Finite Group Databasebeta

← #827340

Subgroups — #827340

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C212yesmaximal
2C2×#2365C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#16565C212yesmaximal
5#15974C214yesmaximal
6#15965C214yesmaximal
7#15156C211yesmaximal, characteristic
8#15143C211yesmaximal, characteristic
9#15128C211yesmaximal, characteristic
10#14881C212yesmaximal
11#14876C212yesmaximal
12#14133C212yesmaximal
13#14053C212yesmaximal
14#14029C212yesmaximal
15#14005C212yesmaximal
16#13918C211yesmaximal, characteristic
17C4×#400C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22#2581—216no—
23C2×#400C2212yes—
24C2×#400C2211yescharacteristic
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#395C2212yes—
28#2365—28no—
29#2365—28no—
30C2×#391C2212yes—
31C2×#391C2214yes—
32C2×#391C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#387C2214yes—
42C2×#387C2214yes—
43C2×#387C2212yes—
44C2×#384C2212yes—
45C2×#382C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#1446C2212yes—
53#1446C2214yes—
54#1445C2211yescharacteristic
55#1445C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1443C2212yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2214yes—
63#1442C2212yes—
64#1442C2211yescharacteristic
65#1442C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2211yescharacteristic
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8Q818yescomplemented
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—44no—
98C2×C4×Q8—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#400—28no—
128#400D418yes—
129#395—28no—
130#395—28no—
131#395—44no—
132#391—28no—
133#391—216no—
134#391—28no—
135#389—28no—
136#389—24no—
137#389—28no—
138#389—28no—
139#389—24no—
140#389—216no—
141#387—28no—
142#387—216no—
143#387—216no—
144#387—28no—
145#387—216no—
146#384—28no—
147#382—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4D418yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—28no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8—44no—
200C22×Q8C2×Q818yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—232no—
219C4⋊Q8—232no—
220C4×Q8—432no—
221C4×Q8—432no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×Q818yes—
271C2×C4⋊C4C2×Q814yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4#4118yes—
274C2×C4⋊C4#4114yes—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C2×D418yes—
298C2×C42C2×D414yes—
299C2×C42C2×Q814yes—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95C2×D418yes—
332#95C2×D414yes—
333#95C2×D418yes—
334#95C2×D418yes—
335C24C2511yescharacteristic
336C2×Q8—216no—
337C2×Q8—216no—
338C2×Q8—216no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—416no—
343C2×Q8—48no—
344C2×Q8—416no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D414yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C2×#4118yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+414yes—
449C22×C42−1+412yes—
450C22×C42−1+414yes—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—88no—
469C4⋊C4C22⋊Q8116yes—
470C42—28no—
471C42—28no—
472C42—216no—
473C42—416no—
474C42—48no—
475C42—48no—
476C42—44no—
477C42—48no—
478C42—48no—
479C23C23×D411yescharacteristic
480C23C23×D411yescharacteristic
481C23C23×Q811yescharacteristic
482C23C22×#4111yescharacteristic
483C23C22×#4112yes—
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×2−1+412yes—
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+412yes—
488C23#58512yes—
489C23#58511yescharacteristic
490Q8—432no—
491Q8—432no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4C2×C22⋊Q818yes—
588C2×C4C2×C22⋊Q814yes—
589C2×C4#52514yes—
590C2×C4#53614yes—
591C2×C4D4214yes—
592C2×C4D4×Q814yes—
593C2×C4#55618yes—
594C2×C4#55718yes—
595C22C22×C22⋊Q811yescharacteristic
596C22C2×#52511yescharacteristic
597C22C2×#53611yescharacteristic
598C22C2×D4211yescharacteristic
599C22C2×D4×Q811yescharacteristic
600C22D4×#4112yes—
601C22D4×#4111yescharacteristic
602C22C2×#55612yes—
603C22C2×#55712yes—
604C22#348711yescharacteristic
605C22#348911yescharacteristic
606C22#349911yescharacteristic
607C22#349912yes—
608C22#350111yescharacteristic
609C22#350212yes—
610C22#351312yes—
611C22#351612yes—
612C22#351812yes—
613C22#352011yescharacteristic
614C22#352312yes—
615C22#355212yes—
616C22#355512yes—
617C22#356512yes—
618C4—24no—
619C4—28no—
620C4—24no—
621C4—42no—
622C4—42no—
623C4—48no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—81no—
634C4#258118yescomplemented
635C2C2×#258111yescharacteristic
636C2#3584511yescharacteristic
637C2#3600711yescharacteristic
638C2D4×C22⋊Q811yescharacteristic
639C2#3610211yescharacteristic
640C2#3936011yescharacteristic
641C2#4289111yescharacteristic
642C2#4528112yes—
643C2#4534112yes—
644C2#5966712yes—
645C2#5971112yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
92. C2×C4⋊Q8Q8semidirect
634. C4#2581semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×#23652
#165652
#159742
#159652
#151562
#151432
#151282
#148812
#148762
#141332
#140532
#140292
#140052
#139182
C4×#4002