Finite Group Databasebeta

← #4416354

Subgroups — #4416354

Order 512.

4052 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#17058C212yesmaximal, complemented
4#16816C214yesmaximal, complemented
5#16392C214yesmaximal, complemented
6#16363C212yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#15014C212yesmaximal, complemented
9#14913C212yesmaximal, complemented
10#14777C212yesmaximal, complemented
11#14718C214yesmaximal, complemented
12#14681C211yesmaximal, characteristic, complemented
13#14674C212yesmaximal, complemented
14#14659C212yesmaximal, complemented
15C25×C4C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23#2847—28no—
24C2×#399C2214yescomplemented
25C2×#394C2218yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#380—28no—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#379C2212yes—
43C22×#95C2212yes—
44#1445C2214yescomplemented
45#1445C2214yescomplemented
46#1444C2214yescomplemented
47#1444C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2218yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1440C2218yes—
53#1440C2218yescomplemented
54#1439C2218yes—
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1438C2218yes—
59#1438C2214yescomplemented
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2214yes—
65C26C2311yescharacteristic
66C23×D4C2312yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4—28no—
71C24×C4—44no—
72C24×C4—44no—
73C24×C4D418yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#121—24no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121—28no—
86C2×#121—44no—
87C2×#121—44no—
88C2×#120—28no—
89C2×#120—48no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2318yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—48no—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2318yes—
116C22×C22⋊C4C2318yes—
117#399—48no—
118#394—416no—
119#386—28no—
120#386—28no—
121#386—28no—
122#386—48no—
123#386—48no—
124#385—48no—
125#381—28no—
126#381—44no—
127#380—24no—
128#380—24no—
129#380—24no—
130#380—28no—
131#380—28no—
132#380—28no—
133#380—28no—
134#379—44no—
135C2×#95—28no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25—42no—
156C25—42no—
157C25C2×D414yes—
158C25C2412yes—
159C25C2411yescharacteristic
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—44no—
177C22×D4—44no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—232no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—416no—
195C23×C4—416no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—464no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4#4118yes—
215C23×C4C2418yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#121—416no—
226#121—416no—
227#121—416no—
228#120—432no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—432no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—48no—
289#95—44no—
290#95—48no—
291#95—48no—
292#95—48no—
293#95—48no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—232no—
299C24—28no—
300C24—28no—
301C24—28no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—28no—
315C24—28no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—44no—
325C24—44no—
326C24—416no—
327C24—416no—
328C24—416no—
329C24—416no—
330C24—416no—
331C24—416no—
332C24C22×D412yes—
333C24C22×D412yes—
334C24C22×D412yes—
335C24C2×#4114yes—
336C24H(5,𝔽2)14yes—
337C24H(5,𝔽2)14yes—
338C24H(5,𝔽2)14yes—
339C242−1+414yes—
340C24C2511yescharacteristic
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—432no—
353C2×D4—44no—
354C2×D4—88no—
355C2×D4—816no—
356C2×D4—816no—
357C22×C4—232no—
358C22×C4—28no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—4128no—
418C22×C4—416no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—816no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4#123116yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4H(5,𝔽2)14yes—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—816no—
444C22⋊C4—48no—
445C22⋊C4—416no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—48no—
450C22⋊C4—416no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—816no—
456C22⋊C4—816no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—232no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—416no—
487C23—416no—
488C23—416no—
489C23—416no—
490C23—416no—
491C23—4128no—
492C23—416no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23C2×#12314yes—
516C23#53414yes—
517C23#53514yes—
518C23#53514yes—
519C23#53614yes—
520C23#562116yes—
521C23C23×D411yescharacteristic
522C23C22×#4112yes—
523C23C2×H(5,𝔽2)12yes—
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)12yes—
527C23C2×2−1+412yes—
528C23#58512yes—
529D4—88no—
530D4—816no—
531D4—816no—
532D4—816no—
533C2×C4—232no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—416no—
549C2×C4—416no—
550C2×C4—416no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—816no—
582C2×C4—816no—
583C2×C4C2×#12314yes—
584C2×C4C2×#12314yes—
585C2×C4#53414yes—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—232no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—464no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—416no—
607C22—416no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—816no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22#281618yes—
619C22C22×#12311yescharacteristic
620C22C2×#53411yescharacteristic
621C22C2×#53411yescharacteristic
622C22C2×#53511yescharacteristic
623C22C2×#53511yescharacteristic
624C22C2×#53611yescharacteristic
625C22#344911yescharacteristic
626C22#349014yes—
627C22#349214yes—
628C22#349414yes—
629C22C2×#56214yes—
630C22#353414yes—
631C22#353714yes—
632C22#354114yes—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—42no—
637C4—82no—
638C4—84no—
639C4—84no—
640C4—82no—
641C4—88no—
642C4—82no—
643C2—28no—
644C2—44no—
645C2—44no—
646C2—162no—
647C2#3499211yescharacteristic
648C2#3522711yescharacteristic
649C2#3526211yescharacteristic
650C2C2×#281611yescharacteristic
651C2#3743811yescharacteristic
652C2#3744611yescharacteristic
653C2#3745011yescharacteristic
654C2#4546218yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#380C2semidirect
3. #17058C2semidirect
4. #16816C2semidirect
5. #16392C2semidirect
6. #16363C2semidirect
7. #15091C2semidirect
8. #15014C2semidirect
9. #14913C2semidirect
10. #14777C2semidirect
11. #14718C2semidirect
12. #14681C2semidirect
13. #14674C2semidirect
14. #14659C2semidirect
24. C2×#399C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
53. #1440C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
59. #1438C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C22×#3802
#170582
#168162
#163922
#163632
#150912
#150142
#149132
#147772
#147182
#146812
#146742
#146592