Finite Group Databasebeta

← #723309

Subgroups — #723309

Order 512.

2246 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#17067C212yesmaximal
4#17061C212yesmaximal
5#15807C214yesmaximal
6#15698C214yesmaximal
7#15225C212yesmaximal
8#15137C212yesmaximal
9#15101C212yesmaximal
10#15101C212yesmaximal
11#14354C211yesmaximal, characteristic
12#13978C212yesmaximal
13#13976C212yesmaximal
14#13954C211yesmaximal, characteristic
15#13926C211yesmaximal, characteristic
16#13850C211yesmaximal, characteristic
17#13786C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#2581—28no—
22C2×#400C2212yes—
23C2×#398C2214yes—
24C2×#395C2212yes—
25Q8×C4⋊C4—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#384C2211yescharacteristic
39C2×#384C2212yes—
40C2×#384C2212yes—
41C2×#382C2211yescharacteristic
42C2×#382C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×#376C2211yescharacteristic
47#1446C2212yes—
48#1445C2212yes—
49#1445C2214yes—
50#1444C2214yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2211yescharacteristic
63#1441C2211yescharacteristic
64#1441C2211yescharacteristic
65#1441C2211yescharacteristic
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C2×C4×Q8—28no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C2×C4×Q8D418yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#400—28no—
126#398—48no—
127#395—28no—
128#391—24no—
129#391—24no—
130#391—44no—
131#391Q8116yescomplemented
132#389—24no—
133#389—24no—
134#389—28no—
135#389—216no—
136#389—44no—
137#389—44no—
138#387—28no—
139#387—28no—
140#384—28no—
141#384—24no—
142#384—44no—
143#382—24no—
144#382—44no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148#376—24no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—41no—
192C22×Q8—41no—
193C22×Q8C2×D414yes—
194C22×Q8C2×Q818yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4×Q8—232no—
213C4×Q8—432no—
214C4×Q8—416no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×Q818yes—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294C2×C42C2×Q818yes—
295C2×C42#4114yes—
296C2×C42#4114yes—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×Q8116yes—
330#95C2×Q8116yes—
331C24C2511yescharacteristic
332C2×Q8—28no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8C4⋊Q8116yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
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—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q818yes—
437C22×C4C22×Q818yes—
438C22×C4C22×Q814yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+414yes—
448C22×C42−1+412yes—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C42—216no—
471C42—216no—
472C42—416no—
473C42—44no—
474C42—44no—
475C42—48no—
476C42—84no—
477C42#123116yes—
478C23C23×D411yescharacteristic
479C23C23×Q812yes—
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C2×H(5,𝔽2)12yes—
483C23C2×H(5,𝔽2)11yescharacteristic
484C23C2×2−1+412yes—
485C23C2×2−1+411yescharacteristic
486C23#58512yes—
487C23#58511yescharacteristic
488C23#58511yescharacteristic
489Q8—232no—
490Q8—416no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
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—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4C2×#12314yes—
589C2×C4C2×#12314yes—
590C2×C4C2×#12314yes—
591C2×C4C2×C4⋊Q818yes—
592C2×C4C2×C4⋊Q814yes—
593C2×C4#53514yes—
594C2×C4#55418yes—
595C2×C4#55418yes—
596C22C22×#12311yescharacteristic
597C22C22×C4⋊Q811yescharacteristic
598C22C2×#53511yescharacteristic
599C22#346112yes—
600C22C2×#55412yes—
601C22C2×#55412yes—
602C22#348611yescharacteristic
603C22#348711yescharacteristic
604C22#348811yescharacteristic
605C22#348911yescharacteristic
606C22#349211yescharacteristic
607C22#349311yescharacteristic
608C22#349411yescharacteristic
609C22#349511yescharacteristic
610C22#350212yes—
611C22#350212yes—
612C22#350312yes—
613C22#350412yes—
614C22#350712yes—
615C22#354612yes—
616C22#355212yes—
617C22#356112yes—
618C22#356212yes—
619C4—24no—
620C4—24no—
621C4—28no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—48no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4—82no—
633C4—84no—
634C4—82no—
635C4#250218yes—
636C2C2×#250211yescharacteristic
637C2#3518211yescharacteristic
638C2#3551611yescharacteristic
639C2#3593711yescharacteristic
640C2#3595711yescharacteristic
641C2#3967312yes—
642C2#3967512yes—
643C2#4281211yescharacteristic
644C2#4288911yescharacteristic
645C2#4492312yes—
646C2#4512112yes—
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
131. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×Q8×C4⋊C42
#170672
#170612
#158072
#156982
#152252
#151372
#151012
#143542
#139782
#139762
#139542
#139262
#138502
#137862