Finite Group Databasebeta

← #13975

Subgroups — #13975

Order 256.

1345 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic
2C2×#386C211yesmaximal, characteristic
3C2×#386C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic, complemented
5C2×#385C211yesmaximal, characteristic, complemented
6C2×#380C211yesmaximal, characteristic, complemented
7C22×#95C211yesmaximal, characteristic, complemented
8#1444C211yesmaximal, characteristic, complemented
9#1442C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1439C211yesmaximal, characteristic, complemented
12#1436C211yesmaximal, characteristic, complemented
13#1436C211yesmaximal, characteristic, complemented
14C22⋊#95C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20C24×C4C2211yescharacteristic
21C2×C4×D4—24no—
22C2×C4×D4—24no—
23C2×C4×D4—24no—
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28C22×C4⋊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×C42C2211yescharacteristic
38#386—24no—
39#386C418yescomplemented
40#385—24no—
41#385—24no—
42#380—24no—
43C2×#95—24no—
44C2×#95C2211yescharacteristic
45C2×#95C2211yescharacteristic
46C2×#95C2211yescharacteristic
47C2×#95C2211yescharacteristic
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
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4C2×C414yes—
67C22×D4D414yes—
68C22×D4Q814yes—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4—41no—
91C23×C4—41no—
92C23×C4—41no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4C2×C414yescomplemented
96C23×C4D414yes—
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C23×C4C2311yescharacteristic
100C23×C4C2311yescharacteristic
101C23×C4C2311yescharacteristic
102C23×C4C2311yescharacteristic
103C23×C4C2311yescharacteristic
104C23×C4C2311yescharacteristic
105C23×C4C2311yescharacteristic
106C23×C4C2311yescharacteristic
107C23×C4C2311yescharacteristic
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110C4×D4—232no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
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—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4C2×C414yescomplemented
147C2×C22⋊C4C2×C414yescomplemented
148C2×C22⋊C4C2×C414yescomplemented
149C2×C22⋊C4D414yes—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—41no—
165#95—41no—
166#95C2×C414yescomplemented
167#95C2×C414yescomplemented
168#95D414yescomplemented
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—21no—
175C24—21no—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24C22×C412yes—
190C24C22×C412yes—
191C24C2×D412yes—
192C24C2×D412yes—
193C24C2×D412yes—
194C24C2×D412yes—
195C24C2×Q812yes—
196C24C2×Q812yes—
197C24#4112yes—
198C24#4112yes—
199C24C2411yescharacteristic
200C2×D4—28no—
201C2×D4—28no—
202C2×D4—28no—
203C2×D4—28no—
204C2×D4—28no—
205C2×D4—28no—
206C2×D4C4⋊C4116yescomplemented
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
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—21no—
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—28no—
272C22×C4—28no—
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—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4C22×C412yes—
310C22×C4C22×C412yes—
311C22×C4C22×C412yes—
312C22×C4C22×C412yes—
313C22×C4C22×C412yes—
314C22×C4C2×D412yes—
315C22×C4C2×D412yes—
316C22×C4C2×D412yes—
317C22×C4C2×D412yes—
318C22×C4C2×D412yes—
319C22×C4C2×D412yes—
320C22×C4C2×D412yes—
321C22×C4C2×D412yes—
322C22×C4C2×Q812yes—
323C22×C4#4112yes—
324C22×C4#4112yes—
325C22×C4#4112yes—
326C4⋊C4—28no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—28no—
334C22⋊C4—28no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C42—28no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
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—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23C2×C4⋊C414yes—
403C23C2×C4⋊C414yes—
404C23C4×D414yes—
405C23C4×D414yes—
406C23#12014yes—
407C23C22⋊Q814yes—
408C23C22⋊Q814yes—
409C23#12314yes—
410C23C23×C411yescharacteristic
411C23C22×D411yescharacteristic
412C23C22×D411yescharacteristic
413C23C22×D411yescharacteristic
414C23C22×D411yescharacteristic
415C23C22×Q811yescharacteristic
416C23C2×#4111yescharacteristic
417C23C2×#4111yescharacteristic
418C23C2×#4111yescharacteristic
419C23C2×#4111yescharacteristic
420C23C2×#4111yescharacteristic
421C23H(5,𝔽2)11yescharacteristic
422C23H(5,𝔽2)11yescharacteristic
423C23H(5,𝔽2)11yescharacteristic
424C232−1+411yescharacteristic
425D4—232no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—416no—
469C2×C4—416no—
470C2×C4—416no—
471C2×C4—416no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4C2×C4⋊C414yes—
519C2×C4C4×D414yescomplemented
520C2×C4#12114yes—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22#38518yescomplemented
551C22C22×C4⋊C411yescharacteristic
552C22C2×C4×D411yescharacteristic
553C22C2×C4×D411yescharacteristic
554C22C2×C4×D411yescharacteristic
555C22C4×#4111yescharacteristic
556C22#51811yescharacteristic
557C22#52011yescharacteristic
558C22C2×#12011yescharacteristic
559C22C2×#12111yescharacteristic
560C22C2×C22⋊Q811yescharacteristic
561C22C2×C22⋊Q811yescharacteristic
562C22C2×#12311yescharacteristic
563C22#52511yescharacteristic
564C22#52911yescharacteristic
565C22#53211yescharacteristic
566C22#53411yescharacteristic
567C22#53711yescharacteristic
568C22#53811yescharacteristic
569C22#53911yescharacteristic
570C22D4211yescharacteristic
571C22D4211yescharacteristic
572C22#54611yescharacteristic
573C22#54611yescharacteristic
574C22#54611yescharacteristic
575C22#54611yescharacteristic
576C22#54611yescharacteristic
577C22#54711yescharacteristic
578C22#54711yescharacteristic
579C22D4×Q811yescharacteristic
580C22#55111yescharacteristic
581C22#55111yescharacteristic
582C22#55211yescharacteristic
583C22#55411yescharacteristic
584C22#55411yescharacteristic
585C22#55411yescharacteristic
586C4—24no—
587C4—24no—
588C4—24no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C2—28no—
600C2—44no—
601C2C2×#38511yescharacteristic
602C2C4×#12111yescharacteristic
603C2#232511yescharacteristic
604C2D4×C4⋊C411yescharacteristic
605C2#234911yescharacteristic
606C2#235811yescharacteristic
607C2#235911yescharacteristic
608C2#241011yescharacteristic
609C2#241711yescharacteristic
610C2#241811yescharacteristic
611C2#241911yescharacteristic
612C2#243011yescharacteristic
613C2#244911yescharacteristic
614C2#245011yescharacteristic
615C2#256411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#385C2semidirect
5. C2×#385C2semidirect
6. C2×#380C2semidirect
7. C22×#95C2semidirect
8. #1444C2semidirect
9. #1442C2semidirect
10. #1439C2semidirect
11. #1439C2semidirect
12. #1436C2semidirect
13. #1436C2semidirect
14. C22⋊#95C2semidirect
15. C22⋊#95C2semidirect
39. #386C4semidirect
95. C23×C4C2×C4semidirect
146. C2×C22⋊C4C2×C4semidirect
147. C2×C22⋊C4C2×C4semidirect
148. C2×C22⋊C4C2×C4semidirect
166. #95C2×C4semidirect
167. #95C2×C4semidirect
168. #95D4semidirect
206. C2×D4C4⋊C4semidirect
519. C2×C4C4×D4semidirect
550. C22#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#3862
C2×#3852
C2×#3802
C22×#952
#14442
#14422
#14392
#14362
C22⋊#952