Finite Group Databasebeta

← #10767939

Subgroups — #10767939

Order 768.

1034 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3491C211yesmaximal, characteristic, complemented
2#83143C212yesmaximal, complemented
3#82273C211yesmaximal, characteristic, complemented
4#71892C211yesmaximal, characteristic, complemented
5#71890C211yesmaximal, characteristic, complemented
6#71888C212yesmaximal, complemented
7#71886C212yesmaximal, complemented
8#71845C211yesmaximal, characteristic
9#71843C211yesmaximal, characteristic, complemented
10#71755C212yesmaximal, complemented
11#71572C211yesmaximal, characteristic, complemented
12#18176—31nomaximal, Sylow
13C3×#552—22no—
14C3×#552—22no—
15C3×#551—22no—
16C3×#551—22no—
17C3×#543C2211yescharacteristic
18C3×#539—21no—
19C3×#539C2212yes—
20C3×#534C2211yescharacteristic
21C3×#529—21no—
22C3×#529C2212yes—
23C2×C3×#123—21no—
24C3×#518—21no—
25C2×C3×#117C2211yescharacteristic
26C2×C4×#67—21no—
27C2×C4×D12C2211yescharacteristic, complemented
28#5066—22no—
29#5057—22no—
30#5046C2212yescomplemented
31#5043C2212yescomplemented
32#5036C2212yes—
33#5033C2212yes—
34#5026—22no—
35#5014—22no—
36#4994—21no—
37#4991C2211yescharacteristic, complemented
38#4981C2211yescharacteristic, complemented
39#4961—21no—
40#4960C2211yescharacteristic
41#4953C2211yescharacteristic
42C4×(C2×D3)⋊C4—21no—
43C4×C4⋊Dic3C2211yescharacteristic, complemented
44#4926C2212yescomplemented
45#4538C2212yescomplemented
46#4538C2212yescomplemented
47#4537C2212yes—
48#4537C2212yes—
49#4528C2212yescomplemented
50#4525C2211yescharacteristic
51#4525C2211yescharacteristic
52#3491D311yescharacteristic, complemented
53#3114—32no—
54#2392—31no—
55#1609—32no—
56#1608—31no—
57#1607—31no—
58#1597—32no—
59#1596—31no—
60#1595—31no—
61#1509—32no—
62#1464—31no—
63C22×C3×D4—21no—
64C23×C3×C4D411yescharacteristic
65C22×#67—21no—
66C22×D12C2311yescharacteristic
67C22×C4×D3—21no—
68C3×#126—22no—
69C3×#126—22no—
70C3×#126—22no—
71C3×#126—22no—
72C3×#125—21no—
73C3×#125—22no—
74C3×#125—22no—
75C3×#125D412yescomplemented
76C3×#124—21no—
77C3×#124D412yescomplemented
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—22no—
82C3×#123—22no—
83C3×#123—22no—
84C3×#123—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8C2312yes—
91C3×#121—21no—
92C3×#121C2312yes—
93C3×#120—22no—
94C3×C4×Q8—21no—
95C3×C4×Q8D412yescomplemented
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—21no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4D412yescomplemented
102C3×#117—22no—
103C3×#117—22no—
104C3×#117—22no—
105C3×#117—22no—
106C3×#117—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4C2311yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×#217—21no—
120C2×#215—22no—
121C2×#213—22no—
122C4×#67—44no—
123C2×(C2×D3)⋊C4—21no—
124C2×(C2×D3)⋊C4—21no—
125C2×(C2×D3)⋊C4—21no—
126C2×(C2×D3)⋊C4—21no—
127C2×C4⋊Dic3—21no—
128C2×C4⋊Dic3C2311yescharacteristic
129C2×Dic3⋊C4—21no—
130C2×C4×Dic3—21no—
131C4×D12—22no—
132C4×D12—22no—
133#835—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95C2311yescharacteristic
138C3⋊#95C2311yescharacteristic
139#830C2312yes—
140#830C2312yes—
141#809—22no—
142#809—22no—
143#808—22no—
144#807—22no—
145C4⋊(C3⋊C8)—22no—
146#803—22no—
147#552—22no—
148#552—22no—
149#551—22no—
150#551—22no—
151#543C2×D311yescharacteristic
152#539—21no—
153#539C2×D312yes—
154#534C2×D311yescharacteristic
155#529—21no—
156#529C2×D312yes—
157C2×#123—21no—
158#518—21no—
159C2×C4×D4—31no—
160C2×C4×D4—61no—
161C2×#117C2×D311yescharacteristic
162#483—32no—
163#480—32no—
164#478—62no—
165#463—62no—
166#450—32no—
167#447—32no—
168#442—62no—
169#440—62no—
170#432—32no—
171#398—31no—
172#397—31no—
173#396—61no—
174#394—31no—
175#394—61no—
176#392—31no—
177C4×C4⋊C4—31no—
178C4×C22⋊C4—61no—
179#328—32no—
180#328—32no—
181#327—32no—
182#327—32no—
183#324—32no—
184#323—31no—
185#323—31no—
186C24×C3—21no—
187C24×C3C2×D411yescharacteristic
188C23×D3—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8C2×D412yes—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—42no—
195C2×C3×D4—42no—
196C2×C3×D4C2×D412yes—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—41no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—42no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C2×D12—22no—
219C2×D12—42no—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—41no—
223C2×C4×D3—41no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4C2×D412yes—
237C3×C4⋊C4C2×D412yes—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C42—22no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—22no—
254C3×C42C2×D412yes—
255C3×C42C2×D412yes—
256#217—42no—
257#215—44no—
258#213—44no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263C4⋊Dic3—22no—
264C4⋊Dic3—42no—
265Dic3⋊C4—42no—
266C4×Dic3—42no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C22×D4—21no—
270C22×D4—31no—
271C22×D4—61no—
272C23×C4—61no—
273C23×C4#6711yescharacteristic
274C2×SD16—62no—
275C2×D8—62no—
276#126—22no—
277#126—22no—
278#126—22no—
279#126—22no—
280#125—21no—
281#125—22no—
282#125—22no—
283#125#6712yescomplemented
284#124—21no—
285#124#6712yescomplemented
286#123—22no—
287#123—22no—
288#123—22no—
289#123—22no—
290#123—22no—
291#123—22no—
292#123—22no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—21no—
297C22⋊Q8—22no—
298C22⋊Q8C22×D312yes—
299#121—21no—
300#121C22×D312yes—
301#120—22no—
302C4×Q8—21no—
303C4×Q8#6712yescomplemented
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—21no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—62no—
310C4×D4—62no—
311C4×D4—124no—
312C4×D4#6712yescomplemented
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4#6711yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4#6711yescharacteristic
335C2×C22⋊C4C22×D311yescharacteristic
336C2×C42—61no—
337C2×C42#6711yescharacteristic
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341C4⋊C8—62no—
342#103—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C22⋊C8—32no—
347C22⋊C8—32no—
348#97—62no—
349#95—31no—
350#95—31no—
351#95—61no—
352#95—61no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—22no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—42no—
359C23×C3—81no—
360C23×C3—81no—
361C23×C3AGL(1,ℤ/8)12yes—
362C23×C3C22×D411yescharacteristic
363C22×D3—22no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C3×Q8—42no—
371C3×Q8—81no—
372C3×D4—42no—
373C3×D4—44no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—81no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—82no—
395C2×C3×C4—81no—
396C2×C3×C4—82no—
397C2×C3×C4—81no—
398C2×C3×C4#12012yes—
399C2×C3×C4#12011yescharacteristic
400C2×C3×C4#12011yescharacteristic
401C2×C3×C4C22×D412yes—
402#67—48no—
403C2×Dic3—22no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412D12—24no—
413D12—42no—
414C4×D3—82no—
415C4×D3—82no—
416C3⋊C8—42no—
417C3⋊C8—42no—
418C24—21no—
419C24—61no—
420C24C2×#6711yescharacteristic
421C2×Q8—21no—
422C2×Q8C2×#6712yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—62no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4—122no—
432C2×D4—122no—
433C2×D4C2×#6712yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—121no—
449C22×C4—121no—
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16—124no—
457D8—124no—
458C2×C8—62no—
459C2×C8—62no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—62no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4C2×#6712yes—
476C4⋊C4C2×#6712yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—122no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—22no—
498C42—122no—
499C42C2×#6712yes—
500C42C2×#6712yes—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—22no—
504C22×C3—42no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—81no—
511C22×C3—162no—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
514C22×C3#57511yescharacteristic
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—42no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C2×D3—81no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—22no—
525C3×C4—22no—
526C3×C4—41no—
527C3×C4—42no—
528C3×C4—41no—
529C3×C4—42no—
530C3×C4—41no—
531C3×C4—42no—
532Dic3—42no—
533Dic3—81no—
534Dic3—81no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—62no—
542C23—81no—
543C23—81no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23#94912yes—
551C23C22×#6711yescharacteristic
552Q8—42no—
553Q8—81no—
554D4—42no—
555D4—44no—
556D4—42no—
557D4—42no—
558D4—64no—
559D4—81no—
560D4—122no—
561D4—128no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—62no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—242no—
593C2×C4—242no—
594C2×C4#95312yes—
595C2×C4#95311yescharacteristic
596C2×C4#95311yescharacteristic
597C2×C4C22×#6712yes—
598C8—122no—
599C8—122no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—42no—
603C2×C3—82no—
604C2×C3#300611yescharacteristic
605C2×C3#301811yescharacteristic
606C2×C3#302511yescharacteristic
607D3—82no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—122no—
619C22—122no—
620C22—122no—
621C22—162no—
622C22—241no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22C2×#94911yescharacteristic
627C22#582211yescharacteristic
628C22C2×#95311yescharacteristic
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—122no—
640C4—241no—
641C4—241no—
642C3#1817611yescharacteristic, complemented, Sylow
643C2—21no—
644C2—21no—
645C2—42no—
646C2—82no—
647C2—242no—
648C2#8408211yescharacteristic
649C2#8412111yescharacteristic
650C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3491C2
socleC22×C3C2×#120
prime radical#71845C2
hypercenter#3491D3
hypocenterC3#18176
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3491

Complements

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

NH ≅ G/Ntype
1. C3×#3491C2semidirect
2. #83143C2semidirect
3. #82273C2semidirect
4. #71892C2semidirect
5. #71890C2semidirect
6. #71888C2semidirect
7. #71886C2semidirect
9. #71843C2semidirect
10. #71755C2semidirect
11. #71572C2semidirect
27. C2×C4×D12C22semidirect
30. #5046C22semidirect
31. #5043C22semidirect
37. #4991C22semidirect
38. #4981C22semidirect
43. C4×C4⋊Dic3C22semidirect
44. #4926C22semidirect
45. #4538C22semidirect
46. #4538C22semidirect
49. #4528C22semidirect
52. #3491D3semidirect
75. C3×#125D4semidirect
77. C3×#124D4semidirect
95. C3×C4×Q8D4semidirect
101. C3×C4×D4D4semidirect
283. #125#67semidirect
285. #124#67semidirect
303. C4×Q8#67semidirect
312. C4×D4#67semidirect
642. C3#18176semidirect

Maximal subgroups

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

subgroupindex
C3×#34912
#831432
#822732
#718922
#718902
#718882
#718862
#718452
#718432
#717552
#715722
#181763