Finite Group Databasebeta

← #14023

Subgroups — #14023

Order 256.

1583 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121C211yesmaximal, characteristic
2C22×C4×D4C211yesmaximal, characteristic, complemented
3C2×#386C211yesmaximal, characteristic, complemented
4C2×#386C211yesmaximal, characteristic, complemented
5C2×#386C211yesmaximal, characteristic, complemented
6C2×#386C211yesmaximal, characteristic, complemented
7C22×#95C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1440C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1438C211yesmaximal, characteristic, complemented
12#1437C211yesmaximal, characteristic, complemented
13#1436C211yesmaximal, characteristic, complemented
14C22⋊#95C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20C24×C4C2211yescharacteristic
21C24×C4C2211yescharacteristic
22C24×C4C2211yescharacteristic
23C2×#121—24no—
24C2×#121—24no—
25C2×#121C418yescomplemented
26C2×C4×D4—24no—
27C2×C4×D4—24no—
28C2×C4×D4—24no—
29C22×C4⋊C4C2211yescharacteristic
30C22×C4⋊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×C22⋊C4C2211yescharacteristic
40C22×C22⋊C4C2211yescharacteristic
41C22×C22⋊C4C2211yescharacteristic
42C22×C42C2211yescharacteristic
43#386—24no—
44#386—24no—
45#386—24no—
46#386—24no—
47C2×#95—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
60C2×#95C2211yescharacteristic
61C2×#95C2211yescharacteristic
62C25C2311yescharacteristic
63C25C2311yescharacteristic
64C25C2311yescharacteristic
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—22no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4C2×C414yes—
82C22×D4C2×C414yescomplemented
83C22×D4C2×C414yescomplemented
84C22×D4D414yes—
85C22×D4D414yes—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4C2×C414yescomplemented
113C23×C4D414yescomplemented
114C23×C4C2311yescharacteristic
115C23×C4C2311yescharacteristic
116C23×C4C2311yescharacteristic
117C23×C4C2311yescharacteristic
118C23×C4C2311yescharacteristic
119C23×C4C2311yescharacteristic
120C23×C4C2311yescharacteristic
121C23×C4C2311yescharacteristic
122C23×C4C2311yescharacteristic
123C23×C4C2311yescharacteristic
124C23×C4C2311yescharacteristic
125C23×C4C2311yescharacteristic
126#121—232no—
127C4×D4—232no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C2×C414yescomplemented
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—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4C2×C414yescomplemented
165C2×C22⋊C4C2×C414yescomplemented
166C2×C22⋊C4D414yescomplemented
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—41no—
182#95—41no—
183#95D414yescomplemented
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—
215C24—21no—
216C24—21no—
217C24—21no—
218C24C22×C412yes—
219C24C22×C412yes—
220C24C22×C412yes—
221C24C2×D412yes—
222C24C2×D412yes—
223C24C2×D412yes—
224C24C2×D412yes—
225C24C2×D412yes—
226C24C2×D412yes—
227C24#4112yes—
228C24#4112yes—
229C24C2411yescharacteristic
230C2×D4—28no—
231C2×D4—28no—
232C2×D4—28no—
233C2×D4—28no—
234C2×D4—28no—
235C2×D4—28no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—28no—
240C2×D4—28no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4C22⋊C4116yescomplemented
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—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
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—21no—
298C22×C4—21no—
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—28no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4C22×C412yes—
348C22×C4C22×C412yes—
349C22×C4C22×C412yes—
350C22×C4C22×C412yes—
351C22×C4C2×D412yes—
352C22×C4C2×D412yes—
353C22×C4C2×D412yes—
354C22×C4C2×D412yes—
355C22×C4C2×D412yes—
356C22×C4C2×D412yes—
357C22×C4C2×D412yes—
358C22×C4C2×D412yes—
359C22×C4C2×D412yes—
360C22×C4#4112yes—
361C22×C4#4112yes—
362C22×C4#4112yes—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C22⋊C4—28no—
366C22⋊C4—28no—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C42—28no—
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—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
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—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23C2×C22⋊C414yes—
475C23C2×C22⋊C414yes—
476C23C4×D414yes—
477C23C4×D414yes—
478C23#12014yes—
479C23#12114yes—
480C23#12114yes—
481C23#12314yes—
482C23C23×C411yescharacteristic
483C23C22×D411yescharacteristic
484C23C22×D411yescharacteristic
485C23C22×D411yescharacteristic
486C23C22×D411yescharacteristic
487C23C22×D411yescharacteristic
488C23C2×#4111yescharacteristic
489C23C2×#4111yescharacteristic
490C23C2×#4111yescharacteristic
491C23C2×#4111yescharacteristic
492C23C2×#4111yescharacteristic
493C23H(5,𝔽2)11yescharacteristic
494C23H(5,𝔽2)11yescharacteristic
495C23H(5,𝔽2)11yescharacteristic
496C232−1+411yescharacteristic
497D4—232no—
498D4—416no—
499D4—416no—
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—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—416no—
542C2×C4—416no—
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×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4C2×C22⋊C414yes—
586C2×C4C4×D414yescomplemented
587C2×C4#12114yescomplemented
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22#38618yescomplemented
633C22C22×C22⋊C411yescharacteristic
634C22C2×C4×D411yescharacteristic
635C22C2×C4×D411yescharacteristic
636C22C2×C4×D411yescharacteristic
637C22C4×#4111yescharacteristic
638C22#51811yescharacteristic
639C22#52011yescharacteristic
640C22C2×#12011yescharacteristic
641C22C2×#12111yescharacteristic
642C22C2×#12111yescharacteristic
643C22C2×#12111yescharacteristic
644C22C2×#12311yescharacteristic
645C22#52511yescharacteristic
646C22#52911yescharacteristic
647C22#53211yescharacteristic
648C22#53411yescharacteristic
649C22#53711yescharacteristic
650C22#53811yescharacteristic
651C22#53911yescharacteristic
652C22D4211yescharacteristic
653C22D4211yescharacteristic
654C22D4211yescharacteristic
655C22#54611yescharacteristic
656C22#54611yescharacteristic
657C22#54611yescharacteristic
658C22#54611yescharacteristic
659C22#54611yescharacteristic
660C22#54611yescharacteristic
661C22#54711yescharacteristic
662C22#54711yescharacteristic
663C22#54711yescharacteristic
664C22#54811yescharacteristic
665C22#55311yescharacteristic
666C22#55311yescharacteristic
667C22#55311yescharacteristic
668C4—24no—
669C4—24no—
670C4—24no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—81no—
679C4—81no—
680C2—28no—
681C2—44no—
682C2—82no—
683C2C2×#38611yescharacteristic
684C2C4×#12111yescharacteristic
685C2#232511yescharacteristic
686C2D4×C22⋊C411yescharacteristic
687C2#233911yescharacteristic
688C2#236111yescharacteristic
689C2#236411yescharacteristic
690C2#241011yescharacteristic
691C2#241611yescharacteristic
692C2#242411yescharacteristic
693C2#242511yescharacteristic
694C2#243011yescharacteristic
695C2#244311yescharacteristic
696C2#244411yescharacteristic
697C2#254311yescharacteristic
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. C22×C4×D4C2semidirect
3. C2×#386C2semidirect
4. C2×#386C2semidirect
5. C2×#386C2semidirect
6. C2×#386C2semidirect
7. C22×#95C2semidirect
8. #1441C2semidirect
9. #1440C2semidirect
10. #1439C2semidirect
11. #1438C2semidirect
12. #1437C2semidirect
13. #1436C2semidirect
14. C22⋊#95C2semidirect
15. C22⋊#95C2semidirect
25. C2×#121C4semidirect
82. C22×D4C2×C4semidirect
83. C22×D4C2×C4semidirect
112. C23×C4C2×C4semidirect
113. C23×C4D4semidirect
133. C2×C4⋊C4C2×C4semidirect
164. C2×C22⋊C4C2×C4semidirect
165. C2×C22⋊C4C2×C4semidirect
166. C2×C22⋊C4D4semidirect
183. #95D4semidirect
250. C2×D4C22⋊C4semidirect
586. C2×C4C4×D4semidirect
587. C2×C4#121semidirect
632. C22#386semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C22×C4×D42
C2×#3862
C22×#952
#14412
#14402
#14392
#14382
#14372
#14362
C22⋊#952