Finite Group Databasebeta

← #951791

Subgroups — #951791

Order 512.

2390 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2489C212yesmaximal, complemented
2#17180C214yesmaximal, complemented
3#16689C214yesmaximal, complemented
4#16495C214yesmaximal
5#16084C214yesmaximal, complemented
6#15182C212yesmaximal, complemented
7#15178C212yesmaximal, complemented
8#15177C212yesmaximal
9#14283C211yesmaximal, characteristic, complemented
10#14274C211yesmaximal, characteristic, complemented
11#14021C212yesmaximal, complemented
12#13875C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2489—216no—
20C2×#400C2214yes—
21C2×#397C2214yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#388C2214yes—
27C2×#384C2211yescharacteristic
28C2×#382C2214yes—
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×#376C2211yescharacteristic
37#1445C2212yes—
38#1445C2214yes—
39#1445C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1439C2212yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1438C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C2×C4⋊Q8—24no—
72C2×C4⋊Q8—42no—
73C2×C4⋊Q8—42no—
74C2×#125—42no—
75C2×#125—42no—
76C2×#125D418yescomplemented
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—44no—
79C2×C22⋊Q8—44no—
80C2×C22⋊Q8—44no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8—44no—
83C2×C4×D4—24no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#400—216no—
112#397—216no—
113#391—28no—
114#391—28no—
115#391—28no—
116#391—44no—
117#388—216no—
118#384—42no—
119#382—216no—
120#382—42no—
121#380—28no—
122#380—44no—
123C4×C4⋊C4—28no—
124C4×C4⋊C4—42no—
125C4×C22⋊C4—28no—
126C4×C22⋊C4—28no—
127#376—42no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C25C2412yes—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—41no—
164C22×D4—41no—
165C22×D4—42no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C4⋊Q8—416no—
186#125—416no—
187C22⋊Q8—432no—
188C22⋊Q8—432no—
189C4×D4—416no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4C2×D418yes—
220C2×C4⋊C4C2×D4116yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—28no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—44no—
260C2×C42C2×D414yes—
261C2×C42#4114yes—
262C2×C42#4114yes—
263#95—24no—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—42no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C242−1+414yes—
303C24C2511yescharacteristic
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×D4—48no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—48no—
315C2×D4—44no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4C22×D414yes—
391C22×C4C22×D418yes—
392C22×C4C22×D412yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4114yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+414yes—
403C4⋊C4—216no—
404C4⋊C4—416no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—416no—
408C4⋊C4—416no—
409C4⋊C4—44no—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—88no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—416no—
421C22⋊C4—416no—
422C22⋊C4—88no—
423C42—28no—
424C42—28no—
425C42—28no—
426C42—416no—
427C42—48no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C23×D411yescharacteristic
464C23C22×#4111yescharacteristic
465C23C22×#4111yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×H(5,𝔽2)11yescharacteristic
470C23C2×2−1+412yes—
471C23C2×2−1+412yes—
472C23#58512yes—
473C23#58512yes—
474Q8—816no—
475D4—416no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
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—816no—
550C2×C4#53414yes—
551C2×C4#53814yes—
552C2×C4#53814yes—
553C2×C4#55014yes—
554C2×C4#55014yes—
555C2×C4#55514yes—
556C2×C4H(3,𝔽4)14yes—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22C2×#53411yescharacteristic
573C22C2×#53811yescharacteristic
574C22C2×#53811yescharacteristic
575C22#346112yes—
576C22C2×#55011yescharacteristic
577C22C2×#55011yescharacteristic
578C22C2×#55511yescharacteristic
579C22#348912yes—
580C22#348912yes—
581C22#349512yes—
582C22#349512yes—
583C22#349512yes—
584C22C2×H(3,𝔽4)11yescharacteristic
585C22#353811yescharacteristic
586C22#353811yescharacteristic
587C22#353911yescharacteristic
588C22#353911yescharacteristic
589C22#354112yes—
590C22#354712yes—
591C22#355212yes—
592C22#355212yes—
593C22#356212yes—
594C22#356212yes—
595C4—42no—
596C4—44no—
597C4—42no—
598C4—44no—
599C4—48no—
600C4—44no—
601C4—84no—
602C4—82no—
603C4—84no—
604C4—82no—
605C4—84no—
606C4#273318yes—
607C2—84no—
608C2#3738011yescharacteristic
609C2#3751911yescharacteristic
610C2#3777412yes—
611C2#3795512yes—
612C2#4292612yes—
613C2#4293612yes—
614C2#4543211yescharacteristic
615C2#4543211yescharacteristic
616C2C2×#273311yescharacteristic
617C2#5303211yescharacteristic
618C2#5307411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2489C2semidirect
2. #17180C2semidirect
3. #16689C2semidirect
5. #16084C2semidirect
6. #15182C2semidirect
7. #15178C2semidirect
9. #14283C2semidirect
10. #14274C2semidirect
11. #14021C2semidirect
12. #13875C2semidirect
76. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24892
#171802
#166892
#164952
#160842
#151822
#151782
#151772
#142832
#142742
#140212
#138752
#138472