Finite Group Databasebeta

← #4414911

Subgroups — #4414911

Order 512.

2534 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2832C212yesmaximal, complemented
2#17077C212yesmaximal, complemented
3#17072C211yesmaximal, characteristic, complemented
4#17068C211yesmaximal, characteristic
5#16480C214yesmaximal, complemented
6#16274C214yesmaximal, complemented
7#15070C212yesmaximal, complemented
8#15009C214yesmaximal, complemented
9#14928C214yesmaximal, complemented
10#14883C212yesmaximal, complemented
11#14700C212yesmaximal, complemented
12#14632C212yesmaximal, complemented
13C4×#391C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×C4×Q8C2211yescharacteristic
16#2832—216no—
17C2×#400C2214yes—
18C2×#397C2214yes—
19C2×#395C2212yes—
20C2×#395C2214yes—
21C2×#394C2212yes—
22C2×#394C2214yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#393C2214yes—
26C2×#393C2214yes—
27C2×#392C2214yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C43C2211yescharacteristic
40#1446C2212yes—
41#1442C2214yes—
42#1442C2214yes—
43#1441C2212yes—
44#1441C2212yes—
45#1441C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65#1434C2212yes—
66#1434C2212yes—
67C4×#95C2214yes—
68C23×Q8C2311yescharacteristic
69C23×Q8C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C2×#124—44no—
74C2×#124—48no—
75C2×C4×Q8—24no—
76C2×C4×Q8—42no—
77C2×C4×Q8—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
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×C22⋊C4C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#400—216no—
108#397—216no—
109#395—28no—
110#395—48no—
111#394—28no—
112#394—28no—
113#394—216no—
114#394—44no—
115#393—216no—
116#393—216no—
117#392—48no—
118#391—24no—
119#391D418yescomplemented
120#390—28no—
121#390—44no—
122#390—44no—
123#390—44no—
124#389—44no—
125#388—216no—
126#388—48no—
127#388—48no—
128C4×C4⋊C4—42no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C43—42no—
149C25C2414yes—
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8C2×D414yes—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—44no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—42no—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191#124—432no—
192C4×Q8—416no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
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—44no—
239C2×C22⋊C4—44no—
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×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—44no—
257C2×C22⋊C4—44no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—42no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—42no—
276C2×C42C2×D414yes—
277C2×C42#4114yes—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—44no—
296#95C2×D4116yes—
297C24—24no—
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—
312C24C2511yescharacteristic
313C2×Q8—28no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—84no—
322C2×Q8—88no—
323C2×Q8—84no—
324C2×Q8—88no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—84no—
340C2×D4—84no—
341C2×D4—88no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
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—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
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—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C2×#4114yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4114yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C4H(5,𝔽2)14yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+414yes—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—84no—
426C22⋊C4—416no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C42—28no—
439C42—44no—
440C42—44no—
441C42—48no—
442C42—48no—
443C42—48no—
444C42—48no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
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—
480C23C23×D411yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4111yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)11yescharacteristic
486C23C2×H(5,𝔽2)12yes—
487C23C2×2−1+412yes—
488C23#58511yescharacteristic
489C23#58511yescharacteristic
490C23#58512yes—
491Q8—416no—
492Q8—816no—
493D4—816no—
494D4—816no—
495D4—816no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
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—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4#53214yes—
568C2×C4#54218yes—
569C2×C4H(3,𝔽4)14yes—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22C2×#53211yescharacteristic
586C22C2×#54212yes—
587C22#345911yescharacteristic
588C22#346011yescharacteristic
589C22#346012yes—
590C22#349011yescharacteristic
591C22#349311yescharacteristic
592C22#349611yescharacteristic
593C22#349611yescharacteristic
594C22#349712yes—
595C22#349812yes—
596C22#351412yes—
597C22#352212yes—
598C22C2×H(3,𝔽4)11yescharacteristic
599C22#352912yes—
600C22#352911yescharacteristic
601C22#353412yes—
602C22#353612yes—
603C22#354411yescharacteristic
604C22#354712yes—
605C22#354911yescharacteristic
606C22#355812yes—
607C22#356412yes—
608C4—24no—
609C4—28no—
610C4—24no—
611C4—44no—
612C4—44no—
613C4—42no—
614C4—44no—
615C4—44no—
616C4—82no—
617C4—84no—
618C4—84no—
619C4—84no—
620C2—164no—
621C2#3500311yescharacteristic
622C2#3519812yes—
623C2#3669411yescharacteristic
624C2#3746112yes—
625C2#3763611yescharacteristic
626C2#4578612yes—
627C2#4595812yes—
628C2#4599611yescharacteristic
629C2#4609911yescharacteristic
630C2#4628711yescharacteristic
631C2#4631211yescharacteristic
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×#2832C2semidirect
2. #17077C2semidirect
3. #17072C2semidirect
5. #16480C2semidirect
6. #16274C2semidirect
7. #15070C2semidirect
8. #15009C2semidirect
9. #14928C2semidirect
10. #14883C2semidirect
11. #14700C2semidirect
12. #14632C2semidirect
13. C4×#391C2semidirect
119. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28322
#170772
#170722
#170682
#164802
#162742
#150702
#150092
#149282
#148832
#147002
#146322
C4×#3912