Finite Group Databasebeta

← #6805674

Subgroups — #6805674

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic, complemented
3#33294C211yesmaximal, characteristic, complemented
4#32456C214yesmaximal, complemented
5#32453C214yesmaximal, complemented
6#30204C211yesmaximal, characteristic
7#30203C211yesmaximal, characteristic
8#29208C211yesmaximal, characteristic, complemented
9#29207C211yesmaximal, characteristic, complemented
10#28717C211yesmaximal, characteristic, complemented
11#28614C211yesmaximal, characteristic
12#28614C211yesmaximal, characteristic
13#28514C211yesmaximal, characteristic
14#24690C214yesmaximal, complemented
15#24346C214yesmaximal, complemented
16#22221C211yesmaximal, characteristic, complemented
17#22218C211yesmaximal, characteristic, complemented
18#21692C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#557C2212yes—
21C2×#556—21no—
22C2×#492C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#478C2214yes—
26C2×#440C2214yes—
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2568C2212yes—
34#2565C2212yes—
35#2565C2212yes—
36#2547—21no—
37#2546—21no—
38#2546—21no—
39#2545—21no—
40#2545—21no—
41#2538—21no—
42#2538C2212yes—
43#2512—21no—
44#2512—21no—
45#2512C2212yes—
46#2512C2212yes—
47#2506—21no—
48#2506—21no—
49#2506—21no—
50#2506—21no—
51#2365—21no—
52#2365—21no—
53#2354—21no—
54#2354C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2092C2214yes—
63#2073C2214yes—
64#2072C2214yes—
65#2071C2214yes—
66#2059C2214yes—
67#2057C2214yes—
68#2055C2214yes—
69#2055C2214yes—
70#2025C2214yes—
71#2023C2214yes—
72#1992C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1985C2212yes—
80#1973C2214yes—
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1957C2211yescharacteristic
84#1956C2211yescharacteristic
85#1949C2214yes—
86#1947C2214yes—
87#1940C2212yes—
88#1928C2214yes—
89#1927C2214yes—
90#1917C2212yes—
91#1917C2212yes—
92#1880C2214yes—
93#1880C2214yes—
94#1865C2214yes—
95#1863C2214yes—
96#1845C2211yescharacteristic
97#1845C2211yescharacteristic
98#1844C2211yescharacteristic
99#1836C2214yes—
100#1803C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108#557—48no—
109#556—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⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×#127C2311yescharacteristic
120C2×#126—22no—
121C2×#126—22no—
122C2×#125—22no—
123C2×#125—22no—
124C2×#125C2312yes—
125C2×#125C2312yes—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124—21no—
130C2×C22⋊Q8—22no—
131C2×C4×Q8—22no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8D411yescharacteristic
135C2×C4×Q8D411yescharacteristic
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4—21no—
142C2×C4×D4C2312yes—
143C2×#117—22no—
144#492—24no—
145#487—24no—
146#487—24no—
147#478—48no—
148#440—48no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167#400—22no—
168#400—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#398—22no—
173#397—22no—
174#397—22no—
175#396—22no—
176#396—22no—
177#396—22no—
178#396—22no—
179#391—22no—
180#391—22no—
181#391—21no—
182#391—21no—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
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#389—21no—
211#389—21no—
212#389—21no—
213#389C2312yes—
214#389C2312yes—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384—21no—
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2312yes—
232#384C2312yes—
233#384C2312yes—
234#382C2312yes—
235#382C2312yes—
236#380—22no—
237#380—22no—
238C4×C4⋊C4—21no—
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⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C4×C4⋊C4C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—22no—
276C22×Q8C2×D411yescharacteristic
277C22×Q8C2412yes—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C2×SD16—28no—
283C2×SD16—28no—
284C2×SD16—44no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—48no—
298C4⋊Q8—44no—
299C4⋊Q8—82no—
300C4⋊Q8#4114yes—
301#127—22no—
302#127—22no—
303#126—48no—
304#126—48no—
305#125—28no—
306#125—44no—
307#125—48no—
308#125—48no—
309#124—44no—
310#124—44no—
311#124—82no—
312#124—82no—
313C22⋊Q8—48no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—28no—
317C4×Q8—42no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—48no—
322C4×Q8—44no—
323C4×D4—28no—
324C4×D4—44no—
325C4×D4—44no—
326#117—48no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
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—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4C2×D411yescharacteristic
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⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C22⋊C4—22no—
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×C22⋊C4—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
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×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—44no—
406#107—44no—
407#106—44no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411#103—28no—
412#103—28no—
413#103—44no—
414#102—44no—
415#102—44no—
416#102—44no—
417#97—24no—
418#97—24no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—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#95—22no—
435#95—22no—
436#95—22no—
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443C24—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—41no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8C2×#4114yes—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
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×C82−1+414yes—
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—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
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—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
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—22no—
580C42—44no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42AGL(1,ℤ/8)14yes—
586C42C2×#4112yes—
587C42H(5,𝔽2)14yes—
588C422−1+412yes—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23C23×D411yescharacteristic
597Q8—28no—
598Q8—44no—
599Q8—44no—
600Q8—88no—
601Q8—84no—
602Q8—84no—
603D4—44no—
604D4—44no—
605D4—84no—
606D4—84no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
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—24no—
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—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
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—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—88no—
661C2×C4#53512yes—
662C2×C4#53512yes—
663C2×C4#54812yes—
664C2×C4#54812yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4#57512yes—
669C2×C4#57712yes—
670C2×C4#57812yes—
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×H(5,𝔽2)12yes—
673C2×C4C2×2−1+412yes—
674C2×C4C2×2−1+411yescharacteristic
675C8—24no—
676C8—42no—
677C8—42no—
678C8—42no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22C2×#53511yescharacteristic
687C22C2×#54811yescharacteristic
688C22#349911yescharacteristic
689C22C22×AGL(1,ℤ/8)11yescharacteristic
690C22C2×#57511yescharacteristic
691C22C2×#57711yescharacteristic
692C22C2×#57811yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—44no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#336814yes—
713C4#337714yes—
714C4#352012yes—
715C4#355012yes—
716C2—162no—
717C2#3916711yescharacteristic
718C2#6080311yescharacteristic
719C2#6082411yescharacteristic
720C2C2×#336811yescharacteristic
721C2C2×#337711yescharacteristic
722C2#6247911yescharacteristic
723C2#6248311yescharacteristic
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
2. #36560C2semidirect
3. #33294C2semidirect
4. #32456C2semidirect
5. #32453C2semidirect
8. #29208C2semidirect
9. #29207C2semidirect
10. #28717C2semidirect
14. #24690C2semidirect
15. #24346C2semidirect
16. #22221C2semidirect
17. #22218C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#365602
#332942
#324562
#324532
#302042
#302032
#292082
#292072
#287172
#286142
#285142
#246902
#243462
#222212
#222182
#216922
#213402