Finite Group Databasebeta

← #10800536

Subgroups — #10800536

Order 768.

1222 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic
2#73863C211yesmaximal, characteristic, complemented
3C4×#877C211yesmaximal, characteristic, complemented
4C3×#1996C211yesmaximal, characteristic, complemented
5#71701C211yesmaximal, characteristic, complemented
6#69528C211yesmaximal, characteristic
7#69456C211yesmaximal, characteristic
8#68954C214yesmaximal, complemented
9#68952C214yesmaximal, complemented
10#21421—31nomaximal, Sylow
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#947C411yescharacteristic, complemented
13C2×Q8×Dic3C411yescharacteristic, complemented
14C2×#930C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16#5624—21no—
17#5624—21no—
18#5624—21no—
19#5624—21no—
20#5624—21no—
21#5624—21no—
22#5624—21no—
23#5624—21no—
24C2×#898C412yescomplemented
25C2×#893C411yescharacteristic, complemented
26C2×#892C412yescomplemented
27C2×#890C411yescharacteristic, complemented
28C2×#886C2212yes—
29C2×#67⋊C4C2212yes—
30C2×#877C2212yes—
31C3×#380C2211yescharacteristic
32C3×C4×C22⋊C4C2211yescharacteristic
33#5212C2211yescharacteristic
34C4×#217C2211yescharacteristic
35#4977—21no—
36Dic3×C4⋊C4—21no—
37C2×#806C2214yes—
38C2×#806C2214yes—
39#4943C2212yes—
40Dic3×C22⋊C4C2212yes—
41Dic3×C22⋊C4C2212yes—
42C4×Dic3⋊C4—21no—
43C42×Dic3—21no—
44#4651—21no—
45#4649—21no—
46#4648—21no—
47#4648—21no—
48C3×#342C2214yes—
49#4526C2214yes—
50C2×#529—31no—
51#2283—31no—
52C4×#117—31no—
53#1996—31no—
54#1996D311yescharacteristic, complemented
55#1964—31no—
56#1964—31no—
57#1899—34no—
58C4×#99—34no—
59C22×C3×Q8C2×C411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67C2312yes—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C22×D12C2×C411yescharacteristic
65C22×C4×D3C2×C412yes—
66C3×C22⋊Q8—21no—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8—21no—
69C3×C22⋊Q8—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2×C411yescharacteristic
72C2×C3×C4⋊C4C2×C412yes—
73C2×C3×C22⋊C4C2312yes—
74C2×C3×C22⋊C4C2312yes—
75C2×C3×C22⋊C4C2312yes—
76C2×C3×C42—21no—
77C2×#217C2312yes—
78C2×#217C2312yes—
79#947—21no—
80#947—21no—
81#947—21no—
82#947—21no—
83Q8×Dic3—21no—
84Q8×Dic3—21no—
85Q8×Dic3—21no—
86Q8×Dic3—21no—
87#930—21no—
88#930—21no—
89#930—21no—
90#930—21no—
91C2×(C2×D3)⋊C4C2×C412yes—
92C2×(C2×D3)⋊C4C2×C412yes—
93C2×(C2×D3)⋊C4C2×C412yes—
94C2×(C2×D3)⋊C4C2×C412yes—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2×C412yes—
97C2×C4⋊Dic3C2×C412yes—
98C2×C4⋊Dic3C2×C411yescharacteristic
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4C2×C412yes—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3C2×C412yes—
106C2×C4×Dic3C2×C411yescharacteristic
107C2×C4×Dic3C2×C411yescharacteristic
108C2×C4×Dic3C2×C411yescharacteristic
109C2×C4×Dic3C2×C411yescharacteristic
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3D411yescharacteristic
118#898—22no—
119#898—22no—
120#898—22no—
121#898—22no—
122#893—21no—
123#893—21no—
124#893—21no—
125#893—21no—
126#892—22no—
127#892—22no—
128#892—22no—
129#892—22no—
130#890—21no—
131#890—21no—
132#890—21no—
133#890—21no—
134#886—22no—
135#886—22no—
136#886—22no—
137#886—22no—
138#67⋊C4—22no—
139#67⋊C4—22no—
140#67⋊C4—22no—
141#67⋊C4—22no—
142#877—24no—
143#877—24no—
144C3×#95—21no—
145C3×#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—22no—
149C3⋊#95—22no—
150C3⋊#95—22no—
151C3⋊#95—22no—
152#806—28no—
153#806—28no—
154#529—61no—
155#529—61no—
156#529—61no—
157#529—61no—
158#529—61no—
159#529—61no—
160#529—61no—
161#529—61no—
162C2×#126—32no—
163C2×#125—31no—
164C2×#124—31no—
165C2×#123—32no—
166C2×C22⋊Q8C2×D311yescharacteristic
167C2×#121—31no—
168C2×C4×Q8—31no—
169C2×C4×D4—31no—
170C2×C4×D4—32no—
171C2×#117—32no—
172C2×#117—32no—
173C22×C42—31no—
174C2×#99—34no—
175C2×#99—34no—
176#383—61no—
177#382—61no—
178#381—32no—
179#380—31no—
180#380C2×D311yescharacteristic
181C4×C4⋊C4—61no—
182C4×C4⋊C4—61no—
183C4×C4⋊C4—61no—
184C4×C22⋊C4—31no—
185C4×C22⋊C4—32no—
186C4×C22⋊C4—32no—
187C4×C22⋊C4C2×D311yescharacteristic
188#376—61no—
189#376—61no—
190C43—61no—
191#342—34no—
192#342C2×D314yes—
193C24×C3C2411yescharacteristic
194C23×D3C22×C412yes—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—42no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—42no—
201C22×C3×C4C22×C411yescharacteristic
202C22×C3×C4C22×C412yes—
203C22×C3×C4C22×C412yes—
204C22×C3×C4C22×C411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C2×#67—22no—
210C2×#67—22no—
211C2×#67—24no—
212C2×#67—24no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—42no—
218C22×Dic3C22×C412yes—
219C22×Dic3C22×C411yescharacteristic
220C22×Dic3C22×C411yescharacteristic
221C22×Dic3C22×C411yescharacteristic
222C22×Dic3C22×C411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—42no—
234C2×C4×D3—22no—
235C2×C4×D3—22no—
236C2×C4×D3—44no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—42no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4#4114yes—
247C3×C42—42no—
248#217—22no—
249#217—24no—
250#217#4114yes—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—22no—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264C4⋊Dic3—22no—
265C4⋊Dic3—42no—
266Dic3⋊C4—28no—
267Dic3⋊C4—22no—
268Dic3⋊C4—22no—
269C4×Dic3—28no—
270C4×Dic3—22no—
271C4×Dic3—22no—
272C4×Dic3—44no—
273C4×Dic3—44no—
274C4×Dic3—42no—
275C4×Dic3C22⋊C411yescharacteristic
276C4×Dic3C22⋊C411yescharacteristic
277C4×Dic3C22⋊C411yescharacteristic
278C4×Dic3C22⋊C411yescharacteristic
279C4×Dic3C22⋊C411yescharacteristic
280C4×Dic3C22⋊C411yescharacteristic
281C4×Dic3C22⋊C411yescharacteristic
282C4×Dic3C22⋊C411yescharacteristic
283C4×Dic3C22⋊C411yescharacteristic
284C4×Dic3C22⋊C411yescharacteristic
285C4×Dic3C22⋊C411yescharacteristic
286C4×Dic3C22⋊C411yescharacteristic
287C4×Dic3C22⋊C411yescharacteristic
288C4×Dic3C22⋊C411yescharacteristic
289C4×Dic3C22⋊C411yescharacteristic
290C4×Dic3C22⋊C411yescharacteristic
291C22×Q8C4×D311yescharacteristic
292C22×D4—32no—
293C22×D4—31no—
294C23×C4—31no—
295C23×C4—31no—
296C23×C4—32no—
297C23×C4C22×D311yescharacteristic
298#126—62no—
299#126—62no—
300#126—62no—
301#126—62no—
302#125—61no—
303#125—61no—
304#125—61no—
305#125—61no—
306#124—61no—
307#124—61no—
308#124—61no—
309#124—61no—
310#123—62no—
311#123—62no—
312#123—62no—
313#123—62no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318#121—61no—
319#121—61no—
320#121—61no—
321#121—61no—
322C4×Q8—61no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—61no—
329C4×D4—61no—
330C4×D4—62no—
331C4×D4—62no—
332C4×D4—62no—
333C4×D4—62no—
334#117—62no—
335#117—62no—
336#117—62no—
337#117—62no—
338#117—64no—
339#117—64no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—62no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4C4×D312yes—
348C2×C4⋊C4C4×D311yescharacteristic
349C2×C22⋊C4—32no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4C22×D312yes—
356C2×C22⋊C4C22×D312yes—
357C2×C22⋊C4C22×D312yes—
358C2×C42—21no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—31no—
366C2×C42—31no—
367C2×C42—31no—
368C2×C42—31no—
369C2×C42—31no—
370C2×C42—31no—
371C2×C42—32no—
372C2×C42—62no—
373C2×C42—62no—
374C2×C42—62no—
375C2×C42—61no—
376#99—68no—
377#99—68no—
378#95—21no—
379#95—21no—
380#95—62no—
381#95—62no—
382#95—62no—
383#95—62no—
384#95—61no—
385#95—61no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3C23×C411yescharacteristic
389C23×C3C22×D411yescharacteristic
390C23×C3C22×D411yescharacteristic
391C23×C3C2×#4112yes—
392C23×C3C2×#4112yes—
393C22×D3—22no—
394C22×D3—24no—
395C22×D3—42no—
396C22×D3—42no—
397C3×Q8—42no—
398C3×Q8—42no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—24no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—42no—
409C2×C3×C4—42no—
410C2×C3×C4—42no—
411C2×C3×C4—42no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4C2×C22⋊C411yescharacteristic
416C2×C3×C4C2×C22⋊C411yescharacteristic
417C2×C3×C4C2×C22⋊C411yescharacteristic
418C2×C3×C4C2×C22⋊C411yescharacteristic
419#67—44no—
420#67—42no—
421#67—42no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—24no—
435C2×Dic3—24no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—42no—
440C2×Dic3—42no—
441C2×Dic3—42no—
442C2×Dic3—42no—
443C2×Dic3—42no—
444C2×Dic3C2×C22⋊C411yescharacteristic
445C2×Dic3C2×C22⋊C411yescharacteristic
446C2×Dic3C2×C22⋊C411yescharacteristic
447C2×Dic3C2×C22⋊C411yescharacteristic
448C2×Dic3C2×C22⋊C411yescharacteristic
449C2×Dic3C2×C22⋊C411yescharacteristic
450C2×Dic3C2×C22⋊C411yescharacteristic
451C2×Dic3C2×C22⋊C411yescharacteristic
452D12—42no—
453D12—42no—
454C4×D3—44no—
455C4×D3—44no—
456C24—32no—
457C24C23×D311yescharacteristic
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—42no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—64no—
464C2×D4—64no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—122no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—32no—
483C22×C4—42no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—124no—
491C22×C4—122no—
492C22×C4C2×C4×D311yescharacteristic
493C22×C4C2×C4×D312yes—
494C22×C4C2×C4×D312yes—
495C22×C4C2×C4×D311yescharacteristic
496C22×C4D3×D411yescharacteristic
497C22×C4D3×D411yescharacteristic
498C22×C4D3×D411yescharacteristic
499C22×C4D3×D411yescharacteristic
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—68no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—122no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—24no—
519C22⋊C4—34no—
520C22⋊C4—62no—
521C22⋊C4—62no—
522C22⋊C4—62no—
523C22⋊C4—62no—
524C22⋊C4—62no—
525C22⋊C4—62no—
526C22⋊C4—62no—
527C22⋊C4—62no—
528C22⋊C4—62no—
529C22⋊C4—64no—
530C22⋊C4#23914yes—
531C42—31no—
532C42—31no—
533C42—31no—
534C42—31no—
535C42—31no—
536C42—31no—
537C42—31no—
538C42—31no—
539C42—31no—
540C42—31no—
541C42—31no—
542C42—31no—
543C42—31no—
544C42—31no—
545C42—31no—
546C42—31no—
547C42—42no—
548C42—62no—
549C42—62no—
550C42—68no—
551C42—122no—
552C42—124no—
553C42—124no—
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—21no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3C22×C22⋊C411yescharacteristic
563C22×C3C4×#4112yes—
564C22×C3#52512yes—
565C22×C3#53211yescharacteristic
566C22×C3#53211yescharacteristic
567C2×D3—42no—
568C2×D3—42no—
569C2×D3—42no—
570C2×D3—84no—
571C3×C4—21no—
572C3×C4—21no—
573C3×C4—21no—
574C3×C4—44no—
575C3×C4—42no—
576C3×C4—42no—
577C3×C4—82no—
578C3×C4#41012yes—
579Dic3—21no—
580Dic3—21no—
581Dic3—21no—
582Dic3—21no—
583Dic3—21no—
584Dic3—21no—
585Dic3—21no—
586Dic3—44no—
587Dic3—42no—
588Dic3—82no—
589Dic3#41012yes—
590C23—41no—
591C23—41no—
592C23—62no—
593C23—64no—
594C23—122no—
595C23—122no—
596C23C22×C4×D311yescharacteristic
597C23C2×D3×D411yescharacteristic
598C23C2×D3×D411yescharacteristic
599C23C2×#23912yes—
600C23D3×#4112yes—
601Q8—42no—
602Q8—42no—
603D4—122no—
604D4—122no—
605D4—122no—
606D4—124no—
607D4—122no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—31no—
615C2×C4—31no—
616C2×C4—31no—
617C2×C4—31no—
618C2×C4—31no—
619C2×C4—31no—
620C2×C4—31no—
621C2×C4—31no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—62no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—64no—
646C2×C4—124no—
647C2×C4—124no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4D3×C22⋊C411yescharacteristic
657C2×C4D3×C22⋊C411yescharacteristic
658C2×C4D3×C22⋊C411yescharacteristic
659C2×C4D3×C22⋊C411yescharacteristic
660C2×C3—22no—
661C2×C3—21no—
662C2×C3—21no—
663C2×C3—22no—
664C2×C3#229711yescharacteristic
665C2×C3C2×#41011yescharacteristic
666C2×C3C2×#41011yescharacteristic
667D3—82no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—42no—
675C22—42no—
676C22—122no—
677C22—122no—
678C22—122no—
679C22—244no—
680C22C2×D3×C22⋊C411yescharacteristic
681C22C4×#23912yes—
682C22#562712yes—
683C22#566811yescharacteristic
684C22#572511yescharacteristic
685C4—21no—
686C4—21no—
687C4—21no—
688C4—32no—
689C4—44no—
690C4—42no—
691C4—42no—
692C4—61no—
693C4—61no—
694C4—61no—
695C4—61no—
696C4—61no—
697C4—61no—
698C4—61no—
699C4—82no—
700C4—124no—
701C4—122no—
702C4—242no—
703C4#474012yes—
704C3#2142111yescharacteristic, complemented, Sylow
705C2—22no—
706C2—21no—
707C2—21no—
708C2—22no—
709C2—242no—
710C2#7647811yescharacteristic
711C2C2×#474011yescharacteristic
712C2D3×#41011yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1996C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#930C22
cosocleC24×C3C24
hypercenter#1996D3
hypocenterC3#21421
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. #73863C2semidirect
3. C4×#877C2semidirect
4. C3×#1996C2semidirect
5. #71701C2semidirect
8. #68954C2semidirect
9. #68952C2semidirect
12. C2×#947C4semidirect
13. C2×Q8×Dic3C4semidirect
24. C2×#898C4semidirect
25. C2×#893C4semidirect
26. C2×#892C4semidirect
27. C2×#890C4semidirect
54. #1996D3semidirect
704. C3#21421semidirect

Maximal subgroups

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

subgroupindex
C2×#56242
#738632
C4×#8772
C3×#19962
#717012
#695282
#694562
#689542
#689522
#214213