Finite Group Databasebeta

← #15325

Subgroups — #15325

Order 256.

1113 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#399C211yesmaximal, characteristic, complemented
3C2×#396C211yesmaximal, characteristic, complemented
4C2×#395C211yesmaximal, characteristic, complemented
5C2×#393C211yesmaximal, characteristic, complemented
6C2×#390C211yesmaximal, characteristic
7C2×#386C211yesmaximal, characteristic
8C2×#385C211yesmaximal, characteristic, complemented
9#1444C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1439C211yesmaximal, characteristic, complemented
12#1439C211yesmaximal, characteristic, complemented
13#1438C211yesmaximal, characteristic
14#1436C211yesmaximal, characteristic, complemented
15C4⋊#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×C4⋊C4C2211yescharacteristic
29C22×C4⋊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×C22⋊C4C2211yescharacteristic
40C22×C22⋊C4C2211yescharacteristic
41C22×C42C2211yescharacteristic
42#399—24no—
43#396—24no—
44#395—24no—
45#393—24no—
46#390—24no—
47#386—24no—
48#385—24no—
49C2×#95C2211yescharacteristic
50C2×#95C2211yescharacteristic
51C2×#95C2211yescharacteristic
52C2×#95C2211yescharacteristic
53C2×#95C2211yescharacteristic
54C2×#95C2211yescharacteristic
55C2×#95C2211yescharacteristic
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C2×#95C2211yescharacteristic
59C2×#95C2211yescharacteristic
60C2×#95C2211yescharacteristic
61C25C2311yescharacteristic
62C25C2311yescharacteristic
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4Q814yescomplemented
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C23×C4C2311yescharacteristic
96C23×C4C2311yescharacteristic
97C22⋊Q8—232no—
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—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4D414yescomplemented
121C2×C4⋊C4D414yescomplemented
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—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4D414yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—41no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—41no—
168#95D414yes—
169#95D414yescomplemented
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—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24C2×D412yes—
197C24C2×Q812yes—
198C24C2×Q812yes—
199C24#4112yes—
200C24C2411yescharacteristic
201C2×Q8—28no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×D4—28no—
209C2×D4—28no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—44no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—28no—
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—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4C2×D412yes—
297C22×C4C2×D412yes—
298C22×C4C2×D412yes—
299C22×C4C2×D412yes—
300C22×C4C2×D412yes—
301C22×C4C2×D412yes—
302C22×C4C2×D412yes—
303C22×C4C2×D412yes—
304C22×C4C2×D412yes—
305C22×C4C2×D412yes—
306C22×C4C2×D412yes—
307C22×C4C2×D412yes—
308C22×C4C2×D412yes—
309C22×C4C2×D412yes—
310C22×C4C2×Q812yes—
311C22×C4#4112yes—
312C22×C4#4112yes—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C22⋊C4—28no—
322C22⋊C4—28no—
323C22⋊C4—28no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C42—44no—
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—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
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—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23C22⋊Q814yes—
403C23C22×D411yescharacteristic
404C23C22×D411yescharacteristic
405C23C22×D411yescharacteristic
406C23C22×D411yescharacteristic
407C23C22×D411yescharacteristic
408C23C22×Q811yescharacteristic
409C23C2×#4111yescharacteristic
410C23C2×#4111yescharacteristic
411C23C2×#4111yescharacteristic
412C23H(5,𝔽2)11yescharacteristic
413C23H(5,𝔽2)11yescharacteristic
414C23H(5,𝔽2)11yescharacteristic
415C232−1+411yescharacteristic
416C232−1+411yescharacteristic
417C232−1+411yescharacteristic
418Q8—416no—
419D4—416no—
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—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—416no—
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—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4#12014yes—
510C2×C4C22⋊Q814yescomplemented
511C2×C4#12714yes—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C2×#12011yescharacteristic
543C22C2×C22⋊Q811yescharacteristic
544C22C2×C22⋊Q811yescharacteristic
545C22#52511yescharacteristic
546C22C2×#12711yescharacteristic
547C22#53211yescharacteristic
548C22#53511yescharacteristic
549C22#53611yescharacteristic
550C22#53611yescharacteristic
551C22#53711yescharacteristic
552C22#53711yescharacteristic
553C22#53911yescharacteristic
554C22#54211yescharacteristic
555C22#54211yescharacteristic
556C22#54311yescharacteristic
557C22D4211yescharacteristic
558C22D4211yescharacteristic
559C22D4211yescharacteristic
560C22D4211yescharacteristic
561C22#54611yescharacteristic
562C22#54611yescharacteristic
563C22#54611yescharacteristic
564C22#54711yescharacteristic
565C22#54711yescharacteristic
566C22#54711yescharacteristic
567C22#54811yescharacteristic
568C22#54811yescharacteristic
569C22#54811yescharacteristic
570C22D4×Q811yescharacteristic
571C22D4×Q811yescharacteristic
572C22D4×Q811yescharacteristic
573C22#55111yescharacteristic
574C22#55411yescharacteristic
575C22#56211yescharacteristic
576C22#56311yescharacteristic
577C4—24no—
578C4—24no—
579C4—24no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—81no—
589C4—81no—
590C2—44no—
591C2—82no—
592C2#240911yescharacteristic
593C2#243011yescharacteristic
594C2#243311yescharacteristic
595C2#243711yescharacteristic
596C2#244911yescharacteristic
597C2#245211yescharacteristic
598C2#246111yescharacteristic
599C2#254011yescharacteristic
600C2#254211yescharacteristic
601C2#267211yescharacteristic
602C2#267411yescharacteristic
603C2#268111yescharacteristic
604C2#268311yescharacteristic
605C2#269211yescharacteristic
606C2#272311yescharacteristic
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×#399C2semidirect
3. C2×#396C2semidirect
4. C2×#395C2semidirect
5. C2×#393C2semidirect
8. C2×#385C2semidirect
9. #1444C2semidirect
10. #1439C2semidirect
11. #1439C2semidirect
12. #1439C2semidirect
14. #1436C2semidirect
15. C4⋊#95C2semidirect
76. C22×D4Q8semidirect
120. C2×C4⋊C4D4semidirect
121. C2×C4⋊C4D4semidirect
169. #95D4semidirect
510. 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×#3992
C2×#3962
C2×#3952
C2×#3932
C2×#3902
C2×#3862
C2×#3852
#14442
#14392
#14382
#14362
C4⋊#952