Finite Group Databasebeta

← #636955

Subgroups — #636955

Order 512.

3208 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×#2854C211yesmaximal, characteristic
2C2×#2504C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4C2×#2283C211yesmaximal, characteristic, complemented
5#16679C214yesmaximal, complemented
6#16518C214yesmaximal, complemented
7#15234C212yesmaximal, complemented
8#15134C212yesmaximal, complemented
9#14912C214yesmaximal, complemented
10#14729C214yesmaximal, complemented
11#14276C212yesmaximal, complemented
12#14260C212yesmaximal, complemented
13#14137C211yesmaximal, characteristic
14#13924C211yesmaximal, characteristic, complemented
15#13880C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22#2854—28no—
23#2504—28no—
24C2×#397C2214yes—
25C2×#396C2218yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389—28no—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#386C2212yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#385C2214yes—
43C2×#384C2212yes—
44#2283—28no—
45C2×#383C2212yes—
46C2×#382C2212yes—
47C2×#382C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×#376C2212yes—
55C22×#95C2212yes—
56#1445C2214yes—
57#1444C2214yes—
58#1442C2218yes—
59#1441C2218yes—
60#1441C2218yes—
61#1441C2218yes—
62#1440C2218yes—
63#1439C2218yes—
64#1439C2214yes—
65#1439C2218yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C22⋊#95C2214yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×#123—28no—
79C2×#123—48no—
80C2×#121—24no—
81C2×#121—24no—
82C2×#121—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—48no—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42D418yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#397—216no—
123#396—416no—
124#390—28no—
125#390—24no—
126#390—24no—
127#390—28no—
128#389—24no—
129#389—24no—
130#389—28no—
131#389—28no—
132#389—28no—
133#389—28no—
134#388—28no—
135#388—28no—
136#386—28no—
137#386—216no—
138#386—48no—
139#385—48no—
140#384—28no—
141#383—28no—
142#382—28no—
143#382—28no—
144#381—28no—
145#380—24no—
146#380—28no—
147C4×C4⋊C4—28no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4D418yes—
150#376—28no—
151C2×#95—28no—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2312yes—
167C25C2412yes—
168C25C2411yescharacteristic
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4#4114yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216#123—432no—
217#121—416no—
218C2×C4⋊C4—232no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—432no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4#4118yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4C2×D418yes—
278C2×C42—232no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—216no—
305#95—24no—
306#95—24no—
307#95—48no—
308#95—48no—
309#95—48no—
310#95—48no—
311#95C2×D418yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24C22×D412yes—
326C24C22×D412yes—
327C24C2×#4112yes—
328C24H(5,𝔽2)14yes—
329C242−1+414yes—
330C24C2511yescharacteristic
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4—48no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
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—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—432no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4#121116yes—
417C22×C4C22×D418yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C2×#4118yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4114yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C42−1+414yes—
432C4⋊C4—216no—
433C4⋊C4—216no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—416no—
443C4⋊C4—44no—
444C4⋊C4—816no—
445C22⋊C4—216no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—816no—
454C22⋊C4—816no—
455C42—28no—
456C42—28no—
457C42—28no—
458C42—28no—
459C42—216no—
460C42—48no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—48no—
490C23C2×#12114yes—
491C23#53514yes—
492C23#53614yes—
493C23#53718yes—
494C23#54218yes—
495C23C23×D411yescharacteristic
496C23C23×D411yescharacteristic
497C23C22×#4111yescharacteristic
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)12yes—
501C23C2×2−1+412yes—
502C23C2×2−1+412yes—
503C23#58512yes—
504D4—816no—
505D4—88no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—232no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—216no—
529C2×C4—416no—
530C2×C4—432no—
531C2×C4—416no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—816no—
571C2×C4C2×#12114yes—
572C2×C4C2×#12114yes—
573C2×C4#52514yes—
574C2×C4#53514yes—
575C2×C4#53614yes—
576C2×C4#54718yes—
577C2×C4#55218yes—
578C22—216no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—216no—
585C22—24no—
586C22—82no—
587C22—88no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—88no—
594C22—82no—
595C22#262018yes—
596C22C22×#12111yescharacteristic
597C22C2×#52511yescharacteristic
598C22C2×#53511yescharacteristic
599C22C2×#53511yescharacteristic
600C22C2×#53611yescharacteristic
601C22C2×#53611yescharacteristic
602C22C2×#53712yes—
603C22#344911yescharacteristic
604C22C2×#54212yes—
605C22C2×#54712yes—
606C22C2×#55212yes—
607C22#348812yes—
608C22#349412yes—
609C22#351614yes—
610C22#351914yes—
611C22#355214yes—
612C22#355914yes—
613C4—24no—
614C4—24no—
615C4—42no—
616C4—42no—
617C4—48no—
618C4—48no—
619C4—42no—
620C4—82no—
621C4—84no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4#241318yes—
626C2—28no—
627C2—162no—
628C2C2×#262011yescharacteristic
629C2C2×#241311yescharacteristic
630C2#3523711yescharacteristic
631C2#3524611yescharacteristic
632C2#3525211yescharacteristic
633C2#3753911yescharacteristic
634C2#3761511yescharacteristic
635C2#3804214yes—
636C2#3806814yes—
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
3. C22×#389C2semidirect
4. C2×#2283C2semidirect
5. #16679C2semidirect
6. #16518C2semidirect
7. #15234C2semidirect
8. #15134C2semidirect
9. #14912C2semidirect
10. #14729C2semidirect
11. #14276C2semidirect
12. #14260C2semidirect
14. #13924C2semidirect
15. #13880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#25042
C22×#3892
C2×#22832
#166792
#165182
#152342
#151342
#149122
#147292
#142762
#142602
#141372
#139242
#138802