Finite Group Databasebeta

← #4312074

Subgroups — #4312074

Order 512.

2830 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3#17007C212yesmaximal, complemented
4#16599C212yesmaximal, complemented
5#16304C214yesmaximal, complemented
6#15334C214yesmaximal, complemented
7#15326C214yesmaximal, complemented
8#14893C212yesmaximal, complemented
9#14822C212yesmaximal, complemented
10#14774C214yesmaximal, complemented
11#14501C211yesmaximal, characteristic, complemented
12#14501C212yesmaximal, complemented
13#13937C212yesmaximal
14C23×#41C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2809—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#398C2214yes—
23#2311—28no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#387C2214yes—
34C2×#386C2214yes—
35C2×#380C2214yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C22⋊C4C2212yes—
39#1444C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2212yes—
45#1441C2214yes—
46#1441C2212yes—
47#1441C2212yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1434C2212yes—
64#1434C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41Q818yes—
76C23×Q8C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C2×#117—24no—
83C2×#117—42no—
84C2×#117—42no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#400—28no—
114#400—44no—
115#400—44no—
116#399—44no—
117#398—216no—
118#391—24no—
119#391—28no—
120#391—28no—
121#391D418yes—
122#390—24no—
123#390—24no—
124#390—44no—
125#390—44no—
126#389—28no—
127#387—216no—
128#386—216no—
129#380—216no—
130#380—44no—
131#380—44no—
132C4×C22⋊C4—44no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C2×#41—232no—
158C2×#41—416no—
159C2×#41—416no—
160C2×#41—416no—
161C2×#41—416no—
162C2×#41—416no—
163C2×#41—416no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8C2×D414yes—
169C22×Q8C2×Q814yes—
170C22×Q8#4114yes—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—42no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—42no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4C2×Q818yes—
184C22×D4C2×Q814yes—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2×Q814yes—
203C23×C4C2×Q818yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215#117—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
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⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C2×D414yes—
290#95—28no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95C2×D4116yes—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24C22×Q814yes—
329C24C22×Q812yes—
330C24H(5,𝔽2)12yes—
331C24H(5,𝔽2)12yes—
332C24C2511yescharacteristic
333#41—4128no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8C22⋊Q8116yes—
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—28no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q814yes—
438C22×C4C22×Q812yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C22⋊C4—48no—
456C22⋊C4—416no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C42—44no—
465C42—44no—
466C42—84no—
467C42—84no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23#54314yes—
508C23C23×D411yescharacteristic
509C23C23×Q811yescharacteristic
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23#58514yes—
516C23#58514yes—
517Q8—416no—
518D4—432no—
519D4—416no—
520C2×C4—232no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—432no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—816no—
589C2×C4C2×C22⋊Q814yes—
590C2×C4C2×C22⋊Q818yes—
591C2×C4#54314yes—
592C22—24no—
593C22—28no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—48no—
605C22—48no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—84no—
610C22—84no—
611C22—82no—
612C22C22×C22⋊Q811yescharacteristic
613C22C2×#54311yescharacteristic
614C22C2×#54311yescharacteristic
615C22#348411yescharacteristic
616C22#348411yescharacteristic
617C22#348712yes—
618C22#349011yescharacteristic
619C22#349011yescharacteristic
620C22#350012yes—
621C22#350414yes—
622C22#352414yes—
623C22#353312yes—
624C22#353914yes—
625C22#354512yes—
626C22#355112yes—
627C22#355112yes—
628C22#356014yes—
629C4—24no—
630C4—24no—
631C4—28no—
632C4—44no—
633C4—44no—
634C4—82no—
635C4—84no—
636C4—82no—
637C4—84no—
638C4—84no—
639C4—84no—
640C2—44no—
641C2—84no—
642C2#3498211yescharacteristic
643C2#3604012yes—
644C2#3742512yes—
645C2#4218012yes—
646C2#4636012yes—
647C2#4717912yes—
648C2#5197811yescharacteristic
649C2#5198412yes—
650C2#5208311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2809C2semidirect
3. #17007C2semidirect
4. #16599C2semidirect
5. #16304C2semidirect
6. #15334C2semidirect
7. #15326C2semidirect
8. #14893C2semidirect
9. #14822C2semidirect
10. #14774C2semidirect
11. #14501C2semidirect
12. #14501C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#23112
#170072
#165992
#163042
#153342
#153262
#148932
#148222
#147742
#145012
#139372