Finite Group Databasebeta

← #10796769

Subgroups — #10796769

Order 768.

1974 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2C2×#5521C211yesmaximal, characteristic, complemented
3C3×#2896C211yesmaximal, characteristic, complemented
4#84072C211yesmaximal, characteristic, complemented
5C22×#823C211yesmaximal, characteristic, complemented
6#71982C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#70343C214yesmaximal, complemented
9#70323C214yesmaximal, complemented
10#20953—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5830—24no—
16#5830C418yescomplemented
17C2×#947C2212yes—
18C2×#940C2212yes—
19C2×#937—22no—
20C2×#930—22no—
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24#5521—22no—
25#5521—22no—
26#5521—22no—
27#5521C2214yes—
28C2×#893—22no—
29C2×D3×C22⋊C4—22no—
30C2×C4×D12—22no—
31C3×#419C2212yescomplemented
32C3×#418C2212yescomplemented
33C3×#417—22no—
34C3×#416—22no—
35#5233—22no—
36#5232—22no—
37#5225C2212yescomplemented
38#5215C2212yescomplemented
39C2×#823—24no—
40C2×#823C2214yes—
41#4555C2214yescomplemented
42#4532C2214yescomplemented
43C2×#535—31no—
44C2×#518—31no—
45#2896—31no—
46#2896D311yescharacteristic, complemented
47C22×#100—31no—
48#1887—34no—
49#1885—34no—
50#1733—31no—
51#1733—31no—
52C24×D3D412yes—
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×#41C2×C414yescomplemented
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C22×#67—24no—
67C22×D12—21no—
68C22×D12—21no—
69C22×D12—21no—
70C22×D12—21no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12C2×C414yes—
74C22×D12D412yes—
75C22×D12D414yes—
76C22×D12C2312yes—
77C22×C4×D3—24no—
78C2×C3×M16—21no—
79C2×C3×M16—22no—
80C2×C3×M16—22no—
81C2×C3×M16C2312yes—
82C22×C3×C8—21no—
83C3×#117—21no—
84C3×#117—21no—
85C3×#117D412yescomplemented
86C3×#117C2312yes—
87C2×C3×C4⋊C4D412yescomplemented
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×#217—21no—
91C2×#217—21no—
92#947—24no—
93#947C2×C418yescomplemented
94#940—24no—
95#940C2×C418yescomplemented
96#937—216no—
97#930—216no—
98C2×(C2×D3)⋊C4—24no—
99C2×(C2×D3)⋊C4—24no—
100#926—21no—
101#926—21no—
102#926—22no—
103#926C2×C414yescomplemented
104#926D412yescomplemented
105#926C2312yes—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3D412yescomplemented
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2312yes—
110C2×#208—21no—
111C2×#208—22no—
112C2×#208—22no—
113C2×#208C2312yes—
114C22×C3⋊C8—21no—
115#893—28no—
116#893—28no—
117D3×C22⋊C4—28no—
118D3×C22⋊C4—28no—
119C4×D12—28no—
120C4×D12—28no—
121C3×#103—24no—
122C3×#102—24no—
123#835—24no—
124#832—24no—
125#823—28no—
126C22×#41C2×D311yescharacteristic
127C23×D4—31no—
128C22×M16—31no—
129C22×M16C2×D311yescharacteristic
130#535—38no—
131#535—64no—
132C2×#127—32no—
133C2×#124—32no—
134C2×#121—62no—
135C2×#120—62no—
136#518—34no—
137#518—62no—
138#518—62no—
139#518—62no—
140C2×C4×D4—62no—
141C2×C4×D4—62no—
142C2×#117—31no—
143C2×#117—31no—
144C2×#117C2×D311yescharacteristic
145C22×C22⋊C4—62no—
146#419—32no—
147#419C2×D312yescomplemented
148#418—32no—
149#418C2×D312yescomplemented
150#417—22no—
151#417—62no—
152#416—22no—
153#416—62no—
154C2×#100—34no—
155C2×#100—64no—
156#339—34no—
157#339—34no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C23×D3—22no—
161C23×D3—22no—
162C23×D3—24no—
163C23×D3—84no—
164C23×D3C22⋊C418yes—
165C23×D3C2×D414yes—
166C3×#41—24no—
167C3×#41—24no—
168C3×#41—24no—
169C3×#41—24no—
170C3×#41—42no—
171C3×#41—42no—
172C3×#41—42no—
173C3×#41—42no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8C22×C412yescomplemented
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—41no—
187C2×C3×D4—41no—
188C2×C3×D4—42no—
189C2×C3×D4C22×C412yescomplemented
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—41no—
194C22×C3×C4—41no—
195C22×C3×C4—42no—
196C22×C3×C4C22×C412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4#4112yes—
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—216no—
207C2×#67—88no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C2×D12—24no—
211C2×D12—24no—
212C2×D12—24no—
213C2×D12—24no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—24no—
217C2×D12—24no—
218C2×D12—24no—
219C2×D12—24no—
220C2×D12—24no—
221C2×D12—22no—
222C2×D12—22no—
223C2×D12—22no—
224C2×D12—22no—
225C2×D12C22⋊C418yes—
226C2×D12C22×C414yes—
227C2×D12C2×D414yes—
228C2×C4×D3—216no—
229C2×C4×D3—88no—
230C3×M16—22no—
231C3×M16—22no—
232C3×M16—42no—
233C2×C3×C8—22no—
234C2×C3×C8—42no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4C2×D414yes—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C42—22no—
240C3×C42—22no—
241#217—24no—
242#217—22no—
243#217—22no—
244(C2×D3)⋊C4—216no—
245(C2×D3)⋊C4—216no—
246C4⋊Dic3—24no—
247C4⋊Dic3—22no—
248C4⋊Dic3C2×D414yes—
249C4×Dic3—22no—
250C4×Dic3—22no—
251C4×Dic3—22no—
252C4×Dic3C22×C414yescomplemented
253#208—22no—
254#208—22no—
255#208—42no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—42no—
258C25—32no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—21no—
262C2×#41—21no—
263C2×#41—22no—
264C2×#41—21no—
265C2×#41—21no—
266C2×#41C4×D314yescomplemented
267C22×Q8C22×D311yescharacteristic
268C22×D4—21no—
269C22×D4—34no—
270C22×D4—32no—
271C22×D4—34no—
272C22×D4—32no—
273C22×D4—64no—
274C22×D4—62no—
275C22×D4—61no—
276C22×D4—61no—
277C22×D4—62no—
278C22×D4—61no—
279C22×D4—61no—
280C22×D4C22×D311yescharacteristic
281C23×C4—21no—
282C23×C4—64no—
283C23×C4C22×D311yescharacteristic
284C2×M16—21no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—32no—
288C2×M16—61no—
289C2×M16—62no—
290C2×M16—62no—
291C2×M16C22×D312yes—
292C22×C8—21no—
293C22×C8—61no—
294#127—38no—
295#127—64no—
296#124—38no—
297#124—64no—
298#121—616no—
299#120—616no—
300C4×D4—68no—
301C4×D4—68no—
302C4×D4—68no—
303C4×D4—68no—
304#117—21no—
305#117—21no—
306#117—34no—
307#117—32no—
308#117—32no—
309#117—61no—
310#117—62no—
311#117—61no—
312#117#6712yescomplemented
313#117C22×D312yes—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4#6712yescomplemented
317C2×C22⋊C4—21no—
318C2×C22⋊C4—68no—
319C2×C22⋊C4—68no—
320C2×C22⋊C4—64no—
321C2×C22⋊C4—64no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—21no—
325C2×C42—32no—
326C2×C42—61no—
327#103—24no—
328#103—64no—
329#102—24no—
330#102—64no—
331#100—68no—
332C23×C3—22no—
333C23×C3—42no—
334C23×C3—44no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C2×C22⋊C412yes—
338C23×C3#12011yescharacteristic
339C23×C3#12011yescharacteristic
340C23×C3#12312yes—
341C23×C3C22×D411yescharacteristic
342C22×D3—24no—
343C22×D3—28no—
344C22×D3—24no—
345C22×D3—28no—
346C22×D3—48no—
347C22×D3—48no—
348C22×D3—88no—
349C22×D3—84no—
350C22×D3—88no—
351C22×D3C2×C22⋊C418yes—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—44no—
359C3×D4—44no—
360C3×D4—41no—
361C3×D4—41no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—44no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—44no—
368C2×C3×C4—44no—
369C2×C3×C4C2×C22⋊C412yes—
370C2×C3×C4C4×D412yes—
371C2×C3×C4C4×D412yes—
372C2×C3×C4C4×D412yes—
373C2×C3×C4C4×D412yes—
374C2×C3×C4#12011yescharacteristic
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12112yes—
377C2×C3×C4#12112yes—
378C2×C3×C4#12112yes—
379C2×C3×C4#12112yes—
380C2×C3×C4#12312yes—
381C2×C3×C4C23×C411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387#67—816no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—44no—
391C2×Dic3—44no—
392D12—24no—
393D12—24no—
394D12—24no—
395D12—24no—
396D12—28no—
397D12—28no—
398C4×D3—816no—
399C3×C8—42no—
400C3⋊C8—42no—
401C24—21no—
402C24—34no—
403C24—38no—
404C24—62no—
405C24—64no—
406C24—62no—
407C24—244no—
408C24D3×D411yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—42no—
414#41—42no—
415#41—42no—
416#41—42no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8C2×C4×D312yescomplemented
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—38no—
430C2×D4—34no—
431C2×D4—34no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—616no—
436C2×D4—64no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—64no—
440C2×D4—64no—
441C2×D4—64no—
442C2×D4—64no—
443C2×D4—62no—
444C2×D4—62no—
445C2×D4—64no—
446C2×D4—64no—
447C2×D4—62no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4—248no—
452C2×D4C2×C4×D312yescomplemented
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—616no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—248no—
463C22×C4C2×C4×D312yes—
464C22×C4#23512yes—
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D412yes—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4#23712yes—
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C23×D311yescharacteristic
473M16—22no—
474M16—22no—
475M16—42no—
476M16—62no—
477M16—62no—
478M16—122no—
479C2×C8—22no—
480C2×C8—42no—
481C2×C8—62no—
482C2×C8—122no—
483C4⋊C4—22no—
484C4⋊C4—34no—
485C4⋊C4—64no—
486C4⋊C4—62no—
487C4⋊C4C2×#6714yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—616no—
491C22⋊C4—616no—
492C22⋊C4—64no—
493C22⋊C4—62no—
494C22⋊C4—62no—
495C42—22no—
496C42—22no—
497C42—34no—
498C42—62no—
499C42—62no—
500C42—62no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—82no—
509C22×C3—82no—
510C22×C3#38612yes—
511C22×C3#38612yes—
512C22×C3C22×C22⋊C411yescharacteristic
513C22×C3C2×#12011yescharacteristic
514C22×C3C2×#12311yescharacteristic
515C2×D3—28no—
516C2×D3—416no—
517C2×D3—48no—
518C2×D3—48no—
519C2×D3—84no—
520C2×D3—88no—
521C3×C4—44no—
522C3×C4—42no—
523C3×C4#38612yes—
524C3×C4#38612yes—
525C3×C4C2×C4×D411yescharacteristic
526C3×C4C2×C4×D411yescharacteristic
527C3×C4C2×#12111yescharacteristic
528C3×C4C2×#12111yescharacteristic
529Dic3—44no—
530Dic3—44no—
531C23—22no—
532C23—38no—
533C23—42no—
534C23—44no—
535C23—41no—
536C23—41no—
537C23—68no—
538C23—64no—
539C23—68no—
540C23—64no—
541C23—128no—
542C23—128no—
543C23—244no—
544C23—248no—
545C23—248no—
546C23D3×C22⋊C412yes—
547C23#88312yes—
548C23#93711yescharacteristic
549C23#93711yescharacteristic
550C23C2×D3×D411yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—41no—
554Q8—41no—
555D4—22no—
556D4—22no—
557D4—44no—
558D4—44no—
559D4—41no—
560D4—41no—
561D4—68no—
562D4—68no—
563D4—64no—
564D4—64no—
565D4—64no—
566D4—64no—
567D4—2416no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4—2416no—
580C2×C4D3×C22⋊C412yes—
581C2×C4#67⋊C412yes—
582C2×C4#67⋊C412yes—
583C2×C4#88312yes—
584C2×C4#88412yes—
585C2×C4#88412yes—
586C2×C4C4×#6712yes—
587C2×C4C4×#6712yes—
588C2×C4#93711yescharacteristic
589C2×C4#93711yescharacteristic
590C2×C4#93812yes—
591C2×C4#93812yes—
592C2×C4C22×C4×D311yescharacteristic
593C2×C4C2×#23511yescharacteristic
594C2×C4C2×D3×D411yescharacteristic
595C2×C4C2×D3×D411yescharacteristic
596C2×C4C2×#23711yescharacteristic
597C2×C4C22×#6711yescharacteristic
598C8—42no—
599C8—122no—
600C2×C3—21no—
601C2×C3—22no—
602C2×C3—21no—
603C2×C3—22no—
604C2×C3—82no—
605C2×C3C2×#38611yescharacteristic
606C2×C3#301912yes—
607D3—48no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—68no—
616C22—82no—
617C22—82no—
618C22—128no—
619C22—1216no—
620C22—128no—
621C22—244no—
622C22—248no—
623C22#495612yes—
624C22#495612yes—
625C22C2×D3×C22⋊C411yescharacteristic
626C22C2×#88311yescharacteristic
627C22C2×#93711yescharacteristic
628C4—44no—
629C4—42no—
630C4—124no—
631C4—124no—
632C4#495612yes—
633C4#495612yes—
634C4C2×#67⋊C411yescharacteristic
635C4C2×#88411yescharacteristic
636C4C2×C4×#6711yescharacteristic
637C4C2×#93811yescharacteristic
638C3#2095311yescharacteristic, complemented, Sylow
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—82no—
644C2—128no—
645C2C2×#495611yescharacteristic
646C2#8400512yes—
subgroupG/H
centerC22C2×#937
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2896C2
socleC22×C3C2×#120
prime radicalC2×#5830C2
cosocleC22×C3×C4C24
hypercenter#2896D3
hypocenterC3#20953
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C2×#5521C2semidirect
3. C3×#2896C2semidirect
4. #84072C2semidirect
5. C22×#823C2semidirect
6. #71982C2semidirect
7. #71574C2semidirect
8. #70343C2semidirect
9. #70323C2semidirect
16. #5830C4semidirect
31. C3×#419C22semidirect
32. C3×#418C22semidirect
37. #5225C22semidirect
38. #5215C22semidirect
41. #4555C22semidirect
42. #4532C22semidirect
46. #2896D3semidirect
60. C2×C3×#41C2×C4semidirect
85. C3×#117D4semidirect
87. C2×C3×C4⋊C4D4semidirect
93. #947C2×C4semidirect
95. #940C2×C4semidirect
103. #926C2×C4semidirect
104. #926D4semidirect
107. C2×C4⋊Dic3D4semidirect
147. #419C2×D3semidirect
149. #418C2×D3semidirect
179. C2×C3×Q8C22×C4semidirect
189. C2×C3×D4C22×C4semidirect
252. C4×Dic3C22×C4semidirect
266. C2×#41C4×D3semidirect
312. #117#67semidirect
316. C2×C4⋊C4#67semidirect
422. C2×Q8C2×C4×D3semidirect
452. C2×D4C2×C4×D3semidirect
638. C3#20953semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C2×#55212
C3×#28962
#840722
C22×#8232
#719822
#715742
#703432
#703232
#209533