Finite Group Databasebeta

← #223157

Subgroups — #223157

Order 512.

979 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36181C211yesmaximal, characteristic
2#22748C211yesmaximal, characteristic
3D4×C4⋊C8C211yesmaximal, characteristic
4#9900C211yesmaximal, characteristic, complemented
5#9899C211yesmaximal, characteristic, complemented
6#9131C211yesmaximal, characteristic, complemented
7#9131C211yesmaximal, characteristic, complemented
8#9130C211yesmaximal, characteristic, complemented
9#9129C211yesmaximal, characteristic, complemented
10#8726C211yesmaximal, characteristic, complemented
11#8723C211yesmaximal, characteristic, complemented
12#8589C211yesmaximal, characteristic, complemented
13C4×#332C211yesmaximal, characteristic, complemented
14#8039C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic, complemented
16C2×#552—21no—
17C2×C8×D4—21no—
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2565—21no—
26#2506—21no—
27#2506C412yes—
28#2503—21no—
29#2499—21no—
30#2496—21no—
31#2496C412yes—
32#2486—21no—
33#2486C412yes—
34#2344—21no—
35#2344C412yes—
36C4×C4⋊Q8C2211yescharacteristic
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1939—21no—
41#1923—21no—
42#1916—21no—
43#1869—21no—
44#1850—21no—
45#1850—21no—
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C8×C22⋊C4—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
71C23×C8—21no—
72C23×C8—21no—
73C23×C8—21no—
74#552—44no—
75#552—44no—
76C2×C4⋊Q8C2311yescharacteristic
77C2×#126—22no—
78C2×#125—22no—
79C2×#125—21no—
80C2×#123—22no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×#117—21no—
88C2×#117—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2×C412yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C8×D4—44no—
95#431—22no—
96#431—41no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8—41no—
107C2×C4⋊C8—41no—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124#400—22no—
125#400—22no—
126#398—22no—
127#398—22no—
128#397—22no—
129#397—22no—
130#391—21no—
131#391—21no—
132#391—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389C2×C412yes—
139#389C2×C412yes—
140#389C2×C412yes—
141#389C2×C412yes—
142#389C2×C412yes—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#384—21no—
148#384—21no—
149#384C2×C412yes—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#382—21no—
154#382—21no—
155#381—22no—
156#381—22no—
157#380—21no—
158#380—21no—
159#380—21no—
160#380C2×C412yes—
161#380C2×C412yes—
162#380C2×C412yes—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C2×C412yes—
166C4×C4⋊C4C2×C412yes—
167C4×C4⋊C4C2×C411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C22⋊C4C2×C411yescharacteristic
171C4×C22⋊C4C2×C411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C4×C22⋊C4D411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175C4×C22⋊C4D411yescharacteristic
176C43C2311yescharacteristic
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#336—21no—
185#332—22no—
186#332—22no—
187#332—41no—
188#332—41no—
189C22×Q8—21no—
190C22×D4C22⋊C411yescharacteristic
191C23×C4—22no—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—81no—
213C22×C8—81no—
214C22×C8—81no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219#126—48no—
220#125—48no—
221#125—42no—
222#125—42no—
223#123—48no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226C22⋊Q8—42no—
227C22⋊Q8—42no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—42no—
233C4×D4D814yes—
234C4×D4SD1614yes—
235#117—42no—
236#117—42no—
237#117—48no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
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—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2411yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
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—42no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8—81no—
316C4⋊C8#4112yes—
317C4⋊C8#4112yes—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—81no—
323C4×C8—42no—
324C4×C8—42no—
325C4×C8—81no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95C22⋊C411yescharacteristic
337#95C22⋊C411yescharacteristic
338#95C22⋊C411yescharacteristic
339#95C22⋊C411yescharacteristic
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342C24C2×C22⋊C411yescharacteristic
343C24C2×C22⋊C411yescharacteristic
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4#10214yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
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—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×D812yes—
397C22×C4C2×D812yes—
398C22×C4C2×SD1612yes—
399C22×C4C2×SD1612yes—
400C22×C4#13512yes—
401C22×C4#13512yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
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—84no—
430C2×C8—84no—
431C2×C8—84no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4#10214yes—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—41no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4#10214yes—
470C22⋊C4#10214yes—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—41no—
490C42—41no—
491C42—44no—
492C42—44no—
493C42C2×D812yes—
494C42C2×SD1612yes—
495C42C2×#4111yescharacteristic
496C42C2×#4111yescharacteristic
497C42H(5,𝔽2)11yescharacteristic
498C422−1+411yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23#41312yes—
510C23C2×#10212yes—
511C23C2×#10212yes—
512C23#41612yes—
513C23C22×C22⋊C411yescharacteristic
514Q8—81no—
515Q8—81no—
516Q8—81no—
517Q8—81no—
518D4—28no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
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—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—88no—
579C2×C4C2×#10212yes—
580C2×C4C2×#10212yes—
581C2×C4C2×#10212yes—
582C2×C4C2×#10212yes—
583C2×C4#41612yes—
584C2×C4C4×#4111yescharacteristic
585C2×C4#52011yescharacteristic
586C2×C4#54611yescharacteristic
587C2×C4#54611yescharacteristic
588C2×C4#54711yescharacteristic
589C2×C4#54711yescharacteristic
590C2×C4C22×D811yescharacteristic
591C2×C4C22×SD1611yescharacteristic
592C2×C4C2×#13511yescharacteristic
593C2×C4C2×#13511yescharacteristic
594C2×C4#57711yescharacteristic
595C2×C4#57711yescharacteristic
596C2×C4#57811yescharacteristic
597C2×C4#57811yescharacteristic
598C8—42no—
599C8—42no—
600C8—81no—
601C8—81no—
602C8—81no—
603C8—81no—
604C8—81no—
605C8—81no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22#154014yes—
620C22#233911yescharacteristic
621C22C2×#41311yescharacteristic
622C22#288911yescharacteristic
623C22C22×#10211yescharacteristic
624C22C2×#41611yescharacteristic
625C22#289811yescharacteristic
626C22#289811yescharacteristic
627C4—22no—
628C4—22no—
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#329411yescharacteristic
644C4#329611yescharacteristic
645C4#329811yescharacteristic
646C4#330011yescharacteristic
647C4#333411yescharacteristic
648C4#333511yescharacteristic
649C4#333611yescharacteristic
650C4#333711yescharacteristic
651C2—24no—
652C2—42no—
653C2C2×#154011yescharacteristic
654C2#1772111yescharacteristic
655C2#1772511yescharacteristic
656C2#1778011yescharacteristic
657C2#2248011yescharacteristic
658C2#2248511yescharacteristic
659C2#2248611yescharacteristic
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. #9900C2semidirect
5. #9899C2semidirect
6. #9131C2semidirect
7. #9131C2semidirect
8. #9130C2semidirect
9. #9129C2semidirect
10. #8726C2semidirect
11. #8723C2semidirect
12. #8589C2semidirect
13. C4×#332C2semidirect
14. #8039C2semidirect
15. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#361812
#227482
D4×C4⋊C82
#99002
#98992
#91312
#91302
#91292
#87262
#87232
#85892
C4×#3322
#80392
C2×#12962