Finite Group Databasebeta

← #15280

Subgroups — #15280

Order 256.

1071 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#396C211yesmaximal, characteristic, complemented
3C2×#393C211yesmaximal, characteristic, complemented
4C2×#391C211yesmaximal, characteristic, complemented
5C2×#390C211yesmaximal, characteristic
6C2×#386C211yesmaximal, characteristic
7#1444C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1439C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1439C211yesmaximal, characteristic, complemented
12#1439C211yesmaximal, characteristic, complemented
13#1438C211yesmaximal, characteristic
14#1436C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C2×C22⋊Q8—24no—
20C2×C22⋊Q8—24no—
21C2×C22⋊Q8—24no—
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊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
40#396—24no—
41#393—24no—
42#391—24no—
43#390—24no—
44#386—24no—
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
59C25C2311yescharacteristic
60C25C2311yescharacteristic
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8D414yescomplemented
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4Q814yescomplemented
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C22⋊Q8—232no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4D414yescomplemented
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
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—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4D414yes—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—41no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95D414yes—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—21no—
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—
189C24—21no—
190C24C2×D412yes—
191C24C2×Q812yes—
192C24C2×Q812yes—
193C24#4112yes—
194C24C2411yescharacteristic
195C2×Q8—28no—
196C2×Q8—28no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×D4—28no—
203C2×D4—28no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
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—21no—
220C22×C4—21no—
221C22×C4—28no—
222C22×C4—21no—
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—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—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4C2×D412yes—
293C22×C4C2×D412yes—
294C22×C4C2×D412yes—
295C22×C4C2×D412yes—
296C22×C4C2×D412yes—
297C22×C4C2×D412yes—
298C22×C4C2×D412yes—
299C22×C4C2×D412yes—
300C22×C4C2×D412yes—
301C22×C4C2×D412yes—
302C22×C4C2×D412yes—
303C22×C4C2×Q812yes—
304C22×C4#4112yes—
305C22×C4#4112yes—
306C22×C4#4112yes—
307C4⋊C4—28no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C22⋊C4—28no—
314C22⋊C4—28no—
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—
324C42—44no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23C22⋊Q814yes—
395C23C22×D411yescharacteristic
396C23C22×D411yescharacteristic
397C23C22×D411yescharacteristic
398C23C22×D411yescharacteristic
399C23C22×Q811yescharacteristic
400C23C2×#4111yescharacteristic
401C23C2×#4111yescharacteristic
402C23C2×#4111yescharacteristic
403C23C2×#4111yescharacteristic
404C23H(5,𝔽2)11yescharacteristic
405C23H(5,𝔽2)11yescharacteristic
406C23H(5,𝔽2)11yescharacteristic
407C23H(5,𝔽2)11yescharacteristic
408C232−1+411yescharacteristic
409C232−1+411yescharacteristic
410Q8—416no—
411D4—416no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
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—416no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
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#12014yes—
502C2×C4#12114yes—
503C2×C4C22⋊Q814yescomplemented
504C22—24no—
505C22—24no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22C2×#12011yescharacteristic
535C22C2×#12111yescharacteristic
536C22C2×C22⋊Q811yescharacteristic
537C22C2×C22⋊Q811yescharacteristic
538C22#52911yescharacteristic
539C22#53211yescharacteristic
540C22#53411yescharacteristic
541C22#53511yescharacteristic
542C22#53611yescharacteristic
543C22#53711yescharacteristic
544C22#53811yescharacteristic
545C22#53911yescharacteristic
546C22#54211yescharacteristic
547C22#54211yescharacteristic
548C22#54311yescharacteristic
549C22D4211yescharacteristic
550C22D4211yescharacteristic
551C22#54611yescharacteristic
552C22#54611yescharacteristic
553C22#54611yescharacteristic
554C22#54611yescharacteristic
555C22#54611yescharacteristic
556C22#54811yescharacteristic
557C22#54811yescharacteristic
558C22#54811yescharacteristic
559C22D4×Q811yescharacteristic
560C22D4×Q811yescharacteristic
561C22#55011yescharacteristic
562C22#55011yescharacteristic
563C22#55111yescharacteristic
564C22#55411yescharacteristic
565C22#55411yescharacteristic
566C22#55611yescharacteristic
567C22#56211yescharacteristic
568C22#56211yescharacteristic
569C4—24no—
570C4—24no—
571C4—24no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C2—44no—
583C2—82no—
584C2#240911yescharacteristic
585C2#241011yescharacteristic
586C2#242611yescharacteristic
587C2#243111yescharacteristic
588C2#243711yescharacteristic
589C2#245011yescharacteristic
590C2#245211yescharacteristic
591C2#253811yescharacteristic
592C2#256011yescharacteristic
593C2#267611yescharacteristic
594C2#268111yescharacteristic
595C2#269211yescharacteristic
596C2#270011yescharacteristic
597C2#273011yescharacteristic
598C2#273211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊Q8C2semidirect
2. C2×#396C2semidirect
3. C2×#393C2semidirect
4. C2×#391C2semidirect
7. #1444C2semidirect
8. #1441C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
11. #1439C2semidirect
12. #1439C2semidirect
14. #1436C2semidirect
15. C22⋊#95C2semidirect
67. C22×Q8D4semidirect
74. C22×D4Q8semidirect
113. C2×C4⋊C4D4semidirect
503. C2×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#3962
C2×#3932
C2×#3912
C2×#3902
C2×#3862
#14442
#14412
#14392
#14382
#14362
C22⋊#952