Finite Group Databasebeta

← #4928979

Subgroups — #4928979

Order 512.

5038 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#2579C212yesmaximal, complemented
3C22×#398C211yesmaximal, characteristic, complemented
4#14891C214yesmaximal, complemented
5#14878C214yesmaximal, complemented
6#14733C214yesmaximal, complemented
7#14728C214yesmaximal, complemented
8#14339C211yesmaximal, characteristic, complemented
9#14128C211yesmaximal, characteristic, complemented
10#14128C211yesmaximal, characteristic, complemented
11#14016C212yesmaximal, complemented
12#14015C212yesmaximal, complemented
13#13956C211yesmaximal, characteristic
14#13956C211yesmaximal, characteristic
15#13872C212yesmaximal, complemented
16C23×#41C2211yescharacteristic
17C2×#520—216no—
18C2×#520—28no—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2212yes—
21C22×C4×D4C2212yes—
22C22×C4×D4C2212yes—
23C22×#117C2212yes—
24C22×#117C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30#2579—216no—
31C2×#398—28no—
32C2×#398C2214yes—
33C2×#398C2214yes—
34C2×#398C2214yes—
35C2×#396C2214yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2214yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1446C2212yes—
53#1446C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1438C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2214yes—
79C22×#41—28no—
80C22×#41—28no—
81C22×#41—24no—
82C22×#41—24no—
83C22×#41Q818yes—
84C23×Q8C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C24×C4C2311yescharacteristic
97#520—2128no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4—28no—
106C2×C4×D4—28no—
107C2×C4×D4—28no—
108C2×C4×D4—28no—
109C2×C4×D4D4116yes—
110C2×#117—24no—
111C2×#117—28no—
112C2×#117—28no—
113C2×#117—28no—
114C2×#117—28no—
115C22×C4⋊C4—28no—
116C22×C4⋊C4—28no—
117C22×C4⋊C4—28no—
118C22×C4⋊C4—28no—
119C22×C4⋊C4—28no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—44no—
124C22×C4⋊C4—44no—
125C22×C4⋊C4D418yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2314yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2312yes—
150#398—216no—
151#398—216no—
152#398—216no—
153#396—216no—
154#387—216no—
155#386—28no—
156#386—28no—
157#386—28no—
158#386—28no—
159#385—216no—
160#385—28no—
161#385—28no—
162#385—48no—
163#381—24no—
164#381—24no—
165#380—28no—
166#380—48no—
167C2×#95—24no—
168C2×#95—24no—
169C2×#95—24no—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C2×#41—232no—
195C2×#41—232no—
196C2×#41—264no—
197C2×#41—264no—
198C2×#41—232no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8C2×Q814yes—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4C2×D418yes—
213C22×D4C2×Q814yes—
214C22×D4C2×Q818yes—
215C22×D4#4118yes—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—44no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4C2×D414yes—
255C23×C4C2×D414yes—
256C23×C4C2×D418yes—
257C23×C4C2×D414yes—
258C23×C4C2×D418yes—
259C23×C4C2×Q818yes—
260C23×C4C2×Q814yes—
261C23×C4#4114yes—
262C23×C4C2414yes—
263C23×C4C2414yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2412yes—
271C23×C4C2412yes—
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2412yes—
275C4×Q8—264no—
276C4×D4—264no—
277C4×D4—264no—
278C4×D4—264no—
279#117—232no—
280#117—264no—
281C2×C4⋊C4—264no—
282C2×C4⋊C4—232no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—416no—
311C2×C4⋊C4—416no—
312C2×C4⋊C4—416no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D418yes—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2×D418yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4C2×D418yes—
346C2×C22⋊C4C2×D418yes—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42C2×D418yes—
356#95—24no—
357#95—24no—
358#95—28no—
359#95—24no—
360#95—24no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—44no—
374#95—44no—
375#95—44no—
376#95—44no—
377#95—44no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24C22×D414yes—
391C24C22×D414yes—
392C24C22×D412yes—
393C24C22×Q814yes—
394C24C22×Q812yes—
395C24C2×#4114yes—
396C24C2×#4112yes—
397C24C2×#4114yes—
398C24C2511yescharacteristic
399#41—2256no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—216no—
403C2×Q8—216no—
404C2×Q8—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—216no—
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—28no—
416C2×D4C22⋊Q8132yes—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—28no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—216no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—216no—
461C22×C4—28no—
462C22×C4—216no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—416no—
490C22×C4—44no—
491C22×C4—416no—
492C22×C4—44no—
493C22×C4—416no—
494C22×C4—416no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—84no—
506C22×C4—84no—
507C22×C4—88no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—88no—
511C22×C4—88no—
512C22×C4—88no—
513C22×C4—88no—
514C22×C4—88no—
515C22×C4#120116yes—
516C22×C4C22⋊Q8116yes—
517C22×C4C22×D414yes—
518C22×C4C22×D414yes—
519C22×C4C22×D414yes—
520C22×C4C22×D414yes—
521C22×C4C22×D414yes—
522C22×C4C22×D414yes—
523C22×C4C22×D412yes—
524C22×C4C22×D414yes—
525C22×C4C22×D412yes—
526C22×C4C22×Q812yes—
527C22×C4C22×Q814yes—
528C22×C4C22×Q812yes—
529C22×C4C2×#4112yes—
530C22×C4C2×#4114yes—
531C22×C4C2×#4112yes—
532C22×C4H(5,𝔽2)14yes—
533C22×C42−1+414yes—
534C4⋊C4—28no—
535C4⋊C4—216no—
536C4⋊C4—28no—
537C4⋊C4—216no—
538C4⋊C4—216no—
539C4⋊C4—216no—
540C4⋊C4—416no—
541C4⋊C4—416no—
542C4⋊C4—416no—
543C4⋊C4—416no—
544C4⋊C4—88no—
545C4⋊C4—88no—
546C22⋊C4—216no—
547C22⋊C4—216no—
548C22⋊C4—216no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—88no—
554C22⋊C4—88no—
555C42—216no—
556C42—216no—
557C42—216no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—28no—
563C23—28no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—28no—
569C23—28no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—28no—
577C23—28no—
578C23—28no—
579C23—28no—
580C23—28no—
581C23—28no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23C2×#12014yes—
593C23C2×C22⋊Q818yes—
594C23C2×C22⋊Q818yes—
595C23C2×C22⋊Q814yes—
596C23#52518yes—
597C23#52514yes—
598C23C23×D411yescharacteristic
599C23C23×D412yes—
600C23C23×Q811yescharacteristic
601C23C22×#4111yescharacteristic
602C23C22×#4112yes—
603C23C2×H(5,𝔽2)12yes—
604C23C2×2−1+412yes—
605C23#58512yes—
606C23#58512yes—
607Q8—232no—
608D4—232no—
609D4—264no—
610C2×C4—264no—
611C2×C4—232no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—416no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—816no—
683C2×C4—88no—
684C2×C4—816no—
685C2×C4—88no—
686C2×C4—88no—
687C2×C4C2×#12018yes—
688C2×C4C2×C22⋊Q814yes—
689C2×C4C2×C22⋊Q818yes—
690C2×C4C2×C22⋊Q814yes—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—216no—
697C22—216no—
698C22—24no—
699C22—28no—
700C22—216no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—48no—
707C22—48no—
708C22—48no—
709C22—44no—
710C22#284818yes—
711C22C22×#12011yescharacteristic
712C22C22×C22⋊Q811yescharacteristic
713C22C22×C22⋊Q812yes—
714C22C2×#52511yescharacteristic
715C22C2×#52512yes—
716C22#348411yescharacteristic
717C22#348511yescharacteristic
718C22#348611yescharacteristic
719C22#348711yescharacteristic
720C22#348914yes—
721C22#349614yes—
722C22#350312yes—
723C22#350412yes—
724C22#351112yes—
725C22#351212yes—
726C22#351314yes—
727C22#351514yes—
728C4—24no—
729C4—28no—
730C4—24no—
731C4—28no—
732C4—44no—
733C4—44no—
734C4—44no—
735C4—44no—
736C4—44no—
737C4—82no—
738C4—82no—
739C4—84no—
740C4—84no—
741C2—28no—
742C2—48no—
743C2#3534711yescharacteristic
744C2#3535611yescharacteristic
745C2#3545311yescharacteristic
746C2#3557411yescharacteristic
747C2#3563911yescharacteristic
748C2#3565911yescharacteristic
749C2#3580614yes—
750C2#3580714yes—
751C2C2×#284811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2579C2semidirect
3. C22×#398C2semidirect
4. #14891C2semidirect
5. #14878C2semidirect
6. #14733C2semidirect
7. #14728C2semidirect
8. #14339C2semidirect
9. #14128C2semidirect
10. #14128C2semidirect
11. #14016C2semidirect
12. #14015C2semidirect
15. #13872C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#25792
C22×#3982
#148912
#148782
#147332
#147282
#143392
#141282
#140162
#140152
#139562
#138722