Finite Group Databasebeta

← #4607312

Subgroups — #4607312

Order 512.

2826 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2544C211yesmaximal, characteristic
3#17089C214yesmaximal, complemented
4#16630C212yesmaximal, complemented
5#16558C212yesmaximal, complemented
6#16285C214yesmaximal, complemented
7#16263C214yesmaximal, complemented
8#16032C212yesmaximal, complemented
9#15951C214yesmaximal, complemented
10#15313C212yesmaximal
11#15098C212yesmaximal, complemented
12#14723C212yesmaximal, complemented
13#14667C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2683—28no—
21#2544—28no—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#395C2212yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#393C2212yes—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#386C2214yes—
33C2×#386C2212yes—
34C2×#386C2214yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#381C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39#1446C2214yes—
40#1446C2214yes—
41#1445C2214yes—
42#1444C2212yes—
43#1444C2214yes—
44#1444C2212yes—
45#1441C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2212yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2212yes—
54#1440C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1434C2212yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×#127—24no—
77C2×#127—44no—
78C2×#124—24no—
79C2×#124—24no—
80C2×#124—42no—
81C2×#124—42no—
82C2×#124—44no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—48no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—48no—
87C2×C22⋊Q8—44no—
88C2×C4×D4—24no—
89C2×C4×D4—42no—
90C2×C4×D4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#397—48no—
118#396—216no—
119#395—28no—
120#394—216no—
121#394—48no—
122#394—48no—
123#393—44no—
124#391—24no—
125#390D418yes—
126#389—28no—
127#386—28no—
128#386—216no—
129#386—48no—
130#385—44no—
131#385—48no—
132#381—44no—
133C4×C4⋊C4—24no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C25C2412yes—
155C25C2412yes—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—28no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—42no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4C2×D414yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204#127—416no—
205#124—416no—
206#124—416no—
207C22⋊Q8—464no—
208C22⋊Q8—432no—
209C4×D4—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×D4116yes—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C2×D414yes—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—42no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300C24—22no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C22×D414yes—
319C24H(5,𝔽2)14yes—
320C24C2511yescharacteristic
321C2×Q8—416no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8—84no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—44no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—88no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—88no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—416no—
405C22×C4—48no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)18yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+414yes—
419C22×C42−1+414yes—
420C4⋊C4—216no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—416no—
424C4⋊C4—416no—
425C4⋊C4—416no—
426C4⋊C4—48no—
427C4⋊C4—84no—
428C4⋊C4—88no—
429C22⋊C4—416no—
430C22⋊C4—48no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—48no—
438C22⋊C4—88no—
439C22⋊C4—84no—
440C22⋊C4—88no—
441C22⋊C4—84no—
442C22⋊C4—88no—
443C42—28no—
444C42—44no—
445C42—44no—
446C23—24no—
447C23—28no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—28no—
452C23—24no—
453C23—28no—
454C23—24no—
455C23—28no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23C23×D411yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)14yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×2−1+412yes—
496C23C2×2−1+412yes—
497C23#58512yes—
498Q8—88no—
499Q8—816no—
500D4—416no—
501D4—88no—
502D4—832no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4#53614yes—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—48no—
572C22—48no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—48no—
577C22—82no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—82no—
586C22—84no—
587C22C2×#53611yescharacteristic
588C22#348411yescharacteristic
589C22#348412yes—
590C22#348411yescharacteristic
591C22#348912yes—
592C22#352911yescharacteristic
593C22#353212yes—
594C22#353612yes—
595C22#353614yes—
596C22#353612yes—
597C22#353712yes—
598C22#354214yes—
599C22#355011yescharacteristic
600C22#355014yes—
601C22#355414yes—
602C22#355912yes—
603C4—24no—
604C4—42no—
605C4—48no—
606C4—44no—
607C4—44no—
608C4—44no—
609C4—42no—
610C4—82no—
611C4—84no—
612C4—84no—
613C4—84no—
614C2—84no—
615C2—84no—
616C2#3671811yescharacteristic
617C2#4248112yes—
618C2#4622811yescharacteristic
619C2#4954612yes—
620C2#5200411yescharacteristic
621C2#5242112yes—
622C2#5800614yes—
623C2#5894412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2683C2semidirect
3. #17089C2semidirect
4. #16630C2semidirect
5. #16558C2semidirect
6. #16285C2semidirect
7. #16263C2semidirect
8. #16032C2semidirect
9. #15951C2semidirect
11. #15098C2semidirect
12. #14723C2semidirect
13. #14667C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#25442
#170892
#166302
#165582
#162852
#162632
#160322
#159512
#153132
#150982
#147232
#146672