Finite Group Databasebeta

← #6623861

Subgroups — #6623861

Order 512.

1640 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic, complemented
2#36573C211yesmaximal, characteristic, complemented
3#33287C211yesmaximal, characteristic, complemented
4#33286C211yesmaximal, characteristic, complemented
5#33226C211yesmaximal, characteristic, complemented
6#29207C211yesmaximal, characteristic, complemented
7#29192C211yesmaximal, characteristic, complemented
8#29191C211yesmaximal, characteristic, complemented
9#28615C211yesmaximal, characteristic
10#28613C211yesmaximal, characteristic, complemented
11#28443C211yesmaximal, characteristic
12#26837C214yesmaximal, complemented
13#26835C214yesmaximal, complemented
14#25298C214yesmaximal, complemented
15#23005C214yesmaximal, complemented
16#21692C211yesmaximal, characteristic, complemented
17#21690C211yesmaximal, characteristic
18C4×#486C211yesmaximal, characteristic, complemented
19C4×#446C211yesmaximal, characteristic, complemented
20C2×#493C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#488C2214yes—
23C2×#486C2211yescharacteristic
24C2×#463C2214yes—
25C2×#446C2212yes—
26C2×#442C2214yes—
27C2×#440C2214yes—
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2858C2211yescharacteristic
34#2592—21no—
35#2592C2212yes—
36#2580—21no—
37#2580—21no—
38#2580—21no—
39#2580C2212yes—
40#2513—21no—
41#2513—21no—
42#2454—21no—
43#2454—21no—
44#2454—21no—
45#2454—21no—
46#2454C2212yes—
47#2454C2212yes—
48#2448—21no—
49#2448C2212yes—
50#2447—21no—
51#2447C2212yes—
52#2367—21no—
53#2367C2212yes—
54#2357—21no—
55#2349—21no—
56#2349—21no—
57#2349C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C4×#127C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C42×Q8C2211yescharacteristic
65#2067C2214yes—
66#2066C2214yes—
67#2059C2214yes—
68#2055C2214yes—
69#2054C2214yes—
70#2012C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1973C2214yes—
75#1972C2214yes—
76#1958C2214yes—
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1949C2214yes—
85#1936C2214yes—
86#1936C2214yes—
87#1935C2214yes—
88#1935C2214yes—
89#1934C2214yes—
90#1879C2214yes—
91#1877C2214yes—
92#1865C2214yes—
93#1862C2214yes—
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1771C2212yes—
98#1771C2212yes—
99#1770C2212yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×#102C2214yes—
104C4×#97C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×D8C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#127C2311yescharacteristic
113C2×#125—21no—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#124—21no—
117C2×#124—21no—
118C2×#124C2312yes—
119C2×#124C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137#493—24no—
138#492—24no—
139#488—48no—
140#486—24no—
141#463—216no—
142#446—28no—
143#442—48no—
144#440—48no—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2311yescharacteristic
168#399—22no—
169#399—22no—
170#399—22no—
171#399—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#396—22no—
177#396—22no—
178#393—22no—
179#393—22no—
180#391—21no—
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#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390C2312yes—
201#390C2312yes—
202#390C2312yes—
203#390C2312yes—
204#390C2312yes—
205#390C2312yes—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#388—22no—
223#388—22no—
224#388—22no—
225#388—22no—
226#388—22no—
227#388—22no—
228#385—22no—
229#385—22no—
230#385—22no—
231#385—22no—
232#385—22no—
233#385—22no—
234#385—22no—
235#385—22no—
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#382—21no—
241#382—21no—
242#382—21no—
243#382—21no—
244#382C2312yes—
245#381—22no—
246#380—22no—
247#380—22no—
248#380—22no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4C2312yes—
259C4×C4⋊C4C2311yescharacteristic
260C4×C4⋊C4C2311yescharacteristic
261C4×C4⋊C4C2311yescharacteristic
262C4×C22⋊C4—22no—
263C2×#95—22no—
264C43C2311yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—21no—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4—22no—
286C23×C4—22no—
287C2×SD16—28no—
288C2×SD16—28no—
289C2×SD16—44no—
290C2×D8—28no—
291C2×D8—28no—
292C2×D8—44no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301#127—22no—
302#127—22no—
303#127—22no—
304#127#4114yes—
305#125—44no—
306#125—82no—
307#125—82no—
308#124—28no—
309#124—44no—
310#124—44no—
311#124—82no—
312C4×Q8—24no—
313C4×Q8—42no—
314C4×Q8—42no—
315C4×Q8—44no—
316C4×Q8—82no—
317C4×D4—28no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—82no—
325C4×D4—82no—
326C4×D4—82no—
327C4×D4—82no—
328C4×D4—82no—
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—21no—
336C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
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⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
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×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C42—22no—
383C2×C42—22no—
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×C42C2×D411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107—44no—
404#106—44no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8#4118yes—
408#103—28no—
409#103—28no—
410#103—44no—
411#102—28no—
412#102—28no—
413#102—28no—
414#102—28no—
415#102—44no—
416#97—24no—
417#97—24no—
418#97—24no—
419#97—24no—
420C4×C8—24no—
421C4×C8—24no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—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#95—22no—
438#95—22no—
439#95—22no—
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446C24—22no—
447C24—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4C2×#4114yes—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C2511yescharacteristic
511SD16—416no—
512D8—416no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8C2×#4114yes—
523C2×C8C2×#4114yes—
524C2×C8H(5,𝔽2)14yes—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
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—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—82no—
597C42—82no—
598C42C2×#4114yes—
599C42C2×#4112yes—
600C422−1+412yes—
601C23—24no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23C23×D411yescharacteristic
616Q8—44no—
617Q8—44no—
618Q8—44no—
619Q8—84no—
620D4—28no—
621D4—48no—
622D4—48no—
623D4—44no—
624D4—44no—
625D4—44no—
626D4—84no—
627D4—84no—
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—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—164no—
681C2×C4—164no—
682C2×C4—164no—
683C2×C4—164no—
684C2×C4#53212yes—
685C2×C4#53212yes—
686C2×C4#54812yes—
687C2×C4#54812yes—
688C2×C4#57512yes—
689C2×C4#57512yes—
690C2×C4#57612yes—
691C2×C4#57712yes—
692C2×C4C22×#4111yescharacteristic
693C2×C4C22×#4112yes—
694C2×C4C2×H(5,𝔽2)12yes—
695C2×C4C2×2−1+411yescharacteristic
696C8—24no—
697C8—24no—
698C8—24no—
699C8—42no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22C2×#53211yescharacteristic
715C22C2×#54811yescharacteristic
716C22#349811yescharacteristic
717C22C2×#57511yescharacteristic
718C22C2×#57511yescharacteristic
719C22C2×#57611yescharacteristic
720C22C2×#57711yescharacteristic
721C4—22no—
722C4—22no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4#337014yes—
740C4#337614yes—
741C4#348112yes—
742C4#352212yes—
743C2—84no—
744C2—162no—
745C2#3649611yescharacteristic
746C2#6069311yescharacteristic
747C2#6072211yescharacteristic
748C2C2×#337011yescharacteristic
749C2C2×#337611yescharacteristic
750C2#6242611yescharacteristic
751C2#6242711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
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. #36609C2semidirect
2. #36573C2semidirect
3. #33287C2semidirect
4. #33286C2semidirect
5. #33226C2semidirect
6. #29207C2semidirect
7. #29192C2semidirect
8. #29191C2semidirect
10. #28613C2semidirect
12. #26837C2semidirect
13. #26835C2semidirect
14. #25298C2semidirect
15. #23005C2semidirect
16. #21692C2semidirect
18. C4×#486C2semidirect
19. C4×#446C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365732
#332872
#332862
#332262
#292072
#291922
#291912
#286152
#286132
#284432
#268372
#268352
#252982
#230052
#216922
#216902
C4×#4862
C4×#4462