Finite Group Databasebeta

← #15311

Subgroups — #15311

Order 256.

1045 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic
2C2×#397C211yesmaximal, characteristic, complemented
3C2×#395C211yesmaximal, characteristic, complemented
4C2×#394C211yesmaximal, characteristic
5C2×#386C211yesmaximal, characteristic
6C2×#380C211yesmaximal, characteristic, complemented
7#1446C211yesmaximal, characteristic, complemented
8#1440C211yesmaximal, characteristic, complemented
9#1440C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1439C211yesmaximal, characteristic, complemented
12#1439C211yesmaximal, characteristic, complemented
13#1436C211yesmaximal, characteristic, complemented
14#1436C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C2×#124—24no—
20C2×#124—24no—
21C2×#124—24no—
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊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×C22⋊C4C2211yescharacteristic
37C22×C42C2211yescharacteristic
38#397—24no—
39#395—24no—
40#394—24no—
41#386—24no—
42#380—24no—
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
58C2×#95C2211yescharacteristic
59C25C2311yescharacteristic
60C25C2311yescharacteristic
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4D414yes—
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
95#124—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—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
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—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—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4D414yes—
140C2×C22⋊C4D414yes—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—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—22no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95D414yes—
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—
185C24—21no—
186C24C2×D412yes—
187C24C2×D412yes—
188C24C2×D412yes—
189C24C2×D412yes—
190C24C2411yescharacteristic
191C2×Q8—28no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×D4—28no—
199C2×D4—28no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—44no—
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—28no—
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—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4C2×D412yes—
291C22×C4C2×D412yes—
292C22×C4C2×D412yes—
293C22×C4C2×D412yes—
294C22×C4C2×D412yes—
295C22×C4C2×D412yes—
296C22×C4C2×D412yes—
297C22×C4C2×D412yes—
298C22×C4#4112yes—
299C22×C4#4112yes—
300C22×C4#4112yes—
301C22×C4#4112yes—
302C22×C4#4112yes—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C22⋊C4—28no—
310C22⋊C4—28no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C42—28no—
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—22no—
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—
387C23—41no—
388C23#12014yes—
389C23C22×D411yescharacteristic
390C23C22×D411yescharacteristic
391C23C22×D411yescharacteristic
392C23C22×D411yescharacteristic
393C23C2×#4111yescharacteristic
394C23C2×#4111yescharacteristic
395C23C2×#4111yescharacteristic
396C23C2×#4111yescharacteristic
397C23C2×#4111yescharacteristic
398C23H(5,𝔽2)11yescharacteristic
399C23H(5,𝔽2)11yescharacteristic
400C23H(5,𝔽2)11yescharacteristic
401C232−1+411yescharacteristic
402C232−1+411yescharacteristic
403C232−1+411yescharacteristic
404Q8—416no—
405D4—416no—
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—416no—
440C2×C4—41no—
441C2×C4—41no—
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×C4—41no—
495C2×C4—41no—
496C2×C4#12114yes—
497C2×C4#12414yescomplemented
498C22—24no—
499C22—24no—
500C22—24no—
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—42no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22C2×#12011yescharacteristic
529C22C2×#12111yescharacteristic
530C22#52511yescharacteristic
531C22C2×#12411yescharacteristic
532C22#52911yescharacteristic
533C22#53211yescharacteristic
534C22#53411yescharacteristic
535C22#53611yescharacteristic
536C22#53611yescharacteristic
537C22#53711yescharacteristic
538C22#53811yescharacteristic
539C22#54211yescharacteristic
540C22#54211yescharacteristic
541C22#54211yescharacteristic
542C22#54211yescharacteristic
543C22D4211yescharacteristic
544C22D4211yescharacteristic
545C22#54611yescharacteristic
546C22#54611yescharacteristic
547C22#54611yescharacteristic
548C22#54611yescharacteristic
549C22#54611yescharacteristic
550C22#54711yescharacteristic
551C22#54711yescharacteristic
552C22#54711yescharacteristic
553C22#54711yescharacteristic
554C22#54711yescharacteristic
555C22#54711yescharacteristic
556C22#54811yescharacteristic
557C22#55111yescharacteristic
558C22#55511yescharacteristic
559C22#55611yescharacteristic
560C22#55611yescharacteristic
561C22#56211yescharacteristic
562C22#56311yescharacteristic
563C4—24no—
564C4—24no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C2—44no—
577C2—82no—
578C2#240611yescharacteristic
579C2#241011yescharacteristic
580C2#242711yescharacteristic
581C2#243011yescharacteristic
582C2#247211yescharacteristic
583C2#247411yescharacteristic
584C2#249111yescharacteristic
585C2#252111yescharacteristic
586C2#254711yescharacteristic
587C2#267211yescharacteristic
588C2#267411yescharacteristic
589C2#268611yescharacteristic
590C2#270011yescharacteristic
591C2#271511yescharacteristic
592C2#271711yescharacteristic
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
2. C2×#397C2semidirect
3. C2×#395C2semidirect
6. C2×#380C2semidirect
7. #1446C2semidirect
8. #1440C2semidirect
9. #1440C2semidirect
10. #1439C2semidirect
11. #1439C2semidirect
12. #1439C2semidirect
13. #1436C2semidirect
14. #1436C2semidirect
15. C22⋊#95C2semidirect
497. C2×C4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3972
C2×#3952
C2×#3942
C2×#3862
C2×#3802
#14462
#14402
#14392
#14362
C22⋊#952