Finite Group Databasebeta

← #825724

Subgroups — #825724

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3#17080C212yesmaximal
4#15682C214yesmaximal
5#15333C214yesmaximal
6#15230C211yesmaximal, characteristic
7#15202C211yesmaximal, characteristic
8#15191C212yesmaximal
9#15128C212yesmaximal
10#15128C211yesmaximal, characteristic
11#14241C212yesmaximal
12#14053C212yesmaximal
13#14046C212yesmaximal
14#14006C212yesmaximal
15#13954C212yesmaximal
16C4×#395C211yesmaximal, characteristic
17#13818C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#2546—28no—
23C2×#398C2214yes—
24C2×#395C2211yescharacteristic
25C2×#395C2212yes—
26#2369—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2214yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#382C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48#1446C2212yes—
49#1446C2212yes—
50#1446C2212yes—
51#1446C2211yescharacteristic
52#1446C2212yes—
53#1446C2214yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2211yescharacteristic
58#1444C2214yes—
59#1444C2214yes—
60#1444C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—48no—
94C2×C4⋊Q8Q818yescomplemented
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#398—216no—
128#395—24no—
129#395—28no—
130#391—28no—
131#391—28no—
132#391—28no—
133#391—48no—
134#389—24no—
135#389—28no—
136#389—28no—
137#389—24no—
138#389—28no—
139#389—48no—
140#389—44no—
141#389—44no—
142#389Q8116yescomplemented
143#382—44no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4D418yes—
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—42no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8C2×Q818yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C4⋊Q8—232no—
216C4⋊Q8—416no—
217C4⋊Q8—432no—
218C4×Q8—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C4⋊C4C2×Q818yes—
270C2×C4⋊C4C2×Q818yes—
271C2×C4⋊C4C2×Q8116yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4C2×Q818yes—
274C2×C4⋊C4#4114yes—
275C2×C4⋊C4#4114yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C2×D414yes—
296C2×C42C2×D418yes—
297C2×C42C2×Q814yes—
298C2×C42C2×Q818yes—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—22no—
315#95—24no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×D418yes—
330#95C2×Q8116yes—
331C24C2511yescharacteristic
332C2×Q8—28no—
333C2×Q8—216no—
334C2×Q8—416no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—44no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—44no—
341C2×Q8—48no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—88no—
345C2×Q8—88no—
346C2×Q8—88no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4C22×D412yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q818yes—
429C22×C4C22×Q814yes—
430C22×C4C22×Q812yes—
431C22×C4C22×Q814yes—
432C22×C4C22×Q814yes—
433C22×C4C22×Q818yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q814yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—416no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C4⋊C4C22⋊Q8116yes—
466C4⋊C4C4⋊Q8116yes—
467C42—28no—
468C42—28no—
469C42—216no—
470C42—48no—
471C42—48no—
472C42—44no—
473C42—48no—
474C42—84no—
475C42—88no—
476C23C23×D411yescharacteristic
477C23C23×Q811yescharacteristic
478C23C23×Q812yes—
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C2×H(5,𝔽2)12yes—
482C23C2×H(5,𝔽2)12yes—
483C23C2×2−1+411yescharacteristic
484C23C2×2−1+411yescharacteristic
485C23#58512yes—
486C23#58511yescharacteristic
487Q8—416no—
488Q8—816no—
489Q8—816no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4C2×C22⋊Q818yes—
581C2×C4C2×C22⋊Q814yes—
582C2×C4C2×C4⋊Q818yes—
583C2×C4C2×C4⋊Q814yes—
584C2×C4#53514yes—
585C2×C4#54814yes—
586C2×C4#55418yes—
587C2×C4#55714yes—
588C2×C4Q8218yes—
589C22C22×C22⋊Q811yescharacteristic
590C22C22×C4⋊Q811yescharacteristic
591C22C2×#53511yescharacteristic
592C22C2×#54811yescharacteristic
593C22C2×#55412yes—
594C22C2×#55711yescharacteristic
595C22C2×Q8212yes—
596C22Q8×#4112yes—
597C22#348011yescharacteristic
598C22#348411yescharacteristic
599C22#348911yescharacteristic
600C22#350011yescharacteristic
601C22#350112yes—
602C22#350112yes—
603C22#350211yescharacteristic
604C22#350212yes—
605C22#350312yes—
606C22#351412yes—
607C22#351912yes—
608C22#352311yescharacteristic
609C22#353612yes—
610C22#354012yes—
611C22#354712yes—
612C4—28no—
613C4—28no—
614C4—24no—
615C4—24no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—44no—
620C4—48no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—84no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4#256718yes—
629C2C2×#256711yescharacteristic
630C2#3570211yescharacteristic
631C2#3592811yescharacteristic
632C2#3647411yescharacteristic
633C2#3657711yescharacteristic
634C2#3912011yescharacteristic
635C2#4299511yescharacteristic
636C2#4536412yes—
637C2#4540912yes—
638C2#5966812yes—
639C2#5977112yes—
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
94. C2×C4⋊Q8Q8semidirect
142. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#23692
#170802
#156822
#153332
#152302
#152022
#151912
#151282
#142412
#140532
#140462
#140062
#139542
C4×#3952
#138182