Finite Group Databasebeta

← #10944026

Subgroups — #10944026

Order 768.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C211yesmaximal, characteristic
2C2×#5239C212yesmaximal
3C2×#5031C211yesmaximal, characteristic, complemented
4C2×#5019C211yesmaximal, characteristic, complemented
5#82076C211yesmaximal, characteristic, complemented
6#71632C212yesmaximal, complemented
7#71626C211yesmaximal, characteristic, complemented
8#71612C212yesmaximal, complemented
9#71604C211yesmaximal, characteristic, complemented
10#71585C212yesmaximal, complemented
11#71530C211yesmaximal, characteristic, complemented
12#32907—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C3×C4×#41—24no—
15C3×C4×#41—24no—
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C42—21no—
25C22×C3×C42C2211yescharacteristic
26C22×#216C2212yes—
27C22×#214C2212yes—
28C2×#924C2212yes—
29C2×#869C2211yescharacteristic
30#5239—28no—
31C2×#835C2212yes—
32C2×#832C2212yes—
33#5212—21no—
34#5212—21no—
35C2×#830C2212yes—
36#5031—24no—
37#5019—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40#4935C2211yescharacteristic
41#4929C2212yes—
42#4928—21no—
43#4928—21no—
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4529C2212yes—
47#4529C2212yes—
48#4524C2212yes—
49C2×C4×#41D311yescharacteristic
50C2×#462—31no—
51C2×#461—31no—
52C2×#452—32no—
53#2857—31no—
54#1878—32no—
55#1877—32no—
56#1873—32no—
57#1803—31no—
58#1802—31no—
59#1797—31no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41D414yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4D411yescharacteristic
73C23×C3×C4D411yescharacteristic
74C23×C3×C4C2311yescharacteristic
75C22×Dic6C2312yes—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8D414yes—
80C3×C4×D4—28no—
81C3×C4×D4—28no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4D414yes—
86C3×#117—28no—
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—28no—
97C2×C3×C42—21no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42—22no—
101C2×C3×C42—22no—
102C2×C3×C42D411yescharacteristic
103C2×C3×C42D411yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×#217—22no—
107C2×#216—24no—
108C2×#216—24no—
109C2×#216—24no—
110C2×#214—24no—
111C2×#214—24no—
112C2×#214—24no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3C2312yes—
118#924—24no—
119#924—24no—
120C2×Dic3⋊C4—22no—
121C22×C3⋊C8C2312yes—
122C22×C3⋊C8C2312yes—
123#869—22no—
124#869—22no—
125#835—24no—
126#832—24no—
127C3⋊#95—22no—
128C3⋊#95—22no—
129#830—24no—
130#810—24no—
131#810—24no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—22no—
134C22×#41C2×D311yescharacteristic
135C22×Q16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139C4×#41—24no—
140C4×#41—24no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4C2×D311yescharacteristic
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—21no—
148C2×#117C2×D311yescharacteristic
149C22×C42—21no—
150C22×C42C2×D311yescharacteristic
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#391—32no—
162#389—61no—
163#389—61no—
164#384—31no—
165#380—61no—
166#380—61no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—21no—
171C24×C3C2×D411yescharacteristic
172C3×#41—28no—
173C3×#41—28no—
174C3×#41—28no—
175C3×#41—28no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8#4112yes—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4C2×D412yes—
189C2×C3×D4C2×D412yes—
190C2×C3×D4#4112yes—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—24no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—24no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—41no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4#4112yes—
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—22no—
222C22×Dic3—22no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4C2×D412yes—
234C3×C4⋊C4C2×D412yes—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C42—24no—
239C3×C42—24no—
240C3×C42—22no—
241C3×C42—24no—
242C3×C42—22no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247#217—44no—
248#216—48no—
249#214—48no—
250C4⋊Dic3—22no—
251C4⋊Dic3—22no—
252C4⋊Dic3—42no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3—42no—
256Dic3⋊C4—44no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41—22no—
268C2×#41D1214yes—
269C22×Q8—32no—
270C22×Q8C22×D311yescharacteristic
271C22×D4—21no—
272C22×D4C22×D311yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4#6711yescharacteristic
277C23×C4#6711yescharacteristic
278C23×C4C22×D311yescharacteristic
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—64no—
285C22×C8—32no—
286C22×C8—32no—
287C4⋊Q8—62no—
288C4⋊Q8—62no—
289C22⋊Q8—64no—
290C22⋊Q8—64no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8#6714yes—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4#6714yes—
301#117—28no—
302#117—22no—
303#117—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—62no—
318C2×C42—28no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42#6711yescharacteristic
325C2×C42#6711yescharacteristic
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C4⋊C8—62no—
329C4⋊C8—62no—
330#103—64no—
331#103—64no—
332#103—64no—
333#102—64no—
334C22⋊C8—64no—
335#95—62no—
336#95—62no—
337C23×C3—21no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3—21no—
341C23×C3—22no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3#12112yes—
351C23×C3C22×D411yescharacteristic
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8#12114yes—
356C3×D4—216no—
357C3×D4—22no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4#12114yes—
361C2×C3×C4—216no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—44no—
381C2×C3×C4—48no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—44no—
387C2×C3×C4—48no—
388C2×C3×C4#12012yes—
389C2×C3×C4#12012yes—
390C2×C3×C4#12112yes—
391C2×C3×C4#13512yes—
392C2×C3×C4#13712yes—
393C2×C3×C4#13712yes—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406Dic6—42no—
407Dic6—42no—
408Dic6—42no—
409Dic6—42no—
410C3⋊C8—42no—
411C3⋊C8—42no—
412C24—21no—
413C24C2×#6711yescharacteristic
414#41—28no—
415#41—28no—
416#41—28no—
417#41—28no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—62no—
422C2×Q8—62no—
423C2×Q8—62no—
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8—62no—
427C2×Q8C2×D1212yes—
428C2×Q8#23512yes—
429C2×Q8C2×#6712yes—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×D1212yes—
437C2×D4#23512yes—
438C2×D4C2×#6712yes—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—41no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4C2×D1212yes—
464C22×C4#23512yes—
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C23×D311yescharacteristic
471Q16—128no—
472SD16—128no—
473C2×C8—62no—
474C2×C8—62no—
475C2×C8—62no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—62no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C4⋊C4—122no—
486C4⋊C4—124no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4C2×#6712yes—
491C4⋊C4C2×#6712yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—124no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—21no—
502C42—21no—
503C42C2×#6712yes—
504C42C2×#6712yes—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—41no—
510C22×C3—42no—
511C22×C3—41no—
512C22×C3—42no—
513C22×C3—41no—
514C22×C3—42no—
515C22×C3—41no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3C2×#12011yescharacteristic
520C22×C3C2×#12111yescharacteristic
521C22×C3#52511yescharacteristic
522C22×C3C2×#13511yescharacteristic
523C22×C3C2×#13711yescharacteristic
524C22×C3C2×#13711yescharacteristic
525C22×C3#57511yescharacteristic
526C3×C4—22no—
527C3×C4—24no—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—41no—
531C3×C4—42no—
532C3×C4—42no—
533C3×C4#45214yescomplemented
534C3×C4C2×#12111yescharacteristic
535C3×C4C2×#12111yescharacteristic
536C3×C4#52511yescharacteristic
537C3×C4#52511yescharacteristic
538Dic3—82no—
539Dic3—82no—
540C23—22no—
541C23—21no—
542C23—22no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23#93012yes—
554C23C22×#6711yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—122no—
559Q8—122no—
560Q8—122no—
561Q8—122no—
562Q8#93014yes—
563D4—216no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4#93014yes—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4#93012yes—
602C2×C4#95012yes—
603C2×C4#95112yes—
604C2×C4#95112yes—
605C2×C4#95312yes—
606C2×C4#95312yes—
607C2×C4C22×D1211yescharacteristic
608C2×C4C2×#23511yescharacteristic
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×#23511yescharacteristic
611C2×C4C22×#6711yescharacteristic
612C2×C4C22×#6711yescharacteristic
613C8—122no—
614C8—122no—
615C2×C3—42no—
616C2×C3—44no—
617C2×C3#284711yescharacteristic
618C2×C3C2×#45211yescharacteristic
619C2×C3#300711yescharacteristic
620C2×C3#303511yescharacteristic
621C2×C3#303911yescharacteristic
622C2×C3#310711yescharacteristic
623C2×C3#311611yescharacteristic
624C22—22no—
625C22—24no—
626C22—22no—
627C22—22no—
628C22—42no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22C2×#93011yescharacteristic
639C22#579111yescharacteristic
640C22C2×#95011yescharacteristic
641C22#582211yescharacteristic
642C22C2×#95111yescharacteristic
643C22C2×#95111yescharacteristic
644C22C2×#95311yescharacteristic
645C4—22no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—242no—
653C4—242no—
654C4#523914yescomplemented
655C4C2×#93011yescharacteristic
656C4C2×#93011yescharacteristic
657C4#579111yescharacteristic
658C4#579111yescharacteristic
659C3#3290711yescharacteristic, complemented, Sylow
660C2—42no—
661C2—44no—
662C2#8252511yescharacteristic
663C2#8253311yescharacteristic
664C2#8253411yescharacteristic
665C2#8253511yescharacteristic
666C2#8403211yescharacteristic
667C2C2×#523911yescharacteristic
668C2#8408411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×#41C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×C4×#41D3
hypocenterC3#32907
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
3. C2×#5031C2semidirect
4. C2×#5019C2semidirect
5. #82076C2semidirect
6. #71632C2semidirect
7. #71626C2semidirect
8. #71612C2semidirect
9. #71604C2semidirect
10. #71585C2semidirect
11. #71530C2semidirect
533. C3×C4#452semidirect
654. C4#5239semidirect
659. C3#32907semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C2×#52392
C2×#50312
C2×#50192
#820762
#716322
#716262
#716122
#716042
#715852
#715302
#329073