Finite Group Databasebeta

← #6373337

Subgroups — #6373337

Order 512.

2352 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×#3110C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic, complemented
3C4×#518C211yesmaximal, characteristic
4#21613C212yesmaximal, complemented
5C4×#474C212yesmaximal, complemented
6#21585C214yesmaximal, complemented
7#21504C212yesmaximal, complemented
8C4×#461C212yesmaximal, complemented
9#21459C214yesmaximal
10#21389C211yesmaximal, characteristic, complemented
11#21364C211yesmaximal, characteristic, complemented
12#21125C214yesmaximal, complemented
13#20732C211yesmaximal, characteristic
14#20504C214yesmaximal, complemented
15#20470C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#518C2212yes—
18C22×C4×D4C2211yescharacteristic
19#3110C4116yescomplemented
20C2×#484C2212yes—
21C2×#474C2212yes—
22C2×#461C2212yes—
23C2×#450C2212yes—
24C2×#440C2214yes—
25C2×#440C2214yes—
26C2×C4×SD16C2214yes—
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2370—22no—
30#2362—22no—
31#2356—21no—
32#2351—21no—
33#2347—21no—
34#2346—21no—
35#2323C2214yes—
36#2320C2212yescomplemented
37C4×C4⋊Q8C2212yescomplemented
38C4×C4⋊Q8C2212yescomplemented
39#2285C2211yescharacteristic
40#2276C2212yescomplemented
41#2275—22no—
42#2275C2212yes—
43#2275C2212yes—
44#2274C2212yes—
45#2273—21no—
46#2273—21no—
47C42×D4—21no—
48C42×D4—21no—
49C42×D4C2212yes—
50C42×D4C2212yes—
51C4×#117C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C2×C4×C22⋊C4—22no—
54#1988C2211yescharacteristic, complemented
55#1988C2211yescharacteristic, complemented
56#1981C2211yescharacteristic, complemented
57#1981C2211yescharacteristic, complemented
58#1948C2212yescomplemented
59#1947C2212yescomplemented
60#1928C2214yescomplemented
61#1927C2214yescomplemented
62#1921C2214yescomplemented
63#1918C2214yescomplemented
64#1877C2214yes—
65#1877C2214yes—
66#1873C2214yes—
67#1864C2214yes—
68#1862C2214yes—
69#1854C2214yes—
70#1823C2212yes—
71#1802C2212yes—
72#1802C2212yes—
73#1786C2212yes—
74#1776C2214yescomplemented
75#1776C2214yescomplemented
76C4×#107C2214yescomplemented
77#1767C2212yescomplemented
78C4×C4⋊C8C2212yescomplemented
79#1763C2214yescomplemented
80#1762C2214yes—
81C4×#103C2214yescomplemented
82C4×#103C2214yescomplemented
83C4×#102C2214yes—
84C4×#102C2214yes—
85#1750C2214yes—
86#1746C2212yescomplemented
87#1726C2212yescomplemented
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91#533—24no—
92#533C2×C418yescomplemented
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#125—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C2312yes—
98#518—28no—
99#518—28no—
100C2×C4×Q8—21no—
101C2×C4×Q8—22no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—24no—
109C2×C4×D4C2×C418yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C2×#117—21no—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42—22no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127#484C2×C4116yescomplemented
128#474C2×C4116yescomplemented
129#461C2×C4116yescomplemented
130#450C2×C4116yes—
131#440—216no—
132#440C2×C4132yescomplemented
133C4×SD16—216no—
134#432C2×C418yescomplemented
135#431—24no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×#97C2314yes—
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#391—22no—
155#391C2314yes—
156#389—24no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#388—24no—
162#388—24no—
163#387—24no—
164#387—24no—
165#384C2312yes—
166#383—22no—
167#382—21no—
168#382—22no—
169#380—21no—
170#380C2312yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D412yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2311yescharacteristic
187C4×C4⋊C4C2311yescharacteristic
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—21no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4D411yescharacteristic
208C4×C22⋊C4D411yescharacteristic
209C4×C22⋊C4D412yes—
210#376—21no—
211#376—22no—
212#376C2311yescharacteristic
213#376C2311yescharacteristic
214C2×#95—22no—
215C2×#95—22no—
216C43—21no—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2314yes—
225#336C2314yes—
226C25—21no—
227C22×Q8—21no—
228C22×Q8C2412yes—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4C22×C414yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—28no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—82no—
244C23×C4—82no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16C22×C4116yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8C22×C414yes—
257C4⋊Q8C22×C414yes—
258#125—28no—
259C22⋊Q8—28no—
260C22⋊Q8—24no—
261C22⋊Q8C22×C418yes—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—28no—
266C4×Q8—84no—
267C4×Q8C22×C418yes—
268C4×D4—216no—
269C4×D4—216no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4C22×C418yes—
281C4×D4C22×C418yes—
282C4×D4#4118yes—
283C4×D4#4118yes—
284#117—28no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—82no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4C22×C414yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—216no—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—84no—
336C2×C42—84no—
337C2×C42C22×C414yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4114yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349#107—28no—
350#107C22×C4116yes—
351#106—28no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8C22×C418yes—
355#103—28no—
356#103—28no—
357#103C22×C4116yes—
358#102—28no—
359#102—28no—
360#102C22×C4116yes—
361C22⋊C8—24no—
362C22⋊C8C22×C418yes—
363#97—28no—
364#97C22×C418yes—
365C4×C8—24no—
366C4×C8—28no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D411yescharacteristic
383#95C2×D411yescharacteristic
384#95C2×D411yescharacteristic
385#95C2×D411yescharacteristic
386C24—28no—
387C24—22no—
388C24—24no—
389C24—81no—
390C24—81no—
391C24—82no—
392C24C22×D411yescharacteristic
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8C23×C414yes—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—216no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4C23×C414yes—
409C2×D4C2×#4114yes—
410C2×D4C2×#4114yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—816no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4C4×D414yes—
445C22×C4C4×D414yes—
446C22×C4C23×C412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4112yes—
457C22×C4C2511yescharacteristic
458SD16—232no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8C23×C418yes—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—84no—
484C4⋊C4C23×C414yes—
485C4⋊C4C23×C414yes—
486C4⋊C4C2×#4114yes—
487C4⋊C4C2×#4114yes—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—82no—
499C42—28no—
500C42—28no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—42no—
506C42—42no—
507C42—48no—
508C42—42no—
509C42—42no—
510C42—82no—
511C42—82no—
512C42—82no—
513C42—82no—
514C42—82no—
515C42—82no—
516C42—84no—
517C42C4×D414yes—
518C42C4×D414yes—
519C42C23×C412yes—
520C42C23×C412yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C42C2×#4114yes—
524C42C2×#4114yes—
525C23—216no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—21no—
532C23—21no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—84no—
538C23—84no—
539C23—88no—
540C23—88no—
541C23C2×C4×D412yes—
542C23C23×D411yescharacteristic
543Q8—28no—
544Q8—42no—
545Q8—42no—
546Q8—48no—
547D4—216no—
548D4—24no—
549D4—24no—
550D4—24no—
551D4C4×#4118yes—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—416no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—88no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4C2×C4×D412yes—
597C2×C4C2×C4×D412yes—
598C2×C4C2×C4×D412yes—
599C2×C4C2×C4×D412yes—
600C2×C4C2×C4×D412yes—
601C2×C4#52514yes—
602C2×C4#52514yes—
603C2×C4#57612yes—
604C2×C4#57712yes—
605C2×C4C24×C411yescharacteristic
606C2×C4C22×#4112yes—
607C2×C4C22×#4112yes—
608C2×C4C22×#4111yescharacteristic
609C8—28no—
610C8—42no—
611C8—42no—
612C22—22no—
613C22—22no—
614C22—28no—
615C22—22no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—816no—
625C22C22×C4×D411yescharacteristic
626C22C2×#52512yes—
627C22C2×#57611yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358612yes—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—24no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—84no—
644C4#294614yes—
645C4C2×C4×#4112yes—
646C4#344411yescharacteristic
647C4#344411yescharacteristic
648C2—42no—
649C2—48no—
650C2C4×#52511yescharacteristic
651C2C2×#294611yescharacteristic
652C2#6016911yescharacteristic
653C2#6059112yes—
654C2#6059612yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34225C2semidirect
4. #21613C2semidirect
5. C4×#474C2semidirect
6. #21585C2semidirect
7. #21504C2semidirect
8. C4×#461C2semidirect
10. #21389C2semidirect
11. #21364C2semidirect
12. #21125C2semidirect
14. #20504C2semidirect
15. #20470C2semidirect
19. #3110C4semidirect
36. #2320C22semidirect
37. C4×C4⋊Q8C22semidirect
38. C4×C4⋊Q8C22semidirect
40. #2276C22semidirect
54. #1988C22semidirect
55. #1988C22semidirect
56. #1981C22semidirect
57. #1981C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect
62. #1921C22semidirect
63. #1918C22semidirect
74. #1776C22semidirect
75. #1776C22semidirect
76. C4×#107C22semidirect
77. #1767C22semidirect
78. C4×C4⋊C8C22semidirect
79. #1763C22semidirect
81. C4×#103C22semidirect
82. C4×#103C22semidirect
86. #1746C22semidirect
87. #1726C22semidirect
92. #533C2×C4semidirect
127. #484C2×C4semidirect
128. #474C2×C4semidirect
129. #461C2×C4semidirect
132. #440C2×C4semidirect
134. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
#342252
C4×#5182
#216132
C4×#4742
#215852
#215042
C4×#4612
#214592
#213892
#213642
#211252
#207322
#205042
#204702