Finite Group Databasebeta

← #2034325

Subgroups — #2034325

Order 512.

2740 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2858C211yesmaximal, characteristic
3#17077C212yesmaximal, complemented
4#16559C214yesmaximal, complemented
5#16074C212yesmaximal, complemented
6#15820C212yesmaximal, complemented
7#15783C212yesmaximal, complemented
8#15631C214yesmaximal, complemented
9#15271C214yesmaximal, complemented
10#15261C212yesmaximal
11#14703C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic, complemented
13#14421C214yesmaximal, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#124C2214yes—
17C22×#123C2212yes—
18#2858—28no—
19#2804—28no—
20C2×#400C2212yes—
21C2×#398C2212yes—
22C2×#394C2214yes—
23C2×#393C2214yes—
24C2×#393C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#390C2214yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×C43C2211yescharacteristic
40#1446C2214yes—
41#1446C2212yes—
42#1442C2214yes—
43#1441C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2212yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2212yes—
58#1438C2214yes—
59#1437C2214yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—42no—
79C2×#127—24no—
80C2×#127—42no—
81C2×#127—42no—
82C2×#124—216no—
83C2×#124—216no—
84C2×#124—48no—
85C2×#123—48no—
86C2×#123—44no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#400—28no—
115#398—44no—
116#394—48no—
117#393—216no—
118#393—48no—
119#391—28no—
120#391—28no—
121#391—44no—
122#391—44no—
123#390—24no—
124#390—24no—
125#390—216no—
126#390—24no—
127#390D418yes—
128#389—44no—
129#388—48no—
130#386—28no—
131#386—44no—
132#385—216no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C43—24no—
153C25C2412yes—
154C25C2412yes—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—28no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—44no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—22no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—44no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—42no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—44no—
185C22×D4—42no—
186C22×D4C2×D414yes—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C4⋊Q8—416no—
204#127—416no—
205#124—464no—
206#123—432no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
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⋊C4C2×D4116yes—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42C2×D414yes—
281C2×C42#4118yes—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—42no—
300#95—44no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C22×D414yes—
319C24C2511yescharacteristic
320C2×Q8—416no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—88no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×D4—28no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—416no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—416no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—88no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—88no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
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—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—48no—
409C22×C4—84no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—88no—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+414yes—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—416no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C22⋊C4—232no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—48no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—84no—
439C22⋊C4—88no—
440C22⋊C4—84no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—28no—
446C42—216no—
447C42—44no—
448C42—48no—
449C42—44no—
450C42—84no—
451C23—24no—
452C23—28no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23C23×D411yescharacteristic
494C23C22×#4112yes—
495C23C2×H(5,𝔽2)12yes—
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)12yes—
499C23C2×2−1+412yes—
500C23#58512yes—
501C23#58512yes—
502Q8—816no—
503Q8—816no—
504D4—88no—
505D4—816no—
506D4—816no—
507D4—88no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
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—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—816no—
572C2×C4#53614yes—
573C2×C4#54618yes—
574C2×C4#55914yes—
575C2×C4H(3,𝔽4)14yes—
576C22—44no—
577C22—48no—
578C22—48no—
579C22—44no—
580C22—44no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—82no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22C2×#53611yescharacteristic
595C22#345512yes—
596C22C2×#54612yes—
597C22C2×#55911yescharacteristic
598C22#348411yescharacteristic
599C22#348511yescharacteristic
600C22#348812yes—
601C22#350612yes—
602C22#350712yes—
603C22#350812yes—
604C22#351112yes—
605C22C2×H(3,𝔽4)11yescharacteristic
606C22#354311yescharacteristic
607C22#354511yescharacteristic
608C22#354712yes—
609C22#354812yes—
610C22#355812yes—
611C22#355812yes—
612C22#356012yes—
613C22#356112yes—
614C22#356112yes—
615C4—24no—
616C4—28no—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—82no—
622C4—84no—
623C4—84no—
624C4—84no—
625C4—84no—
626C2—84no—
627C2—162no—
628C2#3696611yescharacteristic
629C2#3714512yes—
630C2#3769712yes—
631C2#3947012yes—
632C2#3976512yes—
633C2#4597711yescharacteristic
634C2#4601612yes—
635C2#5334712yes—
636C2#5914311yescharacteristic
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×#2804C2semidirect
3. #17077C2semidirect
4. #16559C2semidirect
5. #16074C2semidirect
6. #15820C2semidirect
7. #15783C2semidirect
8. #15631C2semidirect
9. #15271C2semidirect
11. #14703C2semidirect
12. #14445C2semidirect
13. #14421C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#28582
#170772
#165592
#160742
#158202
#157832
#156312
#152712
#152612
#147032
#144452
#144212