Finite Group Databasebeta

← #10811320

Subgroups — #10811320

Order 768.

1696 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5550C211yesmaximal, characteristic
2C22⋊C4×D12C212yesmaximal, complemented
3C3×C4×#99C211yesmaximal, characteristic, complemented
4#71619C211yesmaximal, characteristic, complemented
5#70036C214yesmaximal, complemented
6#69457C212yesmaximal, complemented
7#68969C212yesmaximal, complemented
8D3×#342C212yesmaximal, complemented
9#22476—31nomaximal, Sylow
10C22×#235C411yescharacteristic, complemented
11C23×D12C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×#937C2212yes—
14C2×#930C411yescharacteristic, complemented
15C2×#924C2211yescharacteristic
16#5550—22no—
17#5550—24no—
18#5550—22no—
19C2×#894C412yescomplemented
20C2×#884C412yescomplemented
21C2×D3×C22⋊C4C2212yes—
22C2×D3×C22⋊C4C2214yes—
23C2×#875C411yescharacteristic, complemented
24C2×C4×D12—22no—
25C2×C4×D12C411yescharacteristic, complemented
26C2×C3×#99C2212yes—
27C3×C4×C22⋊C4C2212yes—
28C2×#834C2212yes—
29#5212C2212yes—
30#4955C2214yes—
31#4939—22no—
32#4665—24no—
33C3×#342C2211yescharacteristic
34C3×#342C2211yescharacteristic
35#4527C2214yes—
36#4527C2214yes—
37#4526C2212yes—
38C2×#546—31no—
39D4×C22⋊C4—32no—
40#1996—32no—
41#1898—34no—
42#1819—32no—
43#1799—31no—
44C4×#99D311yescharacteristic, complemented
45C2×#342—32no—
46C24×D3C2312yes—
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4C2×C411yescharacteristic, complemented
49C23×C3×C4C2311yescharacteristic
50C22×#67C2×C412yescomplemented
51C22×#67C2×C412yescomplemented
52C2×#235—22no—
53C2×#235—21no—
54C2×#235—21no—
55C2×#235—44no—
56C22×D12—44no—
57C22×D12C2×C411yescharacteristic
58C22×D12C2×C411yescharacteristic
59C22×D12C2×C412yes—
60C22×D12D412yes—
61C22×D12D414yes—
62C22×D12D412yes—
63C22×C4×D3—24no—
64C22×C4×D3C2×C412yescomplemented
65C22×C4×D3C2×C412yescomplemented
66C22×Dic6C2×C411yescharacteristic, complemented
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
70C2×C3×C22⋊C4C2×C411yescharacteristic, complemented
71C2×C3×C22⋊C4C2312yes—
72C2×C3×C22⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4C2312yes—
74C2×C3×C42—22no—
75C2×C3×C42C2×C411yescharacteristic, complemented
76C2×#217C2314yes—
77C2×#217C2312yes—
78#937—24no—
79#937—24no—
80#930—22no—
81#930—22no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4C2×C412yescomplemented
86C2×(C2×D3)⋊C4C2×C412yescomplemented
87C2×(C2×D3)⋊C4C2×C412yescomplemented
88C2×(C2×D3)⋊C4C2×C412yescomplemented
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3C2×C411yescharacteristic, complemented
91#924—22no—
92#924—22no—
93C2×Dic3⋊C4C2×C412yescomplemented
94#894—24no—
95#894—24no—
96#884—24no—
97#884—24no—
98D3×C22⋊C4—24no—
99D3×C22⋊C4—24no—
100D3×C22⋊C4—28no—
101D3×C22⋊C4—44no—
102#875—22no—
103#875—22no—
104C4×D12—22no—
105C4×D12—21no—
106C4×D12—21no—
107C4×D12—48no—
108C3×#99—24no—
109C3×#95—22no—
110#834—42no—
111C3⋊#95—22no—
112C22×#41—31no—
113C23×D4—31no—
114#546—62no—
115#546—64no—
116#546—62no—
117C2×#124—31no—
118C2×#123—32no—
119C2×C22⋊Q8—31no—
120C2×#121—31no—
121C2×#121—32no—
122C2×#120—32no—
123C2×C4×D4—31no—
124C2×C4×D4—62no—
125C2×C4×D4C2×D311yescharacteristic
126C22×C22⋊C4—32no—
127C22×C22⋊C4—34no—
128C2×#99—32no—
129C2×#99C2×D312yes—
130#390—62no—
131#386—64no—
132#380—32no—
133#379—34no—
134C4×C22⋊C4C2×D312yes—
135#342—32no—
136#342—34no—
137#342—34no—
138#342C2×D311yescharacteristic
139#342C2×D311yescharacteristic
140C24×C3C22×C411yescharacteristic
141C24×C3C2411yescharacteristic
142C23×D3—24no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—44no—
146C23×D3—82no—
147C23×D3C22×C412yes—
148C23×D3C22×C412yes—
149C23×D3C2×D414yes—
150C23×D3C2×D414yes—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C22×C411yescharacteristic
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C22×C411yescharacteristic
165C22×C3×C4C22×C411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C2×#67—22no—
169C2×#67—22no—
170C2×#67—24no—
171C2×#67—44no—
172C2×#67—82no—
173C22×Dic3—24no—
174C22×Dic3C22×C412yes—
175C22×Dic3C22×C412yes—
176#235—48no—
177#235—44no—
178#235—44no—
179C2×D12—28no—
180C2×D12—24no—
181C2×D12—42no—
182C2×D12—44no—
183C2×D12—44no—
184C2×D12—48no—
185C2×D12C22⋊C412yes—
186C2×D12C22⋊C411yescharacteristic
187C2×D12C22⋊C412yes—
188C2×D12C22⋊C414yes—
189C2×D12C22⋊C412yes—
190C2×D12C22⋊C414yes—
191C2×D12C22⋊C411yescharacteristic
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—44no—
197C2×C4×D3—48no—
198C2×C4×D3—48no—
199C2×C4×D3—84no—
200C2×C4×D3—82no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—42no—
204C3×C4⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4#4114yescomplemented
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—44no—
214#217—24no—
215#217—24no—
216#217—24no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—48no—
223(C2×D3)⋊C4—48no—
224(C2×D3)⋊C4—84no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—44no—
228Dic3⋊C4—24no—
229C25—32no—
230C2×#41—61no—
231C2×#41—62no—
232C2×#41—61no—
233C2×#41—124no—
234C22×Q8—31no—
235C22×D4—32no—
236C22×D4—32no—
237C22×D4—31no—
238C22×D4—32no—
239C22×D4—31no—
240C22×D4—32no—
241C22×D4—34no—
242C22×D4—32no—
243C22×D4—124no—
244C22×D4C22×D311yescharacteristic
245C23×C4—32no—
246C23×C4—32no—
247C23×C4—64no—
248C23×C4C4×D311yescharacteristic, complemented
249C23×C4C22×D311yescharacteristic
250#124—62no—
251#124—62no—
252#123—64no—
253#123—64no—
254C22⋊Q8—62no—
255C22⋊Q8—62no—
256#121—64no—
257#121—64no—
258#121—62no—
259#121—62no—
260#120—64no—
261#120—64no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—61no—
265C4×D4—62no—
266C4×D4—61no—
267C4×D4—128no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4C4×D311yescharacteristic, complemented
272C2×C22⋊C4—32no—
273C2×C22⋊C4—32no—
274C2×C22⋊C4—32no—
275C2×C22⋊C4—32no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—64no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—68no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—124no—
285C2×C22⋊C4C4×D311yescharacteristic, complemented
286C2×C22⋊C4C22×D312yes—
287C2×C22⋊C4C22×D311yescharacteristic
288C2×C22⋊C4C22×D312yes—
289C2×C42—22no—
290C2×C42C4×D311yescharacteristic, complemented
291#99—24no—
292#99—122no—
293#95—22no—
294#95—62no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3C23×C411yescharacteristic
303C23×C3C22×D412yes—
304C23×C3C2×#4112yes—
305C23×C3H(5,𝔽2)11yescharacteristic
306C23×C32−1+411yescharacteristic
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—28no—
316C22×D3—24no—
317C22×D3—42no—
318C22×D3—44no—
319C22×D3—44no—
320C22×D3—42no—
321C22×D3—44no—
322C22×D3—42no—
323C22×D3—84no—
324C22×D3—88no—
325C22×D3—84no—
326C22×D3C2×C22⋊C414yes—
327C22×D3C2×C22⋊C412yes—
328C22×D3C2×C22⋊C412yes—
329C3×D4—42no—
330C3×D4—42no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—24no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4C2×C22⋊C412yes—
351C2×C3×C4C2×C22⋊C411yescharacteristic
352C2×C3×C4C2×C22⋊C411yescharacteristic
353#67—44no—
354#67—44no—
355#67—84no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—44no—
361C2×Dic3—44no—
362C2×Dic3—44no—
363D12—28no—
364D12—24no—
365D12—24no—
366D12—22no—
367D12—42no—
368D12—44no—
369D12#9914yescomplemented
370C4×D3—44no—
371C4×D3—44no—
372C4×D3—82no—
373C4×D3—82no—
374C4×D3—816no—
375Dic6—42no—
376Dic6—42no—
377C24—32no—
378C24—34no—
379C24—32no—
380C24—34no—
381C24—64no—
382C24—62no—
383C24—62no—
384C24—124no—
385C24—242no—
386C24C2×C4×D311yescharacteristic
387C24C23×D311yescharacteristic
388#41—124no—
389#41—128no—
390#41—124no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—122no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—31no—
399C2×D4—32no—
400C2×D4—31no—
401C2×D4—34no—
402C2×D4—32no—
403C2×D4—34no—
404C2×D4—32no—
405C2×D4—68no—
406C2×D4—64no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—64no—
410C2×D4—124no—
411C2×D4—122no—
412C2×D4—124no—
413C2×D4—128no—
414C2×D4—124no—
415C2×D4—242no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—64no—
429C22×C4—128no—
430C22×C4—124no—
431C22×C4—128no—
432C22×C4—244no—
433C22×C4—242no—
434C22×C4C2×C4×D311yescharacteristic
435C22×C4C2×C4×D311yescharacteristic
436C22×C4C2×C4×D311yescharacteristic
437C22×C4C2×C4×D311yescharacteristic
438C22×C4C2×C4×D311yescharacteristic
439C22×C4D3×D412yes—
440C22×C4D3×D412yes—
441C4⋊C4—22no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—64no—
445C4⋊C4—124no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—64no—
452C22⋊C4—64no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C22⋊C4—64no—
459C22⋊C4—128no—
460C22⋊C4—128no—
461C22⋊C4—244no—
462C22⋊C4#23514yescomplemented
463C42—21no—
464C42—21no—
465C42—44no—
466C22×C3—22no—
467C22×C3—21no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—42no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3C22×C22⋊C411yescharacteristic
479C22×C3C4×#4111yescharacteristic
480C22×C3#52011yescharacteristic
481C22×C3#54612yes—
482C22×C3#54712yes—
483C2×D3—22no—
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—24no—
487C2×D3—22no—
488C2×D3—24no—
489C2×D3—22no—
490C2×D3—24no—
491C2×D3—48no—
492C2×D3—48no—
493C2×D3—44no—
494C2×D3—44no—
495C2×D3—84no—
496C2×D3—84no—
497C2×D3—82no—
498C2×D3—82no—
499C2×D3C2×#9914yes—
500C3×C4—21no—
501C3×C4—22no—
502C3×C4—44no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—82no—
507C3×C4C2×#9912yes—
508Dic3—42no—
509Dic3—42no—
510Dic3—84no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—32no—
515C23—34no—
516C23—32no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—68no—
522C23—64no—
523C23—62no—
524C23—62no—
525C23—64no—
526C23—64no—
527C23—62no—
528C23—62no—
529C23—64no—
530C23—64no—
531C23—124no—
532C23—122no—
533C23—124no—
534C23—124no—
535C23—122no—
536C23—122no—
537C23—248no—
538C23—244no—
539C23—244no—
540C23C22×C4×D311yescharacteristic
541C23C2×#23512yes—
542C23C2×D3×D412yes—
543C23#100911yescharacteristic
544C23#101011yescharacteristic
545Q8—122no—
546Q8—122no—
547D4—34no—
548D4—42no—
549D4—42no—
550D4—64no—
551D4—68no—
552D4—62no—
553D4—64no—
554D4—122no—
555D4—124no—
556D4—124no—
557D4—124no—
558D4—244no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—124no—
581C2×C4—124no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—124no—
585C2×C4—124no—
586C2×C4—124no—
587C2×C4—242no—
588C2×C4—242no—
589C2×C4—2416no—
590C2×C4D3×C22⋊C411yescharacteristic
591C2×C4D3×C22⋊C412yes—
592C2×C4D3×C22⋊C411yescharacteristic
593C2×C3—22no—
594C2×C3—22no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—81no—
598C2×C3#233911yescharacteristic
599C2×C3C22×#9911yescharacteristic
600C2×C3#288311yescharacteristic
601D3—24no—
602D3—42no—
603D3—44no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—21no—
608C22—34no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—62no—
616C22—62no—
617C22—62no—
618C22—64no—
619C22—64no—
620C22—62no—
621C22—62no—
622C22—64no—
623C22—81no—
624C22—81no—
625C22—128no—
626C22—128no—
627C22—124no—
628C22—124no—
629C22—244no—
630C22—244no—
631C22—242no—
632C22—242no—
633C22C4×#23511yescharacteristic
634C22C2×D3×C22⋊C411yescharacteristic
635C22#552311yescharacteristic
636C22#555012yes—
637C22#555112yes—
638C4—21no—
639C4—22no—
640C4—44no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4—122no—
646C4—122no—
647C4—244no—
648C4D3×#9912yes—
649C3#2247611yescharacteristic, complemented, Sylow
650C2—22no—
651C2—22no—
652C2—21no—
653C2—21no—
654C2—64no—
655C2—81no—
656C2—122no—
657C2—124no—
658C2#7357311yescharacteristic
659C2C2×D3×#9911yescharacteristic
660C2#7869411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×C4×#99C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5550C2
cosocleC24×C3C24
hypercenterC4×#99D3
hypocenterC3#22476
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×D12C2semidirect
3. C3×C4×#99C2semidirect
4. #71619C2semidirect
5. #70036C2semidirect
6. #69457C2semidirect
7. #68969C2semidirect
8. D3×#342C2semidirect
10. C22×#235C4semidirect
14. C2×#930C4semidirect
19. C2×#894C4semidirect
20. C2×#884C4semidirect
23. C2×#875C4semidirect
25. C2×C4×D12C4semidirect
44. C4×#99D3semidirect
48. C23×C3×C4C2×C4semidirect
50. C22×#67C2×C4semidirect
51. C22×#67C2×C4semidirect
64. C22×C4×D3C2×C4semidirect
65. C22×C4×D3C2×C4semidirect
66. C22×Dic6C2×C4semidirect
69. C2×C3×C4⋊C4C2×C4semidirect
70. C2×C3×C22⋊C4C2×C4semidirect
75. C2×C3×C42C2×C4semidirect
85. C2×(C2×D3)⋊C4C2×C4semidirect
86. C2×(C2×D3)⋊C4C2×C4semidirect
87. C2×(C2×D3)⋊C4C2×C4semidirect
88. C2×(C2×D3)⋊C4C2×C4semidirect
90. C2×C4⋊Dic3C2×C4semidirect
93. C2×Dic3⋊C4C2×C4semidirect
210. C3×C22⋊C4#41semidirect
248. C23×C4C4×D3semidirect
271. C2×C4⋊C4C4×D3semidirect
285. C2×C22⋊C4C4×D3semidirect
290. C2×C42C4×D3semidirect
369. D12#99semidirect
462. C22⋊C4#235semidirect
649. C3#22476semidirect

Maximal subgroups

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

subgroupindex
C2×#55502
C22⋊C4×D122
C3×C4×#992
#716192
#700362
#694572
#689692
D3×#3422
#224763