Finite Group Databasebeta

← #10827549

Subgroups — #10827549

Order 768.

1286 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5532C211yesmaximal, characteristic
2C2×C3×#419C211yesmaximal, characteristic, complemented
3#84068C211yesmaximal, characteristic, complemented
4C2×#5227C211yesmaximal, characteristic, complemented
5#71996C211yesmaximal, characteristic, complemented
6#71990C211yesmaximal, characteristic, complemented
7#71575C211yesmaximal, characteristic, complemented
8#70276C212yesmaximal, complemented
9#70272C212yesmaximal, complemented
10#70269C212yesmaximal, complemented
11#70231C212yesmaximal, complemented
12#23977—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#946C2212yes—
17C22×#215C2212yes—
18C2×#930C2211yescharacteristic
19C2×C4×#67C2211yescharacteristic
20#5532—24no—
21#5532—24no—
22C2×#895C2211yescharacteristic
23C2×#893C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×#888C2211yescharacteristic
26C2×#885—21no—
27C2×#884—21no—
28C2×#875—21no—
29C2×#872—21no—
30C3×#419—24no—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33#5231—21no—
34#5230—21no—
35#5229C2212yes—
36#5227—24no—
37C2×#835C2212yes—
38#5213—21no—
39#5212C2211yescharacteristic
40#5210—21no—
41C2×#830C2211yescharacteristic
42C2×#821C2212yes—
43C2×(C2×D3)⋊C8C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#555—31no—
49C2×#448—31no—
50C2×#419D311yescharacteristic, complemented
51#2409—31no—
52#2024—32no—
53#2000—32no—
54#1895—31no—
55#1877—32no—
56#1873—32no—
57#1870—31no—
58#1825—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C22×#67D411yescharacteristic
69C22×D12—21no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×C4×D3C2311yescharacteristic
73C22×Dic6—21no—
74C2×C3×M16—22no—
75C2×C3×M16—42no—
76C22×C3×C8C2312yes—
77C3×#117—22no—
78C3×#117D414yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—21no—
83C2×#217—21no—
84C2×#217D411yescharacteristic
85#946—24no—
86#946—42no—
87C2×#215—24no—
88C2×#215—24no—
89C2×#215—42no—
90#930—22no—
91#930D414yes—
92C4×#67—22no—
93C4×#67—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108#895—22no—
109#895—22no—
110#893—22no—
111#893—22no—
112#892—22no—
113#892—22no—
114#888—22no—
115#888—22no—
116#885—28no—
117#884—28no—
118#875—28no—
119#872—28no—
120C3×#103—24no—
121C3×#103—24no—
122#835—24no—
123C3⋊#95—21no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126#830—22no—
127#821—42no—
128(C2×D3)⋊C8—42no—
129C23×Q8C2×D311yescharacteristic
130C22×SD16—32no—
131C22×M16C2×D311yescharacteristic
132#555—64no—
133#555—64no—
134C2×C4⋊Q8—31no—
135C2×#124—61no—
136C2×#124—61no—
137C2×C22⋊Q8—32no—
138C2×#121—31no—
139C2×#121—31no—
140C2×#121—61no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×#117—31no—
144C2×#117—61no—
145C2×#117C2×D311yescharacteristic
146#448—64no—
147#419—24no—
148C2×#103—32no—
149C2×#103C2×D312yes—
150C2×#103C2×D312yes—
151C2×#102—32no—
152C2×C22⋊C8—31no—
153C2×C22⋊C8—32no—
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—22no—
168C2×C3×Q8—28no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—44no—
174C2×C3×Q8—48no—
175C2×C3×Q8—44no—
176C2×C3×Q8—42no—
177C2×C3×Q8#4114yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—44no—
184C22×C3×C4—42no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2411yescharacteristic
189C2×#67—22no—
190C2×#67—24no—
191C2×#67—22no—
192C2×#67—41no—
193C2×#67—82no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—21no—
202C2×D12—24no—
203C2×D12—82no—
204C2×D12C2×D412yes—
205C2×C4×D3—24no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×Dic6—24no—
213C2×Dic6—82no—
214C3×M16—44no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4C2×D412yes—
221C3×C4⋊C4C2×D412yes—
222C3×C22⋊C4—24no—
223C3×C42—24no—
224#217—22no—
225#217—42no—
226#215—48no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—22no—
232(C2×D3)⋊C4—44no—
233C4⋊Dic3—21no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3C2×D412yes—
237Dic3⋊C4—24no—
238Dic3⋊C4—22no—
239Dic3⋊C4—44no—
240C4×Dic3—24no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C22×Q8—21no—
247C22×Q8—24no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—42no—
251C22×Q8—61no—
252C22×Q8C22×D312yes—
253C22×D4—31no—
254C22×D4—31no—
255C22×D4—61no—
256C22×D4—61no—
257C23×C4—21no—
258C23×C4—31no—
259C23×C4—61no—
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C2×M16—22no—
265C2×M16—42no—
266C22×C8—32no—
267C22×C8C22×D312yes—
268C4⋊Q8—62no—
269C4⋊Q8—62no—
270#124—68no—
271#124—68no—
272C22⋊Q8—64no—
273C22⋊Q8—122no—
274#121—34no—
275#121—68no—
276#121—62no—
277#121—62no—
278#121—62no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283#117—22no—
284#117—68no—
285#117—62no—
286#117—62no—
287#117#6714yes—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—31no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—62no—
303C2×C22⋊C4—61no—
304C2×C42—21no—
305C2×C42—31no—
306C2×C42—61no—
307#103—24no—
308#103—24no—
309#103—64no—
310#102—122no—
311C22⋊C8—62no—
312C22⋊C8—122no—
313#95—62no—
314#95—62no—
315#95—61no—
316C23×C3—22no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3#12012yes—
321C23×C3C22×D411yescharacteristic
322C22×D3—21no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—22no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—42no—
332C3×Q8—24no—
333C3×Q8—24no—
334C3×Q8—48no—
335C3×Q8—42no—
336C3×Q8—42no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—48no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4#12012yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C2×#4112yes—
354C2×C3×C4H(5,𝔽2)12yes—
355#67—44no—
356#67—84no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3#13712yes—
369D12—24no—
370D12—41no—
371D12—41no—
372D12—84no—
373C4×D3—48no—
374C4×D3—41no—
375C4×D3—41no—
376C4×D3—42no—
377Dic6—84no—
378C3×C8—42no—
379C3⋊C8—42no—
380C24—31no—
381C24—61no—
382C24D3×D411yescharacteristic
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—44no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—64no—
394C2×Q8—242no—
395C2×Q8#23514yes—
396C2×D4—32no—
397C2×D4—64no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—64no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—61no—
404C2×D4—121no—
405C2×D4—242no—
406C2×D4—242no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—64no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D412yes—
428C22×C4C2×#6712yes—
429C22×C4C23×D311yescharacteristic
430SD16—128no—
431M16—44no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—62no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—32no—
441C4⋊C4—64no—
442C4⋊C4—62no—
443C4⋊C4—61no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—124no—
447C4⋊C4C2×#6712yes—
448C4⋊C4C2×#6712yes—
449C22⋊C4—24no—
450C22⋊C4—64no—
451C22⋊C4—64no—
452C22⋊C4—64no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—64no—
456C22⋊C4—124no—
457C22⋊C4—122no—
458C42—24no—
459C42—64no—
460C42—61no—
461C42—61no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3C2×#12011yescharacteristic
468C22×C3#54612yes—
469C22×C3C2×#13711yescharacteristic
470C22×C3#57511yescharacteristic
471C22×C3#57712yes—
472C2×D3—42no—
473C2×D3—41no—
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—82no—
481C2×D3—82no—
482C3×C4—22no—
483C3×C4—44no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4#52511yescharacteristic
487C3×C4#53511yescharacteristic
488C3×C4#54612yes—
489Dic3—22no—
490Dic3—41no—
491Dic3—41no—
492Dic3—42no—
493Dic3—82no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—62no—
499C23—61no—
500C23—61no—
501C23—62no—
502C23—62no—
503C23—61no—
504C23—122no—
505C23—121no—
506C23—121no—
507C23—122no—
508C23#93712yes—
509C23C2×D3×D411yescharacteristic
510Q8—24no—
511Q8—24no—
512Q8—48no—
513Q8—42no—
514Q8—42no—
515Q8—244no—
516D4—64no—
517D4—124no—
518D4—121no—
519D4—121no—
520D4—244no—
521D4—244no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—32no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—62no—
543C2×C4—128no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4#93712yes—
552C2×C4C2×#23512yes—
553C2×C4C2×D3×D411yescharacteristic
554C2×C4#100412yes—
555C2×C4C22×#6711yescharacteristic
556C8—42no—
557C8—122no—
558C2×C3—42no—
559C2×C3#240411yescharacteristic
560C2×C3#300711yescharacteristic
561C2×C3#300911yescharacteristic
562C2×C3#332112yes—
563C2×C3#332912yes—
564D3—81no—
565D3—82no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—42no—
570C22—42no—
571C22—122no—
572C22—121no—
573C22—121no—
574C22—122no—
575C22—122no—
576C22—122no—
577C22—121no—
578C22—122no—
579C22—242no—
580C22—242no—
581C22#549012yes—
582C22D3×#13711yescharacteristic
583C22#577711yescharacteristic
584C22#577812yes—
585C22C2×#93711yescharacteristic
586C4—22no—
587C4—42no—
588C4—42no—
589C4—44no—
590C4—62no—
591C4—121no—
592C4—122no—
593C4—121no—
594C4—242no—
595C4#549012yes—
596C4#579111yescharacteristic
597C4#580511yescharacteristic
598C3#2397711yescharacteristic, complemented, Sylow
599C2—42no—
600C2—241no—
601C2—242no—
602C2#7896912yes—
603C2#7897312yes—
604C2#8212511yescharacteristic
605C2#8398111yescharacteristic
606C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC2×#5532C2
hypercenterC2×#419D3
hypocenterC3#23977
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
3. #84068C2semidirect
4. C2×#5227C2semidirect
5. #71996C2semidirect
6. #71990C2semidirect
7. #71575C2semidirect
8. #70276C2semidirect
9. #70272C2semidirect
10. #70269C2semidirect
11. #70231C2semidirect
50. C2×#419D3semidirect
598. C3#23977semidirect

Maximal subgroups

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

subgroupindex
C2×#55322
C2×C3×#4192
#840682
C2×#52272
#719962
#719902
#715752
#702762
#702722
#702692
#702312
#239773