Finite Group Databasebeta

← #10831848

Subgroups — #10831848

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic, complemented
2#82834C212yesmaximal, complemented
3C4×#937C211yesmaximal, characteristic
4#71897C212yesmaximal, complemented
5#71895C212yesmaximal, complemented
6#71701C212yesmaximal, complemented
7#71657C212yesmaximal, complemented
8#71654C212yesmaximal
9#71619C211yesmaximal, characteristic, complemented
10#24350—31nomaximal, Sylow
11C23×C4×D3C2211yescharacteristic
12C3×#554—22no—
13C3×#554—24no—
14C3×D4×Q8—22no—
15C3×#544—21no—
16C3×#543—21no—
17C3×#543C2212yes—
18C3×#533—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2212yes—
21C3×#518C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×#946—22no—
24C2×#937C2211yescharacteristic
25C2×C4×#67—21no—
26#5220C2211yescharacteristic
27C2×#834C2212yes—
28#5212C2212yes—
29C4×#217C2211yescharacteristic
30#4989—21no—
31#4989—24no—
32#4987—22no—
33#4956—21no—
34#4955C2214yes—
35#4955C2211yescharacteristic
36#4946C2214yes—
37#4946C2211yescharacteristic
38C4×(C2×D3)⋊C4—21no—
39#4526C2214yes—
40#4526C2214yes—
41#4526C2212yes—
42#3501D311yescharacteristic, complemented
43#2848—32no—
44C4×#120—31no—
45#2062—32no—
46#2029—32no—
47#1996—32no—
48#1889—32no—
49#1867—32no—
50#1799—31no—
51C24×D3C2311yescharacteristic
52C22×C3×Q8—21no—
53C22×C3×Q8D412yescomplemented
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3C2311yescharacteristic
59C22×C4×D3—21no—
60C22×C4×D3—22no—
61C22×C4×D3—21no—
62C22×C4×D3—22no—
63C22×C4×D3—44no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—24no—
67C3×#125—22no—
68C3×#125—24no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—24no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—24no—
86C3×C4×D4—24no—
87C3×#117—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D412yescomplemented
92C2×C3×C4⋊C4D414yescomplemented
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4C2314yes—
97C2×C3×C42D411yescharacteristic
98C2×#217—21no—
99C2×#217C2314yes—
100C2×#217C2311yescharacteristic
101C2×#217C2311yescharacteristic
102#946—48no—
103#937—22no—
104#937Q814yescomplemented
105C4×#67—44no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—24no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—24no—
111C2×C4⋊Dic3—22no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—24no—
114C2×C4×Dic3—21no—
115#834—24no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—24no—
119C3⋊#95—21no—
120C3⋊#95—22no—
121C24×C4—31no—
122#554—22no—
123#554—24no—
124D4×Q8—22no—
125#544—21no—
126#543—21no—
127#543C2×D312yes—
128#533—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—62no—
131C2×C22⋊Q8C2×D312yes—
132C2×#120—31no—
133#518C2×D312yes—
134C2×C4×D4—61no—
135C2×C4×D4C2×D311yescharacteristic
136C2×#99—32no—
137#386—31no—
138#386—61no—
139#385—31no—
140#385—34no—
141#385—61no—
142#385—64no—
143#380—32no—
144#380—62no—
145#379—34no—
146#379—31no—
147C4×C22⋊C4—31no—
148C4×C22⋊C4—61no—
149#342—32no—
150#342—34no—
151#342—34no—
152C24×C3C2×D411yescharacteristic
153C24×C3C2411yescharacteristic
154C23×D3—22no—
155C23×D3—21no—
156C23×D3—21no—
157C23×D3—22no—
158C23×D3—42no—
159C23×D3—42no—
160C23×D3C2×Q812yescomplemented
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—44no—
166C2×C3×Q8—82no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4C2×Q812yes—
170C2×C3×D4#4112yes—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C2×#67—24no—
187C2×#67—42no—
188C2×#67—42no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—24no—
194C22×Dic3—42no—
195C2×C4×D3—24no—
196C2×C4×D3—28no—
197C2×C4×D3—44no—
198C2×C4×D3—42no—
199C2×C4×D3—42no—
200C2×C4×D3—44no—
201C2×C4×D3—44no—
202C2×C4×D3—42no—
203C2×C4×D3—41no—
204C2×C4×D3—42no—
205C2×C4×D3—41no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—22no—
222C3×C42—24no—
223C3×C42—21no—
224C3×C42—21no—
225#217—21no—
226#217—24no—
227#217—22no—
228#217—24no—
229#217—42no—
230#217C2×Q812yescomplemented
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—48no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—48no—
236C4⋊Dic3—44no—
237Dic3⋊C4—42no—
238Dic3⋊C4—48no—
239C4×Dic3—42no—
240C25—31no—
241C22×Q8—21no—
242C22×Q8#6712yescomplemented
243C22×D4—61no—
244C22×D4C22×D311yescharacteristic
245C23×C4—31no—
246C23×C4—61no—
247C23×C4—62no—
248C23×C4—62no—
249C23×C4—61no—
250C23×C4—124no—
251C23×C4#6711yescharacteristic
252C23×C4C22×D311yescharacteristic
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—24no—
256#125—22no—
257#125—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—128no—
271#120—34no—
272#120—62no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—124no—
280#117—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—64no—
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4#6712yescomplemented
288C2×C4⋊C4#6714yescomplemented
289C2×C22⋊C4—22no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—34no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—64no—
299C2×C22⋊C4#6711yescharacteristic
300C2×C22⋊C4C22×D311yescharacteristic
301C2×C22⋊C4C22×D314yes—
302C2×C42—61no—
303C2×C42#6711yescharacteristic
304#99—64no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—62no—
309#95—64no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3C22⋊Q812yescomplemented
317C23×C3C22×D411yescharacteristic
318C23×C3C22×D412yes—
319C23×C3C22×Q811yescharacteristic
320C23×C3C2×#4111yescharacteristic
321C23×C32−1+412yes—
322C22×D3—24no—
323C22×D3—24no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—24no—
328C22×D3—44no—
329C22×D3—44no—
330C22×D3—48no—
331C22×D3—42no—
332C22×D3—41no—
333C22×D3—42no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—42no—
338C3×Q8—44no—
339C3×Q8—82no—
340C3×Q8—82no—
341C3×Q8—82no—
342C3×D4—24no—
343C3×D4—24no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—21no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—44no—
357C2×C3×C4—44no—
358C2×C3×C4—41no—
359C2×C3×C4—81no—
360C2×C3×C4—82no—
361C2×C3×C4—82no—
362C2×C3×C4#12011yescharacteristic
363C2×C3×C4#12012yes—
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4C22⋊Q812yescomplemented
366#67—48no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—44no—
377C2×Dic3—44no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—44no—
381C4×D3—44no—
382C4×D3—48no—
383C4×D3—44no—
384C24—32no—
385C24—62no—
386C24—61no—
387C24—61no—
388C24—62no—
389C24—122no—
390C24—122no—
391C24C2×#6711yescharacteristic
392C24C23×D311yescharacteristic
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—82no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—64no—
402C2×D4—122no—
403C2×D4—122no—
404C2×D4D3×Q812yes—
405C2×D4#23912yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—64no—
413C22×C4—68no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—64no—
419C22×C4—122no—
420C22×C4—124no—
421C22×C4—122no—
422C22×C4—124no—
423C22×C4—124no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—121no—
427C22×C4—122no—
428C22×C4—121no—
429C22×C4C2×#6711yescharacteristic
430C22×C4C2×#6711yescharacteristic
431C22×C4C2×#6712yes—
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6714yes—
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6714yes—
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6712yes—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—124no—
450C4⋊C4—122no—
451C4⋊C4—128no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—32no—
458C22⋊C4—61no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—62no—
462C22⋊C4—64no—
463C22⋊C4—122no—
464C22⋊C4—128no—
465C22⋊C4—128no—
466C22⋊C4—122no—
467C22⋊C4—122no—
468C42—24no—
469C42—21no—
470C42—21no—
471C42—122no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—81no—
484C22×C3C2×#12011yescharacteristic
485C22×C3C2×C22⋊Q811yescharacteristic
486C22×C3#53611yescharacteristic
487C22×C3#54812yes—
488C22×C3D4×Q812yes—
489C2×D3—24no—
490C2×D3—28no—
491C2×D3—42no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—44no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C3×C4—22no—
501C3×C4—21no—
502C3×C4—22no—
503C3×C4—41no—
504C3×C4—44no—
505C3×C4—44no—
506C3×C4—41no—
507C3×C4B(4,𝔽2)12yes—
508Dic3—42no—
509Dic3—42no—
510Dic3—84no—
511Dic3—81no—
512C23—21no—
513C23—21no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—64no—
519C23—64no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—64no—
524C23—122no—
525C23—121no—
526C23—122no—
527C23—121no—
528C23—124no—
529C23—124no—
530C23—128no—
531C23—121no—
532C23—122no—
533C23—121no—
534C23#94612yescomplemented
535C23C2×D3×Q811yescharacteristic
536C23C2×#23911yescharacteristic
537C23#100712yes—
538C23C22×#6711yescharacteristic
539C23C22×#6712yes—
540Q8—44no—
541Q8—82no—
542Q8—82no—
543Q8—82no—
544D4—24no—
545D4—24no—
546D4—128no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—21no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—62no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—124no—
569C2×C4—128no—
570C2×C4—121no—
571C2×C4—122no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—124no—
576C2×C4—121no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—124no—
582C2×C4#94612yescomplemented
583C2×C4#95311yescharacteristic
584C2×C4#95312yes—
585C2×C4#95311yescharacteristic
586C2×C3—21no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—42no—
591C2×C3#240911yescharacteristic
592C2×C3C2×B(4,𝔽2)11yescharacteristic
593C2×C3#302511yescharacteristic
594D3—44no—
595C22—22no—
596C22—21no—
597C22—22no—
598C22—21no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—64no—
607C22—68no—
608C22—81no—
609C22—122no—
610C22—124no—
611C22—124no—
612C22—122no—
613C22—122no—
614C22—124no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22C2×#94611yescharacteristic
619C22Q8×#6712yes—
620C22#581611yescharacteristic
621C22#581712yes—
622C22C2×#95311yescharacteristic
623C4—22no—
624C4—21no—
625C4—22no—
626C4—41no—
627C4—44no—
628C4—44no—
629C4—41no—
630C4—122no—
631C4—122no—
632C4—244no—
633C4—241no—
634C4#524412yes—
635C3#2435011yescharacteristic, complemented, Sylow
636C2—21no—
637C2—22no—
638C2—21no—
639C2—22no—
640C2—42no—
641C2—124no—
642C2#8406811yescharacteristic
643C2C2×#524411yescharacteristic
644C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3501C2
socleC22×C3C2×#120
prime radicalC2×#937C22
hypercenter#3501D3
hypocenterC3#24350
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. C3×#3501C2semidirect
2. #82834C2semidirect
4. #71897C2semidirect
5. #71895C2semidirect
6. #71701C2semidirect
7. #71657C2semidirect
9. #71619C2semidirect
42. #3501D3semidirect
53. C22×C3×Q8D4semidirect
91. C2×C3×C4⋊C4D4semidirect
92. C2×C3×C4⋊C4D4semidirect
104. #937Q8semidirect
160. C23×D3C2×Q8semidirect
230. #217C2×Q8semidirect
242. C22×Q8#67semidirect
287. C2×C4⋊C4#67semidirect
288. C2×C4⋊C4#67semidirect
316. C23×C3C22⋊Q8semidirect
365. C2×C3×C4C22⋊Q8semidirect
534. C23#946semidirect
582. C2×C4#946semidirect
635. C3#24350semidirect

Maximal subgroups

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

subgroupindex
C3×#35012
#828342
C4×#9372
#718972
#718952
#717012
#716572
#716542
#716192
#243503