semidirect
501. C22×C3#460semidirect
526. Dic3#448semidirect
553. Q8#939semidirect
631. C22#5163semidirect
650. C3#32903semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×C3×#4482
C2×#52272
C2×#51632
#827002
#720482
#716772
#702402
#702212
#699322
#699082
#329033