Finite Group Databasebeta

← #10826979

Subgroups — #10826979

Order 768.

1952 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2C2×#5833C211yesmaximal, characteristic, complemented
3C2×#5218C211yesmaximal, characteristic, complemented
4C2×#5217C211yesmaximal, characteristic, complemented
5C2×#5037C212yesmaximal, complemented
6#72136C211yesmaximal, characteristic, complemented
7#71685C212yesmaximal, complemented
8#71684C212yesmaximal, complemented
9#71677C212yesmaximal, complemented
10#71586C211yesmaximal, characteristic, complemented
11#71585C211yesmaximal, characteristic, complemented
12#23924—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×#235C2211yescharacteristic, complemented
16C3×#535—28no—
17C2×C3×#127—21no—
18C2×C3×#124—21no—
19C2×C3×#121C2212yes—
20C2×C3×#121C2212yes—
21C2×C3×#120—21no—
22C2×C3×#120—21no—
23C2×C3×#117C2211yescharacteristic
24#5833—24no—
25#5833—24no—
26C2×#946—21no—
27C2×#939—21no—
28C2×#939—21no—
29C2×#938—21no—
30C22×#214C2212yes—
31C22×#213C2212yes—
32C2×#930C2211yescharacteristic
33C2×#930C2211yescharacteristic, complemented
34C22×C4⋊Dic3C2211yescharacteristic, complemented
35C2×#924C2211yescharacteristic
36C2×#924C2211yescharacteristic, complemented
37#5218—24no—
38#5217—24no—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×#830C2211yescharacteristic
42C2×#830C2211yescharacteristic
43C2×#830C2212yescomplemented
44C2×#830C2212yescomplemented
45#5037—28no—
46C2×#810C2212yescomplemented
47C2×#809C2212yescomplemented
48#4529C2212yescomplemented
49#4529C2212yescomplemented
50C2×#537—31no—
51C2×#535D311yescharacteristic, complemented
52C2×#449—32no—
53C2×#450—31no—
54C2×#447—31no—
55#2000—32no—
56#1999—32no—
57#1875—31no—
58#1874—32no—
59#1873—31no—
60#1780—31no—
61C25×C3—21no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—24no—
65C2×C3×#41—42no—
66C22×C3×Q8D411yescharacteristic
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—24no—
70C22×C3×D4—21no—
71C22×C3×D4—22no—
72C22×C3×D4—22no—
73C22×C3×D4—42no—
74C22×C3×D4D411yescharacteristic
75C22×C3×D4D411yescharacteristic
76C22×C3×D4D411yescharacteristic
77C22×C3×D4C2312yes—
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×#67—21no—
83C22×#67—21no—
84C23×Dic3—21no—
85C2×#235—22no—
86C2×#235—24no—
87C2×#235—22no—
88C2×#235—42no—
89C22×D12C2311yescharacteristic
90C22×C4×D3—21no—
91C22×Dic6C2311yescharacteristic
92C3×#127—28no—
93C3×#124—28no—
94C3×#121—24no—
95C3×#121—24no—
96C3×#121—24no—
97C3×#121D418yescomplemented
98C3×#120—28no—
99C3×#120—28no—
100C3×#117—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C42—21no—
106C2×#217—21no—
107C2×#217—21no—
108#946—28no—
109#939—28no—
110#939—28no—
111#938—28no—
112C2×#214—24no—
113C2×#214—24no—
114C2×#214—42no—
115C2×#213—24no—
116C2×#213—24no—
117C2×#213—42no—
118#930—22no—
119#930—22no—
120#930—22no—
121#930D414yescomplemented
122C2×(C2×D3)⋊C4—21no—
123C2×(C2×D3)⋊C4—21no—
124C2×C4⋊Dic3—21no—
125C2×C4⋊Dic3—22no—
126C2×C4⋊Dic3—22no—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2311yescharacteristic
129#924—22no—
130#924—22no—
131#924—22no—
132#924D414yescomplemented
133C2×Dic3⋊C4—21no—
134C2×Dic3⋊C4—21no—
135C22×C3⋊C8C2312yes—
136C22×C3⋊C8C2312yes—
137#832—24no—
138#832—24no—
139#830—22no—
140#830—22no—
141#830—24no—
142#830—42no—
143#810—24no—
144#809—24no—
145C22×#41—31no—
146C22×#41C2×D311yescharacteristic
147C23×D4C2×D311yescharacteristic
148C22×SD16—32no—
149C22×D8—32no—
150#537—64no—
151#537—64no—
152#535—28no—
153C2×#127—21no—
154C2×#124—21no—
155C2×C22⋊Q8—31no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—61no—
158C2×#121—31no—
159C2×#121—31no—
160C2×#121—61no—
161C2×#121—61no—
162C2×#121—61no—
163C2×#121C2×D312yes—
164C2×#121C2×D312yes—
165C2×#120—21no—
166C2×#120—21no—
167C2×#117C2×D311yescharacteristic
168C22×C4⋊C4—31no—
169#450—64no—
170#449—68no—
171#447—64no—
172C2×#103—32no—
173C2×#102—32no—
174C2×#102—32no—
175C2×#102—32no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—32no—
178C2×C22⋊C8—31no—
179C2×C22⋊C8—32no—
180#340—32no—
181#340—32no—
182C24×C3—24no—
183C24×C3—22no—
184C24×C3—42no—
185C24×C3—42no—
186C24×C3—82no—
187C24×C3—82no—
188C24×C3C2×D411yescharacteristic
189C24×C3C2×D411yescharacteristic
190C24×C3C2×D411yescharacteristic
191C23×D3—21no—
192C3×#41—48no—
193C3×#41—48no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—41no—
197C2×C3×D4—24no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—28no—
203C2×C3×D4—28no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—22no—
207C2×C3×D4—22no—
208C2×C3×D4—22no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—48no—
212C2×C3×D4—48no—
213C2×C3×D4—41no—
214C2×C3×D4—41no—
215C2×C3×D4—41no—
216C2×C3×D4—42no—
217C2×C3×D4—42no—
218C2×C3×D4—82no—
219C2×C3×D4—82no—
220C2×C3×D4—82no—
221C2×C3×D4—82no—
222C2×C3×D4C2×D414yes—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4C2×D412yes—
234C22×C3×C4C2×D412yes—
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D414yes—
239C22×C3×C4C2411yescharacteristic
240C2×#67—24no—
241C2×#67—24no—
242C2×#67—24no—
243C2×#67—42no—
244C2×#67—42no—
245C2×#67—82no—
246C2×#67—82no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3—21no—
250C22×Dic3—24no—
251C22×Dic3—82no—
252#235—48no—
253#235—48no—
254C2×D12—22no—
255C2×D12—22no—
256C2×D12—21no—
257C2×D12C2×D412yes—
258C2×C4×D3—24no—
259C2×C4×D3—42no—
260C2×C4×D3—42no—
261C2×Dic6—22no—
262C2×Dic6—22no—
263C2×Dic6—21no—
264C2×Dic6C2×D412yes—
265C3×C4⋊C4—22no—
266C3×C4⋊C4C2×D414yes—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—28no—
269C3×C22⋊C4—28no—
270C3×C42—24no—
271#217—24no—
272#217—24no—
273#214—48no—
274#213—48no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—24no—
277C4⋊Dic3—24no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3C2×D412yes—
281C4⋊Dic3C2×D412yes—
282Dic3⋊C4—24no—
283Dic3⋊C4—24no—
284C2×C3⋊C8—22no—
285C2×C3⋊C8—22no—
286C2×C3⋊C8—22no—
287C2×C3⋊C8—22no—
288C2×C3⋊C8—22no—
289C2×C3⋊C8—22no—
290C25—21no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—24no—
294C2×#41—42no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41—64no—
298C2×#41—122no—
299C22×Q8—31no—
300C22×Q8#6711yescharacteristic
301C22×D4—22no—
302C22×D4—22no—
303C22×D4—24no—
304C22×D4—22no—
305C22×D4—22no—
306C22×D4—21no—
307C22×D4—31no—
308C22×D4—42no—
309C22×D4—61no—
310C22×D4—61no—
311C22×D4—61no—
312C22×D4#6711yescharacteristic
313C22×D4#6711yescharacteristic
314C22×D4#6711yescharacteristic
315C22×D4C22×D312yes—
316C23×C4—61no—
317C23×C4—61no—
318C23×C4C22×D311yescharacteristic
319C23×C4C22×D311yescharacteristic
320C23×C4C22×D311yescharacteristic
321C2×SD16—64no—
322C2×SD16—64no—
323C2×SD16—122no—
324C2×D8—64no—
325C2×D8—64no—
326C2×D8—122no—
327C22×C8—32no—
328C22×C8—32no—
329#127—28no—
330#124—28no—
331C22⋊Q8—34no—
332C22⋊Q8—62no—
333C22⋊Q8—62no—
334C22⋊Q8—68no—
335C22⋊Q8—62no—
336#121—24no—
337#121—24no—
338#121—24no—
339#121—34no—
340#121—62no—
341#121—62no—
342#121—68no—
343#121—68no—
344#121—68no—
345#121—62no—
346#121#6718yescomplemented
347#120—28no—
348#120—28no—
349#117—24no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—62no—
353C2×C4⋊C4—62no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4C22×D312yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—61no—
362C2×C22⋊C4—61no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365C2×C42—21no—
366#103—64no—
367#102—64no—
368#102—64no—
369#102—64no—
370C22⋊C8—62no—
371C22⋊C8—62no—
372C22⋊C8—64no—
373C22⋊C8—122no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3—22no—
382C23×C3—21no—
383C23×C3—24no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—22no—
387C23×C3—22no—
388C23×C3—48no—
389C23×C3—42no—
390C23×C3—42no—
391C23×C3—42no—
392C23×C3—42no—
393C23×C3—42no—
394C23×C3—42no—
395C23×C3—84no—
396C23×C3—84no—
397C23×C3—88no—
398C23×C3—88no—
399C23×C3#12012yes—
400C23×C3#12012yes—
401C23×C3C22×D411yescharacteristic
402C22×D3—22no—
403C22×D3—41no—
404C22×D3—41no—
405C22×D3—42no—
406C22×D3—42no—
407C3×Q8—44no—
408C3×D4—24no—
409C3×D4—24no—
410C3×D4—44no—
411C3×D4—44no—
412C3×D4—48no—
413C3×D4—44no—
414C3×D4—42no—
415C3×D4—42no—
416C3×D4—88no—
417C3×D4—88no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—24no—
423C2×C3×C4—24no—
424C2×C3×C4—24no—
425C2×C3×C4—24no—
426C2×C3×C4—21no—
427C2×C3×C4—24no—
428C2×C3×C4—21no—
429C2×C3×C4—21no—
430C2×C3×C4—42no—
431C2×C3×C4—42no—
432C2×C3×C4—42no—
433C2×C3×C4—42no—
434C2×C3×C4#12012yes—
435C2×C3×C4#12012yes—
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D412yes—
439C2×C3×C4H(5,𝔽2)12yes—
440#67—48no—
441#67—84no—
442#67—84no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—42no—
449C2×Dic3—42no—
450C2×Dic3—84no—
451D12—24no—
452D12—42no—
453C4×D3—48no—
454Dic6—24no—
455Dic6—42no—
456C3⋊C8—42no—
457C3⋊C8—42no—
458C24—24no—
459C24—22no—
460C24—42no—
461C24—42no—
462C24—61no—
463C24—82no—
464C24—82no—
465C24C2×#6711yescharacteristic
466C24C2×#6711yescharacteristic
467C24C2×#6711yescharacteristic
468#41—48no—
469#41—48no—
470#41—128no—
471#41—128no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—32no—
475C2×Q8—41no—
476C2×Q8—61no—
477C2×Q8—62no—
478C2×Q8—62no—
479C2×D4—24no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—28no—
485C2×D4—28no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—32no—
494C2×D4—48no—
495C2×D4—48no—
496C2×D4—41no—
497C2×D4—41no—
498C2×D4—41no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—64no—
502C2×D4—64no—
503C2×D4—64no—
504C2×D4—61no—
505C2×D4—62no—
506C2×D4—62no—
507C2×D4—82no—
508C2×D4—82no—
509C2×D4—82no—
510C2×D4—82no—
511C2×D4—122no—
512C2×D4—122no—
513C2×D4—242no—
514C2×D4—242no—
515C2×D4C2×#6714yes—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—64no—
527C22×C4—61no—
528C22×C4—61no—
529C22×C4—61no—
530C22×C4—64no—
531C22×C4—122no—
532C22×C4—122no—
533C22×C4—242no—
534C22×C4D3×D412yes—
535C22×C4D3×D412yes—
536C22×C4C2×#6711yescharacteristic
537C22×C4C2×#6711yescharacteristic
538C22×C4C2×#6711yescharacteristic
539C22×C4C2×#6714yes—
540C22×C4C23×D311yescharacteristic
541SD16—128no—
542D8—128no—
543C2×C8—62no—
544C2×C8—62no—
545C2×C8—62no—
546C2×C8—62no—
547C2×C8—62no—
548C2×C8—62no—
549C4⋊C4—22no—
550C4⋊C4—32no—
551C4⋊C4—32no—
552C4⋊C4—64no—
553C4⋊C4—64no—
554C4⋊C4—64no—
555C4⋊C4—61no—
556C4⋊C4—61no—
557C4⋊C4C2×#6714yes—
558C22⋊C4—24no—
559C22⋊C4—28no—
560C22⋊C4—28no—
561C22⋊C4—64no—
562C22⋊C4—64no—
563C22⋊C4—64no—
564C22⋊C4—64no—
565C42—24no—
566C22×C3—22no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—22no—
570C22×C3—24no—
571C22×C3—42no—
572C22×C3—42no—
573C22×C3—41no—
574C22×C3—41no—
575C22×C3—42no—
576C22×C3—41no—
577C22×C3—44no—
578C22×C3—42no—
579C22×C3—41no—
580C22×C3—41no—
581C22×C3—42no—
582C22×C3—44no—
583C22×C3—44no—
584C22×C3—816no—
585C22×C3—82no—
586C22×C3—82no—
587C22×C3—82no—
588C22×C3—82no—
589C22×C3C2×#12011yescharacteristic
590C22×C3C2×#12011yescharacteristic
591C22×C3#53411yescharacteristic
592C22×C3#57612yes—
593C22×C3#57712yes—
594C2×D3—42no—
595C2×D3—42no—
596C2×D3—42no—
597C2×D3—82no—
598C2×D3—82no—
599C3×C4—22no—
600C3×C4—22no—
601C3×C4—41no—
602C3×C4—44no—
603C3×C4D4212yes—
604C3×C4D4212yes—
605Dic3—42no—
606Dic3—42no—
607Dic3—42no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—22no—
612C23—22no—
613C23—22no—
614C23—22no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—22no—
619C23—22no—
620C23—24no—
621C23—21no—
622C23—48no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23—42no—
628C23—42no—
629C23—62no—
630C23—84no—
631C23—84no—
632C23—88no—
633C23—88no—
634C23—121no—
635C23—122no—
636C23—122no—
637C23—121no—
638C23#93712yes—
639C23#95312yes—
640C23C22×#6711yescharacteristic
641Q8—44no—
642Q8—64no—
643Q8—122no—
644D4—24no—
645D4—24no—
646D4—44no—
647D4—44no—
648D4—48no—
649D4—44no—
650D4—42no—
651D4—42no—
652D4—64no—
653D4—88no—
654D4—88no—
655D4—128no—
656D4—122no—
657D4—244no—
658D4—244no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—24no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—24no—
670C2×C4—21no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—62no—
676C2×C4—62no—
677C2×C4—62no—
678C2×C4—128no—
679C2×C4—121no—
680C2×C4—122no—
681C2×C4—122no—
682C2×C4—121no—
683C2×C4—244no—
684C2×C4#93712yes—
685C2×C4#95312yes—
686C2×C4C2×D3×D411yescharacteristic
687C2×C4C2×D3×D411yescharacteristic
688C2×C4#100412yes—
689C2×C4C22×#6712yes—
690C8—122no—
691C8—122no—
692C2×C3—42no—
693C2×C3—42no—
694C2×C3—81no—
695C2×C3—84no—
696C2×C3#240311yescharacteristic
697C2×C3#300811yescharacteristic
698C2×C3#301011yescharacteristic
699C2×C3#301212yes—
700C2×C3#319812yes—
701D3—82no—
702C22—22no—
703C22—22no—
704C22—24no—
705C22—22no—
706C22—22no—
707C22—42no—
708C22—41no—
709C22—42no—
710C22—42no—
711C22—41no—
712C22—41no—
713C22—44no—
714C22—42no—
715C22—44no—
716C22—44no—
717C22—41no—
718C22—42no—
719C22—41no—
720C22—816no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—122no—
726C22—122no—
727C22—122no—
728C22—242no—
729C22—242no—
730C22C2×#93711yescharacteristic
731C22#580411yescharacteristic
732C22#583512yes—
733C22#583612yes—
734C22C2×#95311yescharacteristic
735C4—22no—
736C4—22no—
737C4—41no—
738C4—44no—
739C4—122no—
740C4—122no—
741C4—122no—
742C4D4×#6712yes—
743C4D4×#6712yes—
744C3#2392411yescharacteristic, complemented, Sylow
745C2—42no—
746C2—42no—
747C2—81no—
748C2—84no—
749C2—242no—
750C2#8338311yescharacteristic
751C2#8338612yes—
752C2#8339011yescharacteristic
753C2#8405211yescharacteristic
754C2#8410212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#535C2
socleC23×C3C22×D4
hypercenterC2×#535D3
hypocenterC3#23924
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. C2×#5833C2semidirect
3. C2×#5218C2semidirect
4. C2×#5217C2semidirect
5. C2×#5037C2semidirect
6. #72136C2semidirect
7. #71685C2semidirect
8. #71684C2semidirect
9. #71677C2semidirect
10. #71586C2semidirect
11. #71585C2semidirect
15. C22×#235C22semidirect
33. C2×#930C22semidirect
34. C22×C4⋊Dic3C22semidirect
36. C2×#924C22semidirect
43. C2×#830C22semidirect
44. C2×#830C22semidirect
46. C2×#810C22semidirect
47. C2×#809C22semidirect
48. #4529C22semidirect
49. #4529C22semidirect
51. C2×#535D3semidirect
97. C3×#121D4semidirect
121. #930D4semidirect
132. #924D4semidirect
346. #121#67semidirect
744. C3#23924semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×#58332
C2×#52182
C2×#52172
C2×#50372
#721362
#716852
#716842
#716772
#715862
#715852
#239243