Finite Group Databasebeta

← #10944028

Subgroups — #10944028

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×C4×#235C211yesmaximal, characteristic
2#82076C211yesmaximal, characteristic, complemented
3C2×#4849C211yesmaximal, characteristic, complemented
4C2×#4841C211yesmaximal, characteristic, complemented
5C2×#4733C212yesmaximal
6C3×#1797C211yesmaximal, characteristic, complemented
7#70485C211yesmaximal, characteristic, complemented
8#70479C211yesmaximal, characteristic, complemented
9#69931C212yesmaximal, complemented
10#68685C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#32907—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×C4×#67—21no—
16C2×C4×#67—21no—
17C2×#926C2211yescharacteristic
18C2×#924C2212yes—
19C22×Dic12C2212yes—
20C22×#204C2212yes—
21C4×#235—24no—
22C4×#235—24no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×#872—21no—
26C2×C42×D3—21no—
27C2×#869C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2212yes—
33#5212—21no—
34#5212—21no—
35C2×#821C2212yes—
36C2×#816C2212yes—
37C2×#816C2212yes—
38C2×#816C2212yes—
39#4935C2211yescharacteristic
40#4929C2212yes—
41#4928—21no—
42#4928—21no—
43#4849—24no—
44#4841—24no—
45#4733—28no—
46C3×#336C2212yes—
47#4551C2212yes—
48#4551C2212yes—
49C2×C4×#41—31no—
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#1797D311yescharacteristic, complemented
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235D414yes—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×C4×D3—21no—
73C22×C4×D3—21no—
74C22×Dic6C2312yes—
75C22×Dic6C2311yescharacteristic
76C22×C3×C8C2312yes—
77C22×C3×C8C2312yes—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85C2×#217—22no—
86C4×#67—28no—
87C4×#67—28no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90#926—22no—
91#926—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99#924—24no—
100#924—24no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—22no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×Dic12—24no—
108C2×Dic12—24no—
109C2×Dic12—24no—
110C2×#204—24no—
111C2×#204—24no—
112C2×#204—24no—
113C4×D12—22no—
114C4×D12—22no—
115C4×D12—22no—
116C4×D12D414yes—
117#872—28no—
118C42×D3—28no—
119#869—22no—
120#869—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6D414yes—
125C3×C4⋊C8—22no—
126C3×C4⋊C8—22no—
127C3×C22⋊C8—24no—
128C3⋊#95—22no—
129C3⋊#95—22no—
130#821—24no—
131#816—24no—
132#816—24no—
133#816—24no—
134C22×#41—31no—
135C22×Q16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139C4×#41—64no—
140C4×#41—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—31no—
148C2×#117—61no—
149C22×C42—61no—
150C22×C42C2×D311yescharacteristic
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8C2×D311yescharacteristic
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#102—32no—
160C2×C22⋊C8C2×D312yes—
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#336C2×D312yes—
170C24×C3C2×D411yescharacteristic
171C23×D3—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—41no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4#4112yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—24no—
191C2×#67—24no—
192C2×#67—24no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198#235—28no—
199#235—28no—
200#235—28no—
201#235—28no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12C2×D412yes—
206C2×D12C2×D412yes—
207C2×D12#4112yes—
208C2×C4×D3—24no—
209C2×C4×D3—24no—
210C2×C4×D3—24no—
211C2×C4×D3—24no—
212C2×C4×D3—24no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6C2×D412yes—
227C2×Dic6C2×D412yes—
228C2×Dic6#4112yes—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42C2×D412yes—
240C3×C42C2×D412yes—
241#217—24no—
242#217—44no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—22no—
248C4⋊Dic3—22no—
249C4⋊Dic3—42no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252C4⋊Dic3—42no—
253C4⋊Dic3C2×D412yes—
254C4⋊Dic3C2×D412yes—
255Dic3⋊C4—24no—
256Dic3⋊C4—24no—
257Dic3⋊C4—44no—
258C4×Dic3—24no—
259C4×Dic3—24no—
260C4×Dic3—24no—
261Dic12—48no—
262#204—48no—
263C2×#41—34no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C2×#41—62no—
269C22×Q8—31no—
270C22×Q8—32no—
271C22×D4—31no—
272C22×D4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C23×C4—61no—
276C23×C4D1211yescharacteristic
277C23×C4D1211yescharacteristic
278C23×C4C22×D311yescharacteristic
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—64no—
285C22×C8C22×D312yes—
286C22×C8C22×D312yes—
287C4⋊Q8—62no—
288C4⋊Q8—62no—
289C22⋊Q8—64no—
290C22⋊Q8—64no—
291C4×Q8—34no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×D4—34no—
296C4×D4—62no—
297C4×D4—62no—
298C4×D4—68no—
299C4×D4—68no—
300C4×D4—62no—
301#117—68no—
302#117—62no—
303#117—62no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—62no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—68no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42D1211yescharacteristic
325C2×C42D1211yescharacteristic
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C4⋊C8—22no—
329C4⋊C8—22no—
330#103—64no—
331#103—64no—
332#103—64no—
333#102—64no—
334C22⋊C8—24no—
335#95—62no—
336#95—62no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3#12112yes—
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—44no—
366C2×C3×C4—44no—
367C2×C3×C4#12012yes—
368C2×C3×C4#12012yes—
369C2×C3×C4#12112yes—
370C2×C3×C4#13512yes—
371C2×C3×C4#13712yes—
372C2×C3×C4#13712yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379#67—216no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395D12—22no—
396D12—22no—
397D12—22no—
398D12#12114yes—
399C4×D3—216no—
400C4×D3—48no—
401C4×D3—48no—
402Dic6—22no—
403Dic6—22no—
404Dic6—22no—
405Dic6—42no—
406Dic6—42no—
407Dic6—42no—
408Dic6—42no—
409Dic6#12114yes—
410C3×C8—42no—
411C3×C8—42no—
412C24—61no—
413C24C2×D1211yescharacteristic
414#41—68no—
415#41—68no—
416#41—68no—
417#41—68no—
418C2×Q8—32no—
419C2×Q8—32no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8—62no—
427C2×Q8—62no—
428C2×Q8—62no—
429C2×Q8—62no—
430C2×D4—32no—
431C2×D4—32no—
432C2×D4—32no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—64no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—64no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4C2×D1211yescharacteristic
464C22×C4C2×D1211yescharacteristic
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4C2×D1211yescharacteristic
468C22×C4D3×D412yes—
469C22×C4#23912yes—
470C22×C4C23×D311yescharacteristic
471Q16—128no—
472SD16—128no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—32no—
480C4⋊C4—32no—
481C4⋊C4—64no—
482C4⋊C4—64no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—62no—
486C4⋊C4—62no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—124no—
491C4⋊C4—122no—
492C22⋊C4—64no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—124no—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—21no—
500C42—64no—
501C42—64no—
502C42—64no—
503C42D3×D412yes—
504C42D3×D412yes—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×#12111yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3C2×#13511yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3#57511yescharacteristic
519C2×D3—24no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C3×C4—22no—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—41no—
530C3×C4#45214yescomplemented
531C3×C4C2×#12111yescharacteristic
532C3×C4C2×#12111yescharacteristic
533C3×C4#52511yescharacteristic
534C3×C4#52511yescharacteristic
535Dic3—24no—
536Dic3—42no—
537Dic3—42no—
538Dic3—82no—
539Dic3—82no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—62no—
547C23—62no—
548C23—62no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23#89512yes—
554C23C22×D1211yescharacteristic
555Q8—34no—
556Q8—62no—
557Q8—62no—
558Q8—62no—
559Q8—122no—
560Q8—122no—
561Q8—122no—
562Q8—122no—
563D4—34no—
564D4—616no—
565D4—62no—
566D4—62no—
567D4—62no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—616no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—128no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—128no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4#88212yes—
602C2×C4#88212yes—
603C2×C4#89512yes—
604C2×C4#90412yes—
605C2×C4#90912yes—
606C2×C4#90912yes—
607C2×C4C2×D3×D411yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×D3×D411yescharacteristic
610C2×C4C2×#23911yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C8—42no—
614C8—42no—
615C2×C3—42no—
616C2×C3#284711yescharacteristic
617C2×C3C2×#45211yescharacteristic
618C2×C3#300711yescharacteristic
619C2×C3#303511yescharacteristic
620C2×C3#303911yescharacteristic
621C2×C3#310711yescharacteristic
622C2×C3#311611yescharacteristic
623D3—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—64no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22C2×#88211yescharacteristic
639C22C2×#89511yescharacteristic
640C22#552411yescharacteristic
641C22C2×#90411yescharacteristic
642C22C2×#90911yescharacteristic
643C22C2×#90911yescharacteristic
644C22#574811yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—64no—
650C4—122no—
651C4—122no—
652C4—242no—
653C4—242no—
654C4#473314yescomplemented
655C4D3×#12111yescharacteristic
656C4D3×#12111yescharacteristic
657C4#562711yescharacteristic
658C4#562711yescharacteristic
659C3#3290711yescharacteristic, complemented, Sylow
660C2—42no—
661C2—124no—
662C2#7404811yescharacteristic
663C2C2×#473311yescharacteristic
664C2#7864011yescharacteristic
665C2#7902111yescharacteristic
666C2#7905711yescharacteristic
667C2#7930211yescharacteristic
668C2#7934811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1797D3
hypocenterC3#32907
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #82076C2semidirect
3. C2×#4849C2semidirect
4. C2×#4841C2semidirect
6. C3×#1797C2semidirect
7. #70485C2semidirect
8. #70479C2semidirect
9. #69931C2semidirect
10. #68685C2semidirect
11. #68672C2semidirect
59. #1797D3semidirect
530. C3×C4#452semidirect
654. C4#4733semidirect
659. C3#32907semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
#820762
C2×#48492
C2×#48412
C2×#47332
C3×#17972
#704852
#704792
#699312
#686852
#686722
#329073