Finite Group Databasebeta

← #10974292

Subgroups — #10974292

Order 768.

4142 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#953C211yesmaximal, characteristic
2C2×#5486C211yesmaximal, characteristic, complemented
3C2×C3×#379C211yesmaximal, characteristic, complemented
4C2×#5247C211yesmaximal, characteristic, complemented
5#82128C218yesmaximal, complemented
6#82121C211yesmaximal, characteristic, complemented
7#82095C211yesmaximal, characteristic, complemented
8#82094C211yesmaximal, characteristic, complemented
9#73853C214yesmaximal, complemented
10#73832C214yesmaximal, complemented
11#73826C218yesmaximal, complemented
12#35350—31nomaximal, Sylow
13C26×C3C2211yescharacteristic
14C23×#67C2211yescharacteristic
15C23×#67C2212yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2214yescomplemented
19C2×#953—28no—
20C2×#953C2214yes—
21C2×#953C2214yes—
22C22×#217C2212yes—
23C22×#217C2211yescharacteristic
24C22×#217C2211yescharacteristic
25C22×#217C2211yescharacteristic
26C22×#217C2214yescomplemented
27C2×C4×#67C2214yescomplemented
28C2×#883C2218yescomplemented
29#5486—28no—
30C2×#67⋊C4C2214yescomplemented
31C2×D3×C22⋊C4C2212yescomplemented
32C2×#877C2212yescomplemented
33C3×#379C2214yes—
34C3×#379C2214yescomplemented
35#5247C2214yes—
36#5247C2214yescomplemented
37#5213C2214yescomplemented
38#5212C2214yescomplemented
39#5210C2214yescomplemented
40#4961C2218yescomplemented
41#4959C2218yescomplemented
42#4958C2214yescomplemented
43#4956C2218yescomplemented
44#4955C2212yescomplemented
45#4952C2218yescomplemented
46#4951C2218yescomplemented
47#4950C2214yescomplemented
48#4950C2214yescomplemented
49#4949C2218yescomplemented
50#4946C2214yescomplemented
51#4945C2218yescomplemented
52#4943C2212yescomplemented
53#4942C2214yescomplemented
54C22×#120—31no—
55C2×#518—31no—
56#2482—34no—
57#2418—38no—
58#2418—38no—
59#2405—34no—
60#2315—31no—
61#2315—31no—
62#2308—31no—
63C2×#379—31no—
64C2×#379D311yescharacteristic, complemented
65C25×C3—24no—
66C25×C3—24no—
67C25×C3—24no—
68C25×C3—48no—
69C25×C3C2312yes—
70C25×C3C2312yes—
71C25×C3C2312yes—
72C25×C3C2311yescharacteristic
73C24×D3C2311yescharacteristic
74C23×C3×C4C2312yes—
75C23×C3×C4C2312yes—
76C22×#67—24no—
77C22×#67—24no—
78C22×#67—28no—
79C22×#67—48no—
80C22×#67D418yes—
81C22×#67C2318yes—
82C22×#67C2312yes—
83C22×#67C2312yes—
84C23×Dic3C2311yescharacteristic
85C23×Dic3C2312yes—
86C23×Dic3C2312yes—
87C23×Dic3C2312yes—
88C22×C4×D3C2314yescomplemented
89C2×C3×C22⋊C4—24no—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4—48no—
92C2×C3×C22⋊C4C2312yes—
93C2×C3×C22⋊C4C2312yes—
94C2×C3×C22⋊C4C2312yes—
95C2×C3×C22⋊C4C2312yes—
96C2×C3×C22⋊C4C2318yes—
97C2×C3×C22⋊C4C2318yes—
98#953—216no—
99#953—216no—
100C2×#217—24no—
101C2×#217—24no—
102C2×#217—24no—
103C2×#217—28no—
104C2×#217—48no—
105C2×#217C2312yes—
106C2×#217C2312yes—
107C2×#217C2318yes—
108C2×#217C2318yes—
109C2×#217C2312yes—
110C2×#217C2312yes—
111C2×#217C2312yes—
112C2×#217C2312yes—
113C2×#217C2318yes—
114C4×#67—216no—
115C2×(C2×D3)⋊C4C2314yescomplemented
116C2×(C2×D3)⋊C4C2318yescomplemented
117C2×C4⋊Dic3C2318yescomplemented
118C2×Dic3⋊C4C2314yescomplemented
119C2×Dic3⋊C4C2318yescomplemented
120C2×C4×Dic3C2314yescomplemented
121#883—416no—
122#67⋊C4—216no—
123D3×C22⋊C4—28no—
124#877—28no—
125C3⋊#95C2318yescomplemented
126C3⋊#95C2318yescomplemented
127C3⋊#95C2318yescomplemented
128C26C2×D311yescharacteristic
129C23×D4—31no—
130C23×D4—32no—
131C2×#123—38no—
132C2×#120—34no—
133C2×#120—34no—
134C2×#120—68no—
135#518—68no—
136C2×C4×D4—34no—
137C2×C4×D4—34no—
138C2×#117—32no—
139C22×C22⋊C4—31no—
140C22×C22⋊C4—31no—
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—34no—
144C22×C22⋊C4—32no—
145C22×C22⋊C4C2×D311yescharacteristic
146C22×C22⋊C4C2×D311yescharacteristic
147C22×C22⋊C4C2×D314yescomplemented
148#399—38no—
149#397—38no—
150#394—38no—
151#394—38no—
152#388—38no—
153#388—38no—
154#386—34no—
155#386—34no—
156#386—38no—
157#385—34no—
158#385—34no—
159#385—34no—
160#385—34no—
161#381—32no—
162#380—34no—
163#380—34no—
164#379—34no—
165#379—34no—
166#379—32no—
167#379C2×D314yes—
168#379C2×D314yescomplemented
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3—28no—
173C24×C3—28no—
174C24×C3—28no—
175C24×C3—24no—
176C24×C3—24no—
177C24×C3—24no—
178C24×C3—24no—
179C24×C3—24no—
180C24×C3—24no—
181C24×C3—44no—
182C24×C3—44no—
183C24×C3—44no—
184C24×C3—44no—
185C24×C3—432no—
186C24×C3—432no—
187C24×C3—432no—
188C24×C3—416no—
189C24×C3C2×D414yes—
190C24×C3#4114yes—
191C24×C3#4114yes—
192C24×C3C2414yes—
193C24×C3C2411yescharacteristic
194C24×C3C2411yescharacteristic
195C24×C3C2411yescharacteristic
196C23×D3—22no—
197C23×D3—22no—
198C23×D3—44no—
199C23×D3C2×D414yes—
200C23×D3C2412yes—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—44no—
204C22×C3×C4—44no—
205C22×C3×C4C2414yes—
206C22×C3×C4C2414yes—
207C2×#67—232no—
208C2×#67—24no—
209C2×#67—24no—
210C2×#67—24no—
211C2×#67—24no—
212C2×#67—216no—
213C2×#67—216no—
214C2×#67—48no—
215C2×#67—432no—
216C2×#67C2×D418yes—
217C2×#67C2×D418yes—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—24no—
221C22×Dic3—24no—
222C22×Dic3—24no—
223C22×Dic3—44no—
224C22×Dic3—44no—
225C22×Dic3—44no—
226C22×Dic3C2×D414yes—
227C22×Dic3C2414yes—
228C22×Dic3C2414yes—
229C22×Dic3C2414yes—
230C22×Dic3C2412yes—
231C2×C4×D3—28no—
232C2×C4×D3—48no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—216no—
238C3×C22⋊C4—48no—
239#217—24no—
240#217—24no—
241#217—24no—
242#217—24no—
243#217—24no—
244#217—24no—
245#217—216no—
246#217—216no—
247#217—48no—
248(C2×D3)⋊C4—28no—
249(C2×D3)⋊C4—48no—
250C4⋊Dic3—48no—
251Dic3⋊C4—28no—
252Dic3⋊C4—48no—
253C4×Dic3—28no—
254C25—24no—
255C25—24no—
256C25—24no—
257C25—31no—
258C25—48no—
259C25C22×D312yes—
260C25C22×D312yes—
261C25C22×D312yes—
262C25C22×D311yescharacteristic
263C22×D4—38no—
264C22×D4—32no—
265C22×D4—32no—
266C22×D4—38no—
267C22×D4—64no—
268C22×D4—64no—
269C22×D4—68no—
270C22×D4—128no—
271C23×C4—31no—
272C23×C4—32no—
273C23×C4—32no—
274C23×C4—32no—
275C23×C4—34no—
276C23×C4C22×D312yes—
277C23×C4C22×D312yes—
278#123—1216no—
279#120—616no—
280#120—616no—
281C4×D4—616no—
282C4×D4—616no—
283#117—68no—
284C2×C4⋊C4—34no—
285C2×C4⋊C4—38no—
286C2×C4⋊C4—38no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—32no—
290C2×C22⋊C4—32no—
291C2×C22⋊C4—32no—
292C2×C22⋊C4—32no—
293C2×C22⋊C4—38no—
294C2×C22⋊C4—38no—
295C2×C22⋊C4—32no—
296C2×C22⋊C4—32no—
297C2×C22⋊C4—34no—
298C2×C22⋊C4—38no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—68no—
302C2×C22⋊C4—68no—
303C2×C22⋊C4—64no—
304C2×C22⋊C4—64no—
305C2×C22⋊C4—64no—
306C2×C22⋊C4—128no—
307C2×C22⋊C4C22×D312yes—
308C2×C22⋊C4C22×D312yes—
309C2×C22⋊C4C22×D312yes—
310C2×C22⋊C4C22×D312yes—
311C2×C22⋊C4C22×D318yes—
312C2×C22⋊C4C22×D318yes—
313C2×C42—34no—
314#95—38no—
315#95—38no—
316#95—38no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—216no—
321C23×C3—216no—
322C23×C3—216no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3—24no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—48no—
330C23×C3—48no—
331C23×C3—48no—
332C23×C3—416no—
333C23×C3—416no—
334C23×C3—416no—
335C23×C3—416no—
336C23×C3—416no—
337C23×C3—416no—
338C23×C3—48no—
339C23×C3—48no—
340C23×C3—48no—
341C23×C3—4128no—
342C23×C3—48no—
343C23×C3—48no—
344C23×C3—48no—
345C23×C3#123116yescomplemented
346C23×C3C22×D412yes—
347C23×C3C22×D412yes—
348C23×C3C22×D412yes—
349C23×C3C2×#4112yes—
350C23×C3C2×#4112yes—
351C23×C3C2×#4112yes—
352C23×C3C2×#4112yes—
353C23×C3C2×#4112yes—
354C23×C3C2×#4112yes—
355C23×C3H(5,𝔽2)18yes—
356C23×C3C2511yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—28no—
360C22×D3—28no—
361C22×D3—48no—
362C22×D3—84no—
363C22×D3—84no—
364C22×D3C22×D414yes—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—28no—
368C2×C3×C4—28no—
369C2×C3×C4—84no—
370C2×C3×C4—84no—
371#67—216no—
372#67—216no—
373#67—432no—
374C2×Dic3—28no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—28no—
381C2×Dic3—28no—
382C2×Dic3—28no—
383C2×Dic3—48no—
384C2×Dic3—84no—
385C2×Dic3—84no—
386C2×Dic3C22×D414yes—
387C4×D3—88no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—28no—
392C24—28no—
393C24—28no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—24no—
400C24—32no—
401C24—34no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24—44no—
406C24—432no—
407C24—432no—
408C24—432no—
409C24—416no—
410C24—62no—
411C24—62no—
412C24—124no—
413C24#23514yes—
414C24D3×D414yes—
415C24#23714yes—
416C24C23×D314yes—
417C24C23×D311yescharacteristic
418C24C23×D311yescharacteristic
419C24C23×D311yescharacteristic
420C2×D4—38no—
421C2×D4—38no—
422C2×D4—632no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—64no—
426C2×D4—64no—
427C2×D4—616no—
428C2×D4—616no—
429C2×D4—128no—
430C2×D4—1232no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—34no—
434C22×C4—32no—
435C22×C4—34no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—68no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—128no—
447C22×C4—124no—
448C22×C4—124no—
449C22×C4—124no—
450C22×C4C23×D314yes—
451C22×C4C23×D314yes—
452C4⋊C4—68no—
453C4⋊C4—128no—
454C4⋊C4—128no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—68no—
462C22⋊C4—616no—
463C22⋊C4—64no—
464C22⋊C4—64no—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C22⋊C4—616no—
470C22⋊C4—128no—
471C22⋊C4—128no—
472C42—68no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—24no—
480C22×C3—24no—
481C22×C3—24no—
482C22×C3—28no—
483C22×C3—28no—
484C22×C3—28no—
485C22×C3—416no—
486C22×C3—432no—
487C22×C3—432no—
488C22×C3—432no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3C2×#12314yes—
494C22×C3C2×#12314yes—
495C22×C3C2×#12314yes—
496C22×C3C23×D411yescharacteristic
497C22×C3C22×#4111yescharacteristic
498C22×C3C22×#4111yescharacteristic
499C22×C3C2×H(5,𝔽2)14yes—
500C2×D3—24no—
501C2×D3—28no—
502C2×D3—44no—
503C2×D3—44no—
504C2×D3—88no—
505C3×C4—44no—
506C3×C4—44no—
507Dic3—24no—
508Dic3—44no—
509Dic3—44no—
510Dic3—44no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—216no—
515C23—216no—
516C23—216no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—34no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—416no—
528C23—416no—
529C23—416no—
530C23—416no—
531C23—416no—
532C23—416no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—4128no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—62no—
541C23—62no—
542C23—68no—
543C23—68no—
544C23—128no—
545C23—244no—
546C23—244no—
547C23#883116yescomplemented
548C23C2×#23512yes—
549C23C2×#23512yes—
550C23C2×#23512yes—
551C23C2×D3×D412yes—
552C23C2×D3×D412yes—
553C23C2×D3×D412yes—
554C23C2×#23712yes—
555C23C2×#23712yes—
556C23C2×#23712yes—
557C23#100418yes—
558C23C24×D311yescharacteristic
559D4—616no—
560D4—616no—
561D4—1232no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—34no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—68no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—68no—
574C2×C4—68no—
575C2×C4—68no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—128no—
579C2×C4—248no—
580C2×C4—244no—
581C2×C4—244no—
582C2×C3—24no—
583C2×C3—24no—
584C2×C3—24no—
585C2×C3—48no—
586C2×C3C22×#12311yescharacteristic
587C2×C3#348412yes—
588C2×C3#349012yes—
589C2×C3#349012yes—
590D3—44no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—416no—
604C22—432no—
605C22—432no—
606C22—432no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—68no—
612C22—64no—
613C22—124no—
614C22—124no—
615C22—248no—
616C22C2×#88314yes—
617C22C2×#88314yes—
618C22C2×#88314yes—
619C22C22×#23511yescharacteristic
620C22C22×D3×D411yescharacteristic
621C22C22×#23711yescharacteristic
622C22C2×#100414yes—
623C4—44no—
624C4—44no—
625C4—64no—
626C4—124no—
627C4—124no—
628C4—124no—
629C3#3535011yescharacteristic, complemented, Sylow
630C2—24no—
631C2—24no—
632C2—24no—
633C2—48no—
634C2—124no—
635C2C22×#88311yescharacteristic
636C2#8569712yes—
637C2#8591312yes—
638C2#8591912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#379C2
prime radicalC22×#953C2
hypercenterC2×#379D3
hypocenterC3#35350
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
2. C2×#5486C2semidirect
3. C2×C3×#379C2semidirect
4. C2×#5247C2semidirect
5. #82128C2semidirect
6. #82121C2semidirect
7. #82095C2semidirect
8. #82094C2semidirect
9. #73853C2semidirect
10. #73832C2semidirect
11. #73826C2semidirect
18. C22×C3×C22⋊C4C22semidirect
26. C22×#217C22semidirect
27. C2×C4×#67C22semidirect
28. C2×#883C22semidirect
30. C2×#67⋊C4C22semidirect
31. C2×D3×C22⋊C4C22semidirect
32. C2×#877C22semidirect
34. C3×#379C22semidirect
36. #5247C22semidirect
37. #5213C22semidirect
38. #5212C22semidirect
39. #5210C22semidirect
40. #4961C22semidirect
41. #4959C22semidirect
42. #4958C22semidirect
43. #4956C22semidirect
44. #4955C22semidirect
45. #4952C22semidirect
46. #4951C22semidirect
47. #4950C22semidirect
48. #4950C22semidirect
49. #4949C22semidirect
50. #4946C22semidirect
51. #4945C22semidirect
52. #4943C22semidirect
53. #4942C22semidirect
64. C2×#379D3semidirect
88. C22×C4×D3C23semidirect
115. C2×(C2×D3)⋊C4C23semidirect
116. C2×(C2×D3)⋊C4C23semidirect
117. C2×C4⋊Dic3C23semidirect
118. C2×Dic3⋊C4C23semidirect
119. C2×Dic3⋊C4C23semidirect
120. C2×C4×Dic3C23semidirect
125. C3⋊#95C23semidirect
126. C3⋊#95C23semidirect
127. C3⋊#95C23semidirect
147. C22×C22⋊C4C2×D3semidirect
168. #379C2×D3semidirect
345. C23×C3#123semidirect
547. C23#883semidirect
629. C3#35350semidirect

Maximal subgroups

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

subgroupindex
C22×#9532
C2×#54862
C2×C3×#3792
C2×#52472
#821282
#821212
#820952
#820942
#738532
#738322
#738262
#353503