Finite Group Databasebeta

← #10822565

Subgroups — #10822565

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5519C211yesmaximal, characteristic, complemented
2C3×#2896C211yesmaximal, characteristic
3#84087C211yesmaximal, characteristic
4#84072C211yesmaximal, characteristic
5C2×#4554C211yesmaximal, characteristic, complemented
6#71982C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#70333C214yesmaximal, complemented
9#70313C214yesmaximal, complemented
10#23462—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×#208C2211yescharacteristic
18#5519—22no—
19#5519—22no—
20#5519—22no—
21#5519C2214yes—
22C2×#890—21no—
23C2×#888—21no—
24C2×#878—22no—
25C2×#870—21no—
26C2×#869—21no—
27C3×#419C2212yes—
28C3×#418C2212yes—
29C3×#417—21no—
30C3×#417—21no—
31C3×#416—22no—
32#5233—22no—
33#5232—21no—
34#5232—21no—
35#5229C2212yes—
36#5225C2212yes—
37#5221C2212yes—
38#5220—22no—
39#5215C2212yes—
40#5212—22no—
41#4555C2214yes—
42#4554C2214yes—
43#4532C2214yes—
44C2×#544—31no—
45#2896—31no—
46#2896D311yescharacteristic
47#2311—31no—
48#2028—34no—
49#2027—34no—
50C2×#343—31no—
51#1733—31no—
52#1733—31no—
53C2×C3×#41—22no—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—21no—
57C2×C3×#41—21no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41Q814yescomplemented
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C23×Dic3D412yes—
67C22×Dic6—22no—
68C2×C3×M16—21no—
69C2×C3×M16—22no—
70C2×C3×M16—22no—
71C2×C3×M16C2312yes—
72C22×C3×C8—21no—
73C3×#117—21no—
74C3×#117—21no—
75C3×#117D412yes—
76C3×#117C2312yes—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—21no—
81C2×#217—21no—
82C2×#217—21no—
83C2×#217—24no—
84#926—21no—
85#926—21no—
86#926—22no—
87#926—22no—
88#926D412yes—
89#926C2312yes—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3D412yes—
97C2×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3D411yescharacteristic
99C2×C4⋊Dic3C2312yes—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3C2312yes—
104C2×#208—21no—
105C2×#208—22no—
106C2×#208—22no—
107C2×#208C2312yes—
108C22×C3⋊C8—21no—
109#890—24no—
110#890—24no—
111#888—24no—
112#888—24no—
113#878—28no—
114#878—28no—
115#870—24no—
116#870—24no—
117#869—24no—
118#869—24no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#102—22no—
122C3×#102—22no—
123#835—22no—
124#835—22no—
125#832—22no—
126#832—22no—
127C3⋊#95—24no—
128C22×#41C2×D311yescharacteristic
129C22×M16—31no—
130C22×M16C2×D311yescharacteristic
131#544—34no—
132#544—62no—
133#544—62no—
134#544—62no—
135C2×C4⋊Q8—61no—
136C2×C4⋊Q8—61no—
137C2×#125—61no—
138C2×#125—61no—
139C2×C22⋊Q8—62no—
140C2×#117—31no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143C22×C4⋊C4—31no—
144#419—32no—
145#419C2×D312yes—
146#418—32no—
147#418C2×D312yes—
148#417—21no—
149#417—21no—
150#417—61no—
151#417—61no—
152#416—22no—
153#416—62no—
154#391—32no—
155#390—32no—
156#386—62no—
157#380—62no—
158#343—34no—
159#339—34no—
160#339—34no—
161C24×C3—21no—
162C24×C3C2×D411yescharacteristic
163C3×#41—24no—
164C3×#41—24no—
165C3×#41—24no—
166C3×#41—24no—
167C3×#41—42no—
168C3×#41—42no—
169C3×#41—42no—
170C3×#41—42no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8C2×Q812yescomplemented
177C2×C3×D4—24no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—41no—
184C2×C3×D4—41no—
185C2×C3×D4—42no—
186C2×C3×D4C2×Q812yescomplemented
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—24no—
191C22×C3×C4—41no—
192C22×C3×C4—41no—
193C22×C3×C4—42no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×Q812yes—
200C22×C3×C4#4112yes—
201C22×C3×C4#4112yes—
202C22×C3×C4#4112yes—
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—82no—
210C22×Dic3C2×D414yes—
211C2×Dic6—28no—
212C2×Dic6—84no—
213C3×M16—22no—
214C3×M16—22no—
215C3×M16—42no—
216C2×C3×C8—22no—
217C2×C3×C8—42no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4C2×D412yes—
221C3×C4⋊C4C2×D412yes—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C42—22no—
225C3×C42—22no—
226#217—22no—
227#217—22no—
228#217—42no—
229#217—84no—
230C4⋊Dic3—28no—
231C4⋊Dic3—21no—
232C4⋊Dic3—22no—
233C4⋊Dic3—21no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C4⋊Dic3—42no—
238C4⋊Dic3C2×D412yes—
239C4⋊Dic3C2×D412yes—
240Dic3⋊C4—28no—
241Dic3⋊C4—28no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246#208—22no—
247#208—22no—
248#208—42no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—42no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—21no—
254C2×#41—21no—
255C2×#41—22no—
256C2×#41—21no—
257C2×#41—21no—
258C2×#41Dic614yescomplemented
259C22×Q8—62no—
260C22×Q8C22×D311yescharacteristic
261C22×D4—21no—
262C22×D4C22×D311yescharacteristic
263C23×C4—21no—
264C23×C4—32no—
265C23×C4C22×D311yescharacteristic
266C2×M16—21no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—32no—
270C2×M16—61no—
271C2×M16—62no—
272C2×M16—62no—
273C2×M16C22×D312yes—
274C22×C8—21no—
275C22×C8—61no—
276C4⋊Q8—64no—
277C4⋊Q8—64no—
278C4⋊Q8—64no—
279C4⋊Q8—64no—
280#125—64no—
281#125—64no—
282#125—64no—
283#125—64no—
284C22⋊Q8—68no—
285C22⋊Q8—68no—
286#117—21no—
287#117—21no—
288#117—32no—
289#117—32no—
290#117—61no—
291#117—61no—
292#117—62no—
293#117—62no—
294#117#6712yes—
295#117C22×D312yes—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—32no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4—62no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4#6711yescharacteristic
309C2×C4⋊C4#6711yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—64no—
314C2×C42—21no—
315C2×C42—32no—
316C2×C42—61no—
317#103—22no—
318#103—22no—
319#103—62no—
320#103—62no—
321#102—22no—
322#102—22no—
323#102—62no—
324#102—62no—
325#95—64no—
326C23×C3—22no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3#12011yescharacteristic
334C23×C3#12011yescharacteristic
335C23×C3C22⋊Q812yes—
336C23×C3#12412yes—
337C23×C3C22×D411yescharacteristic
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—44no—
345C3×D4—44no—
346C3×D4—41no—
347C3×D4—41no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—44no—
356C2×C3×C4—44no—
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4C22⋊Q812yes—
360C2×C3×C4C22⋊Q812yes—
361C2×C3×C4C22⋊Q812yes—
362C2×C3×C4C22⋊Q812yes—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4#12412yes—
365C2×C3×C4#12412yes—
366C2×C3×C4#12412yes—
367C2×C3×C4#12412yes—
368C2×C3×C4#12412yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×Q811yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—24no—
378C2×Dic3—42no—
379C2×Dic3—44no—
380C2×Dic3—42no—
381C2×Dic3—44no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—82no—
386Dic6—88no—
387C3×C8—42no—
388C3⋊C8—42no—
389C24—21no—
390C24D3×D411yescharacteristic
391#41—24no—
392#41—24no—
393#41—24no—
394#41—24no—
395#41—42no—
396#41—42no—
397#41—42no—
398#41—42no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—68no—
405C2×Q8—244no—
406C2×Q8C2×Dic612yescomplemented
407C2×D4—24no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4C2×Dic612yescomplemented
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—34no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—242no—
431C22×C4C2×Dic612yes—
432C22×C4#23512yes—
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4#23712yes—
437C22×C4#23712yes—
438C22×C4C2×#6711yescharacteristic
439C22×C4C2×#6711yescharacteristic
440C22×C4C23×D311yescharacteristic
441M16—22no—
442M16—22no—
443M16—42no—
444M16—62no—
445M16—62no—
446M16—122no—
447C2×C8—22no—
448C2×C8—42no—
449C2×C8—62no—
450C2×C8—122no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—68no—
456C4⋊C4—68no—
457C4⋊C4—68no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—122no—
465C4⋊C4C2×#6712yes—
466C4⋊C4C2×#6712yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—122no—
472C22⋊C4—244no—
473C42—22no—
474C42—22no—
475C42—62no—
476C42—62no—
477C42—62no—
478C42—62no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—82no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3#39312yes—
493C22×C3#39312yes—
494C22×C3C2×#12011yescharacteristic
495C22×C3C2×C22⋊Q811yescharacteristic
496C22×C3C2×#12411yescharacteristic
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4#39312yes—
501C3×C4#39312yes—
502C3×C4C2×C22⋊Q811yescharacteristic
503C3×C4C2×C22⋊Q811yescharacteristic
504C3×C4C2×#12411yescharacteristic
505C3×C4C2×#12411yescharacteristic
506Dic3—42no—
507Dic3—44no—
508Dic3—42no—
509Dic3—82no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23#87812yes—
518C23#88512yes—
519C23#93711yescharacteristic
520C23#93711yescharacteristic
521C23C2×D3×D411yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526Q8—248no—
527D4—22no—
528D4—22no—
529D4—44no—
530D4—44no—
531D4—41no—
532D4—41no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—64no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—124no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—242no—
553C2×C4#87812yes—
554C2×C4#87812yes—
555C2×C4#87812yes—
556C2×C4#88512yes—
557C2×C4#88512yes—
558C2×C4#88512yes—
559C2×C4#92412yes—
560C2×C4#92412yes—
561C2×C4#93612yes—
562C2×C4#93612yes—
563C2×C4#93711yescharacteristic
564C2×C4#93711yescharacteristic
565C2×C4C22×Dic611yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×#23711yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4C22×#6711yescharacteristic
571C8—42no—
572C8—122no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—22no—
576C2×C3—22no—
577C2×C3—82no—
578C2×C3C2×#39311yescharacteristic
579C2×C3#301911yescharacteristic
580C2×C3#302011yescharacteristic
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22#494712yes—
595C22#494712yes—
596C22C2×#87811yescharacteristic
597C22C2×#88511yescharacteristic
598C22C2×#93711yescharacteristic
599C4—42no—
600C4—42no—
601C4—42no—
602C4—124no—
603C4—122no—
604C4—122no—
605C4—242no—
606C4#494712yes—
607C4#494712yes—
608C4C2×#87811yescharacteristic
609C4C2×#88511yescharacteristic
610C4C2×#92411yescharacteristic
611C4C2×#93611yescharacteristic
612C3#2346211yescharacteristic, complemented, Sylow
613C2—21no—
614C2—21no—
615C2—22no—
616C2—22no—
617C2—82no—
618C2C2×#494711yescharacteristic
619C2#8400511yescharacteristic
620C2#8400611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2896C2
socleC22×C3C2×#120
prime radicalC22×C3×#41C22
hypercenter#2896D3
hypocenterC3#23462
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
1. C2×#5519C2semidirect
5. C2×#4554C2semidirect
6. #71982C2semidirect
7. #71574C2semidirect
8. #70333C2semidirect
9. #70313C2semidirect
60. C2×C3×#41Q8semidirect
176. C2×C3×Q8C2×Q8semidirect
186. C2×C3×D4C2×Q8semidirect
258. C2×#41Dic6semidirect
406. C2×Q8C2×Dic6semidirect
416. C2×D4C2×Dic6semidirect
612. C3#23462semidirect

Maximal subgroups

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

subgroupindex
C2×#55192
C3×#28962
#840872
#840722
C2×#45542
#719822
#715742
#703332
#703132
#234623