Finite Group Databasebeta

← #762970

Subgroups — #762970

Order 512.

2850 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2572C212yesmaximal, complemented
2#17077C212yesmaximal, complemented
3#17076C211yesmaximal, characteristic, complemented
4#17068C211yesmaximal, characteristic
5#15107C212yesmaximal, complemented
6#15077C214yesmaximal, complemented
7#14883C212yesmaximal, complemented
8#14730C214yesmaximal, complemented
9#14228C214yesmaximal, complemented
10#14224C212yesmaximal, complemented
11#13913C214yesmaximal, complemented
12C4×#391C211yesmaximal, characteristic, complemented
13C4×#390C212yesmaximal, complemented
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2212yes—
16#2572—216no—
17C2×#400C2212yes—
18C2×#394C2212yes—
19C2×#394C2212yes—
20C2×#394C2212yes—
21C2×#393C2214yes—
22C2×#392C2214yes—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#383C2214yes—
34C2×#381C2214yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2214yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C43C2211yescharacteristic
40#1445C2212yes—
41#1444C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2212yes—
45#1441C2212yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2212yes—
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C4×#95C2214yes—
68C23×Q8C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4C2314yes—
73C2×C4×Q8—24no—
74C2×C4×Q8—42no—
75C2×C4×Q8—42no—
76C2×C4×D4—28no—
77C2×C4×D4—44no—
78C2×C4×D4—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2314yes—
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107#400—28no—
108#394—28no—
109#394—28no—
110#394—44no—
111#393—216no—
112#392—216no—
113#391—24no—
114#391D418yescomplemented
115#390—28no—
116#390—28no—
117#390—44no—
118#390—44no—
119#388—216no—
120#388—48no—
121#386—216no—
122#386—48no—
123#383—216no—
124#381—216no—
125C4×C4⋊C4—44no—
126C4×C4⋊C4—42no—
127C4×C22⋊C4—216no—
128C4×C22⋊C4—216no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C43—42no—
150C25C2414yes—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8C2×D414yes—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4#4118yes—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C4×Q8—416no—
195C4×D4—432no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
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—28no—
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×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×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—42no—
268C2×C42—41no—
269C2×C42—42no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42—41no—
274C2×C42—44no—
275C2×C42C2×D414yes—
276C2×C42#4118yes—
277C2×C42#4114yes—
278#95—28no—
279#95—28no—
280#95—24no—
281#95—28no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—28no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95C2×D4116yes—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24C2×#4118yes—
313C24C2511yescharacteristic
314C2×Q8—28no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×D4—216no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—84no—
337C2×D4—84no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
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—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4C22×D414yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D414yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C4H(5,𝔽2)14yes—
418C22×C4H(5,𝔽2)12yes—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C42—216no—
440C42—28no—
441C42—44no—
442C42—44no—
443C42—48no—
444C42—48no—
445C42—416no—
446C42—84no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23C23×D411yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4112yes—
485C23C22×#4112yes—
486C23C22×#4111yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×H(5,𝔽2)12yes—
489C23C2×H(5,𝔽2)11yescharacteristic
490C23#58511yescharacteristic
491C23#58512yes—
492C23#58511yescharacteristic
493Q8—416no—
494D4—432no—
495D4—816no—
496D4—816no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
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—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—832no—
568C2×C4#52918yes—
569C2×C4#53214yes—
570C2×C4#55018yes—
571C2×C4#55514yes—
572C22—48no—
573C22—48no—
574C22—48no—
575C22—48no—
576C22—48no—
577C22—48no—
578C22—48no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22C2×#52912yes—
588C22C2×#53211yescharacteristic
589C22#345911yescharacteristic
590C22#346012yes—
591C22#346011yescharacteristic
592C22C2×#55012yes—
593C22D4×#4112yes—
594C22C2×#55511yescharacteristic
595C22#347512yes—
596C22#348012yes—
597C22#348112yes—
598C22#348111yescharacteristic
599C22#349012yes—
600C22#349011yescharacteristic
601C22#349112yes—
602C22#349311yescharacteristic
603C22#349611yescharacteristic
604C22#349611yescharacteristic
605C22#350512yes—
606C22#350612yes—
607C22#354512yes—
608C22#354511yescharacteristic
609C22#354911yescharacteristic
610C4—28no—
611C4—24no—
612C4—28no—
613C4—24no—
614C4—42no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—84no—
619C4—84no—
620C4—84no—
621C4—82no—
622C2—88no—
623C2#3493412yes—
624C2#3494311yescharacteristic
625C2#3500311yescharacteristic
626C2#3502212yes—
627C2#3761111yescharacteristic
628C2#5981912yes—
629C2#5982211yescharacteristic
630C2#5983611yescharacteristic
631C2#5988012yes—
632C2#5988511yescharacteristic
633C2#6004511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17068C2
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×#2572C2semidirect
2. #17077C2semidirect
3. #17076C2semidirect
5. #15107C2semidirect
6. #15077C2semidirect
7. #14883C2semidirect
8. #14730C2semidirect
9. #14228C2semidirect
10. #14224C2semidirect
11. #13913C2semidirect
12. C4×#391C2semidirect
13. C4×#390C2semidirect
114. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25722
#170772
#170762
#170682
#151072
#150772
#148832
#147302
#142282
#142242
#139132
C4×#3912
C4×#3902