Finite Group Databasebeta

← #363240

Subgroups — #363240

Order 512.

2930 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×C4×#103C211yesmaximal, characteristic, complemented
7#13906C211yesmaximal, characteristic
8#10970C214yesmaximal, complemented
9#10730C214yesmaximal, complemented
10C2×#554—24no—
11C2×#554—24no—
12C2×#554—24no—
13C2×#554—24no—
14C2×#554—24no—
15C2×#554C418yescomplemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125—21no—
18C22×C22⋊Q8—22no—
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4—21no—
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×C22⋊C8C2214yes—
28C22×C4×C8C2211yescharacteristic
29C2×#390—21no—
30C2×#384C2211yescharacteristic
31C2×#380C2214yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990—24no—
34#1987—24no—
35#1954—24no—
36#1803—24no—
37C4×#103—24no—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42#1439—22no—
43C4⋊#95—21no—
44C23×Q8—21no—
45C23×Q8C2311yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4—22no—
48C24×C4C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51#554—232no—
52#554—232no—
53#554—232no—
54#554—232no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D414yes—
60C2×C4⋊Q8D414yes—
61C2×#125—28no—
62C2×#125—28no—
63C2×#125—28no—
64C2×C22⋊Q8—216no—
65C2×C22⋊Q8—216no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—216no—
72C2×C22⋊Q8C2×C418yes—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8C2×C414yes—
77C2×C4×Q8D414yes—
78C2×C4×Q8D414yes—
79C2×C4×D4—28no—
80C2×C4×D4—28no—
81C2×C4×D4—28no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4C2×C414yes—
87C2×C4×D4D414yes—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2×C418yes—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4D412yes—
108C22×C42—21no—
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×#103—24no—
115C2×#103—24no—
116C2×#103—24no—
117C2×#103—24no—
118C2×#103—44no—
119C2×C22⋊C8—28no—
120C2×C22⋊C8—28no—
121C2×C22⋊C8—44no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125#390—44no—
126#384—22no—
127#384—22no—
128#380—28no—
129#380—44no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C2×#95—24no—
133C2×#95—21no—
134#340—24no—
135#340—28no—
136#340—28no—
137#340—24no—
138C25C2×D412yes—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—82no—
150C22×Q8—82no—
151C22×Q8C22×C412yes—
152C22×Q8C2×D412yes—
153C22×Q8C2×D412yes—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4C22⋊C414yes—
159C22×D4SD1614yes—
160C22×D4Q1614yes—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—28no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—84no—
178C23×C4—84no—
179C23×C4C22×C414yes—
180C23×C4C2×D412yes—
181C23×C4C2×D414yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—24no—
195C22×C8—24no—
196C22×C8#4114yes—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8C22⋊C4116yescomplemented
201#125—264no—
202C22⋊Q8—216no—
203C22⋊Q8—216no—
204C22⋊Q8—2128no—
205C22⋊Q8—216no—
206C22⋊Q8—216no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8C22⋊C4116yescomplemented
211C4×D4—264no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—28no—
215C4×D4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—216no—
219C2×C4⋊C4—216no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4C22⋊C414yes—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4#4114yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—84no—
270C2×C22⋊C4C22⋊C418yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C4⋊C8—44no—
284#103—216no—
285#103—48no—
286C22⋊C8—416no—
287C4×C8—44no—
288#95—48no—
289#95—42no—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24C2×C22⋊C414yes—
299C24C2×SD1614yes—
300C24C2×Q1614yes—
301C24C22×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—216no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—88no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—88no—
324C2×Q8C2×C22⋊C414yes—
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4#103116yescomplemented
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—232no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—816no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—816no—
388C22×C4C2×C22⋊C414yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4#12012yes—
395C22×C4C2×SD1612yes—
396C22×C4C2×Q1612yes—
397C22×C4#13712yes—
398C22×C4#13712yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+412yes—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—232no—
422C4⋊C4—232no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4C2×C22⋊C414yes—
442C4⋊C4C2×C22⋊C418yes—
443C22⋊C4—232no—
444C22⋊C4—216no—
445C22⋊C4—816no—
446C42—216no—
447C42—84no—
448C42C2×C22⋊C414yes—
449C42C2×C22⋊C414yes—
450C42#12014yes—
451C42#12014yes—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—24no—
456C23—28no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—24no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23C2×#10318yes—
475C23C22×C22⋊C411yescharacteristic
476C23C2×#12011yescharacteristic
477C23C2×#12011yescharacteristic
478C23#54612yes—
479C23#54712yes—
480C23C22×SD1611yescharacteristic
481C23C22×Q1611yescharacteristic
482C23C2×#13711yescharacteristic
483C23C2×#13711yescharacteristic
484C23#57612yes—
485C23#57712yes—
486Q8—24no—
487Q8—24no—
488Q8—44no—
489Q8—44no—
490Q8—42no—
491Q8—42no—
492Q8—816no—
493D4—232no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—48no—
520C2×C4—48no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—832no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4#37914yes—
538C2×C4#37914yes—
539C2×C4C2×#10314yes—
540C2×C4#41914yes—
541C2×C4#44814yes—
542C2×C4#44814yes—
543C2×C4#45114yes—
544C2×C4#45114yes—
545C2×C4C22×C22⋊C411yescharacteristic
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C4×#4112yes—
548C2×C4#52012yes—
549C2×C4C2×#12011yescharacteristic
550C2×C4C2×#12011yescharacteristic
551C2×C4#52512yes—
552C2×C4#53211yescharacteristic
553C2×C4#53211yescharacteristic
554C2×C4#53511yescharacteristic
555C2×C4#53611yescharacteristic
556C2×C4#53712yes—
557C2×C4#54612yes—
558C2×C4#54712yes—
559C2×C4#54812yes—
560C2×C4#54812yes—
561C2×C4#55012yes—
562C2×C4#55012yes—
563C8—44no—
564C8—82no—
565C22—28no—
566C22—216no—
567C22—48no—
568C22—48no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—48no—
576C22C2×#37911yescharacteristic
577C22#233912yes—
578C22#240712yes—
579C22#243111yescharacteristic
580C22#243211yescharacteristic
581C22C22×#10311yescharacteristic
582C22C2×#41911yescharacteristic
583C22#289712yes—
584C22C2×#44811yescharacteristic
585C22C2×#44811yescharacteristic
586C22C2×#45111yescharacteristic
587C22C2×#45111yescharacteristic
588C22#300811yescharacteristic
589C22#300911yescharacteristic
590C22#301212yes—
591C22#329512yes—
592C22#329912yes—
593C22#333112yes—
594C22#333312yes—
595C22#333512yes—
596C22#333812yes—
597C22#335612yes—
598C22#335712yes—
599C4—24no—
600C4—44no—
601C4—44no—
602C4—44no—
603C4—48no—
604C4—84no—
605C4—162no—
606C4#178718yes—
607C4#229711yescharacteristic
608C4#231111yescharacteristic
609C4#234112yes—
610C4#240712yes—
611C4#243111yescharacteristic
612C4#243211yescharacteristic
613C2—48no—
614C2#1393711yescharacteristic
615C2C2×#178711yescharacteristic
616C2#2060211yescharacteristic
617C2#2248712yes—
618C2#2250212yes—
619C2#2411812yes—
620C2#2412212yes—
621C2#2556911yescharacteristic
622C2#2557311yescharacteristic
623C2#2570211yescharacteristic
624C2#2570611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1987C2semidirect
5. C2×#1803C2semidirect
6. C2×C4×#103C2semidirect
8. #10970C2semidirect
9. #10730C2semidirect
15. C2×#554C4semidirect
200. C4⋊Q8C22⋊C4semidirect
210. C4×Q8C22⋊C4semidirect
331. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#19902
C2×#19872
C2×#19542
C2×#18032
C2×C4×#1032
#139062
#109702
#107302