Finite Group Databasebeta

← #11002000

Subgroups — #11002000

Order 768.

2326 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5519C211yesmaximal, characteristic, complemented
2C22×#878C211yesmaximal, characteristic
3C3×#2482C211yesmaximal, characteristic
4#82254C211yesmaximal, characteristic, complemented
5#82113C211yesmaximal, characteristic
6#78061C214yesmaximal, complemented
7#77573C214yesmaximal, complemented
8#75672C214yesmaximal, complemented
9#75667C214yesmaximal, complemented
10#74502C212yesmaximal, complemented
11#74412C212yesmaximal, complemented
12#74074C211yesmaximal, characteristic, complemented
13#74054C211yesmaximal, characteristic, complemented
14#73846C212yesmaximal
15#73835C212yesmaximal
16#36708—31nomaximal, Sylow
17C24×Dic3C2211yescharacteristic
18C23×Dic6C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×#217C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2211yescharacteristic
26#5519—28no—
27C2×#890C2212yescomplemented
28C2×#888C2212yescomplemented
29C2×#888C2214yescomplemented
30C2×#878—28no—
31C2×#878C2214yes—
32C2×#878C2212yes—
33C2×#878C2212yes—
34C2×#878C2214yes—
35C2×#878C2214yes—
36C2×#870C2212yescomplemented
37C2×#869C2212yescomplemented
38C3×#399C2212yes—
39C3×#397C2212yes—
40C3×#388C2214yes—
41C3×#385C2212yes—
42C3×#381C2211yescharacteristic
43C3×#379C2212yes—
44#5247C2212yes—
45#5211C2211yescharacteristic
46#5210C2212yes—
47#4985C2212yescomplemented
48#4983C2214yescomplemented
49#4983C2212yescomplemented
50#4979C2214yescomplemented
51#4979C2214yescomplemented
52#4953C2212yes—
53#4952C2214yes—
54#4951C2214yes—
55#4950C2214yes—
56#4950C2212yes—
57#4949C2212yes—
58#4947C2214yes—
59#4946C2212yes—
60#4946C2214yes—
61#4944C2214yes—
62#4944C2214yes—
63#4943C2212yes—
64#4943C2212yes—
65#4942C2214yes—
66#4942C2212yes—
67Dic3×C22⋊C4C2212yes—
68#4933C2214yescomplemented
69#4661C2212yescomplemented
70#4659C2212yescomplemented
71#4658C2214yescomplemented
72#4657C2214yescomplemented
73#4656C2214yescomplemented
74#4654C2214yescomplemented
75#4653C2214yescomplemented
76#4653C2214yescomplemented
77#4646C2214yescomplemented
78C2×#544—31no—
79C22×C22⋊Q8—31no—
80#2848—32no—
81#2690—34no—
82#2690—34no—
83#2677—34no—
84#2677—34no—
85#2627—32no—
86#2561—32no—
87#2484—31no—
88#2484—31no—
89#2482—31no—
90#2482D311yescharacteristic
91#2434—32no—
92#2313—31no—
93C25×C3C2311yescharacteristic
94C23×C3×C4C2311yescharacteristic
95C23×C3×C4C2311yescharacteristic
96C23×Dic3—24no—
97C23×Dic3—44no—
98C23×Dic3C2311yescharacteristic
99C23×Dic3C2311yescharacteristic
100C23×Dic3C2314yes—
101C23×Dic3C2311yescharacteristic
102C23×Dic3C2311yescharacteristic
103C22×Dic6—24no—
104C22×Dic6—44no—
105C22×Dic6C2314yes—
106C22×Dic6C2312yes—
107C22×Dic6C2312yes—
108C3×#117—24no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C22⋊C4—24no—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2314yes—
115C2×C3×C22⋊C4C2312yes—
116C2×C3×C22⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C42C2312yes—
120C2×#217—24no—
121C2×#217C2312yes—
122C2×#217C2314yes—
123C2×#217C2312yes—
124C2×#217C2314yes—
125C2×#217C2314yes—
126#926—24no—
127C2×C4⋊Dic3D418yes—
128C2×C4⋊Dic3C2314yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2312yes—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×Dic3⋊C4—24no—
136C2×Dic3⋊C4—24no—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2314yes—
149#890—28no—
150#888—28no—
151#888—48no—
152#878—28no—
153#878—28no—
154#878—216no—
155#878—216no—
156#878—48no—
157#870—28no—
158#869—28no—
159C3×#95C2314yes—
160C3×#95C2312yes—
161C3×#95C2312yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C3⋊#95C2314yes—
173C23×Q8—31no—
174C24×C4—31no—
175#544—68no—
176C2×C4⋊Q8—32no—
177C2×C4⋊Q8—34no—
178C2×C4⋊Q8—32no—
179C2×#125—32no—
180C2×#125—32no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—32no—
183C2×C22⋊Q8—34no—
184C2×C22⋊Q8—34no—
185C2×C22⋊Q8—34no—
186C2×C22⋊Q8—68no—
187C2×#117—31no—
188C2×#117C2×D311yescharacteristic
189C22×C4⋊C4—31no—
190C22×C4⋊C4—31no—
191C22×C4⋊C4—31no—
192C22×C22⋊C4—31no—
193C22×C22⋊C4C2×D311yescharacteristic
194#400—32no—
195#400—34no—
196#400—32no—
197#399—32no—
198#399C2×D312yes—
199#398—34no—
200#398—32no—
201#398—34no—
202#398—32no—
203#397—34no—
204#397—32no—
205#397C2×D312yes—
206#395—34no—
207#395—34no—
208#395—34no—
209#393—34no—
210#391—34no—
211#388—34no—
212#388—34no—
213#388—34no—
214#388C2×D314yes—
215#387—34no—
216#387—34no—
217#387—34no—
218#387—34no—
219#385—32no—
220#385—34no—
221#385—32no—
222#385—34no—
223#385—32no—
224#385C2×D312yes—
225#381—31no—
226#381—32no—
227#381—32no—
228#381C2×D311yescharacteristic
229#380—32no—
230#380—34no—
231#379—32no—
232#379C2×D312yes—
233C4×C22⋊C4—32no—
234C24×C3—22no—
235C24×C3—24no—
236C24×C3—24no—
237C24×C3—42no—
238C24×C3C2412yes—
239C24×C3C2411yescharacteristic
240C22×C3×C4—24no—
241C22×C3×C4—22no—
242C22×C3×C4—42no—
243C22×C3×C4C2×D414yes—
244C22×C3×C4C2414yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2412yes—
247C22×C3×C4C2412yes—
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2411yescharacteristic
250C22×Dic3—22no—
251C22×Dic3—24no—
252C22×Dic3—22no—
253C22×Dic3—24no—
254C22×Dic3—28no—
255C22×Dic3—28no—
256C22×Dic3—42no—
257C22×Dic3—42no—
258C22×Dic3—44no—
259C22×Dic3—416no—
260C22×Dic3C2×D414yes—
261C22×Dic3C2×D414yes—
262C22×Dic3C2414yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C22×Dic3C2412yes—
267C22×Dic3C2412yes—
268C22×Dic3C2412yes—
269C2×Dic6—28no—
270C2×Dic6—28no—
271C2×Dic6—24no—
272C2×Dic6—24no—
273C2×Dic6—44no—
274C2×Dic6—416no—
275C2×Dic6—44no—
276C2×Dic6—44no—
277C3×C4⋊C4—22no—
278C3×C4⋊C4—22no—
279C3×C4⋊C4—44no—
280C3×C22⋊C4—24no—
281C3×C22⋊C4—28no—
282C3×C22⋊C4—24no—
283C3×C22⋊C4—24no—
284C3×C22⋊C4—24no—
285C3×C22⋊C4—44no—
286C3×C42—24no—
287#217—24no—
288#217—28no—
289#217—24no—
290#217—28no—
291#217—44no—
292C4⋊Dic3—22no—
293C4⋊Dic3—22no—
294C4⋊Dic3—24no—
295C4⋊Dic3—24no—
296C4⋊Dic3—28no—
297C4⋊Dic3C2×D418yes—
298C4⋊Dic3C2×D418yes—
299Dic3⋊C4—24no—
300Dic3⋊C4—24no—
301Dic3⋊C4—24no—
302Dic3⋊C4—24no—
303Dic3⋊C4—24no—
304Dic3⋊C4—24no—
305Dic3⋊C4—28no—
306Dic3⋊C4—28no—
307Dic3⋊C4—44no—
308Dic3⋊C4—44no—
309C4×Dic3—24no—
310C4×Dic3—44no—
311C25C22×D311yescharacteristic
312C22×Q8—34no—
313C22×Q8—32no—
314C22×Q8—32no—
315C22×Q8—64no—
316C22×Q8—124no—
317C23×C4—31no—
318C23×C4—31no—
319C23×C4—31no—
320C23×C4—31no—
321C23×C4—34no—
322C23×C4—64no—
323C23×C4—124no—
324C23×C4C22×D311yescharacteristic
325C23×C4C22×D311yescharacteristic
326C4⋊Q8—68no—
327C4⋊Q8—68no—
328C4⋊Q8—128no—
329#125—68no—
330#125—68no—
331C22⋊Q8—616no—
332C22⋊Q8—616no—
333C22⋊Q8—68no—
334C22⋊Q8—68no—
335C22⋊Q8—128no—
336#117—24no—
337#117—64no—
338C2×C4⋊C4—38no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—34no—
356C2×C4⋊C4—64no—
357C2×C4⋊C4—64no—
358C2×C4⋊C4C22×D311yescharacteristic
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C4⋊C4C22×D314yes—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—64no—
368C2×C22⋊C4C22×D312yes—
369C2×C22⋊C4C22×D312yes—
370C2×C22⋊C4C22×D312yes—
371C2×C22⋊C4C22×D312yes—
372C2×C22⋊C4C22×D314yes—
373C2×C22⋊C4C22×D314yes—
374C2×C42—32no—
375C2×C42—34no—
376C2×C42C22×D312yes—
377#95—34no—
378#95—34no—
379#95—34no—
380#95—34no—
381#95—32no—
382#95—32no—
383#95—34no—
384#95—34no—
385#95—34no—
386#95—34no—
387#95—34no—
388#95C22×D312yes—
389#95C22×D312yes—
390#95C22×D314yes—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3—28no—
394C23×C3—22no—
395C23×C3—24no—
396C23×C3—22no—
397C23×C3—24no—
398C23×C3—416no—
399C23×C3—44no—
400C23×C3—44no—
401C23×C3C22×D412yes—
402C23×C32−1+414yes—
403C23×C32−1+414yes—
404C23×C3C2511yescharacteristic
405C2×C3×C4—28no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—24no—
412C2×C3×C4—24no—
413C2×C3×C4—24no—
414C2×C3×C4—24no—
415C2×C3×C4—24no—
416C2×C3×C4—24no—
417C2×C3×C4—44no—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D412yes—
420C2×Dic3—28no—
421C2×Dic3—22no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—24no—
429C2×Dic3—24no—
430C2×Dic3—24no—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—44no—
435C2×Dic3—44no—
436C2×Dic3—416no—
437C2×Dic3—44no—
438C2×Dic3C22×D414yes—
439C2×Dic3C22×D414yes—
440C2×Dic3H(5,𝔽2)14yes—
441Dic6—416no—
442Dic6—84no—
443Dic6—84no—
444C24—22no—
445C24—24no—
446C24—24no—
447C24—42no—
448C24C23×D312yes—
449C24C23×D311yescharacteristic
450C2×Q8—68no—
451C2×Q8—68no—
452C2×Q8—64no—
453C2×Q8—64no—
454C2×Q8—124no—
455C2×Q8—124no—
456C2×Q8—124no—
457C2×Q8—1216no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—32no—
461C22×C4—32no—
462C22×C4—32no—
463C22×C4—34no—
464C22×C4—34no—
465C22×C4—34no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—32no—
469C22×C4—42no—
470C22×C4—64no—
471C22×C4—68no—
472C22×C4—68no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—64no—
476C22×C4—122no—
477C22×C4—124no—
478C22×C4—122no—
479C22×C4—1216no—
480C22×C4D3×D414yes—
481C22×C4C23×D314yes—
482C22×C4C23×D312yes—
483C22×C4C23×D312yes—
484C22×C4C23×D312yes—
485C22×C4C23×D311yescharacteristic
486C22×C4C23×D311yescharacteristic
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—38no—
490C4⋊C4—38no—
491C4⋊C4—44no—
492C4⋊C4—68no—
493C4⋊C4—64no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—68no—
500C4⋊C4—68no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—124no—
506C4⋊C4—124no—
507C22⋊C4—24no—
508C22⋊C4—28no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—68no—
514C22⋊C4—64no—
515C22⋊C4—68no—
516C22⋊C4—64no—
517C22⋊C4—124no—
518C42—24no—
519C42—64no—
520C42—124no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—28no—
524C22×C3—24no—
525C22×C3—28no—
526C22×C3—416no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—44no—
530C22×C3#53614yes—
531C22×C3C23×D411yescharacteristic
532C22×C3C2×H(5,𝔽2)12yes—
533C22×C3C2×2−1+412yes—
534C22×C3C2×2−1+412yes—
535C3×C4—22no—
536C3×C4—22no—
537C3×C4—42no—
538C3×C4—42no—
539C3×C4—42no—
540C3×C4—44no—
541Dic3—24no—
542Dic3—24no—
543Dic3—24no—
544Dic3—42no—
545Dic3—42no—
546Dic3—42no—
547Dic3—44no—
548C23—22no—
549C23—22no—
550C23—28no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—416no—
556C23—44no—
557C23—44no—
558C23C2×D3×D412yes—
559C23#101014yes—
560C23#101014yes—
561C23C24×D311yescharacteristic
562Q8—1216no—
563Q8—244no—
564Q8—244no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—34no—
578C2×C4—34no—
579C2×C4—34no—
580C2×C4—44no—
581C2×C4—68no—
582C2×C4—62no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—124no—
596C2×C4—124no—
597C2×C4—1216no—
598C2×C4—124no—
599C2×C4C2×D3×D412yes—
600C2×C4C2×D3×D412yes—
601C2×C3—24no—
602C2×C3—44no—
603C2×C3C2×#53611yescharacteristic
604C2×C3#348411yescharacteristic
605C2×C3#348611yescharacteristic
606C2×C3#353814yes—
607C22—22no—
608C22—28no—
609C22—22no—
610C22—24no—
611C22—28no—
612C22—416no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22#564814yes—
617C22C22×D3×D411yescharacteristic
618C22C2×#101012yes—
619C22C2×#101012yes—
620C22D3×H(5,𝔽2)12yes—
621C4—22no—
622C4—22no—
623C4—42no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—64no—
628C4—64no—
629C4—64no—
630C4—122no—
631C4—124no—
632C4—122no—
633C4—122no—
634C3#3670811yescharacteristic, complemented, Sylow
635C2—24no—
636C2—44no—
637C2C2×#564811yescharacteristic
638C2#8602211yescharacteristic
639C2#8633011yescharacteristic
640C2#8680314yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2482C2
prime radicalC25×C3C23
hypercenter#2482D3
hypocenterC3#36708
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2482

Complements

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

NH ≅ G/Ntype
1. C2×#5519C2semidirect
4. #82254C2semidirect
6. #78061C2semidirect
7. #77573C2semidirect
8. #75672C2semidirect
9. #75667C2semidirect
10. #74502C2semidirect
11. #74412C2semidirect
12. #74074C2semidirect
13. #74054C2semidirect
27. C2×#890C22semidirect
28. C2×#888C22semidirect
29. C2×#888C22semidirect
36. C2×#870C22semidirect
37. C2×#869C22semidirect
47. #4985C22semidirect
48. #4983C22semidirect
49. #4983C22semidirect
50. #4979C22semidirect
51. #4979C22semidirect
68. #4933C22semidirect
69. #4661C22semidirect
70. #4659C22semidirect
71. #4658C22semidirect
72. #4657C22semidirect
73. #4656C22semidirect
74. #4654C22semidirect
75. #4653C22semidirect
76. #4653C22semidirect
77. #4646C22semidirect
634. C3#36708semidirect

Maximal subgroups

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

subgroupindex
C2×#55192
C22×#8782
C3×#24822
#822542
#821132
#780612
#775732
#756722
#756672
#745022
#744122
#740742
#740542
#738462
#738352
#367083