Finite Group Databasebeta

← #14421

Subgroups — #14421

Order 256.

1063 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic
2C2×#391C211yesmaximal, characteristic, complemented
3C2×#388C211yesmaximal, characteristic, complemented
4C2×#386C211yesmaximal, characteristic
5C2×#385C211yesmaximal, characteristic, complemented
6#1446C211yesmaximal, characteristic, complemented
7#1440C211yesmaximal, characteristic, complemented
8#1439C211yesmaximal, characteristic, complemented
9#1439C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1438C211yesmaximal, characteristic
12#1437C211yesmaximal, characteristic, complemented
13C4⋊#95C211yesmaximal, characteristic, complemented
14#1434C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C2×#124—24no—
20C2×#124—24no—
21C2×#124C418yescomplemented
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊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×C22⋊C4C2211yescharacteristic
36C22×C42C2211yescharacteristic
37C22×C42C2211yescharacteristic
38#391—24no—
39#388—24no—
40#386—24no—
41#385—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
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C25C2311yescharacteristic
59C25C2311yescharacteristic
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8C2×C414yescomplemented
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—22no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4C2×C414yes—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94#124—232no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
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—41no—
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⋊C4C2×C414yes—
132C2×C22⋊C4C2×C414yes—
133C2×C22⋊C4C2×C414yescomplemented
134C2×C22⋊C4C2×C414yescomplemented
135C2×C22⋊C4D414yes—
136C2×C22⋊C4D414yes—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—41no—
142C2×C42C2×C414yescomplemented
143#95—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95—22no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95D414yes—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
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—
185C24C22×C412yes—
186C24C22×C412yes—
187C24C2×D412yes—
188C24C2×D412yes—
189C24C2411yescharacteristic
190C2×Q8—28no—
191C2×Q8—28no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×D4—28no—
198C2×D4—28no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
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—21no—
220C22×C4—21no—
221C22×C4—21no—
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—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4C22×C412yes—
288C22×C4C22×C412yes—
289C22×C4C22×C412yes—
290C22×C4C22×C412yes—
291C22×C4C22×C412yes—
292C22×C4C2×D412yes—
293C22×C4C2×D412yes—
294C22×C4C2×D412yes—
295C22×C4C2×D412yes—
296C22×C4C2×D412yes—
297C22×C4C2×D412yes—
298C22×C4C2×D412yes—
299C22×C4#4112yes—
300C22×C4#4112yes—
301C22×C4#4112yes—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C22⋊C4—28no—
306C22⋊C4—28no—
307C22⋊C4—28no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4C22⋊C4116yescomplemented
316C42—28no—
317C42—44no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
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—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
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—
387C23C2×C22⋊C414yes—
388C23C23×C411yescharacteristic
389C23C22×D411yescharacteristic
390C23C22×D411yescharacteristic
391C23C22×D411yescharacteristic
392C23C2×#4111yescharacteristic
393C23C2×#4111yescharacteristic
394C23C2×#4111yescharacteristic
395C23H(5,𝔽2)11yescharacteristic
396C23H(5,𝔽2)11yescharacteristic
397C23H(5,𝔽2)11yescharacteristic
398C23H(5,𝔽2)11yescharacteristic
399C23H(5,𝔽2)11yescharacteristic
400C232−1+411yescharacteristic
401C232−1+411yescharacteristic
402C232−1+411yescharacteristic
403Q8—416no—
404D4—416no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
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—416no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
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×C4C2×C22⋊C414yes—
495C2×C4C2×C22⋊C414yes—
496C2×C4C4×D414yes—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
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—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22C22×C22⋊C411yescharacteristic
528C22C2×C4×D411yescharacteristic
529C22C4×#4111yescharacteristic
530C22#51811yescharacteristic
531C22#52011yescharacteristic
532C22#52011yescharacteristic
533C22#52011yescharacteristic
534C22#53511yescharacteristic
535C22#53511yescharacteristic
536C22#53611yescharacteristic
537C22#53611yescharacteristic
538C22#53711yescharacteristic
539C22#53711yescharacteristic
540C22#53811yescharacteristic
541C22#53811yescharacteristic
542C22#54211yescharacteristic
543C22#54211yescharacteristic
544C22#54211yescharacteristic
545C22#54211yescharacteristic
546C22D4211yescharacteristic
547C22D4211yescharacteristic
548C22#54611yescharacteristic
549C22#54611yescharacteristic
550C22#54611yescharacteristic
551C22#54611yescharacteristic
552C22#54611yescharacteristic
553C22#54711yescharacteristic
554C22#54711yescharacteristic
555C22#54811yescharacteristic
556C22#55611yescharacteristic
557C22#55911yescharacteristic
558C22H(3,𝔽4)11yescharacteristic
559C22#56211yescharacteristic
560C22#56211yescharacteristic
561C22#56311yescharacteristic
562C4—24no—
563C4—24no—
564C4—24no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—44no—
576C2—82no—
577C2#231111yescharacteristic
578C2#232611yescharacteristic
579C2#233711yescharacteristic
580C2D4×C22⋊C411yescharacteristic
581C2#233911yescharacteristic
582C2#236711yescharacteristic
583C2#237911yescharacteristic
584C2#267011yescharacteristic
585C2#267211yescharacteristic
586C2#267411yescharacteristic
587C2#267611yescharacteristic
588C2#270011yescharacteristic
589C2#271011yescharacteristic
590C2#271211yescharacteristic
591C2#271511yescharacteristic
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
2. C2×#391C2semidirect
3. C2×#388C2semidirect
5. C2×#385C2semidirect
6. #1446C2semidirect
7. #1440C2semidirect
8. #1439C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
12. #1437C2semidirect
13. C4⋊#95C2semidirect
14. #1434C2semidirect
15. C22⋊#95C2semidirect
21. C2×#124C4semidirect
66. C22×Q8C2×C4semidirect
133. C2×C22⋊C4C2×C4semidirect
134. C2×C22⋊C4C2×C4semidirect
142. C2×C42C2×C4semidirect
315. C22⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3912
C2×#3882
C2×#3862
C2×#3852
#14462
#14402
#14392
#14382
#14372
C4⋊#952
#14342
C22⋊#952