Finite Group Databasebeta

← #10967484

Subgroups — #10967484

Order 768.

3114 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic, complemented
2C22×#893C211yesmaximal, characteristic, complemented
3C3×#2502C211yesmaximal, characteristic, complemented
4#82761C211yesmaximal, characteristic, complemented
5#82180C211yesmaximal, characteristic, complemented
6#77812C214yesmaximal, complemented
7#77737C214yesmaximal, complemented
8#76533C212yesmaximal, complemented
9#76480C212yesmaximal, complemented
10#74602C214yesmaximal, complemented
11#74379C214yesmaximal, complemented
12#74339C212yesmaximal, complemented
13#74319C212yesmaximal, complemented
14#73587C211yesmaximal, characteristic
15C4×#877C211yesmaximal, characteristic, complemented
16#34965—31nomaximal, Sylow
17C23×D12C2211yescharacteristic
18C23×C4×D3C2212yes—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×#947C2212yes—
22C2×#947C2211yescharacteristic, complemented
23C2×Q8×Dic3C2211yescharacteristic, complemented
24C2×#930C2211yescharacteristic
25C22×(C2×D3)⋊C4C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27#5624—28no—
28C2×#898C2212yescomplemented
29C2×#893—28no—
30C2×#893C2212yes—
31C2×#893C2212yes—
32C2×#893C2212yes—
33C2×#893C2212yes—
34C2×#893C2211yescharacteristic, complemented
35C2×#893C2218yescomplemented
36C2×#892C2212yescomplemented
37C2×#890C2211yescharacteristic, complemented
38C2×#886C2212yes—
39C2×#882C2214yes—
40C2×#67⋊C4C2212yes—
41C2×#877C2212yes—
42C2×#874C2212yes—
43C3×#398C2214yescomplemented
44C3×#391C2212yescomplemented
45C3×#385C2214yes—
46C3×#384C2212yescomplemented
47C3×C4×C22⋊C4C2211yescharacteristic
48#5213C2214yes—
49C4×#217C2211yescharacteristic
50#4994C2218yescomplemented
51#4992C2214yescomplemented
52#4991C2214yes—
53#4989C2218yescomplemented
54#4988C2212yes—
55#4988C2212yescomplemented
56#4988C2218yescomplemented
57#4977C2212yescomplemented
58#4955C2214yes—
59Dic3×C22⋊C4C2212yes—
60#4939C2212yes—
61C4×Dic3⋊C4C2212yescomplemented
62C42×Dic3C2212yescomplemented
63#4672C2218yescomplemented
64#4671C2214yes—
65#4670C2218yescomplemented
66#4666C2218yescomplemented
67#4665C2218yescomplemented
68#4649C2212yescomplemented
69#4648C2212yescomplemented
70C2×#529—31no—
71C22×C4×D4—31no—
72#2603—34no—
73#2566—32no—
74#2544—32no—
75#2504—31no—
76#2502D311yescharacteristic, complemented
77#2435—31no—
78#2433—31no—
79#2411—34no—
80#2392—34no—
81C4×#127—32no—
82C4×#124—32no—
83C4×#120—34no—
84C4×#117—31no—
85C24×D3C2312yes—
86C22×C3×Q8C2311yescharacteristic
87C23×C3×C4C2311yescharacteristic
88C23×C3×C4C2312yes—
89C22×#67C2312yes—
90C23×Dic3C2311yescharacteristic
91C23×Dic3C2311yescharacteristic
92C22×D12—24no—
93C22×D12—28no—
94C22×D12C2312yes—
95C22×D12C2311yescharacteristic
96C22×D12C2318yes—
97C22×D12C2312yes—
98C22×C4×D3—28no—
99C22×C4×D3—48no—
100C22×C4×D3C2314yes—
101C22×C4×D3C2312yes—
102C22×C4×D3C2314yes—
103C3×C22⋊Q8—24no—
104C2×C3×C4⋊C4—24no—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2318yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C42C2312yes—
113C2×#217C2312yes—
114#947—28no—
115#947—24no—
116Q8×Dic3—24no—
117#930—24no—
118C2×(C2×D3)⋊C4—28no—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2318yes—
121C2×(C2×D3)⋊C4C2318yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2318yes—
128C2×(C2×D3)⋊C4C2318yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×C4×Dic3D418yescomplemented
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143#898—28no—
144#893—28no—
145#893—28no—
146#893—28no—
147#893—28no—
148#893—24no—
149#893—416no—
150#892—28no—
151#890—24no—
152#886—28no—
153#882—216no—
154#67⋊C4—28no—
155#877—28no—
156#874—28no—
157C3×#95C2312yes—
158C3×#95C2318yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2318yes—
162C23×D4—31no—
163C24×C4—32no—
164C2×#127—32no—
165#529—68no—
166C2×#126—32no—
167C2×#125—31no—
168C2×#124—31no—
169C2×#124—32no—
170C2×#123—32no—
171C2×C22⋊Q8C2×D311yescharacteristic
172C2×#121—31no—
173C2×#120—34no—
174C2×C4×Q8—31no—
175C2×C4×D4—31no—
176C2×C4×D4—32no—
177C2×C4×D4—32no—
178C2×C4×D4—32no—
179C2×C4×D4—32no—
180C2×C4×D4—38no—
181C2×C4×D4—32no—
182C2×C4×D4—68no—
183C2×#117—32no—
184C2×#117—32no—
185C22×C4⋊C4C2×D311yescharacteristic
186C22×C22⋊C4—32no—
187C22×C42—31no—
188#398C2×D314yescomplemented
189#397—34no—
190#396—38no—
191#394—34no—
192#394—38no—
193#392—34no—
194#391C2×D312yescomplemented
195#390—32no—
196#390—38no—
197#390—32no—
198#390—32no—
199#388—38no—
200#386—34no—
201#386—38no—
202#385—38no—
203#385C2×D314yes—
204#384C2×D312yescomplemented
205#382—32no—
206#379—34no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—32no—
209C4×C22⋊C4—31no—
210C4×C22⋊C4—32no—
211C4×C22⋊C4—38no—
212C4×C22⋊C4C2×D311yescharacteristic
213#376—32no—
214C43—32no—
215C24×C3C2411yescharacteristic
216C23×D3—24no—
217C23×D3—28no—
218C23×D3—28no—
219C23×D3—44no—
220C23×D3C2412yes—
221C23×D3C2414yes—
222C2×C3×Q8—22no—
223C2×C3×Q8—24no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—44no—
227C22×C3×C4C2×D414yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2412yes—
231C22×C3×C4C2411yescharacteristic
232C22×C3×C4C2414yes—
233C22×C3×C4C2411yescharacteristic
234C2×#67—24no—
235C2×#67—28no—
236C22×Dic3—24no—
237C22×Dic3—24no—
238C22×Dic3C2×D414yes—
239C22×Dic3C2×D414yes—
240C22×Dic3C2414yes—
241C22×Dic3C2412yes—
242C22×Dic3C2411yescharacteristic
243C22×Dic3C2411yescharacteristic
244C22×Dic3C2411yescharacteristic
245C22×Dic3C2411yescharacteristic
246C2×D12—232no—
247C2×D12—28no—
248C2×D12—24no—
249C2×D12—24no—
250C2×D12—24no—
251C2×D12—216no—
252C2×D12—22no—
253C2×D12—24no—
254C2×D12—216no—
255C2×D12—48no—
256C2×D12#4118yes—
257C2×C4×D3—24no—
258C2×C4×D3—28no—
259C2×C4×D3—28no—
260C2×C4×D3—28no—
261C2×C4×D3—216no—
262C2×C4×D3—48no—
263C2×C4×D3—432no—
264C3×C4⋊C4—22no—
265C3×C4⋊C4—24no—
266C3×C4⋊C4—24no—
267C3×C4⋊C4—24no—
268C3×C4⋊C4—48no—
269C3×C22⋊C4—24no—
270C3×C22⋊C4#4118yes—
271C3×C42—24no—
272#217—24no—
273(C2×D3)⋊C4—216no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—216no—
276(C2×D3)⋊C4—24no—
277(C2×D3)⋊C4—24no—
278(C2×D3)⋊C4—24no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—28no—
281(C2×D3)⋊C4—48no—
282(C2×D3)⋊C4—48no—
283C4⋊Dic3—22no—
284C4⋊Dic3—24no—
285C4⋊Dic3—24no—
286Dic3⋊C4—28no—
287Dic3⋊C4—24no—
288C4×Dic3—28no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C4×Dic3—28no—
292C4×Dic3C2×D414yes—
293C4×Dic3C2×D414yes—
294C4×Dic3C2×D414yes—
295C4×Dic3C2×D414yes—
296C25—32no—
297C22×Q8C22×D311yescharacteristic
298C22×D4—32no—
299C22×D4—32no—
300C22×D4—32no—
301C22×D4—31no—
302C22×D4—38no—
303C22×D4—68no—
304C22×D4—64no—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—32no—
308C23×C4—34no—
309C23×C4—34no—
310C23×C4—68no—
311C23×C4—128no—
312C23×C4C22×D311yescharacteristic
313C23×C4C22×D312yes—
314#127—68no—
315#126—68no—
316#125—64no—
317#124—64no—
318#124—68no—
319#123—68no—
320C22⋊Q8—24no—
321#121—64no—
322#120—616no—
323C4×Q8—64no—
324C4×D4—68no—
325C4×D4—68no—
326C4×D4—68no—
327C4×D4—68no—
328C4×D4—64no—
329C4×D4—68no—
330C4×D4—1216no—
331#117—68no—
332#117—68no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—34no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4C22×D312yes—
340C2×C4⋊C4C22×D312yes—
341C2×C4⋊C4C22×D312yes—
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D318yes—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—32no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4—38no—
352C2×C22⋊C4—38no—
353C2×C22⋊C4—38no—
354C2×C22⋊C4—38no—
355C2×C22⋊C4—68no—
356C2×C22⋊C4C22×D312yes—
357C2×C22⋊C4C22×D312yes—
358C2×C42—38no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—32no—
364C2×C42—34no—
365C2×C42—34no—
366C2×C42—32no—
367C2×C42C22×D312yes—
368#95—34no—
369#95—32no—
370#95—38no—
371#95C22×D312yes—
372#95C22×D318yes—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3C22×D412yes—
376C23×C3C2×#4114yes—
377C23×C3C2511yescharacteristic
378C22×D3—216no—
379C22×D3—24no—
380C22×D3—28no—
381C22×D3—24no—
382C22×D3—28no—
383C22×D3—22no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—432no—
388C22×D3—48no—
389C22×D3—48no—
390C22×D3C2×#4118yes—
391C3×Q8—44no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—24no—
395C2×C3×C4—28no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—24no—
399C2×C3×C4—24no—
400C2×C3×C4—24no—
401C2×C3×C4—28no—
402C2×C3×C4—48no—
403C2×C3×C4C22×D412yes—
404C2×C3×C4C22×D412yes—
405C2×C3×C4C2×#4118yes—
406C2×C3×C4C2×#4114yes—
407#67—48no—
408C2×Dic3—28no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—28no—
413C2×Dic3—22no—
414C2×Dic3—24no—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—24no—
418C2×Dic3C22×D412yes—
419C2×Dic3C22×D412yes—
420C2×Dic3C22×D412yes—
421C2×Dic3C22×D412yes—
422C2×Dic3H(5,𝔽2)14yes—
423D12—216no—
424D12—216no—
425D12—432no—
426D12—44no—
427C4×D3—416no—
428C4×D3—48no—
429C4×D3—416no—
430C24—34no—
431C24—32no—
432C24—68no—
433C24—64no—
434C24—68no—
435C24—124no—
436C24C23×D311yescharacteristic
437C2×Q8—22no—
438C2×Q8—24no—
439C2×D4—38no—
440C2×D4—68no—
441C2×D4—632no—
442C2×D4—616no—
443C2×D4—64no—
444C2×D4—68no—
445C2×D4—64no—
446C2×D4—64no—
447C2×D4—62no—
448C2×D4—616no—
449C2×D4—64no—
450C2×D4—64no—
451C2×D4—128no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—31no—
455C22×C4—34no—
456C22×C4—34no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—34no—
461C22×C4—32no—
462C22×C4—44no—
463C22×C4—64no—
464C22×C4—68no—
465C22×C4—68no—
466C22×C4—68no—
467C22×C4—616no—
468C22×C4—64no—
469C22×C4—64no—
470C22×C4—128no—
471C22×C4—1232no—
472C22×C4D3×D414yes—
473C22×C4C23×D314yes—
474C22×C4C23×D312yes—
475C22×C4C23×D312yes—
476C22×C4C23×D311yescharacteristic
477C22×C4C23×D314yes—
478C22×C4C23×D311yescharacteristic
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—48no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—62no—
488C4⋊C4—68no—
489C22⋊C4—24no—
490C22⋊C4—68no—
491C22⋊C4—68no—
492C22⋊C4—64no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—616no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—616no—
499C22⋊C4—128no—
500C22⋊C4—128no—
501C22⋊C4#23918yes—
502C42—24no—
503C42—34no—
504C42—34no—
505C42—34no—
506C42—34no—
507C42—64no—
508C42—68no—
509C42—68no—
510C42—64no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—28no—
514C22×C3#53214yes—
515C22×C3C23×D411yescharacteristic
516C22×C3C22×#4112yes—
517C22×C3C22×#4112yes—
518C22×C3C2×H(5,𝔽2)12yes—
519C2×D3—28no—
520C2×D3—216no—
521C2×D3—432no—
522C2×D3—44no—
523C2×D3—44no—
524C2×D3—48no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—48no—
529C3×C4—22no—
530C3×C4—28no—
531C3×C4—42no—
532C3×C4—44no—
533C3×C4—42no—
534C3×C4#52514yes—
535Dic3—22no—
536Dic3—22no—
537Dic3—22no—
538Dic3—42no—
539Dic3—44no—
540Dic3#53514yes—
541C23—22no—
542C23—22no—
543C23—38no—
544C23—616no—
545C23—64no—
546C23—68no—
547C23—64no—
548C23—68no—
549C23—62no—
550C23—64no—
551C23—64no—
552C23—64no—
553C23—1232no—
554C23—128no—
555C23—128no—
556C23C2×D3×D412yes—
557C23C2×#23914yes—
558C23C24×D311yescharacteristic
559Q8—44no—
560D4—616no—
561D4—616no—
562D4—1232no—
563D4—128no—
564D4—124no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—32no—
576C2×C4—34no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—48no—
581C2×C4—68no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—68no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—1216no—
592C2×C4—128no—
593C2×C4—1216no—
594C2×C4C2×D3×D412yes—
595C2×C4C2×D3×D412yes—
596C2×C4C2×#23918yes—
597C2×C4D3×#4114yes—
598C2×C3—24no—
599C2×C3C2×#52511yescharacteristic
600C2×C3C2×#53211yescharacteristic
601C2×C3C2×#53511yescharacteristic
602C2×C3#348114yes—
603D3—48no—
604D3—82no—
605C22—22no—
606C22—28no—
607C22—22no—
608C22—616no—
609C22—68no—
610C22—1232no—
611C22—124no—
612C22—128no—
613C22—124no—
614C22—122no—
615C22—128no—
616C22—122no—
617C22—122no—
618C22#572514yes—
619C22C22×D3×D411yescharacteristic
620C22C22×#23912yes—
621C22C2×D3×#4112yes—
622C22D3×H(5,𝔽2)12yes—
623C4—28no—
624C4—22no—
625C4—34no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—62no—
630C4—62no—
631C4—62no—
632C4—124no—
633C4—122no—
634C4#565614yes—
635C3#3496511yescharacteristic, complemented, Sylow
636C2—24no—
637C2—128no—
638C2—242no—
639C2C2×#565611yescharacteristic
640C2C2×#572511yescharacteristic
641C2D3×#53511yescharacteristic
642C2#8692414yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2502C2
prime radical#73587C2
hypercenter#2502D3
hypocenterC3#34965
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2502

Complements

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

NH ≅ G/Ntype
1. C2×#5624C2semidirect
2. C22×#893C2semidirect
3. C3×#2502C2semidirect
4. #82761C2semidirect
5. #82180C2semidirect
6. #77812C2semidirect
7. #77737C2semidirect
8. #76533C2semidirect
9. #76480C2semidirect
10. #74602C2semidirect
11. #74379C2semidirect
12. #74339C2semidirect
13. #74319C2semidirect
15. C4×#877C2semidirect
22. C2×#947C22semidirect
23. C2×Q8×Dic3C22semidirect
28. C2×#898C22semidirect
34. C2×#893C22semidirect
35. C2×#893C22semidirect
36. C2×#892C22semidirect
37. C2×#890C22semidirect
43. C3×#398C22semidirect
44. C3×#391C22semidirect
46. C3×#384C22semidirect
50. #4994C22semidirect
51. #4992C22semidirect
53. #4989C22semidirect
55. #4988C22semidirect
56. #4988C22semidirect
57. #4977C22semidirect
61. C4×Dic3⋊C4C22semidirect
62. C42×Dic3C22semidirect
63. #4672C22semidirect
65. #4670C22semidirect
66. #4666C22semidirect
67. #4665C22semidirect
68. #4649C22semidirect
69. #4648C22semidirect
76. #2502D3semidirect
134. C2×C4×Dic3D4semidirect
188. #398C2×D3semidirect
194. #391C2×D3semidirect
204. #384C2×D3semidirect
635. C3#34965semidirect

Maximal subgroups

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

subgroupindex
C2×#56242
C22×#8932
C3×#25022
#827612
#821802
#778122
#777372
#765332
#764802
#746022
#743792
#743392
#743192
#735872
C4×#8772
#349653