Finite Group Databasebeta

← #6805785

Subgroups — #6805785

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic
4#30829C214yesmaximal, complemented
5#30824C214yesmaximal, complemented
6#30205C211yesmaximal, characteristic, complemented
7#30203C211yesmaximal, characteristic
8#29206C211yesmaximal, characteristic, complemented
9#28750C211yesmaximal, characteristic
10#28656C211yesmaximal, characteristic, complemented
11#28655C211yesmaximal, characteristic
12#28512C211yesmaximal, characteristic, complemented
13#28509C211yesmaximal, characteristic
14#26476C214yesmaximal, complemented
15#26328C214yesmaximal, complemented
16#22219C211yesmaximal, characteristic
17#22218C211yesmaximal, characteristic, complemented
18#21726C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×Q82—21no—
21C2×#500C2211yescharacteristic
22C2×#489C2214yes—
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#475C2214yes—
26C2×#476C2214yes—
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2592—21no—
34#2592C2212yes—
35#2580—21no—
36#2567—21no—
37#2567—21no—
38#2567—21no—
39#2551—21no—
40#2551—21no—
41#2551—21no—
42#2551C2212yes—
43#2551C2212yes—
44#2551C2212yes—
45#2513—21no—
46#2513—21no—
47#2448—21no—
48#2448C2212yes—
49#2447—21no—
50#2447C2212yes—
51#2367—21no—
52#2367C2212yes—
53#2357—21no—
54Q8×C4⋊C4—21no—
55Q8×C4⋊C4C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C42×Q8C2211yescharacteristic
63#2096C2214yes—
64#2094C2214yes—
65#2072C2214yes—
66#2071C2214yes—
67#2068C2214yes—
68#2067C2214yes—
69#2058C2214yes—
70#2042C2214yes—
71#2024C2214yes—
72#2023C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1985C2212yes—
78#1983C2214yes—
79#1959C2214yes—
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1948C2214yes—
87#1943C2214yes—
88#1940C2212yes—
89#1928C2214yes—
90#1927C2214yes—
91#1926C2214yes—
92#1925C2214yes—
93#1917C2212yes—
94#1917C2212yes—
95#1865C2214yes—
96#1864C2214yes—
97#1844C2211yescharacteristic
98#1844C2211yescharacteristic
99#1844C2211yescharacteristic
100#1776C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107Q82—48no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2312yes—
123C2×#127C2311yescharacteristic
124C2×#124—21no—
125C2×#124—21no—
126C2×#124C2312yes—
127C2×#124C2312yes—
128C2×C4×Q8—22no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8D411yescharacteristic
134C2×C4×Q8D411yescharacteristic
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4C2312yes—
140#500—24no—
141#489—48no—
142#487—24no—
143#486—24no—
144#476—48no—
145#475—48no—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167#399—22no—
168#397—22no—
169#397—22no—
170#397—22no—
171#397—22no—
172#395—22no—
173#395—22no—
174#395—22no—
175#395—22no—
176#395—22no—
177#395—22no—
178#393—22no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390C2312yes—
197#390C2312yes—
198#390C2312yes—
199#389—22no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389—22no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#388—22no—
221#385—22no—
222#385—22no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2312yes—
234#384C2312yes—
235#384C2312yes—
236#382—21no—
237#382—21no—
238#381—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⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2311yescharacteristic
256C4×C4⋊C4C2311yescharacteristic
257C4×C4⋊C4C2311yescharacteristic
258C4×C22⋊C4—22no—
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
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×Q8—22no—
277C22×Q8—22no—
278C22×Q8C2×D411yescharacteristic
279C22×Q8C2412yes—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—44no—
298C4⋊Q8—48no—
299C4⋊Q8—48no—
300C4⋊Q8—48no—
301C4⋊Q8—44no—
302C4⋊Q8—82no—
303C4⋊Q8—82no—
304C4⋊Q8—82no—
305C4⋊Q8#4114yes—
306#127—22no—
307#127—22no—
308#124—28no—
309#124—44no—
310#124—44no—
311#124—82no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×Q8—48no—
319C4×Q8—44no—
320C4×Q8—82no—
321C4×D4—44no—
322C4×D4—44no—
323C4×D4—82no—
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—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
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⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
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×C22⋊C4—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
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×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—28no—
401#106—28no—
402#106—44no—
403#106—44no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407#103—28no—
408#103—28no—
409#103—44no—
410#103—44no—
411#102—28no—
412#102—28no—
413#102—44no—
414#102—44no—
415#97—24no—
416#97—24no—
417C4×C8—22no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—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#95—22no—
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441C24—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—82no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×Q8—82no—
462C2×Q8—82no—
463C2×Q8—82no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—82no—
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×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C2511yescharacteristic
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×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8H(5,𝔽2)14yes—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4C2×#4114yes—
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—
563C22⋊C4—82no—
564C22⋊C4—82no—
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—82no—
584C42—82no—
585C42AGL(1,ℤ/8)14yes—
586C42#13714yes—
587C42C2×#4112yes—
588C422−1+414yes—
589C422−1+412yes—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23C23×D411yescharacteristic
598Q8—28no—
599Q8—48no—
600Q8—44no—
601Q8—44no—
602Q8—84no—
603Q8—164no—
604Q8—164no—
605Q8—164no—
606D4—44no—
607D4—44no—
608D4—84no—
609D4—84no—
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—24no—
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—21no—
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—42no—
661C2×C4—164no—
662C2×C4#53612yes—
663C2×C4#53612yes—
664C2×C4#54812yes—
665C2×C4#54812yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C2×#13712yes—
670C2×C4C2×#13712yes—
671C2×C4C2×#13712yes—
672C2×C4#57712yes—
673C2×C4#57712yes—
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×H(5,𝔽2)12yes—
676C2×C4C2×2−1+412yes—
677C2×C4C2×2−1+411yescharacteristic
678C8—24no—
679C8—42no—
680C8—42no—
681C8—42no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22C2×#53611yescharacteristic
690C22C2×#54811yescharacteristic
691C22#349811yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C22×#13711yescharacteristic
694C22C2×#57711yescharacteristic
695C22C2×#57711yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—44no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#336914yes—
716C4#337314yes—
717C4#352212yes—
718C4#355012yes—
719C2—162no—
720C2#3916811yescharacteristic
721C2#6086011yescharacteristic
722C2#6088111yescharacteristic
723C2C2×#337311yescharacteristic
724C2C2×#336911yescharacteristic
725C2#6242111yescharacteristic
726C2#6242211yescharacteristic
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. Q8×C4⋊Q8C2semidirect
4. #30829C2semidirect
5. #30824C2semidirect
6. #30205C2semidirect
8. #29206C2semidirect
10. #28656C2semidirect
12. #28512C2semidirect
14. #26476C2semidirect
15. #26328C2semidirect
17. #22218C2semidirect
18. #21726C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#366092
Q8×C4⋊Q82
#332302
#308292
#308242
#302052
#302032
#292062
#287502
#286562
#286552
#285122
#285092
#264762
#263282
#222192
#222182
#217262
#213402