Finite Group Databasebeta

← #6854527

Subgroups — #6854527

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31549C214yesmaximal, complemented
5#31544C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#30304C211yesmaximal, characteristic, complemented
8#29192C211yesmaximal, characteristic
9#29190C211yesmaximal, characteristic
10#29144C211yesmaximal, characteristic, complemented
11#29141C211yesmaximal, characteristic, complemented
12#29139C211yesmaximal, characteristic
13#28515C211yesmaximal, characteristic, complemented
14#24824C214yesmaximal, complemented
15#24349C214yesmaximal, complemented
16#22268C211yesmaximal, characteristic
17#22268C211yesmaximal, characteristic
18#21692C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#554C2212yes—
21C2×#492C2211yescharacteristic
22C2×#479C2214yes—
23C2×#440C2214yes—
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2582C2212yes—
31#2567—21no—
32#2566C2212yes—
33#2566C2212yes—
34#2563C2212yes—
35#2552—21no—
36#2550—21no—
37#2550—21no—
38#2538—21no—
39#2538—21no—
40#2507—21no—
41#2507—21no—
42#2452—21no—
43#2452—21no—
44#2452—21no—
45#2452—21no—
46#2451C2212yes—
47#2451C2212yes—
48#2369—21no—
49#2355—21no—
50Q8×C4⋊C4C2212yes—
51D4×C4⋊C4—21no—
52D4×C4⋊C4—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2097C2214yes—
61#2085C2214yes—
62#2083C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2059C2214yes—
66#2057C2214yes—
67#2026C2214yes—
68#2024C2214yes—
69#2023C2214yes—
70#1993C2214yes—
71#1992C2214yes—
72#1985C2212yes—
73#1973C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1940C2212yes—
81#1917C2212yes—
82#1917C2212yes—
83#1880C2214yes—
84#1880C2214yes—
85#1877C2214yes—
86#1877C2214yes—
87#1864C2214yes—
88#1862C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1844C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1805C2214yes—
99#1803C2214yes—
100#1802C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108#554—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×#127C2311yescharacteristic
117C2×#125—21no—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×#125C2312yes—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8D411yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C22×C4⋊C4—22no—
140#492Q818yescomplemented
141#479—48no—
142#440—48no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#396—22no—
166#396—22no—
167#396—22no—
168#396—22no—
169#395—22no—
170#395—22no—
171#393—22no—
172#393—22no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#389—22no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—22no—
208#389—22no—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#387—22no—
214#387—22no—
215#385—22no—
216#385—22no—
217#385—22no—
218#385—22no—
219#384—22no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2312yes—
231#384C2312yes—
232#382—21no—
233#382—21no—
234#382C2312yes—
235#382C2312yes—
236#380—22no—
237#380—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—22no—
271C22×Q8—22no—
272C22×Q8C2×D411yescharacteristic
273C22×Q8C2412yes—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4C2412yes—
277C23×C4—22no—
278C23×C4—22no—
279C23×C4—22no—
280C2×SD16—44no—
281C2×SD16—44no—
282C2×SD16C2×Q8116yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—22no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293C4⋊Q8—82no—
294C4⋊Q8C2×Q814yes—
295#127—22no—
296#127C2×Q814yes—
297#125—28no—
298#125—44no—
299#125—82no—
300#125—82no—
301C22⋊Q8—48no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—28no—
307C4×Q8—42no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×D4—28no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—82no—
316C4×D4—82no—
317C4×D4—82no—
318C4×D4—82no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—48no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—44no—
398#107—44no—
399#106—44no—
400C4⋊C8—24no—
401C4⋊C8—42no—
402C4⋊C8—42no—
403#103—44no—
404#103—44no—
405#103—44no—
406#102—44no—
407#102—44no—
408#102—44no—
409#97—24no—
410#97—42no—
411C4×C8—22no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8C2×Q814yes—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435C24—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—41no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8C22×Q814yes—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C22×Q814yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2511yescharacteristic
490SD16—416no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8C22×Q814yes—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—44no—
575C42—82no—
576C42—82no—
577C42#13714yes—
578C42C22×Q812yes—
579C42H(5,𝔽2)14yes—
580C422−1+412yes—
581C23—24no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23C23×D411yescharacteristic
589Q8—28no—
590Q8—44no—
591Q8—44no—
592Q8—164no—
593Q8—164no—
594D4—28no—
595D4—48no—
596D4—48no—
597D4—44no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—24no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—88no—
652C2×C4—164no—
653C2×C4—164no—
654C2×C4#53512yes—
655C2×C4#53512yes—
656C2×C4D4×Q812yes—
657C2×C4D4×Q812yes—
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4C2×#13712yes—
661C2×C4#57512yes—
662C2×C4#57612yes—
663C2×C4#57712yes—
664C2×C4C23×Q811yescharacteristic
665C2×C4C2×H(5,𝔽2)12yes—
666C2×C4C2×2−1+411yescharacteristic
667C2×C4#58512yes—
668C8—24no—
669C8—42no—
670C8—42no—
671C8—42no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22C2×#53511yescharacteristic
680C22C2×D4×Q811yescharacteristic
681C22#350011yescharacteristic
682C22C22×#13711yescharacteristic
683C22C2×#57511yescharacteristic
684C22C2×#57611yescharacteristic
685C22C2×#57711yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—44no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4#338514yes—
706C4#338814yes—
707C4#350112yes—
708C4#354712yes—
709C2—84no—
710C2#3961811yescharacteristic
711C2#6080411yescharacteristic
712C2#6082211yescharacteristic
713C2C2×#338511yescharacteristic
714C2C2×#338811yescharacteristic
715C2#6254011yescharacteristic
716C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36563C2semidirect
3. #33297C2semidirect
4. #31549C2semidirect
5. #31544C2semidirect
6. #30307C2semidirect
7. #30304C2semidirect
10. #29144C2semidirect
11. #29141C2semidirect
13. #28515C2semidirect
14. #24824C2semidirect
15. #24349C2semidirect
19. #21340C2semidirect
140. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#365592
#332972
#315492
#315442
#303072
#303042
#291922
#291902
#291442
#291412
#291392
#285152
#248242
#243492
#222682
#216922
#213402