Finite Group Databasebeta

← #224004

Subgroups — #224004

Order 512.

951 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#22797C211yesmaximal, characteristic
3#22751C211yesmaximal, characteristic
4#9905C211yesmaximal, characteristic, complemented
5#9898C211yesmaximal, characteristic, complemented
6#9140C211yesmaximal, characteristic, complemented
7#9138C211yesmaximal, characteristic, complemented
8#9137C211yesmaximal, characteristic, complemented
9#9132C211yesmaximal, characteristic, complemented
10#8725C211yesmaximal, characteristic, complemented
11#8724C211yesmaximal, characteristic, complemented
12#8589C211yesmaximal, characteristic, complemented
13#8236C211yesmaximal, characteristic, complemented
14#8043C211yesmaximal, characteristic, complemented
15#8040C211yesmaximal, characteristic, complemented
16C2×#554—21no—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2567—21no—
25#2552—21no—
26#2552C412yes—
27#2502—21no—
28#2502—21no—
29#2440—21no—
30#2440—21no—
31#2440C412yes—
32#2440C412yes—
33D4×C4⋊C4—21no—
34D4×C4⋊C4C412yes—
35C4×C4⋊Q8C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1955—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1917—21no—
44#1870—21no—
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1771—21no—
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1753—21no—
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72#554—44no—
73#554—44no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8C2×C412yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×#125—22no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C4×Q8—21no—
84C2×C4×D4—22no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×C4×D4D411yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2×C412yes—
93C22×C4⋊C4C2×C412yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#432—22no—
97#432—22no—
98#432—41no—
99#432—41no—
100#431—22no—
101#431—41no—
102#423—22no—
103#423—41no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127#398—22no—
128#398—22no—
129#398—22no—
130#398—22no—
131#395—22no—
132#395—22no—
133#391—21no—
134#391—21no—
135#391—21no—
136#389—22no—
137#389—22no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389C2×C412yes—
142#389C2×C412yes—
143#389C2×C412yes—
144#385—22no—
145#385—22no—
146#385—22no—
147#385—22no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384—21no—
153#384—21no—
154#384C2×C412yes—
155#384C2×C412yes—
156#384C2×C412yes—
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#380—21no—
161#380—21no—
162#380C2×C412yes—
163#380C2×C412yes—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C2×C412yes—
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C22⋊C4C2×C411yescharacteristic
170C4×C22⋊C4C2×C411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C4×C22⋊C4D411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175C43C2311yescharacteristic
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#332—22no—
185#332—22no—
186#332—41no—
187#332—41no—
188C22×Q8—22no—
189C22×Q8—21no—
190C22×D4C22⋊C411yescharacteristic
191C23×C4—22no—
192C23×C4—22no—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—42no—
200C2×M16—42no—
201C2×M16—81no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—81no—
217C4⋊Q8—81no—
218C4⋊Q8—82no—
219#125—48no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
222C22⋊Q8—42no—
223C22⋊Q8—42no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226C4×Q8—42no—
227C4×Q8—42no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—24no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—48no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—81no—
258C2×C4⋊C4—81no—
259C2×C4⋊C4—81no—
260C2×C4⋊C4—81no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316C4⋊C8—81no—
317C4⋊C8#4112yes—
318C4⋊C8#4112yes—
319C22⋊C8—42no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—42no—
323C22⋊C8—81no—
324C22⋊C8—81no—
325C22⋊C8—81no—
326#97—42no—
327#97—42no—
328#97—42no—
329C4×C8—42no—
330C4×C8—81no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95C22⋊C411yescharacteristic
338#95C22⋊C411yescharacteristic
339#95C22⋊C411yescharacteristic
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343C24C2×C22⋊C411yescharacteristic
344C24C2×C22⋊C411yescharacteristic
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—44no—
352C2×Q8—82no—
353C2×Q8—82no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4#13712yes—
408C22×C4#13712yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412M16—84no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4#10114yes—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—22no—
498C42—41no—
499C42—44no—
500C42AGL(1,ℤ/8)12yes—
501C42AGL(1,ℤ/8)12yes—
502C42C2×#4111yescharacteristic
503C42C2×#4111yescharacteristic
504C42H(5,𝔽2)11yescharacteristic
505C422−1+411yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C2×#10112yes—
519C23#41912yes—
520C23C22×C22⋊C411yescharacteristic
521Q8—81no—
522Q8—81no—
523Q8—81no—
524Q8—81no—
525Q8—164no—
526D4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—164no—
587C2×C4—164no—
588C2×C4C2×#10112yes—
589C2×C4C2×#10112yes—
590C2×C4#41812yes—
591C2×C4#41812yes—
592C2×C4#41912yes—
593C2×C4C4×#4111yescharacteristic
594C2×C4#52011yescharacteristic
595C2×C4#54611yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54711yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
600C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
601C2×C4C2×#13711yescharacteristic
602C2×C4C2×#13711yescharacteristic
603C2×C4#57611yescharacteristic
604C2×C4#57711yescharacteristic
605C2×C4#57711yescharacteristic
606C2×C4#57811yescharacteristic
607C8—42no—
608C8—42no—
609C8—81no—
610C8—81no—
611C8—81no—
612C8—81no—
613C8—81no—
614C8—81no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22#233911yescharacteristic
630C22C22×#10111yescharacteristic
631C22#288811yescharacteristic
632C22C2×#41811yescharacteristic
633C22C2×#41911yescharacteristic
634C22#289711yescharacteristic
635C22#289811yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—21no—
640C4—21no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4#331411yescharacteristic
653C4#331611yescharacteristic
654C4#333011yescharacteristic
655C4#333211yescharacteristic
656C4#334411yescharacteristic
657C4#334611yescharacteristic
658C4#335311yescharacteristic
659C4#335411yescharacteristic
660C2—42no—
661C2—42no—
662C2#1774211yescharacteristic
663C2#1775811yescharacteristic
664C2#1776011yescharacteristic
665C2#1777411yescharacteristic
666C2#2248111yescharacteristic
667C2#2249711yescharacteristic
668C2#2250011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9905C2semidirect
5. #9898C2semidirect
6. #9140C2semidirect
7. #9138C2semidirect
8. #9137C2semidirect
9. #9132C2semidirect
10. #8725C2semidirect
11. #8724C2semidirect
12. #8589C2semidirect
13. #8236C2semidirect
14. #8043C2semidirect
15. #8040C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#227972
#227512
#99052
#98982
#91402
#91382
#91372
#91322
#87252
#87242
#85892
#82362
#80432
#80402