Finite Group Databasebeta

← #11711697

Subgroups — #11711697

Order 768.

1325 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 1325 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×S4C211yesmaximal, characteristic
2C2×A4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×(C2×S4)⋊C4C212yesmaximal
4C2×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5A4⋊#95C212yesmaximal, complemented
6#13872—31nomaximal, Sylow
7C23×S4C2211yescharacteristic
8C22×C4×A4C2212yes—
9C22×C4×A4C2211yescharacteristic
10C22×SL(2,ℤ/4)C2212yes—
11C22×SL(2,ℤ/4)C2211yescharacteristic
12C2×C4×S4—22no—
13C2×C4×S4—22no—
14C2×C4×S4C414yescomplemented
15A4×C4⋊C4—22no—
16(C2×S4)⋊C4—24no—
17C4⋊SL(2,ℤ/4)—22no—
18#4987—41nomaximal
19C22×C4×D4—31no—
20C23×C4⋊C4D311yescharacteristic, complemented
21C2×#386—32no—
22C2×#380—31no—
23C2×#380—31no—
24C2×#380—31no—
25#1441—32no—
26#1439—32no—
27C4⋊#95—31no—
28C4⋊#95—31no—
29C22⋊#95—32no—
30C23×A4C2311yescharacteristic
31C22×S4—21no—
32C22×S4—21no—
33C22×S4—21no—
34C22×S4—21no—
35C22×S4C2×C412yes—
36C22×S4D412yes—
37C22×S4Q812yes—
38C22×C4×D3—41no—
39C2×C4×A4—22no—
40C2×C4×A4—22no—
41C2×C4×A4—22no—
42C2×C4×A4C2×C412yes—
43C2×C4×A4D412yes—
44C2×C4×A4D412yes—
45C2×SL(2,ℤ/4)—21no—
46C2×SL(2,ℤ/4)—21no—
47C2×SL(2,ℤ/4)—22no—
48C2×SL(2,ℤ/4)—22no—
49C2×SL(2,ℤ/4)—22no—
50C2×SL(2,ℤ/4)C2×C412yes—
51C4×S4—28no—
52C2×C3×C4⋊C4—41no—
53C2×(C2×D3)⋊C4—42no—
54C2×C4⋊Dic3—41no—
55C3⋊#95—42no—
56C23×D4—31no—
57C24×C4—31no—
58C24×C4C2×D312yes—
59C24×C4C2×D311yescharacteristic
60C2×C4×D4—34no—
61C2×C4×D4—34no—
62C2×C4×D4—62no—
63C2×C4×D4—62no—
64C2×C4×D4—62no—
65C2×C4×D4—62no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—31no—
68C22×C4⋊C4—31no—
69C22×C4⋊C4—31no—
70C22×C4⋊C4—32no—
71C22×C4⋊C4—31no—
72C22×C4⋊C4—31no—
73C22×C4⋊C4—62no—
74C22×C22⋊C4—31no—
75C22×C22⋊C4—31no—
76C22×C22⋊C4—32no—
77C22×C22⋊C4—32no—
78C22×C22⋊C4—32no—
79C22×C42—31no—
80#386—64no—
81#386—64no—
82#380—64no—
83#380—61no—
84#380—62no—
85#380—62no—
86#380—62no—
87#380—61no—
88C2×#95—32no—
89C2×#95—32no—
90C2×#95—32no—
91C2×#95—32no—
92C2×#95—32no—
93C2×#95—32no—
94C2×#95—32no—
95C23×D3—41no—
96C22×A4C22×C411yescharacteristic
97C22×A4C2×D411yescharacteristic
98C22×A4C2×D411yescharacteristic
99C22×A4C2×D411yescharacteristic
100C22×A4C2×Q811yescharacteristic
101C22×A4#4111yescharacteristic
102C22×A4#4111yescharacteristic
103C2×S4—22no—
104C2×S4—22no—
105C2×S4—22no—
106C2×S4—22no—
107C2×S4—22no—
108C2×S4—22no—
109C2×S4C4⋊C414yescomplemented
110C22×C3×C4—41no—
111C22×C3×C4—42no—
112C22×Dic3—41no—
113C22×Dic3—42no—
114C2×C4×D3—44no—
115C2×C4×D3—82no—
116C2×C4×D3—82no—
117C4×A4—42no—
118C4×A4C22⋊C414yescomplemented
119SL(2,ℤ/4)—22no—
120SL(2,ℤ/4)—42no—
121C3×C4⋊C4—82no—
122(C2×D3)⋊C4—84no—
123C4⋊Dic3—82no—
124C25—31no—
125C25C22×D311yescharacteristic
126C22×D4—32no—
127C22×D4—32no—
128C22×D4—32no—
129C22×D4—32no—
130C22×D4—32no—
131C22×D4—32no—
132C22×D4—61no—
133C22×D4—61no—
134C22×D4—61no—
135C22×D4—61no—
136C22×D4—61no—
137C22×D4—61no—
138C22×D4—61no—
139C22×D4—61no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4—32no—
146C23×C4—32no—
147C23×C4—32no—
148C23×C4—31no—
149C23×C4—32no—
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—34no—
153C23×C4—32no—
154C23×C4—32no—
155C23×C4—32no—
156C23×C4—62no—
157C23×C4—62no—
158C23×C4—62no—
159C23×C4—62no—
160C23×C4—62no—
161C23×C4—61no—
162C23×C4—62no—
163C23×C4—61no—
164C23×C4—62no—
165C23×C4—62no—
166C23×C4C4×D312yes—
167C23×C4D1212yes—
168C23×C4#6712yes—
169C4×D4—68no—
170C4×D4—68no—
171C4×D4—68no—
172C4×D4—68no—
173C2×C4⋊C4—34no—
174C2×C4⋊C4—34no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—64no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4S411yescharacteristic, complemented
193C2×C22⋊C4—34no—
194C2×C22⋊C4—32no—
195C2×C22⋊C4—32no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—64no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—64no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—122no—
214C2×C22⋊C4—122no—
215C2×C42—34no—
216C2×C42—62no—
217C2×C42—62no—
218#95—64no—
219#95—64no—
220#95—62no—
221#95—64no—
222#95—64no—
223#95—62no—
224#95—122no—
225#95—122no—
226C23×C3—41no—
227C22×D3—42no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—81no—
231C22×D3—81no—
232C22×D3—81no—
233C22×D3—81no—
234C2×A4C2×C22⋊C411yescharacteristic
235C2×A4C2×C4⋊C411yescharacteristic
236C2×A4#11711yescharacteristic
237C2×A4#12111yescharacteristic
238C2×A4#12111yescharacteristic
239C2×A4C22⋊Q811yescharacteristic
240C2×A4C22⋊Q811yescharacteristic
241S4—24no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—82no—
246C2×C3×C4—82no—
247C2×C3×C4—82no—
248C2×Dic3—42no—
249C2×Dic3—81no—
250C2×Dic3—81no—
251C2×Dic3—82no—
252C2×Dic3—82no—
253C2×Dic3—82no—
254C4×D3—88no—
255C24—31no—
256C24—31no—
257C24—31no—
258C24—31no—
259C24—31no—
260C24—31no—
261C24—31no—
262C24—31no—
263C24—32no—
264C24—32no—
265C24—32no—
266C24—61no—
267C24—61no—
268C24—61no—
269C24—61no—
270C24—61no—
271C24—61no—
272C24—61no—
273C24—61no—
274C24—61no—
275C24—61no—
276C24—61no—
277C24—61no—
278C24C2×C4×D311yescharacteristic
279C24C2×D1211yescharacteristic
280C24D3×D411yescharacteristic
281C24#23711yescharacteristic
282C24D3×Q811yescharacteristic
283C24#23911yescharacteristic
284C24C2×#6711yescharacteristic
285C2×D4—34no—
286C2×D4—34no—
287C2×D4—34no—
288C2×D4—34no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—62no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—62no—
306C2×D4—62no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—62no—
310C2×D4—62no—
311C2×D4—62no—
312C2×D4—62no—
313C22×C4—31no—
314C22×C4—32no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—32no—
323C22×C4—34no—
324C22×C4—34no—
325C22×C4—34no—
326C22×C4—32no—
327C22×C4—32no—
328C22×C4—32no—
329C22×C4—34no—
330C22×C4—32no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—42no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—64no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—68no—
345C22×C4—64no—
346C22×C4—68no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—64no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—122no—
384C22×C4—122no—
385C22×C4—124no—
386C22×C4—124no—
387C22×C4—124no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4—122no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4(C2×D3)⋊C414yescomplemented
394C22×C4C2×S412yes—
395C22×C4C2×S411yescharacteristic
396C4⋊C4—22no—
397C4⋊C4—64no—
398C4⋊C4—64no—
399C4⋊C4—68no—
400C4⋊C4—62no—
401C4⋊C4—122no—
402C4⋊C4—124no—
403C4⋊C4—122no—
404C22⋊C4—68no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—62no—
408C22⋊C4—62no—
409C22⋊C4—124no—
410C22⋊C4—128no—
411C22⋊C4—124no—
412C22⋊C4—122no—
413C22⋊C4—122no—
414C22⋊C4—122no—
415C22⋊C4—122no—
416C42—68no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C2×D3—44no—
425C2×D3—82no—
426C2×D3—82no—
427C2×D3—82no—
428C2×D3—82no—
429C2×D3—82no—
430C2×D3—82no—
431A4#38011yescharacteristic, complemented
432C3×C4—44no—
433C3×C4—162no—
434Dic3—82no—
435Dic3—162no—
436C23—31no—
437C23—31no—
438C23—31no—
439C23—31no—
440C23—31no—
441C23—31no—
442C23—31no—
443C23—31no—
444C23—31no—
445C23—31no—
446C23—31no—
447C23—31no—
448C23—31no—
449C23—31no—
450C23—31no—
451C23—31no—
452C23—31no—
453C23—31no—
454C23—31no—
455C23—31no—
456C23—31no—
457C23—31no—
458C23—31no—
459C23—31no—
460C23—31no—
461C23—31no—
462C23—31no—
463C23—31no—
464C23—31no—
465C23—31no—
466C23—31no—
467C23—31no—
468C23—31no—
469C23—31no—
470C23—31no—
471C23—34no—
472C23—61no—
473C23—61no—
474C23—61no—
475C23—61no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—62no—
480C23—62no—
481C23—62no—
482C23—62no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—62no—
487C23—62no—
488C23—62no—
489C23—62no—
490C23—62no—
491C23—62no—
492C23—62no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—62no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23—121no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—121no—
512C23—121no—
513C23D3×C4⋊C411yescharacteristic
514C23#89211yescharacteristic
515C23#89511yescharacteristic
516C23#89711yescharacteristic
517C23C2×(C2×D3)⋊C411yescharacteristic
518C23#93811yescharacteristic
519C23#94611yescharacteristic
520C23C22×S411yescharacteristic
521D4—64no—
522D4—64no—
523D4—64no—
524D4—64no—
525D4—64no—
526D4—64no—
527D4—64no—
528D4—64no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—34no—
536C2×C4—34no—
537C2×C4—34no—
538C2×C4—34no—
539C2×C4—34no—
540C2×C4—34no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—64no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—128no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—128no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4C4×S412yes—
592C2×C4C4⋊S412yes—
593C2×C4GL(2,ℤ/4)12yes—
594C2×C3—41no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—41no—
600C2×C3—41no—
601D3—84no—
602C22—31no—
603C22—31no—
604C22—31no—
605C22—31no—
606C22—31no—
607C22—31no—
608C22—31no—
609C22—31no—
610C22—31no—
611C22—31no—
612C22—31no—
613C22—31no—
614C22—31no—
615C22—31no—
616C22—31no—
617C22—31no—
618C22—31no—
619C22—31no—
620C22—31no—
621C22—31no—
622C22—31no—
623C22—31no—
624C22—31no—
625C22—31no—
626C22—31no—
627C22—31no—
628C22—31no—
629C22—31no—
630C22—31no—
631C22—31no—
632C22—31no—
633C22—31no—
634C22—31no—
635C22—31no—
636C22—31no—
637C22—61no—
638C22—61no—
639C22—61no—
640C22—61no—
641C22—61no—
642C22—61no—
643C22—61no—
644C22—64no—
645C22—64no—
646C22—64no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22—122no—
659C22#498711yescharacteristic, complemented
660C22C2×C4×S411yescharacteristic
661C22C2×C4⋊S411yescharacteristic
662C22D4×S411yescharacteristic
663C22#591411yescharacteristic
664C22Q8×S411yescharacteristic
665C22#591911yescharacteristic
666C22C2×GL(2,ℤ/4)11yescharacteristic
667C4—34no—
668C4—42no—
669C4—64no—
670C4—122no—
671C4—122no—
672C4—122no—
673C4—242no—
674C4—242no—
675C4(C2×S4)⋊C414yescomplemented
676C3—41noSylow
677C2—31no—
678C2—31no—
679C2—31no—
680C2—31no—
681C2—31no—
682C2—31no—
683C2—31no—
684C2—31no—
685C2—124no—
686C2C4⋊C4×S411yescharacteristic
687C2#8530711yescharacteristic
688C2#8531011yescharacteristic
689C2#8531211yescharacteristic
690C2C2×(C2×S4)⋊C411yescharacteristic
691C2#8549411yescharacteristic
692C2#8550611yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutatorC22×A4C22×C4
FittingC23×C4⋊C4D3
socleC25C22×D3
prime radicalC23×S4C22
cosocleC23×A4C23
hypercenterC2×C4⋊C4S4
hypocenterA4#380
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×A4×C4⋊C4C2semidirect
4. C2×C4⋊SL(2,ℤ/4)C2semidirect
5. A4⋊#95C2semidirect
14. C2×C4×S4C4semidirect
20. C23×C4⋊C4D3semidirect
109. C2×S4C4⋊C4semidirect
118. C4×A4C22⋊C4semidirect
192. C2×C4⋊C4S4semidirect
393. C22×C4(C2×D3)⋊C4semidirect
431. A4#380semidirect
659. C22#4987semidirect
675. C4(C2×S4)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×S42
C2×A4×C4⋊C42
C2×(C2×S4)⋊C42
C2×C4⋊SL(2,ℤ/4)2
A4⋊#952
#138723
#49874