Finite Group Databasebeta

← #2036857

Subgroups — #2036857

Order 512.

2742 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3#17079C212yesmaximal, complemented
4#16316C214yesmaximal, complemented
5#15984C212yesmaximal
6#15944C212yesmaximal, complemented
7#15784C212yesmaximal, complemented
8#15671C212yesmaximal, complemented
9#15323C214yesmaximal, complemented
10#15308C214yesmaximal, complemented
11#14697C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic, complemented
13#14364C214yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2214yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2850—28no—
20#2683—28no—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#386C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#382C2211yescharacteristic
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40#1446C2214yes—
41#1446C2214yes—
42#1445C2214yes—
43#1445C2212yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2214yes—
60#1437C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×#124—24no—
77C2×#124—44no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—42no—
82C2×C22⋊Q8—48no—
83C2×C22⋊Q8—42no—
84C2×#121—24no—
85C2×#121—42no—
86C2×#121—42no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42—24no—
109C22×C42—44no—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#397—48no—
115#396—216no—
116#395—28no—
117#393—44no—
118#391—24no—
119#391—44no—
120#390D418yes—
121#389—28no—
122#389—44no—
123#388—216no—
124#388—48no—
125#386—28no—
126#386—216no—
127#386—44no—
128#385—48no—
129#382—24no—
130#381—44no—
131#380—28no—
132#380—44no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C25C2412yes—
155C25C2411yescharacteristic
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—28no—
159C22×Q8—24no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—44no—
175C22×D4C2×D414yes—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204#124—416no—
205C22⋊Q8—464no—
206C22⋊Q8—416no—
207#121—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4#4118yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4C2×D4116yes—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—416no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42C2×D414yes—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—44no—
298#95—44no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24C22×D414yes—
315C24H(5,𝔽2)12yes—
316C242−1+414yes—
317C24C2511yescharacteristic
318C2×Q8—416no—
319C2×Q8—48no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—88no—
327C2×Q8—84no—
328C2×D4—28no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—416no—
333C2×D4—48no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—88no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
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—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C2×#4118yes—
417C22×C4C2×#4114yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C42−1+414yes—
422C4⋊C4—216no—
423C4⋊C4—44no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—416no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—88no—
432C22⋊C4—232no—
433C22⋊C4—416no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—416no—
438C22⋊C4—84no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C42—28no—
444C42—44no—
445C42—48no—
446C42—84no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23#56314yes—
489C23C23×D411yescharacteristic
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×2−1+412yes—
496C23C2×2−1+412yes—
497C23#58512yes—
498Q8—88no—
499Q8—816no—
500D4—88no—
501D4—816no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—416no—
523C2×C4—432no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
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—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—816no—
570C2×C4#53614yes—
571C2×C4#54618yes—
572C22—28no—
573C22—24no—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—48no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22C2×#53611yescharacteristic
591C22#346012yes—
592C22C2×#54612yes—
593C22#348111yescharacteristic
594C22#348411yescharacteristic
595C22#348411yescharacteristic
596C22#348912yes—
597C22#349212yes—
598C22#351812yes—
599C22#351912yes—
600C22#352312yes—
601C22C2×#56311yescharacteristic
602C22#353412yes—
603C22#353612yes—
604C22#353612yes—
605C22#353612yes—
606C22#354511yescharacteristic
607C22#354911yescharacteristic
608C22#355212yes—
609C22#355412yes—
610C22#356112yes—
611C4—24no—
612C4—28no—
613C4—42no—
614C4—44no—
615C4—48no—
616C4—44no—
617C4—82no—
618C4—84no—
619C4—82no—
620C4—84no—
621C4—84no—
622C2—44no—
623C2—84no—
624C2#3671811yescharacteristic
625C2#3720912yes—
626C2#3774512yes—
627C2#3876312yes—
628C2#3891112yes—
629C2#4609112yes—
630C2#4613611yescharacteristic
631C2#5320412yes—
632C2#5904111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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. #17079C2semidirect
4. #16316C2semidirect
6. #15944C2semidirect
7. #15784C2semidirect
8. #15671C2semidirect
9. #15323C2semidirect
10. #15308C2semidirect
11. #14697C2semidirect
12. #14445C2semidirect
13. #14364C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#28502
#170792
#163162
#159842
#159442
#157842
#156712
#153232
#153082
#146972
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