Finite Group Databasebeta

← #803726

Subgroups — #803726

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C212yesmaximal
2C2×#2362C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#17060C212yesmaximal
5#15895C214yesmaximal, complemented
6#15840C214yesmaximal
7#15242C211yesmaximal, characteristic, complemented
8#15195C211yesmaximal, characteristic
9#15152C212yesmaximal, complemented
10#15149C211yesmaximal, characteristic, complemented
11#15081C212yesmaximal
12#14223C212yesmaximal, complemented
13#14006C212yesmaximal, complemented
14#13951C212yesmaximal
15#13935C211yesmaximal, characteristic, complemented
16#13931C212yesmaximal, complemented
17C4×#397C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2421—216no—
23C2×#400C2214yes—
24C2×#398C2212yes—
25C2×#397C2211yescharacteristic
26C2×#397C2214yes—
27#2362—28no—
28Q8×C4⋊C4—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#389C2214yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#384C2212yes—
42C2×#384C2211yescharacteristic
43C2×#382C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2211yescharacteristic
54#1445C2214yes—
55#1443C2212yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2211yescharacteristic
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1440C2212yes—
67#1439C2211yescharacteristic
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1436C2214yes—
73C4⋊#95C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8D418yes—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8D418yes—
96C2×#117—24no—
97C2×#117—28no—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#400—216no—
130#398—28no—
131#397—24no—
132#397—216no—
133#391—24no—
134#391—28no—
135#389—216no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—24no—
140#388—28no—
141#388—216no—
142#388—28no—
143#387—28no—
144#387—216no—
145#384—24no—
146#384—28no—
147#382—28no—
148#380—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—24no—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C25C2411yescharacteristic
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8C2×D414yes—
191C22×Q8C2×D414yes—
192C22×Q8#4114yes—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4C2×D414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C22⋊Q8—232no—
215C4×Q8—232no—
216C4×Q8—432no—
217#117—416no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4114yes—
259C2×C4⋊C4#4118yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D418yes—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42C2×D414yes—
299C2×C42C2×D418yes—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—22no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—44no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24C22×D412yes—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—216no—
347C2×Q8—216no—
348C2×Q8#121116yes—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4118yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C42−1+412yes—
446C22×C42−1+412yes—
447C22×C42−1+414yes—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—416no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C4⋊C4#123116yes—
464C22⋊C4—216no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C42—216no—
472C42—28no—
473C42—48no—
474C42—416no—
475C42—48no—
476C42—48no—
477C42—44no—
478C42—48no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23C23×D411yescharacteristic
503C23C23×D411yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4112yes—
506C23C22×#4111yescharacteristic
507C23C2×2−1+412yes—
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23#58511yescharacteristic
511C23#58512yes—
512C23#58512yes—
513Q8—232no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—416no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4C2×#12118yes—
602C2×C4C2×#12114yes—
603C2×C4C2×#12318yes—
604C2×C4C2×#12314yes—
605C2×C4#53614yes—
606C2×C4#54718yes—
607C2×C4#54814yes—
608C2×C4#55218yes—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—81no—
615C22—82no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—82no—
620C22C22×#12111yescharacteristic
621C22C22×#12311yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#54712yes—
624C22C2×#54811yescharacteristic
625C22D4×#4111yescharacteristic
626C22C2×#55212yes—
627C22#348012yes—
628C22#348211yescharacteristic
629C22#348711yescharacteristic
630C22#348712yes—
631C22#348911yescharacteristic
632C22#349911yescharacteristic
633C22#350011yescharacteristic
634C22#351512yes—
635C22#351612yes—
636C22#351812yes—
637C22#351811yescharacteristic
638C22#352112yes—
639C22#352412yes—
640C22#355512yes—
641C22#356312yes—
642C22#356512yes—
643C4—28no—
644C4—24no—
645C4—28no—
646C4—44no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—81no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—84no—
658C4#248618yes—
659C2—82no—
660C2C2×#248611yescharacteristic
661C2#3601811yescharacteristic
662C2#3606711yescharacteristic
663C2#3644411yescharacteristic
664C2#3657811yescharacteristic
665C2#3917211yescharacteristic
666C2#4288811yescharacteristic
667C2#4523812yes—
668C2#4529312yes—
669C2#5911812yes—
670C2#5950512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×Q8×C4⋊C4C2semidirect
5. #15895C2semidirect
7. #15242C2semidirect
9. #15152C2semidirect
10. #15149C2semidirect
12. #14223C2semidirect
13. #14006C2semidirect
15. #13935C2semidirect
16. #13931C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24212
C2×#23622
C2×Q8×C4⋊C42
#170602
#158952
#158402
#152422
#151952
#151522
#151492
#150812
#142232
#140062
#139512
#139352
#139312
C4×#3972