Finite Group Databasebeta

← #10848771

Subgroups — #10848771

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic, complemented
2C3×#2856C211yesmaximal, characteristic, complemented
3C2×#5074C211yesmaximal, characteristic, complemented
4C2×#5071C211yesmaximal, characteristic, complemented
5#71685C212yesmaximal, complemented
6#71684C212yesmaximal, complemented
7#71529C211yesmaximal, characteristic, complemented
8#69660C212yesmaximal, complemented
9#69657C212yesmaximal, complemented
10#69583C211yesmaximal, characteristic, complemented
11#69581C211yesmaximal, characteristic, complemented
12#25605—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#121—21no—
15C2×C3×#121—21no—
16C2×C3×#121C2212yes—
17C22×C3×C42C2211yescharacteristic
18C22×#214C2212yes—
19C22×#213C2212yes—
20C2×#930C2211yescharacteristic
21C2×#929—21no—
22C2×C4×#67—21no—
23C2×#924C2211yescharacteristic
24#5479—24no—
25#5479—24no—
26C2×#876—21no—
27C2×#875C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×#872—21no—
30C2×#870C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C3×#390—21no—
33C3×#390—21no—
34C3×#390C2212yes—
35C3×#384C2211yescharacteristic
36C2×#832C2212yes—
37C2×#832C2212yes—
38C2×#830C2212yes—
39#5074—24no—
40#5071—24no—
41C2×#810C2212yes—
42C2×#809C2212yes—
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#529—31no—
49C2×#496—31no—
50C2×#490—31no—
51#2856D311yescharacteristic, complemented
52#2000—32no—
53#1999—32no—
54#1992—31no—
55#1959—31no—
56#1880—32no—
57#1879—32no—
58#1749—31no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2312yes—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C22×#67—21no—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×Dic6C2311yescharacteristic
70C3×#127—22no—
71C3×#127—22no—
72C3×#121—42no—
73C3×#121—42no—
74C3×#121—44no—
75C3×#121—44no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—22no—
83C2×C3×C42—22no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#214—24no—
90C2×#214—24no—
91C2×#214—42no—
92C2×#213—24no—
93C2×#213—24no—
94C2×#213—42no—
95#930—22no—
96#930—22no—
97#929—28no—
98C4×#67—28no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105#924—22no—
106#924—22no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114#876—28no—
115#875—22no—
116#875D414yescomplemented
117C4×D12—22no—
118C4×D12—22no—
119#872—28no—
120#870—22no—
121#870D414yescomplemented
122C4×Dic6—22no—
123C4×Dic6—22no—
124#832—24no—
125#832—24no—
126#830—24no—
127#810—42no—
128#809—42no—
129#803—22no—
130#803—22no—
131C22×SD16—32no—
132C22×D8—32no—
133C2×#127C2×D311yescharacteristic
134#529—64no—
135#529—64no—
136C2×#126—61no—
137C2×#125—31no—
138C2×#124—31no—
139C2×#123—61no—
140C2×C22⋊Q8—31no—
141C2×#121—21no—
142C2×#121—21no—
143C2×#121—31no—
144C2×#121C2×D312yes—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C2×#117—61no—
149C22×C42C2×D311yescharacteristic
150#496—64no—
151#490—64no—
152C2×#103—32no—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102—32no—
156C2×C22⋊C8—32no—
157C2×#97—31no—
158C2×#97—31no—
159#390—21no—
160#390—21no—
161#390C2×D312yes—
162#384C2×D311yescharacteristic
163#340—32no—
164#340—32no—
165#336—32no—
166C24×C3—22no—
167C24×C3C2×D411yescharacteristic
168C23×D3—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4—44no—
180C2×C3×D4—44no—
181C2×C3×D4—44no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—41no—
194C22×C3×C4—41no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C2×#67—24no—
202C2×#67—42no—
203C2×#67—42no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—21no—
210C2×D12C2×D412yes—
211C2×C4×D3—24no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6C2×D412yes—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C22⋊C4—44no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—44no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42C2×D412yes—
230C3×C42C2×D412yes—
231#217—24no—
232#214—48no—
233#213—48no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3C2×D412yes—
241C4⋊Dic3C2×D412yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—24no—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246C4×Dic3—24no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8#4114yes—
253C22×Q8—31no—
254C22×D4—22no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—31no—
258C22×D4—61no—
259C22×D4C22×D312yes—
260C23×C4—61no—
261C23×C4#6711yescharacteristic
262C23×C4#6711yescharacteristic
263C23×C4C22×D311yescharacteristic
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×D8—64no—
268C2×D8—64no—
269C2×D8—122no—
270C22×C8—32no—
271C22×C8—32no—
272#127—22no—
273#127—22no—
274#126—68no—
275#125—34no—
276#125—62no—
277#124—34no—
278#124—62no—
279#123—68no—
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282#121—42no—
283#121—42no—
284#121—44no—
285#121—44no—
286#121—62no—
287#121—62no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—68no—
293#117—68no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4C22×D312yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—61no—
314C2×C42#6711yescharacteristic
315C2×C42#6711yescharacteristic
316C2×C42C22×D311yescharacteristic
317C2×C42C22×D311yescharacteristic
318#103—122no—
319#102—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—64no—
323#97—62no—
324#97—62no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3AGL(1,ℤ/8)12yes—
339C23×C3C22×D411yescharacteristic
340C22×D3—22no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—84no—
352C3×D4—84no—
353C3×D4—88no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—44no—
367C2×C3×C4—44no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4#12012yes—
374C2×C3×C4#12012yes—
375C2×C3×C4AGL(1,ℤ/8)12yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4112yes—
379C2×C3×C4H(5,𝔽2)12yes—
380#67—48no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390D12—24no—
391D12—42no—
392C4×D3—48no—
393Dic6—24no—
394Dic6—42no—
395C3⋊C8—24no—
396C3⋊C8—42no—
397C24—22no—
398C24—61no—
399C24C2×#6711yescharacteristic
400C2×Q8—32no—
401C2×Q8—61no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—32no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—64no—
421C2×D4—61no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—122no—
425C2×D4—122no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—64no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C23×D311yescharacteristic
449SD16—128no—
450D8—128no—
451C2×C8—34no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—32no—
460C4⋊C4—32no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—64no—
464C4⋊C4—64no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C22⋊C4—64no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—64no—
482C42D3×D412yes—
483C42D3×D412yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3C2×#12011yescharacteristic
499C22×C3#53211yescharacteristic
500C22×C3#53511yescharacteristic
501C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
502C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
503C22×C3#57511yescharacteristic
504C22×C3#57511yescharacteristic
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—82no—
518C3×C4#54612yes—
519C3×C4#55012yes—
520Dic3—42no—
521Dic3—42no—
522Dic3—42no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—62no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23#93212yes—
544C23C22×#6711yescharacteristic
545Q8—64no—
546Q8—122no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—64no—
552D4—84no—
553D4—84no—
554D4—88no—
555D4—128no—
556D4—122no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—128no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4#93212yes—
587C2×C4#93712yes—
588C2×C4#93712yes—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4#100412yes—
592C2×C4D3×#4112yes—
593C8—64no—
594C8—122no—
595C2×C3—42no—
596C2×C3—162no—
597C2×C3#243111yescharacteristic
598C2×C3#300611yescharacteristic
599C2×C3#300611yescharacteristic
600C2×C3#317111yescharacteristic
601C2×C3#318011yescharacteristic
602C2×C3#321711yescharacteristic
603C2×C3#321911yescharacteristic
604D3—82no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—122no—
620C22—122no—
621C22—122no—
622C22—241no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22C2×#93211yescharacteristic
627C22C2×#93211yescharacteristic
628C22C2×#93711yescharacteristic
629C22#580511yescharacteristic
630C22#582211yescharacteristic
631C22#582211yescharacteristic
632C22#583211yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4—122no—
640C4—122no—
641C4—122no—
642C4#559912yes—
643C4#561212yes—
644C3#2560511yescharacteristic, complemented, Sylow
645C2—42no—
646C2—162no—
647C2—242no—
648C2#8262411yescharacteristic
649C2#8262411yescharacteristic
650C2#8265211yescharacteristic
651C2#8316711yescharacteristic
652C2#8329211yescharacteristic
653C2#8329311yescharacteristic
654C2#8330911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2856C2
socleC23×C3C22×D4
hypercenter#2856D3
hypocenterC3#25605
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×#5479C2semidirect
2. C3×#2856C2semidirect
3. C2×#5074C2semidirect
4. C2×#5071C2semidirect
5. #71685C2semidirect
6. #71684C2semidirect
7. #71529C2semidirect
8. #69660C2semidirect
9. #69657C2semidirect
10. #69583C2semidirect
11. #69581C2semidirect
51. #2856D3semidirect
116. #875D4semidirect
121. #870D4semidirect
644. C3#25605semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C3×#28562
C2×#50742
C2×#50712
#716852
#716842
#715292
#696602
#696572
#695832
#695812
#256053