Finite Group Databasebeta

← #6433216

Subgroups — #6433216

Order 512.

2084 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3097C212yesmaximal
2#35316C211yesmaximal, characteristic
3#34250C211yesmaximal, characteristic
4#29766C214yesmaximal, complemented
5#29764C214yesmaximal, complemented
6#29759C212yesmaximal, complemented
7#29759C212yesmaximal, complemented
8#29757C212yesmaximal, complemented
9#29756C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23656C211yesmaximal, characteristic, complemented
12#23615C212yesmaximal, complemented
13#23613C212yesmaximal, complemented
14#23507C211yesmaximal, characteristic, complemented
15C2×#2043C211yesmaximal, characteristic, complemented
16#20913C212yesmaximal
17#20630C211yesmaximal, characteristic
18C2×#536C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23#3097—216no—
24C2×#484C2214yes—
25C2×#485C2214yes—
26C2×#485C2214yes—
27C2×#483C2214yes—
28C2×#428C2212yes—
29C2×#427C2212yes—
30C2×#424C2212yes—
31C2×#419C2212yes—
32C2×#419C2212yes—
33C22×#103C2212yes—
34C22×#103C2212yes—
35C2×#408C2212yes—
36#2857C2212yes—
37#2857C2212yes—
38#2621C2214yes—
39#2547—21no—
40#2547—21no—
41#2546—21no—
42#2546—21no—
43#2486—22no—
44#2486—22no—
45#2351—21no—
46Q8×C4⋊C4—21no—
47#2349—22no—
48#2349—21no—
49D4×C4⋊C4—22no—
50D4×C4⋊C4—21no—
51#2317C2211yescharacteristic
52#2317C2212yes—
53#2316C2211yescharacteristic
54#2316C2212yes—
55#2310C2212yes—
56#2296C2212yes—
57#2296C2211yescharacteristic
58C2×#389C2212yes—
59C2×#389C2211yescharacteristic
60#2285C2211yescharacteristic
61#2085C2212yes—
62#2085C2214yes—
63#2085C2212yes—
64#2084C2211yescharacteristic
65#2084C2212yes—
66#2084C2211yescharacteristic
67#2083C2211yescharacteristic
68#2083C2212yes—
69#2083C2211yescharacteristic
70#2068C2214yes—
71#2068C2214yes—
72#2068C2214yes—
73#2068C2214yes—
74#2066C2214yes—
75#2066C2214yes—
76#2066C2214yes—
77#2066C2214yes—
78#2043C2212yes—
79#2043C2212yes—
80#2043C2212yes—
81#2043C2212yes—
82#2043C2212yes—
83#2043C2214yes—
84#2043C2212yes—
85#1873C2214yes—
86#1873C2214yes—
87#1872C2214yes—
88#1872C2214yes—
89#1798C2214yes—
90#1797C2212yes—
91#1797C2212yes—
92#1736C2212yes—
93#1735C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8C2312yes—
98C22×M16C2312yes—
99C23×C8C2312yes—
100#536—28no—
101#536—28no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2314yes—
112#520—24no—
113#520—44no—
114#520D418yes—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4D412yes—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4D411yescharacteristic
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2312yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145#485—216no—
146#485—216no—
147#484—216no—
148#483—216no—
149#428—28no—
150#427—28no—
151C2×#107C2312yes—
152C2×#107C2314yes—
153C2×#107C2312yes—
154C2×#106C2312yes—
155C2×#106C2314yes—
156C2×#106C2312yes—
157#424—44no—
158C2×C4⋊C8C2314yes—
159C2×C4⋊C8C2314yes—
160#419—28no—
161#419—44no—
162C2×#103—28no—
163C2×#103—44no—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172#408—28no—
173C2×C22⋊C8C2314yes—
174C2×C22⋊C8C2314yes—
175#398—24no—
176#398—24no—
177#397—24no—
178#397—24no—
179#391—22no—
180#391—22no—
181#391C2314yes—
182#391C2314yes—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—21no—
192#389—22no—
193#389—21no—
194#389—22no—
195#389C2312yes—
196#389C2311yescharacteristic
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2311yescharacteristic
202#389C2311yescharacteristic
203#389C2312yes—
204#389C2311yescharacteristic
205#385—21no—
206#385—22no—
207#385—21no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#384—21no—
213#384—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#382—21no—
217#382—22no—
218#382C2312yes—
219#381—24no—
220#380—22no—
221#380—22no—
222#380—22no—
223#380—22no—
224#380—21no—
225#380—21no—
226#380—22no—
227#380—22no—
228#380—22no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C2×#95—21no—
236C2×#95—22no—
237#340C2312yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#340C2312yes—
243#336C2311yescharacteristic
244#336C2312yes—
245#336C2311yescharacteristic
246#336C2312yes—
247#336C2312yes—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41C2×D414yes—
251C2×#41#4114yes—
252C22×Q8—22no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8—24no—
256C22×Q8—24no—
257C22×Q8—24no—
258C22×Q8—24no—
259C22×Q8C2×D411yescharacteristic
260C22×Q8C2414yes—
261C22×D4C2×D412yes—
262C22×D4C2×D411yescharacteristic
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—21no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2×D412yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C2×M16—24no—
275C2×M16—24no—
276C2×M16—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—48no—
288#124—48no—
289#123—416no—
290C22⋊Q8—28no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C4×Q8—28no—
295C4×Q8—42no—
296C4×Q8—42no—
297C4×Q8—44no—
298C4×D4—216no—
299C4×D4—28no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—48no—
305C4×D4—44no—
306#117—24no—
307#117—22no—
308#117—24no—
309#117—24no—
310#117—24no—
311#117—24no—
312#117—44no—
313#117—44no—
314#117C2×D414yes—
315#117C2×D418yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—82no—
347C2×C4⋊C4—84no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D418yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—24no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—44no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D412yes—
382C2×C42C2×D412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386#107—24no—
387#107—24no—
388#107—28no—
389#106—24no—
390#106—24no—
391#106—28no—
392C4⋊C8—44no—
393C4⋊C8—44no—
394#103—28no—
395#103—28no—
396#103—28no—
397#103—28no—
398#103—44no—
399#103—44no—
400#103—44no—
401#103—44no—
402C22⋊C8—28no—
403C22⋊C8—28no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—21no—
410#95—22no—
411#95—21no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—21no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420C24C22×D412yes—
421C24C22×D411yescharacteristic
422#41—216no—
423#41—216no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8#12114yes—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4#12118yes—
445C2×D4#12114yes—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—81no—
477C22×C4—82no—
478C22×C4—81no—
479C22×C4—82no—
480C22×C4—81no—
481C22×C4—82no—
482C22×C4—81no—
483C22×C4—82no—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C42−1+414yes—
498C22×C4C2511yescharacteristic
499M16—48no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—48no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—81no—
542C4⋊C4—82no—
543C4⋊C4—81no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4C22×D414yes—
549C4⋊C4C22×D412yes—
550C4⋊C4C22×D412yes—
551C22⋊C4—28no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—42no—
561C42—28no—
562C42—24no—
563C42—22no—
564C42—22no—
565C42—24no—
566C42—22no—
567C42—22no—
568C42—42no—
569C42—42no—
570C42—44no—
571C42—44no—
572C42—42no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23C2×#12114yes—
584C23C2×#12112yes—
585C23#53412yes—
586C23#53612yes—
587C23C23×D411yescharacteristic
588Q8—28no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—44no—
593Q8—88no—
594D4—216no—
595D4—28no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4—88no—
650C2×C4—162no—
651C2×C4—164no—
652C2×C4C2×#12114yes—
653C2×C4C2×#12112yes—
654C2×C4#53412yes—
655C2×C4#53612yes—
656C2×C4#53718yes—
657C2×C4#53918yes—
658C2×C4#57712yes—
659C2×C4#57812yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×2−1+412yes—
664C8—42no—
665C8—42no—
666C8—44no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22#261814yes—
681C22C22×#12111yescharacteristic
682C22C2×#53411yescharacteristic
683C22C2×#53611yescharacteristic
684C22C2×#57711yescharacteristic
685C22C2×#57811yescharacteristic
686C22#358611yescharacteristic
687C22#358611yescharacteristic
688C4—22no—
689C4—22no—
690C4—24no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—41no—
697C4—42no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—81no—
702C4—81no—
703C4—84no—
704C4#261814yes—
705C4C2×#53712yes—
706C4C2×#53912yes—
707C2—24no—
708C2—44no—
709C2C2×#261811yescharacteristic
710C2#6042911yescharacteristic
711C2#6043211yescharacteristic
712C2#6075811yescharacteristic
713C2#6075911yescharacteristic
714C2#6087811yescharacteristic
715C2#6088411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #29766C2semidirect
5. #29764C2semidirect
6. #29759C2semidirect
7. #29759C2semidirect
8. #29757C2semidirect
9. #29756C2semidirect
10. #23657C2semidirect
11. #23656C2semidirect
12. #23615C2semidirect
13. #23613C2semidirect
14. #23507C2semidirect
15. C2×#2043C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30972
#353162
#342502
#297662
#297642
#297592
#297572
#297562
#236572
#236562
#236152
#236132
#235072
C2×#20432
#209132
#206302