Finite Group Databasebeta

← #7505404

Subgroups — #7505404

Order 512.

2128 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#35554C211yesmaximal, characteristic, complemented
2#35529C211yesmaximal, characteristic, complemented
3#31665C214yesmaximal, complemented
4#31658C214yesmaximal, complemented
5#31641C214yesmaximal, complemented
6#31634C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23540C211yesmaximal, characteristic, complemented
10#23456C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12C2×#2011C211yesmaximal, characteristic
13#21203C211yesmaximal, characteristic, complemented
14#21095C211yesmaximal, characteristic, complemented
15#21022C211yesmaximal, characteristic, complemented
16#20934C211yesmaximal, characteristic, complemented
17#20934C211yesmaximal, characteristic, complemented
18#20934C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C23×D8C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×#127C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2566C2212yescomplemented
37#2566C2212yescomplemented
38#2563C2212yescomplemented
39#2542C2212yescomplemented
40#2454—22no—
41#2452—22no—
42#2445—22no—
43#2445—22no—
44#2445C2214yescomplemented
45#2445C2214yescomplemented
46#2437—24no—
47#2405—24no—
48#2317C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2285C2211yescharacteristic
57#2101C2212yescomplemented
58#2101C2212yescomplemented
59#2100C2212yescomplemented
60#2099C2212yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2054C2214yescomplemented
64#2054C2214yescomplemented
65#2025C2214yescomplemented
66#2025C2214yescomplemented
67#2023C2214yescomplemented
68#2023C2214yescomplemented
69#2011C2214yes—
70#2011C2214yes—
71#1961C2212yescomplemented
72#1961C2212yescomplemented
73#1957C2212yescomplemented
74#1957C2212yescomplemented
75#1936C2214yescomplemented
76#1934C2214yescomplemented
77#1917C2214yescomplemented
78#1916C2214yescomplemented
79#1879C2214yescomplemented
80#1879C2214yescomplemented
81#1879C2214yescomplemented
82#1879C2214yescomplemented
83#1879C2214yescomplemented
84#1879C2214yescomplemented
85#1804C2214yescomplemented
86#1802C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×D4—22no—
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C22×D8—24no—
102C22×D8—24no—
103C22×D8—24no—
104C22×D8—42no—
105C22×D8—42no—
106C22×D8—42no—
107C22×D8C2314yes—
108C22×D8C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—42no—
114#544—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2312yes—
117C2×#127—24no—
118C2×#127—22no—
119C2×#127—42no—
120C2×#127C2312yes—
121C2×#127C2312yes—
122C2×#125—21no—
123C2×#125C2312yes—
124C2×#123—24no—
125C2×#123—24no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—22no—
128#518—24no—
129#518—42no—
130#518—42no—
131#518—42no—
132C2×C4×D4—22no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2314yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—22no—
146C22×C22⋊C4—22no—
147C22×C22⋊C4—22no—
148C22×C42C2311yescharacteristic
149#424—42no—
150C2×C4⋊C8C2314yes—
151#418—24no—
152#418—42no—
153#418—42no—
154#418—42no—
155#418—42no—
156#418—42no—
157C2×#102—42no—
158C2×#102—42no—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C4×M16—42no—
172C2×#97C2312yes—
173C2×C4×C8C2312yes—
174#399—24no—
175#396—24no—
176#396—24no—
177#394—24no—
178#394—24no—
179#394—24no—
180#394—24no—
181#393—24no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—22no—
187#390—22no—
188#390—22no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2314yes—
193#390C2314yes—
194#390C2314yes—
195#390C2312yes—
196#389—22no—
197#389D412yes—
198#389D412yes—
199#389C2312yes—
200#389C2312yes—
201#386—24no—
202#386—24no—
203#385—22no—
204#385—22no—
205#385—24no—
206#385—24no—
207#385D414yes—
208#384C2311yescharacteristic
209#384C2312yes—
210#384C2311yescharacteristic
211#382C2311yescharacteristic
212#382C2312yes—
213#382C2311yescharacteristic
214#380—22no—
215#380—24no—
216#379—22no—
217#379—22no—
218#379—22no—
219#379—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C25—21no—
241C25—21no—
242C22×Q8—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—24no—
247C22×D4—24no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—48no—
265C22×D4—84no—
266C22×D4—84no—
267C22×D4#4114yes—
268C22×D4#4114yes—
269C22×D4C2412yes—
270C22×D4C2412yes—
271C22×D4C2412yes—
272C23×C4—21no—
273C23×C4—22no—
274C23×C4—24no—
275C23×C4—22no—
276C23×C4C2×D412yes—
277C23×C4C2411yescharacteristic
278C2×D8—28no—
279C2×D8—28no—
280C2×D8—28no—
281C2×D8—44no—
282C2×D8—44no—
283C2×D8—44no—
284C2×D8—416no—
285C2×D8—416no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C4⋊Q8—42no—
303C4⋊Q8—42no—
304C4⋊Q8—44no—
305#127—24no—
306#127—24no—
307#127—42no—
308#127—42no—
309#127—48no—
310#125—42no—
311#125—42no—
312#125—44no—
313#123—88no—
314#123—88no—
315C22⋊Q8—48no—
316C22⋊Q8—88no—
317C4×D4—28no—
318C4×D4—44no—
319C4×D4—42no—
320C4×D4—42no—
321C4×D4—42no—
322C4×D4—42no—
323C4×D4—48no—
324C4×D4—44no—
325C4×D4—44no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
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—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—48no—
379C2×C22⋊C4—48no—
380C2×C22⋊C4—84no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C2×D412yes—
390C2×C42C2412yes—
391C2×C42C2412yes—
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C4⋊C8—44no—
395#102—28no—
396#102—44no—
397#102—44no—
398#102—44no—
399#102—42no—
400#102—42no—
401#102—42no—
402#102—42no—
403#102—42no—
404#102—42no—
405#102—42no—
406#102—42no—
407#97—42no—
408C4×C8—42no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—24no—
414#95C2×D414yes—
415#95C2×D414yes—
416C24—22no—
417C24—22no—
418C24—24no—
419C24—24no—
420C24—22no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—82no—
429C24—82no—
430C24—82no—
431C24—82no—
432C24—82no—
433C24—82no—
434C24C22×D411yescharacteristic
435C2×Q8—44no—
436C2×Q8—84no—
437C2×D4—28no—
438C2×D4—28no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—48no—
460C2×D4—48no—
461C2×D4—48no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—48no—
478C2×D4—84no—
479C2×D4—84no—
480C2×D4—832no—
481C2×D4C2×#4114yes—
482C2×D4C2×#4114yes—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—48no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—42no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4—84no—
500C22×C4—84no—
501C22×C4—82no—
502C22×C4C22×D412yes—
503C22×C4C22×D414yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4H(5,𝔽2)12yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C42−1+412yes—
514C22×C4C2511yescharacteristic
515D8—416no—
516D8—416no—
517M16—44no—
518M16—44no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—44no—
532C2×C8—44no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—28no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—82no—
560C4⋊C4—84no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—48no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—48no—
571C22⋊C4—84no—
572C22⋊C4—84no—
573C22⋊C4—84no—
574C22⋊C4—84no—
575C22⋊C4—84no—
576C22⋊C4—84no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C22⋊C4—84no—
580C42—24no—
581C42—21no—
582C42—21no—
583C42—22no—
584C42—21no—
585C42—21no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—42no—
590C42—42no—
591C42—42no—
592C42—42no—
593C23—22no—
594C23—22no—
595C23—24no—
596C23—24no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—44no—
604C23—44no—
605C23—48no—
606C23—48no—
607C23—44no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—44no—
612C23—88no—
613C23—88no—
614C23—88no—
615C23—84no—
616C23—162no—
617C23—162no—
618C23—162no—
619C23—162no—
620C23—162no—
621C23—162no—
622C23—164no—
623C23—164no—
624C23—164no—
625C23#53512yes—
626C23#54612yes—
627C23#54812yes—
628C23C23×D411yescharacteristic
629Q8—164no—
630D4—24no—
631D4—24no—
632D4—24no—
633D4—24no—
634D4—42no—
635D4—42no—
636D4—42no—
637D4—42no—
638D4—48no—
639D4—48no—
640D4—42no—
641D4—42no—
642D4—42no—
643D4—42no—
644D4—816no—
645D4—816no—
646D4—84no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—48no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—168no—
677C2×C4—164no—
678C2×C4—164no—
679C2×C4—162no—
680C2×C4#53512yes—
681C2×C4#53814yes—
682C2×C4#53914yes—
683C2×C4#54612yes—
684C2×C4#54812yes—
685C2×C4#55114yes—
686C2×C4#56214yes—
687C2×C4#57614yes—
688C2×C4C22×#4111yescharacteristic
689C2×C4C22×#4111yescharacteristic
690C2×C4C2×H(5,𝔽2)11yescharacteristic
691C2×C4C2×H(5,𝔽2)11yescharacteristic
692C2×C4C2×H(5,𝔽2)11yescharacteristic
693C2×C4C2×2−1+411yescharacteristic
694C8—42no—
695C8—42no—
696C8—42no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—24no—
701C22—24no—
702C22—44no—
703C22—44no—
704C22—48no—
705C22—48no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—84no—
711C22—84no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—84no—
716C22—84no—
717C22—168no—
718C22—168no—
719C22—168no—
720C22#269614yes—
721C22C2×#53511yescharacteristic
722C22C2×#54611yescharacteristic
723C22C2×#54811yescharacteristic
724C22C2×#57612yes—
725C22#358512yes—
726C4—24no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—81no—
732C4—82no—
733C4—81no—
734C4—81no—
735C4—81no—
736C4—82no—
737C4—82no—
738C4#269614yes—
739C4C2×#53811yescharacteristic
740C4C2×#53911yescharacteristic
741C4C2×#55111yescharacteristic
742C4C2×#56211yescharacteristic
743C2—24no—
744C2—84no—
745C2—84no—
746C2—162no—
747C2C2×#269611yescharacteristic
748C2#6081711yescharacteristic
749C2#6082511yescharacteristic
750C2#6118312yes—
751C2#6133912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35554C2semidirect
2. #35529C2semidirect
3. #31665C2semidirect
4. #31658C2semidirect
5. #31641C2semidirect
6. #31634C2semidirect
7. #23744C2semidirect
8. #23632C2semidirect
9. #23540C2semidirect
10. #23456C2semidirect
11. #23456C2semidirect
13. #21203C2semidirect
14. #21095C2semidirect
15. #21022C2semidirect
16. #20934C2semidirect
17. #20934C2semidirect
18. #20934C2semidirect
19. #20650C2semidirect
36. #2566C22semidirect
37. #2566C22semidirect
38. #2563C22semidirect
39. #2542C22semidirect
44. #2445C22semidirect
45. #2445C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
63. #2054C22semidirect
64. #2054C22semidirect
65. #2025C22semidirect
66. #2025C22semidirect
67. #2023C22semidirect
68. #2023C22semidirect
71. #1961C22semidirect
72. #1961C22semidirect
73. #1957C22semidirect
74. #1957C22semidirect
75. #1936C22semidirect
76. #1934C22semidirect
77. #1917C22semidirect
78. #1916C22semidirect
79. #1879C22semidirect
80. #1879C22semidirect
81. #1879C22semidirect
82. #1879C22semidirect
83. #1879C22semidirect
84. #1879C22semidirect
85. #1804C22semidirect
86. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#355542
#355292
#316652
#316582
#316412
#316342
#237442
#236322
#235402
#234562
C2×#20112
#212032
#210952
#210222
#209342
#206502