Finite Group Databasebeta

← #641719

Subgroups — #641719

Order 512.

4868 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2628C212yesmaximal, complemented
3C2×#2290C211yesmaximal, characteristic
4#16603C212yesmaximal, complemented
5#16602C212yesmaximal, complemented
6#16560C212yesmaximal, complemented
7#16511C212yesmaximal, complemented
8#15128C212yesmaximal, complemented
9#15125C212yesmaximal, complemented
10#14874C212yesmaximal, complemented
11#14873C212yesmaximal
12#14843C212yesmaximal, complemented
13#14777C212yesmaximal
14#14682C211yesmaximal, characteristic, complemented
15#14282C212yesmaximal, complemented
16#14130C212yesmaximal, complemented
17#13872C212yesmaximal
18C24×Q8C2212yes—
19C22×C4⋊Q8—28no—
20C22×C4⋊Q8—216no—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2212yes—
24C22×C4⋊Q8C2212yes—
25C22×C4×D4C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2212yes—
29C23×C42C2211yescharacteristic
30#2628—216no—
31C2×#400C2214yes—
32C2×#399C2214yes—
33C2×#395C2214yes—
34C2×#395C2214yes—
35C2×#393C2214yes—
36C2×#393C2214yes—
37C2×#389C2214yes—
38#2290—28no—
39C2×#388C2211yescharacteristic
40C2×#388C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#387C2211yescharacteristic
43C2×#386C2211yescharacteristic
44C2×#386C2211yescharacteristic
45C2×#386C2214yes—
46C2×#385C2211yescharacteristic
47C2×#385C2211yescharacteristic
48C2×#385C2214yes—
49C2×#382C2212yes—
50C2×#380C2214yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×#376C2211yescharacteristic
56#1446C2214yes—
57#1446C2214yes—
58#1446C2214yes—
59#1446C2214yes—
60#1445C2214yes—
61#1445C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73#1434C2212yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2212yes—
79C23×Q8—216no—
80C23×Q8—28no—
81C23×Q8—48no—
82C23×Q8—48no—
83C23×Q8C2314yes—
84C23×Q8C2314yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—2128no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—216no—
94C2×C4⋊Q8—216no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8D4116yescomplemented
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4Q818yescomplemented
104C2×#117—28no—
105C2×#117—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—216no—
109C22×C4⋊C4—216no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2318yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2318yes—
131C22×C42—28no—
132C22×C42D418yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#400—216no—
141#399—216no—
142#395—216no—
143#395—48no—
144#393—216no—
145#393—48no—
146#389—216no—
147#388—24no—
148#388—24no—
149#387—24no—
150#387—24no—
151#386—24no—
152#386—24no—
153#386—48no—
154#385—24no—
155#385—24no—
156#385—48no—
157#382—28no—
158#380—28no—
159#380—216no—
160C4×C4⋊C4—24no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163#376—24no—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2318yes—
182C2×#95C2318yes—
183C2×#95C2318yes—
184C2×#95C2318yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C22×Q8—264no—
188C22×Q8—216no—
189C22×Q8—28no—
190C22×Q8—28no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—28no—
194C22×Q8—48no—
195C22×Q8—48no—
196C22×Q8—48no—
197C22×Q8—4128no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×Q8C2×D4116yes—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4C2×Q814yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—216no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—28no—
222C23×C4—48no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×Q814yes—
227C23×C4C2×Q814yescomplemented
228C23×C4#4114yes—
229C23×C4C2411yescharacteristic
230C23×C4C2418yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2418yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C4⋊Q8—264no—
245C4⋊Q8—264no—
246C4⋊Q8—264no—
247C4⋊Q8—264no—
248C4×D4—232no—
249#117—416no—
250C2×C4⋊C4—264no—
251C2×C4⋊C4—264no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4C2×D4116yes—
277C2×C4⋊C4C2×D4116yes—
278C2×C4⋊C4C2×Q814yescomplemented
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4C2×Q814yes—
302C2×C22⋊C4C2×Q814yescomplemented
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—232no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D418yes—
318C2×C42C2×Q814yescomplemented
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—216no—
330#95—28no—
331#95—28no—
332#95—216no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—48no—
337#95—48no—
338#95—48no—
339#95—48no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24C22×D412yes—
353C24C22×Q812yes—
354C24C22×Q812yes—
355C24C2×#4112yes—
356C24H(5,𝔽2)14yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—232no—
360C2×Q8—232no—
361C2×Q8—432no—
362C2×Q8—432no—
363C2×Q8—432no—
364C2×Q8—4256no—
365C2×Q8—416no—
366C2×Q8—416no—
367C2×Q8—416no—
368C2×Q8—432no—
369C2×Q8—416no—
370C2×D4—28no—
371C2×D4—216no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—48no—
375C22×C4—232no—
376C22×C4—232no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—232no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—432no—
442C22×C4—432no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4C22⋊Q8116yes—
449C22×C4C22×D4116yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C42−1+414yes—
468C22×C42−1+414yes—
469C4⋊C4—232no—
470C4⋊C4—232no—
471C4⋊C4—232no—
472C4⋊C4—232no—
473C4⋊C4—28no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—416no—
479C4⋊C4—816no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—416no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—816no—
490C42—28no—
491C42—28no—
492C42—28no—
493C42—28no—
494C42—44no—
495C42—44no—
496C42#120116yes—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23C2×C22⋊Q814yes—
527C23#53514yes—
528C23#53614yes—
529C23#54218yes—
530C23#54418yes—
531C23C23×D411yescharacteristic
532C23C23×D412yes—
533C23C23×Q811yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×H(5,𝔽2)12yes—
537C23C2×2−1+412yes—
538C23C2×2−1+412yes—
539C23C2×2−1+412yes—
540Q8—464no—
541Q8—464no—
542D4—416no—
543C2×C4—232no—
544C2×C4—232no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—232no—
562C2×C4—216no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—216no—
569C2×C4—216no—
570C2×C4—24no—
571C2×C4—464no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—416no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4C2×#12014yes—
610C2×C4C2×#12014yes—
611C2×C4C2×#12014yes—
612C2×C4C2×C22⋊Q814yes—
613C2×C4C2×C22⋊Q814yes—
614C2×C4#53514yes—
615C2×C4#53614yes—
616C2×C4#53614yes—
617C2×C4#54818yes—
618C2×C4D4×Q818yes—
619C22—24no—
620C22—216no—
621C22—216no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—48no—
629C22—42no—
630C22—42no—
631C22—81no—
632C22—84no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22#262818yescomplemented
637C22C22×#12011yescharacteristic
638C22C22×C22⋊Q811yescharacteristic
639C22C2×#53511yescharacteristic
640C22C2×#53511yescharacteristic
641C22C2×#53611yescharacteristic
642C22C2×#53611yescharacteristic
643C22C2×#53611yescharacteristic
644C22C2×#54212yes—
645C22C2×#54412yes—
646C22C2×#54812yes—
647C22C2×D4×Q812yes—
648C22#348414yes—
649C22#348614yes—
650C22#350312yes—
651C22#351112yes—
652C22#354414yes—
653C22#355214yes—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—24no—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—416no—
662C4—42no—
663C4—81no—
664C4—88no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4#240918yes—
669C2—28no—
670C2—82no—
671C2C2×#262811yescharacteristic
672C2C2×#240911yescharacteristic
673C2#3522211yescharacteristic
674C2#3524211yescharacteristic
675C2#3527911yescharacteristic
676C2#3761711yescharacteristic
677C2#3763211yescharacteristic
678C2#3768614yes—
679C2#3770414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊Q8C2semidirect
2. C2×#2628C2semidirect
4. #16603C2semidirect
5. #16602C2semidirect
6. #16560C2semidirect
7. #16511C2semidirect
8. #15128C2semidirect
9. #15125C2semidirect
10. #14874C2semidirect
12. #14843C2semidirect
14. #14682C2semidirect
15. #14282C2semidirect
16. #14130C2semidirect
100. C2×C4⋊Q8D4semidirect
103. C2×C4×D4Q8semidirect
227. C23×C4C2×Q8semidirect
278. C2×C4⋊C4C2×Q8semidirect
302. C2×C22⋊C4C2×Q8semidirect
318. C2×C42C2×Q8semidirect
636. C22#2628semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#26282
C2×#22902
#166032
#166022
#165602
#165112
#151282
#151252
#148742
#148732
#148432
#147772
#146822
#142822
#141302
#138722