Finite Group Databasebeta

← #14100

Subgroups — #14100

Order 256.

1345 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121C211yesmaximal, characteristic
2C2×#390C211yesmaximal, characteristic, complemented
3C2×#388C211yesmaximal, characteristic, complemented
4C2×#386C211yesmaximal, characteristic, complemented
5C2×#386C211yesmaximal, characteristic, complemented
6C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1438C211yesmaximal, characteristic, complemented
10#1438C211yesmaximal, characteristic, complemented
11#1438C211yesmaximal, characteristic, complemented
12#1437C211yesmaximal, characteristic, complemented
13#1437C211yesmaximal, characteristic, complemented
14#1436C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20C24×C4C2211yescharacteristic
21C2×#121—24no—
22C2×#121—24no—
23C2×#121C418yescomplemented
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2211yescharacteristic
32C22×C22⋊C4C2211yescharacteristic
33C22×C22⋊C4C2211yescharacteristic
34C22×C22⋊C4C2211yescharacteristic
35C22×C22⋊C4C2211yescharacteristic
36C22×C22⋊C4C2211yescharacteristic
37C22×C22⋊C4C2211yescharacteristic
38C22×C22⋊C4C2211yescharacteristic
39C22×C42C2211yescharacteristic
40C22×C42C2211yescharacteristic
41C22×C42C2211yescharacteristic
42#390—24no—
43#388—24no—
44#386—24no—
45#386—24no—
46C4×C22⋊C4—24no—
47C4×C22⋊C4—24no—
48C2×#95C2211yescharacteristic
49C2×#95C2211yescharacteristic
50C2×#95C2211yescharacteristic
51C2×#95C2211yescharacteristic
52C2×#95C2211yescharacteristic
53C2×#95C2211yescharacteristic
54C2×#95C2211yescharacteristic
55C2×#95C2211yescharacteristic
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C2×#95C2211yescharacteristic
59C2×#95C2211yescharacteristic
60C25C2311yescharacteristic
61C25C2311yescharacteristic
62C25C2311yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4—41no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4C2×C414yes—
82C22×D4C2×C414yes—
83C22×D4C2×C414yescomplemented
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4C2×C414yescomplemented
98C23×C4C2311yescharacteristic
99C23×C4C2311yescharacteristic
100C23×C4C2311yescharacteristic
101C23×C4C2311yescharacteristic
102C23×C4C2311yescharacteristic
103C23×C4C2311yescharacteristic
104C23×C4C2311yescharacteristic
105C23×C4C2311yescharacteristic
106C23×C4C2311yescharacteristic
107C23×C4C2311yescharacteristic
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110#121—232no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2×C414yescomplemented
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4C2×C414yes—
152C2×C22⋊C4C2×C414yescomplemented
153C2×C22⋊C4D414yes—
154C2×C22⋊C4D414yes—
155C2×C22⋊C4D414yescomplemented
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—41no—
175#95—41no—
176#95D414yescomplemented
177C24—21no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24C22×C412yes—
216C24C22×C412yes—
217C24C22×C412yes—
218C24C2×D412yes—
219C24C2×D412yes—
220C24C2×D412yes—
221C24#4112yes—
222C24C2411yescharacteristic
223C2×D4—28no—
224C2×D4—28no—
225C2×D4—28no—
226C2×D4—28no—
227C2×D4—28no—
228C2×D4—28no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—28no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4C22×C412yes—
325C22×C4C22×C412yes—
326C22×C4C22×C412yes—
327C22×C4C22×C412yes—
328C22×C4C2×D412yes—
329C22×C4C2×D412yes—
330C22×C4C2×D412yes—
331C22×C4C2×D412yes—
332C22×C4C2×D412yes—
333C22×C4C2×D412yes—
334C22×C4C2×D412yes—
335C22×C4C2×D412yes—
336C22×C4C2×D412yes—
337C22×C4#4112yes—
338C22×C4#4112yes—
339C22×C4#4112yes—
340C22×C4#4112yes—
341C22×C4#4112yes—
342C4⋊C4—28no—
343C4⋊C4—44no—
344C22⋊C4—28no—
345C22⋊C4—28no—
346C22⋊C4—28no—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4C22⋊C4116yescomplemented
357C42—44no—
358C42—44no—
359C42—44no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C2×C22⋊C414yes—
464C23C4×D414yes—
465C23C23×C411yescharacteristic
466C23C22×D411yescharacteristic
467C23C22×D411yescharacteristic
468C23C22×D411yescharacteristic
469C23C22×D411yescharacteristic
470C23C2×#4111yescharacteristic
471C23C2×#4111yescharacteristic
472C23C2×#4111yescharacteristic
473C23C2×#4111yescharacteristic
474C23C2×#4111yescharacteristic
475C23C2×#4111yescharacteristic
476C23H(5,𝔽2)11yescharacteristic
477C23H(5,𝔽2)11yescharacteristic
478C23H(5,𝔽2)11yescharacteristic
479C23H(5,𝔽2)11yescharacteristic
480D4—416no—
481D4—416no—
482D4—416no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—416no—
526C2×C4—416no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4C2×C22⋊C414yes—
566C2×C4C2×C22⋊C414yes—
567C2×C4C4×D414yes—
568C2×C4#12114yescomplemented
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C22×C22⋊C411yescharacteristic
615C22C2×C4×D411yescharacteristic
616C22C2×C4×D411yescharacteristic
617C22C4×#4111yescharacteristic
618C22C4×#4111yescharacteristic
619C22#51811yescharacteristic
620C22#51811yescharacteristic
621C22C2×#12111yescharacteristic
622C22#52511yescharacteristic
623C22#52511yescharacteristic
624C22#52511yescharacteristic
625C22#52911yescharacteristic
626C22#53211yescharacteristic
627C22#53211yescharacteristic
628C22#53211yescharacteristic
629C22#53811yescharacteristic
630C22#53811yescharacteristic
631C22#54011yescharacteristic
632C22#54011yescharacteristic
633C22D4211yescharacteristic
634C22D4211yescharacteristic
635C22D4211yescharacteristic
636C22D4211yescharacteristic
637C22#54611yescharacteristic
638C22#54611yescharacteristic
639C22#54611yescharacteristic
640C22#54611yescharacteristic
641C22#54611yescharacteristic
642C22#54611yescharacteristic
643C22#54611yescharacteristic
644C22#55011yescharacteristic
645C22#55311yescharacteristic
646C22#55311yescharacteristic
647C22#55311yescharacteristic
648C22#55511yescharacteristic
649C4—24no—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—81no—
661C2—44no—
662C2—44no—
663C2—82no—
664C2#229711yescharacteristic
665C2C4×#12111yescharacteristic
666C2#233311yescharacteristic
667C2D4×C22⋊C411yescharacteristic
668C2D4×C22⋊C411yescharacteristic
669C2#236311yescharacteristic
670C2#236411yescharacteristic
671C2#242411yescharacteristic
672C2#242811yescharacteristic
673C2#245611yescharacteristic
674C2#246411yescharacteristic
675C2#246411yescharacteristic
676C2#247611yescharacteristic
677C2#255511yescharacteristic
678C2#255911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC22×#121C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#390C2semidirect
3. C2×#388C2semidirect
4. C2×#386C2semidirect
5. C2×#386C2semidirect
6. C2×C4×C22⋊C4C2semidirect
7. C2×C4×C22⋊C4C2semidirect
8. #1441C2semidirect
9. #1438C2semidirect
10. #1438C2semidirect
11. #1438C2semidirect
12. #1437C2semidirect
13. #1437C2semidirect
14. #1436C2semidirect
15. C22⋊#95C2semidirect
23. C2×#121C4semidirect
83. C22×D4C2×C4semidirect
97. C23×C4C2×C4semidirect
116. C2×C4⋊C4C2×C4semidirect
152. C2×C22⋊C4C2×C4semidirect
155. C2×C22⋊C4D4semidirect
176. #95D4semidirect
356. C22⋊C4C22⋊C4semidirect
568. C2×C4#121semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#3902
C2×#3882
C2×#3862
C2×C4×C22⋊C42
#14412
#14382
#14372
#14362
C22⋊#952