Finite Group Databasebeta

← #11711757

Subgroups — #11711757

Order 768.

861 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 861 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×A4×C4⋊C4C211yesmaximal, characteristic
2C2×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
4C2×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
5A4⋊#95C211yesmaximal, characteristic
6A4⋊#95C211yesmaximal, characteristic
7A4⋊#95C211yesmaximal, characteristic
8#14728—31nomaximal, Sylow
9C22×C4×A4C2211yescharacteristic
10C22×C4×A4C2211yescharacteristic
11C22×C4×A4C2211yescharacteristic
12C22×SL(2,ℤ/4)C2211yescharacteristic
13C22×SL(2,ℤ/4)C2211yescharacteristic
14C22×SL(2,ℤ/4)C2211yescharacteristic
15C22×SL(2,ℤ/4)C2211yescharacteristic
16A4×C4⋊C4—22no—
17C4⋊SL(2,ℤ/4)—22no—
18SL(2,ℤ/4)⋊C4—22no—
19SL(2,ℤ/4)⋊C4—22no—
20#4982—41nomaximal
21C23×C4⋊C4D311yescharacteristic
22C2×#398—31no—
23C2×#398—31no—
24C2×#385—31no—
25C2×#385—31no—
26C2×#380—31no—
27#1444—31no—
28#1444—31no—
29#1442—31no—
30#1442—31no—
31#1441—31no—
32#1441—31no—
33C22⋊#95—31no—
34C22⋊#95—31no—
35C22⋊#95—31no—
36C23×A4C2311yescharacteristic
37C2×C4×A4—21no—
38C2×C4×A4—21no—
39C2×C4×A4—21no—
40C2×C4×A4—21no—
41C2×C4×A4—21no—
42C2×C4×A4—21no—
43C2×C4×A4—21no—
44C2×C4×A4D412yes—
45C2×C4×A4Q812yes—
46C2×SL(2,ℤ/4)—21no—
47C2×SL(2,ℤ/4)—21no—
48C2×SL(2,ℤ/4)—21no—
49C2×SL(2,ℤ/4)—21no—
50C2×SL(2,ℤ/4)—21no—
51C2×SL(2,ℤ/4)—21no—
52C2×SL(2,ℤ/4)—21no—
53C2×SL(2,ℤ/4)—21no—
54C2×SL(2,ℤ/4)—21no—
55C2×SL(2,ℤ/4)—21no—
56C2×SL(2,ℤ/4)D412yes—
57C2×SL(2,ℤ/4)Q812yes—
58C2×C3×C4⋊C4—41no—
59C2×C4⋊Dic3—41no—
60C2×Dic3⋊C4—41no—
61C2×Dic3⋊C4—41no—
62C3⋊#95—41no—
63C3⋊#95—41no—
64C3⋊#95—41no—
65C24×C4C2×D311yescharacteristic
66C24×C4C2×D311yescharacteristic
67C24×C4C2×D311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—31no—
70C22×C4⋊C4—31no—
71C22×C4⋊C4—31no—
72C22×C4⋊C4—31no—
73C22×C4⋊C4—31no—
74C22×C4⋊C4—31no—
75C22×C4⋊C4—31no—
76C22×C4⋊C4—31no—
77C22×C4⋊C4—31no—
78C22×C4⋊C4—31no—
79C22×C4⋊C4—62no—
80C22×C22⋊C4—31no—
81C22×C22⋊C4—31no—
82C22×C22⋊C4—31no—
83C22×C22⋊C4—31no—
84#398—64no—
85#398—61no—
86#398—61no—
87#398—61no—
88#398—61no—
89#385—62no—
90#385—62no—
91#385—62no—
92#385—62no—
93#380—62no—
94#380—62no—
95C2×#95—31no—
96C2×#95—31no—
97C2×#95—31no—
98C2×#95—31no—
99C2×#95—31no—
100C2×#95—31no—
101C2×#95—31no—
102C2×#95—31no—
103C2×#95—31no—
104C2×#95—31no—
105C2×#95—31no—
106C2×#95—31no—
107C2×#95—31no—
108C2×#95—31no—
109C2×#95—31no—
110C2×#95—31no—
111C2×#95—31no—
112C2×#95—31no—
113C22×A4C2×D411yescharacteristic
114C22×A4C2×D411yescharacteristic
115C22×A4C2×Q811yescharacteristic
116C22×A4C2×Q811yescharacteristic
117C22×A4#4111yescharacteristic
118C22×A4#4111yescharacteristic
119C22×A4#4111yescharacteristic
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×Dic3—41no—
124C22×Dic3—41no—
125C22×Dic3—41no—
126C22×Dic3—41no—
127C4×A4—22no—
128C4×A4—22no—
129C4×A4—41no—
130SL(2,ℤ/4)—22no—
131SL(2,ℤ/4)—22no—
132SL(2,ℤ/4)—41no—
133SL(2,ℤ/4)—41no—
134C3×C4⋊C4—82no—
135C4⋊Dic3—82no—
136Dic3⋊C4—82no—
137Dic3⋊C4—82no—
138C25C22×D311yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—31no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C23×C4—32no—
152C23×C4—32no—
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156C23×C4—31no—
157C23×C4—31no—
158C23×C4—31no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—31no—
162C23×C4—61no—
163C23×C4—61no—
164C23×C4—61no—
165C23×C4—61no—
166C23×C4—61no—
167C23×C4—61no—
168C23×C4—61no—
169C23×C4Dic612yes—
170C23×C4#6712yes—
171C2×C4⋊C4—34no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—62no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—121no—
199C2×C4⋊C4—121no—
200C2×C4⋊C4—121no—
201C2×C4⋊C4—121no—
202C2×C4⋊C4—121no—
203C2×C4⋊C4—121no—
204C2×C4⋊C4S411yescharacteristic
205C2×C22⋊C4—32no—
206C2×C22⋊C4—32no—
207C2×C22⋊C4—32no—
208C2×C22⋊C4—32no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—61no—
223C2×C22⋊C4—61no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229#95—62no—
230#95—62no—
231#95—62no—
232#95—62no—
233#95—62no—
234#95—62no—
235#95—62no—
236#95—62no—
237#95—62no—
238#95—62no—
239#95—62no—
240#95—62no—
241#95—61no—
242#95—61no—
243#95—62no—
244#95—61no—
245#95—61no—
246#95—61no—
247#95—61no—
248#95—121no—
249#95—121no—
250C23×C3—41no—
251C2×A4#12111yescharacteristic
252C2×A4C22⋊Q811yescharacteristic
253C2×A4C22⋊Q811yescharacteristic
254C2×A4#12311yescharacteristic
255C2×A4#12511yescharacteristic
256C2×A4#12511yescharacteristic
257C2×A4C4⋊Q811yescharacteristic
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—81no—
261C2×C3×C4—81no—
262C2×C3×C4—81no—
263C2×C3×C4—81no—
264C2×C3×C4—81no—
265C2×C3×C4—81no—
266C2×C3×C4—81no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—81no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C24—31no—
280C24—31no—
281C24—31no—
282C24—31no—
283C24—31no—
284C24—31no—
285C24—31no—
286C24—31no—
287C24C2×Dic611yescharacteristic
288C24#23511yescharacteristic
289C24D3×D411yescharacteristic
290C24#23711yescharacteristic
291C24D3×Q811yescharacteristic
292C24#23911yescharacteristic
293C24C2×#6711yescharacteristic
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—32no—
309C22×C4—32no—
310C22×C4—32no—
311C22×C4—31no—
312C22×C4—41no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4—61no—
330C22×C4—64no—
331C22×C4—61no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—61no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
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—61no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—122no—
400C22×C4—122no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4—121no—
404C22×C4C2×S411yescharacteristic
405C22×C4C2×S411yescharacteristic
406C22×C4C2×S411yescharacteristic
407C4⋊C4—22no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—64no—
411C4⋊C4—62no—
412C4⋊C4—122no—
413C4⋊C4—122no—
414C4⋊C4—122no—
415C4⋊C4—124no—
416C4⋊C4—122no—
417C4⋊C4—124no—
418C4⋊C4—122no—
419C4⋊C4—124no—
420C4⋊C4—122no—
421C22⋊C4—62no—
422C22⋊C4—62no—
423C22⋊C4—62no—
424C22⋊C4—62no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C22⋊C4—121no—
430C22⋊C4—121no—
431C22⋊C4—121no—
432C22⋊C4—121no—
433C22⋊C4—121no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C22⋊C4—121no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444A4#39811yescharacteristic, complemented
445C3×C4—82no—
446C3×C4—82no—
447C3×C4—161no—
448Dic3—82no—
449Dic3—82no—
450Dic3—161no—
451Dic3—161no—
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—31no—
472C23—31no—
473C23—31no—
474C23—31no—
475C23—31no—
476C23—31no—
477C23—31no—
478C23—31no—
479C23—31no—
480C23—31no—
481C23—31no—
482C23—31no—
483C23—31no—
484C23—31no—
485C23—31no—
486C23—31no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23#88811yescharacteristic
495C23#88911yescharacteristic
496C23#89011yescharacteristic
497C23#89411yescharacteristic
498C23#92411yescharacteristic
499C23#93911yescharacteristic
500C23#94611yescharacteristic
501C23C22×S411yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—34no—
510C2×C4—32no—
511C2×C4—32no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—61no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
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—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4A4⋊Q812yes—
599C2×C4GL(2,ℤ/4)12yes—
600C2×C3—41no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—41no—
604C2×C3—41no—
605C2×C3—41no—
606C2×C3—41no—
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—31no—
638C22—31no—
639C22—31no—
640C22—31no—
641C22—31no—
642C22—61no—
643C22—61no—
644C22—61no—
645C22—61no—
646C22—61no—
647C22—61no—
648C22—61no—
649C22#498211yescharacteristic, complemented
650C22C2×A4⋊Q811yescharacteristic
651C22#591211yescharacteristic
652C22D4×S411yescharacteristic
653C22#591411yescharacteristic
654C22Q8×S411yescharacteristic
655C22#591911yescharacteristic
656C22C2×GL(2,ℤ/4)11yescharacteristic
657C4—22no—
658C4—22no—
659C4—41no—
660C4—62no—
661C4—62no—
662C4—122no—
663C4—122no—
664C4—122no—
665C4—122no—
666C4—121no—
667C4—241no—
668C4—241no—
669C4—241no—
670C4—241no—
671C3—41noSylow
672C2—31no—
673C2—31no—
674C2—31no—
675C2—31no—
676C2—31no—
677C2—31no—
678C2—31no—
679C2—31no—
680C2#8530311yescharacteristic
681C2#8530411yescharacteristic
682C2#8530511yescharacteristic
683C2#8530911yescharacteristic
684C2#8548011yescharacteristic
685C2#8549511yescharacteristic
686C2#8550611yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutator, prime radical, cosocleC23×A4C23
FittingC23×C4⋊C4D3
socleC25C22×D3
hypercenterC2×C4⋊C4S4
hypocenterA4#398
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
444. A4#398semidirect
649. C22#4982semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C4⋊C42
C2×C4⋊SL(2,ℤ/4)2
C2×SL(2,ℤ/4)⋊C42
A4⋊#952
#147283
#49824