Finite Group Databasebeta

← #6854528

Subgroups — #6854528

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31546C214yesmaximal, complemented
5#31543C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#30304C211yesmaximal, characteristic, complemented
8#29193C211yesmaximal, characteristic, complemented
9#29191C211yesmaximal, characteristic, complemented
10#29143C211yesmaximal, characteristic
11#29141C211yesmaximal, characteristic, complemented
12#29139C211yesmaximal, characteristic
13#28514C211yesmaximal, characteristic
14#24825C214yesmaximal, complemented
15#24346C214yesmaximal, complemented
16#22268C211yesmaximal, characteristic
17#22268C211yesmaximal, characteristic
18#21692C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#557C2212yes—
21C2×#554—21no—
22C2×#492C2211yescharacteristic
23C2×#478C2214yes—
24C2×#440C2214yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2568C2212yes—
32#2566—21no—
33#2566—21no—
34#2565C2212yes—
35#2565C2212yes—
36#2563—21no—
37#2546—21no—
38#2546—21no—
39#2538—21no—
40#2538C2212yes—
41#2506—21no—
42#2506—21no—
43#2506—21no—
44#2506—21no—
45#2452—21no—
46#2452—21no—
47#2452C2212yes—
48#2452C2212yes—
49#2365—21no—
50#2365—21no—
51D4×C4⋊C4—21no—
52D4×C4⋊C4C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2092C2214yes—
61#2085C2214yes—
62#2083C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2059C2214yes—
66#2057C2214yes—
67#2025C2214yes—
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#1865C2214yes—
88#1863C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1844C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1804C2214yes—
99#1803C2214yes—
100#1802C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108#557—48no—
109#554—48no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2312yes—
118C2×#127C2311yescharacteristic
119C2×#125—22no—
120C2×#125—22no—
121C2×#125—21no—
122C2×#125—21no—
123C2×#125C2312yes—
124C2×#125C2312yes—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—22no—
128C2×C4×Q8—22no—
129C2×C4×Q8D411yescharacteristic
130C2×C4×Q8D411yescharacteristic
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C22×C4⋊C4—22no—
145#492Q818yescomplemented
146#478—48no—
147#440—48no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166#400—22no—
167#400—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#393—22no—
177#393—22no—
178#391—22no—
179#391—22no—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
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—21no—
208#389—21no—
209#389—21no—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#385—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2312yes—
233#384C2312yes—
234#382—21no—
235#382—21no—
236#382C2312yes—
237#382C2312yes—
238#380—22no—
239#380—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—22no—
272C22×Q8—22no—
273C22×Q8C2×D411yescharacteristic
274C22×Q8C2412yes—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—22no—
281C2×SD16—44no—
282C2×SD16—44no—
283C2×SD16C2×Q8116yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—22no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—48no—
294C4⋊Q8—44no—
295C4⋊Q8—82no—
296C4⋊Q8C2×Q814yes—
297#127—22no—
298#127C2×Q814yes—
299#125—28no—
300#125—44no—
301#125—44no—
302#125—48no—
303#125—48no—
304#125—44no—
305C22⋊Q8—48no—
306C22⋊Q8—48no—
307C22⋊Q8—48no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—28no—
311C4×Q8—42no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—48no—
315C4×D4—28no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—82no—
323C4×D4—82no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—48no—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—44no—
403#107—44no—
404#106—44no—
405C4⋊C8—24no—
406C4⋊C8—42no—
407C4⋊C8—42no—
408#103—44no—
409#103—44no—
410#103—44no—
411#102—44no—
412#102—44no—
413#102—44no—
414#97—24no—
415#97—42no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—22no—
419C4×C8C2×Q814yes—
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#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440C24—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—41no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×Q8C22×Q814yes—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4C22×Q814yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
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×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C2511yescharacteristic
495SD16—416no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8C22×Q814yes—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—44no—
580C42—44no—
581C42—44no—
582C42AGL(1,ℤ/8)14yes—
583C42C22×Q812yes—
584C42H(5,𝔽2)14yes—
585C422−1+412yes—
586C23—24no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23C23×D411yescharacteristic
594Q8—28no—
595Q8—44no—
596Q8—44no—
597Q8—88no—
598Q8—164no—
599D4—28no—
600D4—48no—
601D4—48no—
602D4—44no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—24no—
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—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
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—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—88no—
657C2×C4—164no—
658C2×C4—164no—
659C2×C4#53512yes—
660C2×C4#53512yes—
661C2×C4D4×Q812yes—
662C2×C4D4×Q812yes—
663C2×C4C2×AGL(1,ℤ/8)12yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4#57512yes—
667C2×C4#57712yes—
668C2×C4#57812yes—
669C2×C4C23×Q811yescharacteristic
670C2×C4C2×H(5,𝔽2)12yes—
671C2×C4C2×2−1+411yescharacteristic
672C2×C4#58512yes—
673C8—24no—
674C8—42no—
675C8—42no—
676C8—42no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22C2×#53511yescharacteristic
685C22C2×D4×Q811yescharacteristic
686C22#350011yescharacteristic
687C22C22×AGL(1,ℤ/8)11yescharacteristic
688C22C2×#57511yescharacteristic
689C22C2×#57711yescharacteristic
690C22C2×#57811yescharacteristic
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—44no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4#338614yes—
711C4#338814yes—
712C4#350112yes—
713C4#354712yes—
714C2—84no—
715C2#3961811yescharacteristic
716C2#6080311yescharacteristic
717C2#6082411yescharacteristic
718C2C2×#338611yescharacteristic
719C2C2×#338811yescharacteristic
720C2#6253911yescharacteristic
721C2#6254511yescharacteristic
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. #36560C2semidirect
3. #33297C2semidirect
4. #31546C2semidirect
5. #31543C2semidirect
6. #30307C2semidirect
7. #30304C2semidirect
8. #29193C2semidirect
9. #29191C2semidirect
11. #29141C2semidirect
14. #24825C2semidirect
15. #24346C2semidirect
19. #21340C2semidirect
145. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#365602
#365592
#332972
#315462
#315432
#303072
#303042
#291932
#291912
#291432
#291412
#291392
#285142
#248252
#243462
#222682
#216922
#213402