Finite Group Databasebeta

← #14520

Subgroups — #14520

Order 256.

1091 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#392C211yesmaximal, characteristic
2C2×#390C211yesmaximal, characteristic, complemented
3C2×#388C211yesmaximal, characteristic, complemented
4C2×#386C211yesmaximal, characteristic, complemented
5C2×#386C211yesmaximal, characteristic, complemented
6#1442C211yesmaximal, characteristic, complemented
7#1440C211yesmaximal, characteristic, complemented
8#1438C211yesmaximal, characteristic, complemented
9#1438C211yesmaximal, characteristic, complemented
10#1438C211yesmaximal, characteristic, complemented
11#1437C211yesmaximal, characteristic, complemented
12#1437C211yesmaximal, characteristic, complemented
13#1437C211yesmaximal, characteristic, complemented
14#1434C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊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×C42C2211yescharacteristic
36C22×C42C2211yescharacteristic
37#392C418yescomplemented
38#390—24no—
39#388—24no—
40#386—24no—
41#386—24no—
42C2×#95C2211yescharacteristic
43C2×#95C2211yescharacteristic
44C2×#95C2211yescharacteristic
45C2×#95C2211yescharacteristic
46C2×#95C2211yescharacteristic
47C2×#95C2211yescharacteristic
48C2×#95C2211yescharacteristic
49C2×#95C2211yescharacteristic
50C2×#95C2211yescharacteristic
51C2×#95C2211yescharacteristic
52C2×#95C2211yescharacteristic
53C2×#95C2211yescharacteristic
54C2×#95C2211yescharacteristic
55C2×#95C2211yescharacteristic
56C25C2311yescharacteristic
57C25C2311yescharacteristic
58C25C2311yescharacteristic
59C22×D4—22no—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4C2×C414yes—
78C22×D4C2×C414yes—
79C22×D4C2×C414yescomplemented
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C23×C4C2311yescharacteristic
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
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—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4C2×C414yes—
141C2×C22⋊C4C2×C414yescomplemented
142C2×C22⋊C4C2×C414yescomplemented
143C2×C22⋊C4D414yescomplemented
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42—41no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—22no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95C2×C414yescomplemented
163#95D414yescomplemented
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—
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—
204C24C22×C412yes—
205C24C22×C412yes—
206C24C22×C412yes—
207C24C2×D412yes—
208C24#4112yes—
209C24C2411yescharacteristic
210C2×D4—28no—
211C2×D4—28no—
212C2×D4—28no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—44no—
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—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—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
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×C4#4112yes—
320C22×C4#4112yes—
321C22×C4#4112yes—
322C4⋊C4—44no—
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—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C42—44no—
338C42—44no—
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—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—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—
402C23—41no—
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—
443C23C4×D414yes—
444C23C23×C411yescharacteristic
445C23C22×D411yescharacteristic
446C23C22×D411yescharacteristic
447C23C2×#4111yescharacteristic
448C23C2×#4111yescharacteristic
449C23C2×#4111yescharacteristic
450C23C2×#4111yescharacteristic
451C23H(5,𝔽2)11yescharacteristic
452C23H(5,𝔽2)11yescharacteristic
453C23H(5,𝔽2)11yescharacteristic
454C23H(5,𝔽2)11yescharacteristic
455C23H(5,𝔽2)11yescharacteristic
456C23H(5,𝔽2)11yescharacteristic
457C23H(5,𝔽2)11yescharacteristic
458C23H(5,𝔽2)11yescharacteristic
459D4—416no—
460D4—416no—
461D4—416no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
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—416no—
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×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
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×C4C4×D414yes—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22C2×C4×D411yescharacteristic
593C22C2×C4×D411yescharacteristic
594C22C4×#4111yescharacteristic
595C22#51811yescharacteristic
596C22#51811yescharacteristic
597C22#51811yescharacteristic
598C22#51811yescharacteristic
599C22#53411yescharacteristic
600C22#53511yescharacteristic
601C22#53511yescharacteristic
602C22#53511yescharacteristic
603C22#53811yescharacteristic
604C22#53811yescharacteristic
605C22#53811yescharacteristic
606C22#53811yescharacteristic
607C22#54011yescharacteristic
608C22#54011yescharacteristic
609C22#54011yescharacteristic
610C22#54011yescharacteristic
611C22D4211yescharacteristic
612C22#54611yescharacteristic
613C22#54611yescharacteristic
614C22#54611yescharacteristic
615C22#54611yescharacteristic
616C22#54611yescharacteristic
617C22#55011yescharacteristic
618C22#55111yescharacteristic
619C22#55311yescharacteristic
620C22#55311yescharacteristic
621C22#55311yescharacteristic
622C22#55511yescharacteristic
623C22#56011yescharacteristic
624C22#56011yescharacteristic
625C22#56011yescharacteristic
626C22H(3,𝔽4)11yescharacteristic
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—81no—
638C4—81no—
639C2—44no—
640C2—82no—
641C2—82no—
642C2#232111yescharacteristic
643C2#232811yescharacteristic
644C2#233311yescharacteristic
645C2#235811yescharacteristic
646C2#236311yescharacteristic
647C2#236411yescharacteristic
648C2#237511yescharacteristic
649C2#267011yescharacteristic
650C2#267311yescharacteristic
651C2#269911yescharacteristic
652C2#270311yescharacteristic
653C2#270311yescharacteristic
654C2#271411yescharacteristic
655C2#275311yescharacteristic
656C2#275611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC2×#392C2
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. #1442C2semidirect
7. #1440C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
10. #1438C2semidirect
11. #1437C2semidirect
12. #1437C2semidirect
13. #1437C2semidirect
14. #1434C2semidirect
15. C22⋊#95C2semidirect
37. #392C4semidirect
79. C22×D4C2×C4semidirect
141. C2×C22⋊C4C2×C4semidirect
142. C2×C22⋊C4C2×C4semidirect
143. C2×C22⋊C4D4semidirect
162. #95C2×C4semidirect
163. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3922
C2×#3902
C2×#3882
C2×#3862
#14422
#14402
#14382
#14372
#14342
C22⋊#952