Finite Group Databasebeta

← #373736

Subgroups — #373736

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117—21no—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18#2344—24no—
19#2344—24no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#387—21no—
24C2×#387—21no—
25C2×#385—21no—
26C2×#381—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1852—24no—
33#1823C418yescomplemented
34C2×#340—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1442—21no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—42no—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C2×#117—28no—
59C2×#117—44no—
60C2×#117—44no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2×C414yes—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4Q814yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—42no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2×C414yes—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—24no—
101#389—42no—
102#387—28no—
103#387—28no—
104#385—44no—
105#381—28no—
106#380—22no—
107#380—22no—
108#380C2×C414yes—
109#380D414yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—42no—
132#340C2×C418yescomplemented
133#340C2×C418yescomplemented
134#336—24no—
135#336—42no—
136C25C2×D411yescharacteristic
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—82no—
159C23×C4C22×C412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C22×C8—28no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—48no—
185C22×C8—48no—
186C22×C8—44no—
187C22×C8—44no—
188C22×C8C22×C414yes—
189C22×C8#4114yes—
190#117—432no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C4⋊C4116yescomplemented
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×Q812yes—
242C2×C4⋊C4C2×Q812yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4D814yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270#107—48no—
271C22⋊C8—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—44no—
281#95—44no—
282#95—82no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24C4×D412yes—
288C24#12012yes—
289C24C22⋊Q812yes—
290C24C2×D812yes—
291C24#13512yes—
292C24AGL(1,ℤ/8)12yes—
293C24#13712yes—
294C24C22×D411yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—216no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4C2×C4⋊C414yes—
380C22×C4C4×D414yes—
381C22×C4C4×D412yes—
382C22×C4#12012yes—
383C22×C4C22⋊Q814yes—
384C22×C4C22⋊Q812yes—
385C22×C4#12314yes—
386C22×C4C2×D812yes—
387C22×C4C2×D812yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×Q811yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+412yes—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—416no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—216no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—88no—
438C4⋊C4C2×C4⋊C414yes—
439C4⋊C4C2×C4⋊C414yes—
440C22⋊C4—28no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C42—48no—
445C42—48no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—42no—
451C42—42no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23#38514yes—
469C23C4×D814yes—
470C23#44014yes—
471C23#44714yes—
472C23#45214yes—
473C23#47414yes—
474C23#47814yes—
475C23C2×C4×D411yescharacteristic
476C23C2×#12011yescharacteristic
477C23C2×C22⋊Q811yescharacteristic
478C23#54612yes—
479C23#54712yes—
480C23C22×D811yescharacteristic
481C23C2×#13511yescharacteristic
482C23C2×AGL(1,ℤ/8)11yescharacteristic
483C23C2×#13711yescharacteristic
484C23#57712yes—
485C23#57812yes—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—432no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—816no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#38514yes—
557C2×C4C22×C4⋊C411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C4×#4112yes—
560C2×C4#52012yes—
561C2×C4C2×C22⋊Q811yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52511yescharacteristic
565C2×C4#53312yes—
566C2×C4#53511yescharacteristic
567C2×C4#53611yescharacteristic
568C2×C4#54412yes—
569C2×C4#54612yes—
570C2×C4#54712yes—
571C2×C4#55112yes—
572C2×C4#55112yes—
573C2×C4#55212yes—
574C2×C4#55212yes—
575C8—44no—
576C8—44no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—28no—
581C22—28no—
582C22—24no—
583C22—28no—
584C22—28no—
585C22—24no—
586C22—24no—
587C22#186218yes—
588C22C2×#38511yescharacteristic
589C22#235912yes—
590C22#240411yescharacteristic
591C22#240611yescharacteristic
592C22#243812yes—
593C22C2×C4×D811yescharacteristic
594C22C2×#44011yescharacteristic
595C22#294712yes—
596C22C2×#44711yescharacteristic
597C22C2×#45211yescharacteristic
598C22#301011yescharacteristic
599C22#301111yescharacteristic
600C22C2×#47411yescharacteristic
601C22C2×#47811yescharacteristic
602C22#308212yes—
603C22#329412yes—
604C22#330012yes—
605C22#330612yes—
606C22#331212yes—
607C22#333412yes—
608C22#333712yes—
609C22#334312yes—
610C22#334412yes—
611C4—24no—
612C4—42no—
613C4—44no—
614C4—44no—
615C4—42no—
616C4—82no—
617C4—81no—
618C4—84no—
619C4—82no—
620C4—84no—
621C4—81no—
622C4#229611yescharacteristic
623C4#231711yescharacteristic
624C4#235912yes—
625C4#243812yes—
626C4#248211yescharacteristic
627C4#248411yescharacteristic
628C2—28no—
629C2#1412811yescharacteristic
630C2C2×#186211yescharacteristic
631C2#2087711yescharacteristic
632C2#2291612yes—
633C2#2293412yes—
634C2#2396011yescharacteristic
635C2#2396711yescharacteristic
636C2#2408811yescharacteristic
637C2#2409511yescharacteristic
638C2#2598812yes—
639C2#2599812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11328C2semidirect
7. #11292C2semidirect
8. #10907C2semidirect
9. #10884C2semidirect
33. #1823C4semidirect
132. #340C2×C4semidirect
133. #340C2×C4semidirect
233. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C2×#18522
C2×#18232
C22×#3402
#141302
#113282
#112922
#109072
#108842
#107212
#107142