Finite Group Databasebeta

← #14359

Subgroups — #14359

Order 256.

1085 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×#123C211yesmaximal, characteristic
2C2×#388C211yesmaximal, characteristic, complemented
3C2×#386C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic, complemented
5C2×#380C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1440C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1438C211yesmaximal, characteristic
12#1437C211yesmaximal, characteristic, complemented
13C4⋊#95C211yesmaximal, characteristic, complemented
14C22⋊#95C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C2×#123—24no—
20C2×#123—24no—
21C2×#123C418yescomplemented
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊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#388—24no—
39#386—24no—
40#385—24no—
41#380—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×D4—22no—
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4C2×C414yes—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4C2×C414yescomplemented
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#123—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—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×C4⋊C4C2×C414yescomplemented
109C2×C4⋊C4C2×C414yescomplemented
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—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⋊C4C2×C414yes—
135C2×C22⋊C4C2×C414yes—
136C2×C22⋊C4C2×C414yescomplemented
137C2×C22⋊C4D414yes—
138C2×C22⋊C4D414yes—
139C2×C22⋊C4D414yes—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—41no—
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—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
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—
183C24C22×C412yes—
184C24C22×C412yes—
185C24C2×D412yes—
186C24C2×D412yes—
187C24C2×D412yes—
188C24#4112yes—
189C24C2411yescharacteristic
190C2×D4—28no—
191C2×D4—28no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—28no—
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—28no—
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—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4C22×C412yes—
290C22×C4C22×C412yes—
291C22×C4C22×C412yes—
292C22×C4C22×C412yes—
293C22×C4C22×C412yes—
294C22×C4C2×D412yes—
295C22×C4C2×D412yes—
296C22×C4C2×D412yes—
297C22×C4C2×D412yes—
298C22×C4C2×D412yes—
299C22×C4C2×D412yes—
300C22×C4#4112yes—
301C22×C4#4112yes—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4C22⋊C4116yescomplemented
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—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
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—
386C23C2×C22⋊C414yes—
387C23C4×D414yes—
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
403D4—416no—
404C2×C4—22no—
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—416no—
438C2×C4—416no—
439C2×C4—41no—
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×C4C2×C22⋊C414yes—
496C2×C4C2×C22⋊C414yes—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—24no—
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—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22C22×C22⋊C411yescharacteristic
528C22C2×C4×D411yescharacteristic
529C22C4×#4111yescharacteristic
530C22#51811yescharacteristic
531C22#52011yescharacteristic
532C22#52011yescharacteristic
533C22#52011yescharacteristic
534C22#53411yescharacteristic
535C22#53511yescharacteristic
536C22#53611yescharacteristic
537C22#53611yescharacteristic
538C22#53711yescharacteristic
539C22#53711yescharacteristic
540C22#53811yescharacteristic
541C22#53911yescharacteristic
542C22#53911yescharacteristic
543C22#53911yescharacteristic
544C22#54011yescharacteristic
545C22#54211yescharacteristic
546C22D4211yescharacteristic
547C22D4211yescharacteristic
548C22#54611yescharacteristic
549C22#54611yescharacteristic
550C22#54611yescharacteristic
551C22#54611yescharacteristic
552C22#54611yescharacteristic
553C22#54711yescharacteristic
554C22#54711yescharacteristic
555C22#54711yescharacteristic
556C22#55111yescharacteristic
557C22#55211yescharacteristic
558C22#56011yescharacteristic
559C22#56211yescharacteristic
560C22#56211yescharacteristic
561C22#56311yescharacteristic
562C4—24no—
563C4—24no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—44no—
576C2—44no—
577C2#231111yescharacteristic
578C2#232211yescharacteristic
579C2#233511yescharacteristic
580C2D4×C22⋊C411yescharacteristic
581C2#233911yescharacteristic
582C2#235911yescharacteristic
583C2#237611yescharacteristic
584C2#267111yescharacteristic
585C2#267211yescharacteristic
586C2#267811yescharacteristic
587C2#268011yescharacteristic
588C2#268711yescharacteristic
589C2#269311yescharacteristic
590C2#269511yescharacteristic
591C2#270611yescharacteristic
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×#388C2semidirect
4. C2×#385C2semidirect
5. C2×#380C2semidirect
6. #1445C2semidirect
7. #1441C2semidirect
8. #1441C2semidirect
9. #1440C2semidirect
10. #1439C2semidirect
12. #1437C2semidirect
13. C4⋊#95C2semidirect
14. C22⋊#95C2semidirect
15. C22⋊#95C2semidirect
21. C2×#123C4semidirect
80. C23×C4C2×C4semidirect
108. C2×C4⋊C4C2×C4semidirect
109. C2×C4⋊C4C2×C4semidirect
136. C2×C22⋊C4C2×C4semidirect
316. C22⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#3882
C2×#3862
C2×#3852
C2×#3802
#14452
#14412
#14402
#14392
#14382
#14372
C4⋊#952
C22⋊#952