Finite Group Databasebeta

← #10686999

Subgroups — #10686999

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5634C211yesmaximal, characteristic
2C3×#1872C211yesmaximal, characteristic, complemented
3C2×#4538C211yesmaximal, characteristic, complemented
4#71570C211yesmaximal, characteristic
5#71559C211yesmaximal, characteristic, complemented
6C2×#4467C211yesmaximal, characteristic, complemented
7C2×#4468C211yesmaximal, characteristic
8#8405—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#946—21no—
12C22×(C2×D3)⋊C4—21no—
13C22×(C2×D3)⋊C4C2211yescharacteristic
14C2×#924C2211yescharacteristic
15#5634—22no—
16#5634—22no—
17#5634—22no—
18#5634C414yescomplemented
19C2×#897—21no—
20C2×#896—21no—
21C2×#896—21no—
22C2×D3×C22⋊C4—21no—
23C2×#878—21no—
24C2×C3×#103—21no—
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3⋊#95—21no—
27C2×C3⋊#95C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#810—21no—
30C3×#340—21no—
31#4538—22no—
32#4529—21no—
33#4468—22no—
34#4467—22no—
35C2×#543—31no—
36#1872—31no—
37#1872D311yescharacteristic, complemented
38C2×#328—31no—
39C2×#328—31no—
40C2×#323—31no—
41#1439—31no—
42C24×D3D411yescharacteristic
43C22×C3×Q8D411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4C2311yescharacteristic
46C23×Dic3—21no—
47C23×Dic3D411yescharacteristic
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×Dic6—21no—
51C22×Dic6D411yescharacteristic
52C22×C3×C8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8C2×C412yes—
56C3×C22⋊Q8D412yes—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—42no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×#217—21no—
64#946—24no—
65#946—24no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—42no—
73C2×(C2×D3)⋊C4—42no—
74C2×(C2×D3)⋊C4—42no—
75C2×(C2×D3)⋊C4C2×C412yes—
76C2×(C2×D3)⋊C4D412yes—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3D411yescharacteristic
79#924—21no—
80#924—21no—
81#924C2×C412yes—
82#924D412yes—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C22×C3⋊C8—21no—
87#897—24no—
88#897—24no—
89#896—24no—
90#896—24no—
91#896—24no—
92#896—24no—
93D3×C22⋊C4—24no—
94D3×C22⋊C4—24no—
95#878—24no—
96#878—24no—
97C3×#103—42no—
98C3×C22⋊C8—21no—
99C3×C22⋊C8—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—42no—
103#830—21no—
104#830—21no—
105#810—42no—
106#543—34no—
107#543—62no—
108#543—62no—
109#543—62no—
110C2×C22⋊Q8—31no—
111C2×C22⋊Q8—61no—
112C2×C22⋊Q8—61no—
113C2×C22⋊Q8—61no—
114C2×C22⋊Q8—61no—
115C2×C22⋊Q8—61no—
116C2×C22⋊Q8C2×D311yescharacteristic
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C22⋊C4—31no—
119C22×C22⋊C4—61no—
120C22×C22⋊C4—61no—
121C2×#103—21no—
122C2×#103—61no—
123C2×C22⋊C8—31no—
124C2×C22⋊C8C2×D311yescharacteristic
125C2×#95—31no—
126C2×#95—61no—
127#340—21no—
128#340—61no—
129#328—62no—
130#328—62no—
131#323—62no—
132C24×C3C2×D411yescharacteristic
133C23×D3—21no—
134C23×D3—21no—
135C23×D3—21no—
136C23×D3—21no—
137C23×D3—41no—
138C23×D3—41no—
139C23×D3—41no—
140C23×D3—41no—
141C23×D3C22⋊C412yes—
142C23×D3SD1612yes—
143C23×D3Q1612yes—
144C2×C3×Q8—21no—
145C2×C3×Q8—41no—
146C2×C3×Q8—41no—
147C2×C3×Q8C22⋊C412yescomplemented
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4C22×C411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—41no—
171C22×Dic3—41no—
172C22×Dic3—41no—
173C22×Dic3—41no—
174C22×Dic3—41no—
175C22×Dic3—41no—
176C22×Dic3—41no—
177C22×Dic3—41no—
178C22×Dic3C22⋊C412yes—
179C2×C4×D3—24no—
180C2×C4×D3—24no—
181C2×C4×D3—81no—
182C2×C4×D3—81no—
183C2×C4×D3—81no—
184C2×C4×D3—81no—
185C2×Dic6—24no—
186C2×Dic6—21no—
187C2×Dic6—41no—
188C2×Dic6—41no—
189C2×Dic6—81no—
190C2×Dic6—81no—
191C2×Dic6C22⋊C412yescomplemented
192C2×C3×C8—41no—
193C2×C3×C8—41no—
194C2×C3×C8—41no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4C22⋊C412yescomplemented
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205#217—22no—
206#217—22no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—22no—
210(C2×D3)⋊C4—22no—
211(C2×D3)⋊C4—22no—
212(C2×D3)⋊C4—22no—
213(C2×D3)⋊C4—48no—
214C4⋊Dic3—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3C22⋊C412yescomplemented
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219Dic3⋊C4—22no—
220Dic3⋊C4—22no—
221C2×C3⋊C8—41no—
222C2×C3⋊C8—41no—
223C2×C3⋊C8—41no—
224C25—31no—
225C22×Q8—31no—
226C22×Q8—61no—
227C22×Q8#6711yescharacteristic
228C23×C4—21no—
229C23×C4—31no—
230C23×C4—61no—
231C23×C4—61no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234C22×C8—21no—
235C22×C8—61no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—32no—
239C22⋊Q8—32no—
240C22⋊Q8—64no—
241C22⋊Q8—64no—
242C22⋊Q8—64no—
243C22⋊Q8—64no—
244C22⋊Q8—64no—
245C22⋊Q8—64no—
246C22⋊Q8—64no—
247C22⋊Q8—64no—
248C22⋊Q8—64no—
249C22⋊Q8—61no—
250C22⋊Q8—61no—
251C22⋊Q8—64no—
252C22⋊Q8C4×D312yes—
253C22⋊Q8D1212yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4#6711yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—32no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—122no—
277C2×C22⋊C4—122no—
278C2×C22⋊C4—122no—
279#103—42no—
280#103—122no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—61no—
284C22⋊C8—61no—
285#95—61no—
286#95—61no—
287#95—122no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3C2×C22⋊C411yescharacteristic
293C23×C3#12011yescharacteristic
294C23×C3#12011yescharacteristic
295C23×C3C2×SD1611yescharacteristic
296C23×C3C2×Q1611yescharacteristic
297C23×C3#13711yescharacteristic
298C23×C3#13711yescharacteristic
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—81no—
316C22×D3—81no—
317C22×D3—81no—
318C22×D3—81no—
319C22×D3—81no—
320C22×D3—81no—
321C22×D3#10314yescomplemented
322C3×Q8—42no—
323C3×Q8—81no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—81no—
342C2×C3×C4—81no—
343C2×C3×C4—81no—
344C2×C3×C4—81no—
345C2×C3×C4C2×C22⋊C411yescharacteristic
346C2×C3×C4C2×C22⋊C411yescharacteristic
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—81no—
363C2×Dic3—81no—
364C2×Dic3—81no—
365C2×Dic3—81no—
366C2×Dic3—81no—
367C4×D3—84no—
368C4×D3—84no—
369Dic6—42no—
370Dic6—81no—
371Dic6—84no—
372C3×C8—81no—
373C3⋊C8—81no—
374C24—32no—
375C24—32no—
376C24—32no—
377C24—61no—
378C24—61no—
379C24—61no—
380C24—61no—
381C24—121no—
382C24—121no—
383C24—121no—
384C24—121no—
385C24D3×D411yescharacteristic
386C2×Q8—21no—
387C2×Q8—32no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×Q8—64no—
391C2×Q8—61no—
392C2×Q8—121no—
393C2×Q8—121no—
394C2×Q8—241no—
395C2×Q8—241no—
396C2×Q8(C2×D3)⋊C412yescomplemented
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—32no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4—121no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4—121no—
422C22×C4—121no—
423C22×C4—121no—
424C22×C4—241no—
425C22×C4—241no—
426C22×C4—241no—
427C22×C4—241no—
428C22×C4C2×C4×D311yescharacteristic
429C22×C4C2×D1211yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—121no—
438C2×C8—121no—
439C2×C8—121no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—32no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4(C2×D3)⋊C412yescomplemented
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—64no—
458C22⋊C4—64no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—62no—
462C22⋊C4—62no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C22⋊C4—128no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3#37911yescharacteristic
479C22×C3C2×#10311yescharacteristic
480C22×C3#41911yescharacteristic
481C22×C3#44811yescharacteristic
482C22×C3#44811yescharacteristic
483C22×C3#45111yescharacteristic
484C22×C3#45111yescharacteristic
485C22×C3#45311yescharacteristic
486C22×C3#45511yescharacteristic
487C22×C3B(4,𝔽2)11yescharacteristic
488C22×C3#45811yescharacteristic
489C2×D3—24no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—44no—
497C2×D3—82no—
498C2×D3—82no—
499C2×D3—82no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—82no—
507Dic3—42no—
508Dic3—42no—
509Dic3—42no—
510Dic3—82no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—34no—
516C23—62no—
517C23—62no—
518C23—62no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—241no—
533C23—241no—
534C23—241no—
535C23—241no—
536C23—241no—
537C23—241no—
538C23D3×C22⋊C411yescharacteristic
539C23#88211yescharacteristic
540C23D3×SD1611yescharacteristic
541C23#91511yescharacteristic
542C23D3×Q1611yescharacteristic
543C23#91811yescharacteristic
544C23#93711yescharacteristic
545Q8—42no—
546Q8—81no—
547Q8—122no—
548Q8—241no—
549Q8—244no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—244no—
585C2×C4—244no—
586C2×C4—241no—
587C2×C4—241no—
588C2×C4—241no—
589C2×C4—241no—
590C2×C4—241no—
591C2×C4D3×C22⋊C411yescharacteristic
592C2×C4#88211yescharacteristic
593C2×C4C2×(C2×D3)⋊C411yescharacteristic
594C2×C4#93711yescharacteristic
595C8—81no—
596C8—241no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3#159611yescharacteristic
602C2×C3#160211yescharacteristic
603C2×C3#160811yescharacteristic
604C2×C3#161411yescharacteristic
605C2×C3#178711yescharacteristic
606C2×C3#179011yescharacteristic
607C2×C3#179211yescharacteristic
608D3—44no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—64no—
622C22—124no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—122no—
627C22—122no—
628C22—122no—
629C22—242no—
630C22—242no—
631C22—242no—
632C22D3×#10311yescharacteristic
633C22#480211yescharacteristic
634C22#480611yescharacteristic
635C22#480911yescharacteristic
636C22#495511yescharacteristic
637C22#503211yescharacteristic
638C22#505311yescharacteristic
639C22#517011yescharacteristic
640C22#518611yescharacteristic
641C22#519811yescharacteristic
642C22#520211yescharacteristic
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—122no—
651C4—122no—
652C4—122no—
653C4—242no—
654C3#840511yescharacteristic, complemented, Sylow
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—124no—
660C2#6959811yescharacteristic
661C2#7026611yescharacteristic
662C2#7030511yescharacteristic
663C2#7097611yescharacteristic
664C2#7098011yescharacteristic
665C2#7117211yescharacteristic
666C2#7117611yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1872C2
socleC23×C3#120
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1872D3
hypocenterC3#8405
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C3×#1872C2semidirect
3. C2×#4538C2semidirect
5. #71559C2semidirect
6. C2×#4467C2semidirect
18. #5634C4semidirect
37. #1872D3semidirect
147. C2×C3×Q8C22⋊C4semidirect
191. C2×Dic6C22⋊C4semidirect
202. C3×C4⋊C4C22⋊C4semidirect
216. C4⋊Dic3C22⋊C4semidirect
321. C22×D3#103semidirect
396. C2×Q8(C2×D3)⋊C4semidirect
454. C4⋊C4(C2×D3)⋊C4semidirect
654. C3#8405semidirect

Maximal subgroups

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

subgroupindex
C2×#56342
C3×#18722
C2×#45382
#715702
#715592
C2×#44672
C2×#44682
#84053