Finite Group Databasebeta

← #7687956

Subgroups — #7687956

Order 512.

5157 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59918C218yesmaximal, complemented
2#59871C214yesmaximal, complemented
3#59832C211yesmaximal, characteristic, complemented
4#59322C212yesmaximal, complemented
5#45364C214yesmaximal, complemented
6#43126C218yesmaximal, complemented
7#39626C214yesmaximal, complemented
8#39624C218yesmaximal, complemented
9#36563C212yesmaximal, complemented
10Q8×C22⋊Q8C214yesmaximal, complemented
11#36261C214yesmaximal
12#36180C211yesmaximal, characteristic
13#35950C214yesmaximal
14#34950C211yesmaximal, characteristic, complemented
15C2×#2582C212yesmaximal, complemented
16#34334C214yesmaximal
17C4×D4×Q8C212yesmaximal
18C2×Q82C2212yes—
19C2×#554C2212yes—
20C2×#554C2214yes—
21C2×D4×Q8C2212yes—
22C22×C4×Q8C2214yes—
23#2868C2211yescharacteristic
24#2859C2214yes—
25#2857C2212yes—
26#2857C2218yes—
27#2811C2218yes—
28#2806C22116yes—
29#2806C22116yes—
30#2806C22116yes—
31#2792C22116yes—
32#2792C2218yes—
33#2734C2214yes—
34#2660C2214yes—
35#2659C2218yes—
36#2586C2218yes—
37#2586C2214yes—
38#2584C2218yes—
39#2584C2214yes—
40#2583C22116yes—
41#2583C2218yes—
42#2582C2218yes—
43#2582C2218yes—
44#2582C2211yescharacteristic
45#2582C2212yes—
46#2582C2214yes—
47#2582C2218yes—
48#2582C2218yes—
49#2567C2218yes—
50#2567C2212yes—
51#2567C2212yes—
52#2567C2218yes—
53#2567C2214yes—
54#2552C2214yes—
55#2552C2218yes—
56#2552C2214yes—
57#2551C2211yescharacteristic
58#2551C2214yes—
59#2550C2218yes—
60#2550C2218yes—
61#2549C22116yes—
62#2549C22116yes—
63#2546C2214yes—
64#2507C2214yes—
65#2507C22116yes—
66#2507C2218yes—
67#2507C2218yes—
68#2502C2218yes—
69#2502C2214yes—
70#2451C2218yes—
71#2451C2218yes—
72#2451C22116yes—
73#2451C2214yes—
74#2451C2218yes—
75#2440C2218yes—
76#2440C2218yes—
77#2440C22116yes—
78#2440C2218yes—
79#2399C2214yes—
80C2×#398C2218yes—
81#2369C2214yes—
82#2369C2218yes—
83#2368C22116yes—
84#2365C2214yes—
85#2365C2214yes—
86#2365C2212yes—
87#2360C2214yes—
88#2360C2218yes—
89#2360C2218yes—
90#2355C2214yes—
91Q8×C4⋊C4C2218yes—
92Q8×C4⋊C4C2212yes—
93Q8×C4⋊C4C2214yes—
94#2349C2218yes—
95D4×C4⋊C4C2214yes—
96D4×C4⋊C4C2214yes—
97D4×C4⋊C4C2212yes—
98Q8×C22⋊C4C2218yes—
99Q8×C22⋊C4C2214yes—
100#2331C2218yes—
101#2329C22116yes—
102C4×C4⋊Q8C2214yes—
103C4×C4⋊Q8C2211yescharacteristic
104C4×C4⋊Q8C2212yes—
105C4×C4⋊Q8C2214yes—
106C4×C4⋊Q8C2214yes—
107C4×C22⋊Q8C2218yes—
108C4×C22⋊Q8C2218yes—
109C4×C22⋊Q8C2214yes—
110#2295C2214yes—
111C2×#391C2218yes—
112C2×#389C2218yes—
113C2×#384C2212yes—
114#2275C2214yes—
115C42×Q8C2212yes—
116C42×D4C2214yes—
117C42×D4C2211yescharacteristic
118C23×Q8C2314yes—
119Q82—48no—
120#554—48no—
121#554—416no—
122D4×Q8Q8132yescomplemented
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2314yes—
125C2×C4⋊Q8C2318yes—
126C2×C4⋊Q8C2314yes—
127C2×C4⋊Q8C2312yes—
128C2×C4⋊Q8C2314yes—
129C2×C4⋊Q8C2318yes—
130C2×C4⋊Q8C2312yes—
131C2×C4⋊Q8C2314yes—
132C2×#125C2318yes—
133C2×#125C2314yes—
134C2×C22⋊Q8C23116yes—
135C2×C22⋊Q8C2318yes—
136C2×C22⋊Q8C2314yes—
137C2×C22⋊Q8C2318yes—
138C2×C22⋊Q8C2318yes—
139C2×C22⋊Q8C2314yes—
140C2×C4×Q8—416no—
141C2×C4×Q8C2314yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2314yes—
144C2×C4×Q8C2318yes—
145C2×C4×Q8C2318yes—
146C2×C4×Q8C2312yes—
147C2×C4×Q8C2312yes—
148C2×C4×Q8C2314yes—
149C2×C4×Q8C2314yes—
150C2×C4×Q8C2318yes—
151C2×C4×Q8C2318yes—
152C2×C4×Q8C2312yes—
153C2×C4×D4C2311yescharacteristic
154C2×C4×D4C2314yes—
155C2×C4×D4C2314yes—
156C2×C4×D4C2312yes—
157C2×C4×D4C2314yes—
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2318yes—
160C22×C4⋊C4C2318yes—
161C22×C4⋊C4C2314yes—
162C22×C4⋊C4C2318yes—
163C22×C42C2312yes—
164C22×C42C2318yes—
165#398C2314yes—
166#398C2318yes—
167#398C2318yes—
168#398C2318yes—
169#398C23116yes—
170#398C23116yes—
171#398C23116yes—
172#398C23116yes—
173#398C2318yes—
174#398C23116yes—
175#398C23116yes—
176#398C2314yes—
177#398C2318yes—
178#398C2318yes—
179#398C23116yes—
180#395C2318yes—
181#395C23116yes—
182#395C23116yes—
183#395C2318yes—
184#395C2318yes—
185#395C23116yes—
186#395C23116yes—
187#395C2314yes—
188#395C23116yes—
189#395C2314yes—
190#391C2318yes—
191#391C2312yes—
192#391C2318yes—
193#391C2314yes—
194#391C2318yes—
195#391C2314yes—
196#391C2318yes—
197#391C2318yes—
198#391C23116yes—
199#391C2314yes—
200#391C2314yes—
201#391C2314yes—
202#391C23116yes—
203#391C23116yes—
204#391C23116yes—
205#391C23116yes—
206#391C2318yes—
207#391C2318yes—
208#391C23116yes—
209#389C2314yes—
210#389C2318yes—
211#389C2314yes—
212#389C2318yes—
213#389C2314yes—
214#389C2314yes—
215#389C2318yes—
216#389C2314yes—
217#389C2314yes—
218#389C2318yes—
219#389C2314yes—
220#389C2314yes—
221#389C2314yes—
222#389C2318yes—
223#389C2318yes—
224#389C23116yes—
225#389C2318yes—
226#389C23116yes—
227#389C2318yes—
228#389C23116yes—
229#389C2318yes—
230#389C2318yes—
231#389C23116yes—
232#389C23116yes—
233#389C2318yes—
234#389C23116yes—
235#389C23116yes—
236#389C2318yes—
237#389C2318yes—
238#389C2318yes—
239#389C2314yes—
240#387C2318yes—
241#387C23116yes—
242#387C23116yes—
243#387C23116yes—
244#387C2318yes—
245#387C23116yes—
246#387C2318yes—
247#387C2318yes—
248#387C2318yes—
249#385C23116yes—
250#385C2318yes—
251#385C2318yes—
252#385C23116yes—
253#385C23116yes—
254#385C2318yes—
255#384C2314yes—
256#384C2311yescharacteristic
257#384C2312yes—
258#384C2314yes—
259#384C2314yes—
260#384C2318yes—
261#384C2318yes—
262#384C2314yes—
263#384C23116yes—
264#384C2312yes—
265#384C2312yes—
266#382C2318yes—
267#380C23116yes—
268#380C2314yes—
269#380C2318yes—
270#380C2318yes—
271C4×C4⋊C4C2312yes—
272C4×C4⋊C4C2314yes—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2314yes—
275C4×C4⋊C4C2318yes—
276C4×C4⋊C4C2314yes—
277C4×C4⋊C4C2318yes—
278C4×C4⋊C4C2318yes—
279C4×C4⋊C4C2318yes—
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2318yes—
283C4×C4⋊C4C2314yes—
284C4×C4⋊C4C2314yes—
285C4×C4⋊C4C2314yes—
286C4×C4⋊C4C23116yes—
287C4×C4⋊C4C2314yes—
288C4×C4⋊C4C2314yes—
289C4×C4⋊C4C2312yes—
290C4×C4⋊C4C2314yes—
291C4×C22⋊C4C2318yes—
292C4×C22⋊C4C2318yes—
293C4×C22⋊C4C2318yes—
294C4×C22⋊C4C23116yes—
295C4×C22⋊C4C2318yes—
296C4×C22⋊C4C2314yes—
297C4×C22⋊C4C2312yes—
298#376C2314yes—
299C2×#95C2318yes—
300C2×#95C23116yes—
301C43C2314yes—
302C43C2311yescharacteristic
303C22×Q8—48no—
304C22×Q8—48no—
305C22×Q8C2×Q8132yescomplemented
306C22×Q8C2414yes—
307C22×Q8C2412yes—
308C22×Q8C2412yes—
309C22×Q8C2414yes—
310C22×Q8C2418yes—
311C22×Q8C2414yes—
312C22×Q8C2418yes—
313C22×D4C2411yescharacteristic
314C23×C4C2412yes—
315C23×C4C2414yes—
316C23×C4C2418yes—
317C23×C4C2418yes—
318C23×C4C2418yes—
319C4⋊Q8—24no—
320C4⋊Q8—48no—
321C4⋊Q8—48no—
322C4⋊Q8—48no—
323C4⋊Q8—44no—
324C4⋊Q8—416no—
325C4⋊Q8—416no—
326C4⋊Q8C2×Q8116yescomplemented
327C4⋊Q8C2×Q8132yescomplemented
328#125—416no—
329#125—48no—
330C22⋊Q8—432no—
331C22⋊Q8—416no—
332C22⋊Q8—48no—
333C22⋊Q8—416no—
334C22⋊Q8C2×Q8132yescomplemented
335C22⋊Q8C2×Q8164yescomplemented
336C4×Q8—28no—
337C4×Q8—216no—
338C4×Q8—416no—
339C4×Q8—48no—
340C4×Q8—44no—
341C4×Q8—416no—
342C4×Q8—48no—
343C4×Q8—48no—
344C4×Q8—416no—
345C4×Q8—44no—
346C4×Q8—416no—
347C4×Q8C2×Q8116yescomplemented
348C4×D4—216no—
349C4×D4—216no—
350C4×D4—24no—
351C4×D4C2×Q8116yes—
352C4×D4C2×Q8132yes—
353C2×C4⋊C4—416no—
354C2×C4⋊C4—48no—
355C2×C4⋊C4—416no—
356C2×C4⋊C4—48no—
357C2×C4⋊C4—416no—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2414yes—
362C2×C4⋊C4C2418yes—
363C2×C4⋊C4C2418yes—
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2418yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C24116yes—
376C2×C4⋊C4C2414yes—
377C2×C4⋊C4C2418yes—
378C2×C4⋊C4C24116yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2414yes—
381C2×C4⋊C4C24116yes—
382C2×C4⋊C4C2414yes—
383C2×C4⋊C4C24116yes—
384C2×C4⋊C4C2418yes—
385C2×C4⋊C4C24116yes—
386C2×C4⋊C4C2418yes—
387C2×C4⋊C4C2418yes—
388C2×C4⋊C4C2418yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C2414yes—
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2414yes—
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2414yes—
395C2×C4⋊C4C2414yes—
396C2×C4⋊C4C2418yes—
397C2×C22⋊C4C2412yes—
398C2×C22⋊C4C2418yes—
399C2×C22⋊C4C2418yes—
400C2×C22⋊C4C2414yes—
401C2×C22⋊C4C2418yes—
402C2×C42—28no—
403C2×C42—416no—
404C2×C42C2414yes—
405C2×C42C2414yes—
406C2×C42C2411yescharacteristic
407C2×C42C2414yes—
408C2×C42C2412yes—
409C2×C42C2418yes—
410C2×C42C2414yes—
411C2×C42C2414yes—
412C2×C42C2412yes—
413C2×C42C2418yes—
414C2×C42C2418yes—
415C2×C42C2418yes—
416C2×C42C2414yes—
417C2×C42C24116yes—
418C2×C42C2411yescharacteristic
419C2×C42C2414yes—
420C2×C42C2412yes—
421C2×C42C2414yes—
422C2×C42C2411yescharacteristic
423#95C2414yes—
424#95C2418yes—
425#95C2418yes—
426#95C24116yes—
427#95C24116yes—
428#95C24116yes—
429#95C24116yes—
430#95C2414yes—
431#95C2418yes—
432#95C2418yes—
433#95C2418yes—
434#95C2412yes—
435#95C2414yes—
436#95C2418yes—
437#95C2414yes—
438#95C2414yes—
439#95C2414yes—
440#95C2418yes—
441#95C2418yes—
442#95C2418yes—
443#95C24116yes—
444#95C2418yes—
445#95C24116yes—
446#95C2418yes—
447C24C2512yes—
448C2×Q8—28no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—28no—
452C2×Q8—24no—
453C2×Q8—48no—
454C2×Q8—48no—
455C2×Q8—42no—
456C2×Q8—44no—
457C2×Q8—48no—
458C2×Q8—42no—
459C2×Q8—48no—
460C2×Q8—44no—
461C2×Q8—48no—
462C2×Q8—48no—
463C2×Q8—48no—
464C2×Q8—464no—
465C2×Q8C22×Q818yes—
466C2×Q8C22×Q8116yes—
467C2×Q8C22×Q8132yes—
468C2×D4C22×Q818yes—
469C2×D4H(5,𝔽2)14yes—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—24no—
473C22×C4—216no—
474C22×C4—216no—
475C22×C4—24no—
476C22×C4—48no—
477C22×C4—42no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—48no—
482C22×C4C22×Q8116yes—
483C22×C4C22×Q8132yes—
484C22×C4C2511yescharacteristic
485C22×C4C2514yes—
486C22×C4C2514yes—
487C22×C4C2514yes—
488C22×C4C2518yes—
489C22×C4C2518yes—
490C22×C4C2514yes—
491C22×C4C2512yes—
492C22×C4C2514yes—
493C22×C4C2514yes—
494C22×C4C2512yes—
495C22×C4C2514yes—
496C22×C4C2518yes—
497C22×C4C2511yescharacteristic
498C22×C4C2511yescharacteristic
499C22×C4C2512yes—
500C4⋊C4—28no—
501C4⋊C4—24no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—216no—
510C4⋊C4—24no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—416no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—416no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—416no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—44no—
528C4⋊C4—416no—
529C4⋊C4—416no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4C22×Q818yes—
534C4⋊C4C22×Q8116yes—
535C4⋊C4C22×Q8116yes—
536C4⋊C4C22×Q818yes—
537C4⋊C4C22×Q8132yes—
538C4⋊C4C22×Q8116yes—
539C22⋊C4—216no—
540C22⋊C4—216no—
541C22⋊C4—24no—
542C22⋊C4C22×Q8116yes—
543C22⋊C4C22×Q8132yes—
544C42—24no—
545C42—28no—
546C42—28no—
547C42—216no—
548C42—28no—
549C42—28no—
550C42—22no—
551C42—28no—
552C42—24no—
553C42—48no—
554C42—48no—
555C42—41no—
556C42—44no—
557C42—416no—
558C42—44no—
559C42—48no—
560C42C22×Q8116yes—
561C42C22×Q818yes—
562C42H(5,𝔽2)14yes—
563C23—22no—
564C23—24no—
565C23—22no—
566C23C23×Q818yes—
567C23C2×H(5,𝔽2)14yes—
568C23C2611yescharacteristic
569Q8—216no—
570Q8—232no—
571Q8—432no—
572Q8—84no—
573Q8—88no—
574Q8—164no—
575Q8—168no—
576D4Q82116yescomplemented
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—232no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—216no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—432no—
612C2×C4—48no—
613C2×C4—816no—
614C2×C4—816no—
615C2×C4C23×Q818yes—
616C2×C4C23×Q814yes—
617C2×C4C23×Q818yes—
618C2×C4C23×Q818yes—
619C2×C4C23×Q8116yes—
620C2×C4C23×Q814yes—
621C2×C4C23×Q814yes—
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4C2×H(5,𝔽2)14yes—
625C2×C4#58514yes—
626C22—28no—
627C22—28no—
628C22—24no—
629C22—24no—
630C22C2×Q8218yes—
631C22C24×Q812yes—
632C22C22×H(5,𝔽2)11yescharacteristic
633C22C22×H(5,𝔽2)11yescharacteristic
634C22C2×#58512yes—
635C222−1+611yescharacteristic
636C4—28no—
637C4—24no—
638C4—28no—
639C4—28no—
640C4—216no—
641C4—22no—
642C4—44no—
643C4—48no—
644C4—44no—
645C4—41no—
646C4—42no—
647C4—44no—
648C4—48no—
649C4—44no—
650C4C2×Q8214yes—
651C4#350118yes—
652C2—28no—
653C2C22×Q8211yescharacteristic
654C2C2×#350112yes—
655C2#6291411yescharacteristic
656C2#6300412yes—
657C2#6302111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC22×D4C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59918C2semidirect
2. #59871C2semidirect
3. #59832C2semidirect
4. #59322C2semidirect
5. #45364C2semidirect
6. #43126C2semidirect
7. #39626C2semidirect
8. #39624C2semidirect
9. #36563C2semidirect
10. Q8×C22⋊Q8C2semidirect
14. #34950C2semidirect
15. C2×#2582C2semidirect
122. D4×Q8Q8semidirect
305. C22×Q8C2×Q8semidirect
326. C4⋊Q8C2×Q8semidirect
327. C4⋊Q8C2×Q8semidirect
334. C22⋊Q8C2×Q8semidirect
335. C22⋊Q8C2×Q8semidirect
347. C4×Q8C2×Q8semidirect
576. D4Q82semidirect

Maximal subgroups

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

subgroupindex
#599182
#598712
#598322
#593222
#453642
#431262
#396262
#396242
#365632
Q8×C22⋊Q82
#362612
#361802
#359502
#349502
C2×#25822
#343342
C4×D4×Q82