Finite Group Databasebeta

← C2×#72213835 #72226263

Subgroups — C2×#72213835 #72226263

Order 3072.

6400 conjugacy classes of subgroups; only the 273 normal subgroups are shown.

normal subgroups only

This group has 6400 conjugacy classes of subgroups — too many to draw. Only its 273 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×#14710⋊C3C213yesmaximal, complemented
2C2×#14711⋊C3C213yesmaximal
3C2×#6342420C311yesmaximal, characteristic, complemented, Sylow
4#72213835C218yesmaximal, direct product factor
5C2×C42×C42⋊C3C211yesmaximal, characteristic, complemented
6C23×#126⋊C3C2211yescharacteristic
7C2×A4×#126—41nomaximal
8C22×#383⋊C3C2213yes—
9C22×C4×C42⋊C3C2213yes—
10#14711⋊C3C22112yes—
11#14710⋊C3C22112yescomplemented
12C42×C42⋊C3C2214yescomplemented
13#6342420—38no—
14#6342420C2×C318yescomplemented
15C2×#14711—31no—
16C2×#14711—33no—
17C2×#14711C2×C313yes—
18C2×#14710—31no—
19C2×#14710—33no—
20C2×#14710C2×C313yescomplemented
21C2×C44C2×C311yescharacteristic, complemented
22A4×#126—48no—
23C22×#126⋊C3—26no—
24C22×#126⋊C3C2312yes—
25C2×A4×C4⋊C4—43no—
26C2×A4×C22⋊C4—43no—
27C2×C42×A4—41no—
28C23×C42⋊C3C2311yescharacteristic
29C2×#383⋊C3—26no—
30C2×#383⋊C3C2316yes—
31C2×C4×C42⋊C3—26no—
32C2×C4×C42⋊C3C2316yes—
33C23×#126—31no—
34C23×#126A411yescharacteristic
35C23×#126C22×C311yescharacteristic
36C22×#383—33no—
37C22×#383—31no—
38C22×#383C22×C313yes—
39C22×C43—31no—
40C22×C43—33no—
41C22×C43C22×C313yes—
42#14711—324no—
43#14711—312no—
44#14711—324no—
45#14711—38no—
46#14711—38no—
47#14711C22×C3112yes—
48#14710—38no—
49#14710—38no—
50#14710—324no—
51#14710—312no—
52#14710—324no—
53#14710C22×C3112yescomplemented
54C2×#1437—31no—
55C2×#1437—33no—
56C2×#1437—33no—
57C2×#1437—33no—
58C2×#1436—33no—
59C2×#1436—33no—
60C2×#1436—33no—
61C2×#1436—33no—
62C2×#1436—33no—
63C2×#1436—33no—
64C2×#1436—33no—
65C2×#1436—33no—
66C2×#1436—33no—
67C2×#1436—33no—
68C44—34no—
69C44C22×C314yescomplemented
70C24×A4—41no—
71C22×C4×A4—43no—
72C22×C4×A4—43no—
73C2×C3×#126—161no—
74C2×#126⋊C3—26no—
75C2×#126⋊C3—44no—
76A4×C4⋊C4—412no—
77A4×C22⋊C4—412no—
78C42×A4—44no—
79C22×C42⋊C3#4116yes—
80C22×C42⋊C3C2411yescharacteristic
81#383⋊C3—26no—
82C4×C42⋊C3—26no—
83C22×#126—26no—
84C22×#126—38no—
85C22×#126—32no—
86C22×#126—61no—
87C22×#126—624no—
88C22×#126—63no—
89C22×#126—64no—
90C22×#126—66no—
91C22×#126C2×A418yes—
92C22×#126C23×C312yes—
93C23×C4⋊C4—31no—
94C23×C4⋊C4—33no—
95C23×C4⋊C4C2×A413yes—
96C23×C22⋊C4—31no—
97C23×C22⋊C4—33no—
98C23×C22⋊C4C2×A413yes—
99C23×C42—31no—
100C23×C42—31no—
101C23×C42—33no—
102C23×C42—33no—
103C23×C42—33no—
104C23×C42C2×A411yescharacteristic
105C23×C42C23×C311yescharacteristic
106C2×#383—26no—
107C2×#383—324no—
108C2×#383—38no—
109C2×#383—36no—
110C2×#383—624no—
111C2×#383—61no—
112C2×#383—63no—
113C2×#383—64no—
114C2×#383—66no—
115C2×#383C23×C316yes—
116C22×#95—33no—
117C22×#95—33no—
118C22×#95—33no—
119C22×#95—33no—
120C22×#95—33no—
121C22×#95—33no—
122C22×#95—33no—
123C22×#95—33no—
124C22×#95—33no—
125C22×#95—33no—
126C2×C43—26no—
127C2×C43—36no—
128C2×C43—312no—
129C2×C43—312no—
130C2×C43—34no—
131C2×C43—34no—
132C2×C43—66no—
133C2×C43—624no—
134C2×C43—61no—
135C2×C43—64no—
136C2×C43—63no—
137C2×C43C23×C316yes—
138#1437—38no—
139#1437—312no—
140#1437—312no—
141#1437—324no—
142#1437—312no—
143#1437—312no—
144#1436—324no—
145#1436—324no—
146#1436—312no—
147#1436—312no—
148#1436—312no—
149#1436—312no—
150#1436—312no—
151#1436—324no—
152#1436—312no—
153#1436—312no—
154#1436—312no—
155#1436—312no—
156#1436—312no—
157#1436—312no—
158#1436—312no—
159#1436—324no—
160C23×A4—41no—
161C23×A4—42no—
162C23×A4—86no—
163C2×C4×A4—46no—
164C2×C4×A4—46no—
165C2×C4×A4—86no—
166C2×C4×A4—86no—
167C3×#126—168no—
168C2×C3×C4⋊C4—163no—
169C2×C3×C22⋊C4—163no—
170C2×C3×C42—161no—
171#126⋊C3—42no—
172C2×C42⋊C3#12614yescomplemented
173C2×C42⋊C3C2×#4113yes—
174C26C22×A411yescharacteristic
175C24×C4—31no—
176C24×C4—33no—
177C24×C4—33no—
178C24×C4—31no—
179C24×C4C22×A413yes—
180C24×C4C22×A413yes—
181C2×#126—26no—
182C2×#126—41no—
183C2×#126—44no—
184C2×#126—68no—
185C2×#126—624no—
186C2×#126—612no—
187C2×#126—696no—
188C2×#126—612no—
189C2×#126—66no—
190C2×#126—123no—
191C2×#126—124no—
192C2×#126—1212no—
193C2×#126—122no—
194C2×#126—128no—
195C2×#126—1264no—
196C2×#126—1212no—
197C22×C4⋊C4—32no—
198C22×C4⋊C4—312no—
199C22×C4⋊C4—32no—
200C22×C4⋊C4—63no—
201C22×C4⋊C4—66no—
202C22×C4⋊C4—66no—
203C22×C4⋊C4—612no—
204C22×C4⋊C4—612no—
205C22×C4⋊C4—612no—
206C22×C4⋊C4—66no—
207C22×C4⋊C4—63no—
208C22×C4⋊C4C22×A4112yes—
209C22×C22⋊C4—312no—
210C22×C22⋊C4—32no—
211C22×C22⋊C4—32no—
212C22×C22⋊C4—63no—
213C22×C22⋊C4—66no—
214C22×C22⋊C4—66no—
215C22×C22⋊C4—612no—
216C22×C22⋊C4—612no—
217C22×C22⋊C4—612no—
218C22×C22⋊C4—66no—
219C22×C22⋊C4—63no—
220C22×C22⋊C4C22×A4112yes—
221C22×C42—31no—
222C22×C42—36no—
223C22×C42—34no—
224C22×C42—324no—
225C22×C42—31no—
226C22×C42—33no—
227C22×C42—34no—
228C22×C42—36no—
229C22×C42—34no—
230C22×C42—36no—
231C22×C42—36no—
232C22×C42—312no—
233C22×C42—36no—
234C22×C42—66no—
235C22×C42—66no—
236C22×C42—66no—
237C22×C42—66no—
238C22×C42—612no—
239C22×C42—612no—
240C22×C42—612no—
241C22×C42—612no—
242C22×C42—612no—
243C22×C42—612no—
244C22×C42—63no—
245C22×C42—63no—
246C22×C42—66no—
247C22×C42—63no—
248C22×C42—66no—
249C22×C42—63no—
250C22×C42C3×#4116yes—
251C22×C42C22×A414yes—
252C22×C42C24×C311yescharacteristic
253#383—26no—
254#383—612no—
255#383—696no—
256#383—68no—
257#383—624no—
258#383—612no—
259#383—66no—
260C2×#95—36no—
261C2×#95—36no—
262C2×#95—312no—
263C2×#95—312no—
264C2×#95—312no—
265C2×#95—36no—
266C2×#95—36no—
267C2×#95—312no—
268C2×#95—312no—
269C2×#95—312no—
270C2×#95—312no—
271C2×#95—312no—
272C2×#95—66no—
273C2×#95—66no—
274C2×#95—66no—
275C2×#95—66no—
276C2×#95—612no—
277C2×#95—612no—
278C2×#95—612no—
279C2×#95—612no—
280C2×#95—612no—
281C2×#95—612no—
282C2×#95—63no—
283C2×#95—63no—
284C2×#95—66no—
285C2×#95—66no—
286C2×#95—63no—
287C2×#95—63no—
288C43—26no—
289C43—66no—
290C43—612no—
291C43—612no—
292C43—648no—
293C43—648no—
294C43—64no—
295C43—64no—
296C43—612no—
297C43—612no—
298C24×C3—161no—
299C22×A4—46no—
300C22×A4—41no—
301C22×A4—86no—
302C22×A4—164no—
303C22×C3×C4—163no—
304C22×C3×C4—163no—
305C4×A4—86no—
306C4×A4—86no—
307C3×C4⋊C4—1612no—
308C3×C22⋊C4—1612no—
309C3×C42—164no—
310C42⋊C3C2×#12611yescharacteristic, complemented
311C25—26no—
312C25—61no—
313C25—61no—
314C25—66no—
315C25C23×A412yes—
316C25C23×A411yescharacteristic
317C23×C4—26no—
318C23×C4—26no—
319C23×C4—312no—
320C23×C4—312no—
321C23×C4—312no—
322C23×C4—36no—
323C23×C4—36no—
324C23×C4—36no—
325C23×C4—31no—
326C23×C4—31no—
327C23×C4—33no—
328C23×C4—33no—
329C23×C4—36no—
330C23×C4—36no—
331C23×C4—32no—
332C23×C4—63no—
333C23×C4—66no—
334C23×C4—61no—
335C23×C4—66no—
336C23×C4—66no—
337C23×C4—63no—
338C23×C4—61no—
339C23×C4—63no—
340C23×C4—66no—
341C23×C4—66no—
342C23×C4—66no—
343C23×C4—66no—
344C23×C4—66no—
345C23×C4—66no—
346C23×C4—66no—
347C23×C4—63no—
348C23×C4—66no—
349C23×C4—61no—
350C23×C4—63no—
351C23×C4—61no—
352C23×C4—66no—
353C23×C4—66no—
354C23×C4—63no—
355C23×C4—66no—
356C23×C4C23×A416yes—
357C23×C4C23×A416yes—
358#126—48no—
359#126—42no—
360#126—1224no—
361#126—1216no—
362#126—124no—
363#126—1224no—
364#126—12128no—
365#126—126no—
366#126—128no—
367C2×C4⋊C4—43no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—66no—
370C2×C4⋊C4—66no—
371C2×C4⋊C4—624no—
372C2×C4⋊C4—624no—
373C2×C4⋊C4—624no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—126no—
377C2×C4⋊C4—123no—
378C2×C4⋊C4—124no—
379C2×C4⋊C4—1212no—
380C2×C4⋊C4—126no—
381C2×C4⋊C4—1248no—
382C2×C22⋊C4—43no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—66no—
385C2×C22⋊C4—66no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4—624no—
388C2×C22⋊C4—624no—
389C2×C22⋊C4—624no—
390C2×C22⋊C4—612no—
391C2×C22⋊C4—126no—
392C2×C22⋊C4—123no—
393C2×C22⋊C4—124no—
394C2×C22⋊C4—1212no—
395C2×C22⋊C4—126no—
396C2×C22⋊C4—1248no—
397C2×C42—36no—
398C2×C42—33no—
399C2×C42—312no—
400C2×C42—348no—
401C2×C42—36no—
402C2×C42—364no—
403C2×C42—32no—
404C2×C42—38no—
405C2×C42—312no—
406C2×C42—33no—
407C2×C42—34no—
408C2×C42—34no—
409C2×C42—312no—
410C2×C42—66no—
411C2×C42—66no—
412C2×C42—612no—
413C2×C42—66no—
414C2×C42—66no—
415C2×C42—612no—
416C2×C42—612no—
417C2×C42—624no—
418C2×C42—612no—
419C2×C42—612no—
420C2×C42—612no—
421C2×C42—624no—
422C2×C42—624no—
423C2×C42—612no—
424C2×C42—612no—
425C2×C42—66no—
426C2×C42—612no—
427C2×C42—66no—
428C2×C42—66no—
429C2×C42—612no—
430C2×C42—66no—
431C2×C42—612no—
432C2×C42—612no—
433C2×C42—624no—
434C2×C42—648no—
435C2×C42—648no—
436C2×C42—648no—
437C2×C42—632no—
438C2×C42—624no—
439C2×C42—612no—
440C2×C42—624no—
441C2×C42—612no—
442C2×C42—624no—
443C2×C42#126⋊C311yescharacteristic, complemented
444C2×C42C3×#12614yescomplemented
445C2×C42C2×C3×#4113yes—
446#95—66no—
447#95—612no—
448#95—66no—
449#95—66no—
450#95—66no—
451#95—624no—
452#95—624no—
453#95—624no—
454#95—624no—
455#95—624no—
456#95—624no—
457#95—612no—
458#95—612no—
459#95—612no—
460#95—612no—
461#95—612no—
462C23×C3—162no—
463C23×C3—161no—
464C23×C3—326no—
465C2×A4—44no—
466C2×A4—43no—
467C2×A4—162no—
468C2×C3×C4—166no—
469C2×C3×C4—166no—
470C2×C3×C4—326no—
471C2×C3×C4—326no—
472C24—26no—
473C24—36no—
474C24—31no—
475C24—31no—
476C24—41no—
477C24—44no—
478C24—62no—
479C24—66no—
480C24—121no—
481C24—126no—
482C24—126no—
483C24—126no—
484C24—1212no—
485C24—124no—
486C24—126no—
487C24—124no—
488C24A4×#4116yes—
489C24C24×A411yescharacteristic
490C22×C4—26no—
491C22×C4—26no—
492C22×C4—312no—
493C22×C4—312no—
494C22×C4—312no—
495C22×C4—348no—
496C22×C4—36no—
497C22×C4—33no—
498C22×C4—34no—
499C22×C4—312no—
500C22×C4—33no—
501C22×C4—33no—
502C22×C4—36no—
503C22×C4—36no—
504C22×C4—36no—
505C22×C4—36no—
506C22×C4—43no—
507C22×C4—66no—
508C22×C4—66no—
509C22×C4—66no—
510C22×C4—66no—
511C22×C4—66no—
512C22×C4—63no—
513C22×C4—63no—
514C22×C4—66no—
515C22×C4—61no—
516C22×C4—61no—
517C22×C4—612no—
518C22×C4—624no—
519C22×C4—624no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—66no—
523C22×C4—612no—
524C22×C4—66no—
525C22×C4—624no—
526C22×C4—612no—
527C22×C4—612no—
528C22×C4—612no—
529C22×C4—612no—
530C22×C4—612no—
531C22×C4—66no—
532C22×C4—66no—
533C22×C4—66no—
534C22×C4—612no—
535C22×C4—66no—
536C22×C4—66no—
537C22×C4—66no—
538C22×C4—66no—
539C22×C4—66no—
540C22×C4—66no—
541C22×C4—66no—
542C22×C4—62no—
543C22×C4—66no—
544C22×C4—66no—
545C22×C4—66no—
546C22×C4—66no—
547C22×C4—66no—
548C22×C4—66no—
549C22×C4—123no—
550C22×C4—126no—
551C22×C4—126no—
552C22×C4—1212no—
553C22×C4—126no—
554C22×C4—1212no—
555C22×C4—1212no—
556C22×C4—126no—
557C22×C4—1212no—
558C22×C4—126no—
559C22×C4—1212no—
560C22×C4—126no—
561C22×C4—126no—
562C22×C4—126no—
563C22×C4—126no—
564C22×C4—126no—
565C22×C4—126no—
566C22×C4—124no—
567C22×C4—126no—
568C22×C4C2×#126⋊C313yes—
569C4⋊C4—412no—
570C4⋊C4—1212no—
571C4⋊C4—122no—
572C4⋊C4—126no—
573C4⋊C4—1248no—
574C4⋊C4—128no—
575C22⋊C4—412no—
576C22⋊C4—1212no—
577C22⋊C4—122no—
578C22⋊C4—126no—
579C22⋊C4—128no—
580C22⋊C4—1248no—
581C42—34no—
582C42—33no—
583C42—364no—
584C42—312no—
585C42—32no—
586C42—38no—
587C42—312no—
588C42—612no—
589C42—612no—
590C42—624no—
591C42—648no—
592C42—624no—
593C42—624no—
594C42—648no—
595C42—632no—
596C42—612no—
597C42—612no—
598C42—624no—
599C42—648no—
600C42C2×#126⋊C314yescomplemented
601C42C2×C3×#12611yescharacteristic, complemented
602C22×C3—166no—
603C22×C3—161no—
604C22×C3—326no—
605C22×C3—644no—
606A4—41no—
607C3×C4—326no—
608C3×C4—326no—
609C23—33no—
610C23—312no—
611C23—36no—
612C23—34no—
613C23—34no—
614C23—36no—
615C23—36no—
616C23—36no—
617C23—31no—
618C23—31no—
619C23—42no—
620C23—42no—
621C23—86no—
622C23—126no—
623C23—126no—
624C23—126no—
625C23—122no—
626C23—122no—
627C23—246no—
628C23—244no—
629C23—2412no—
630C23—244no—
631C23—2412no—
632C23C22×#126⋊C311yescharacteristic
633C23A4×#12614yes—
634C23C2×A4×#4113yes—
635C2×C4—26no—
636C2×C4—324no—
637C2×C4—31no—
638C2×C4—34no—
639C2×C4—33no—
640C2×C4—36no—
641C2×C4—46no—
642C2×C4—66no—
643C2×C4—66no—
644C2×C4—612no—
645C2×C4—612no—
646C2×C4—612no—
647C2×C4—63no—
648C2×C4—63no—
649C2×C4—66no—
650C2×C4—612no—
651C2×C4—612no—
652C2×C4—612no—
653C2×C4—612no—
654C2×C4—63no—
655C2×C4—63no—
656C2×C4—66no—
657C2×C4—64no—
658C2×C4—612no—
659C2×C4—648no—
660C2×C4—648no—
661C2×C4—64no—
662C2×C4—612no—
663C2×C4—612no—
664C2×C4—66no—
665C2×C4—66no—
666C2×C4—66no—
667C2×C4—86no—
668C2×C4—126no—
669C2×C4—126no—
670C2×C4—126no—
671C2×C4—126no—
672C2×C4—126no—
673C2×C4—126no—
674C2×C4—126no—
675C2×C4—126no—
676C2×C4—126no—
677C2×C4—126no—
678C2×C4—126no—
679C2×C4—122no—
680C2×C4—246no—
681C2×C4—244no—
682C2×C4—2412no—
683C2×C4—2412no—
684C2×C4—244no—
685C2×C4C22×#126⋊C316yes—
686C2×C3—164no—
687C2×C3—163no—
688C2×C3—642no—
689C22—33no—
690C22—34no—
691C22—31no—
692C22—31no—
693C22—312no—
694C22—312no—
695C22—34no—
696C22—33no—
697C22—33no—
698C22—36no—
699C22—86no—
700C22—164no—
701C22—242no—
702C22—246no—
703C22—484no—
704C22#36641⋊C316yes—
705C22C2×A4×#12611yescharacteristic
706C22C23×#126⋊C311yescharacteristic
707C4—26no—
708C4—624no—
709C4—61no—
710C4—64no—
711C4—63no—
712C4—66no—
713C4—86no—
714C4—246no—
715C4—242no—
716C3—161noSylow
717C2—33no—
718C2—31no—
719C2—34no—
720C2—162no—
721C2#7221383514yesdirect product factor
722C2C2×#36641⋊C313yes—
subgroupG/H
centerC23C22×#126⋊C3
commutator, cosocleC22×C42C24×C3
FrattiniC24C24×A4
FittingC2×#6342420C3
socleC25C23×A4
prime radicalC23×#126⋊C3C22
hypercenterC2×C42#126⋊C3
hypocenterC42C2×C3×#126
LCSG ▷ C22×C42 ▷ C42
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×#14710⋊C3C2semidirect
3. C2×#6342420C3semidirect
4. #72213835C2direct (N × H)
5. C2×C42×C42⋊C3C2semidirect
11. #14710⋊C3C22semidirect
12. C42×C42⋊C3C22semidirect
14. #6342420C2×C3semidirect
20. C2×#14710C2×C3semidirect
21. C2×C44C2×C3semidirect
53. #14710C22×C3semidirect
69. C44C22×C3semidirect
172. C2×C42⋊C3#126semidirect
310. C42⋊C3C2×#126semidirect
443. C2×C42#126⋊C3semidirect
444. C2×C42C3×#126semidirect
600. C42C2×#126⋊C3semidirect
601. C42C2×C3×#126semidirect
721. C2#72213835direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#14710⋊C32
C2×#14711⋊C32
C2×#63424203
#722138352
C2×C42×C42⋊C32
C2×A4×#1264