Finite Group Databasebeta

← #536512

Subgroups — #536512

Order 512.

2930 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3C2×#2331C211yesmaximal, characteristic, complemented
4#14575C214yesmaximal, complemented
5#14370C214yesmaximal, complemented
6#14220C214yesmaximal, complemented
7#14155C211yesmaximal, characteristic, complemented
8#14150C212yesmaximal, complemented
9#14047C214yesmaximal, complemented
10#14030C212yesmaximal, complemented
11#14018C212yesmaximal
12#13848C211yesmaximal, characteristic, complemented
13#13820C212yesmaximal
14#13804C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2563C4116yescomplemented
20C2×#393C2212yes—
21C2×#393C2212yes—
22#2349—28no—
23#2331—28no—
24C2×#391C2214yes—
25C2×#391C2214yes—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#387C2214yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#382C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2214yes—
45C2×#376C2211yescharacteristic
46C22×#95C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1444C2212yes—
50#1444C2214yes—
51#1443C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71#1434C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8C2×C418yescomplemented
85C2×C4×D4—28no—
86C2×C4×D4—28no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—44no—
90C2×C4×D4C2×C418yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117#393C2×C4116yescomplemented
118#393C2×C4116yescomplemented
119#391—216no—
120#391—48no—
121#390—24no—
122#390C2×C4116yes—
123#390Q818yescomplemented
124#389—28no—
125#389—28no—
126#387—216no—
127#385—216no—
128#385—48no—
129#382C2×C4116yescomplemented
130#382D418yes—
131#380—28no—
132#380C2×C4132yescomplemented
133C4×C4⋊C4—28no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—44no—
137C4×C4⋊C4—44no—
138C4×C4⋊C4C2×C418yescomplemented
139C4×C22⋊C4—48no—
140#376—24no—
141C2×#95—28no—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25C2412yes—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8C22×C418yes—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4C22×C414yes—
182C22×D4C2×Q814yes—
183C22×D4#4114yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4C22×C418yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C4⋊Q8—232no—
211C4×D4—232no—
212C4×D4—432no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4C2×Q814yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4C22×C418yes—
264C2×C22⋊C4C22×C4116yes—
265C2×C22⋊C4C22×C4116yes—
266C2×C22⋊C4C2×Q8116yes—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42C22×C418yes—
286C2×C42C22×C414yes—
287C2×C42C2×D414yes—
288C2×C42C2×Q814yes—
289C2×C42#4114yes—
290#95—22no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C22×C414yes—
314#95C22×C414yes—
315#95C22×C4116yes—
316#95C22×C4116yes—
317#95C2×D418yes—
318#95C2×D418yes—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24C23×C414yes—
328C24C22×Q814yes—
329C24C2×#4114yes—
330C24C2511yescharacteristic
331C2×Q8—216no—
332C2×Q8—416no—
333C2×Q8—88no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4C4×Q8116yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—216no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—88no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4C23×C414yes—
423C22×C4C23×C414yes—
424C22×C4C23×C414yes—
425C22×C4C23×C418yes—
426C22×C4C23×C412yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C22×D418yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C22⋊C4—216no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—88no—
461C42—216no—
462C42—28no—
463C42—44no—
464C42—48no—
465C42—84no—
466C42—88no—
467C42—88no—
468C42C4×D4116yes—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23C2×C4×Q818yes—
492C23C24×C411yescharacteristic
493C23C23×D411yescharacteristic
494C23C23×Q811yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)12yes—
499C23C2×2−1+411yescharacteristic
500C23#58512yes—
501C23#58512yes—
502C23#58512yes—
503Q8—816no—
504D4—232no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—816no—
585C2×C4—832no—
586C2×C4C2×C4×D414yes—
587C2×C4C2×C4×D414yes—
588C2×C4C2×C4×D414yes—
589C2×C4C2×C4×Q814yes—
590C2×C4#52014yes—
591C2×C4#54614yes—
592C2×C4D4×Q814yes—
593C2×C4#55214yes—
594C2×C4#55414yes—
595C22—216no—
596C22—28no—
597C22—44no—
598C22—44no—
599C22—48no—
600C22—44no—
601C22—44no—
602C22—48no—
603C22—44no—
604C22—44no—
605C22—48no—
606C22C22×C4×D411yescharacteristic
607C22C22×C4×Q811yescharacteristic
608C22C2×#52011yescharacteristic
609C22#342812yes—
610C22C4×H(5,𝔽2)12yes—
611C22C2×#54611yescharacteristic
612C22C2×D4×Q811yescharacteristic
613C22C2×#55211yescharacteristic
614C22C2×#55411yescharacteristic
615C22#348812yes—
616C22#350012yes—
617C22#350112yes—
618C22#350412yes—
619C22#350811yescharacteristic
620C22#351011yescharacteristic
621C22#351011yescharacteristic
622C22#351011yescharacteristic
623C22#351312yes—
624C22#352112yes—
625C22#354012yes—
626C22#354212yes—
627C22#354712yes—
628C22#356512yes—
629C4—24no—
630C4—24no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—48no—
637C4—48no—
638C4—44no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4#236018yes—
643C2—48no—
644C2C2×#236011yescharacteristic
645C2#3421111yescharacteristic
646C2#3426911yescharacteristic
647C2#3433312yes—
648C2#3446712yes—
649C2#4493912yes—
650C2#4494711yescharacteristic
651C2#4504912yes—
652C2#4505611yescharacteristic
653C2#4523111yescharacteristic
654C2#4542511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2331C2semidirect
4. #14575C2semidirect
5. #14370C2semidirect
6. #14220C2semidirect
7. #14155C2semidirect
8. #14150C2semidirect
9. #14047C2semidirect
10. #14030C2semidirect
12. #13848C2semidirect
14. #13804C2semidirect
19. #2563C4semidirect
84. C2×C4⋊Q8C2×C4semidirect
117. #393C2×C4semidirect
118. #393C2×C4semidirect
123. #390Q8semidirect
129. #382C2×C4semidirect
132. #380C2×C4semidirect
138. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23492
C2×#23312
#145752
#143702
#142202
#141552
#141502
#140472
#140302
#140182
#138482
#138202
#138042
#137852