Finite Group Databasebeta

← #7502009

Subgroups — #7502009

Order 512.

2050 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#490C211yesmaximal, characteristic
2C2×#3081C211yesmaximal, characteristic, complemented
3#35640C211yesmaximal, characteristic, complemented
4#35554C211yesmaximal, characteristic
5#30835C214yesmaximal, complemented
6#30833C214yesmaximal, complemented
7#30827C214yesmaximal, complemented
8#30825C214yesmaximal, complemented
9#23735C211yesmaximal, characteristic, complemented
10#23632C211yesmaximal, characteristic, complemented
11#23612C211yesmaximal, characteristic, complemented
12#23553C211yesmaximal, characteristic, complemented
13#23505C211yesmaximal, characteristic
14#23456C211yesmaximal, characteristic, complemented
15#21154C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic, complemented
17#21063C211yesmaximal, characteristic, complemented
18#21021C211yesmaximal, characteristic, complemented
19#20876C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C22×#127C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#490—28no—
25C2×#490C2214yes—
26C2×#490C2214yes—
27#3081—28no—
28C2×#478C2212yescomplemented
29C2×#478C2212yescomplemented
30C2×#476C2212yescomplemented
31C2×#474C2212yescomplemented
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#427C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C2×#424C2211yescharacteristic
37C2×#418C2211yescharacteristic
38C2×#418C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#97C2211yescharacteristic
46#2583—22no—
47#2552—22no—
48#2552C2214yescomplemented
49#2550—22no—
50#2550C2214yescomplemented
51#2502—22no—
52#2445—22no—
53#2445—22no—
54#2445C2214yes—
55#2445C2214yes—
56#2405—22no—
57#2405—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2309C2211yescharacteristic
61#2309C2211yescharacteristic
62#2309C2211yescharacteristic
63#2296C2211yescharacteristic
64#2296C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2094C2212yescomplemented
68#2093C2212yescomplemented
69#2092C2212yescomplemented
70#2092C2212yescomplemented
71#2073C2214yescomplemented
72#2071C2214yescomplemented
73#2067C2214yescomplemented
74#2065C2214yescomplemented
75#2059C2214yescomplemented
76#2059C2214yescomplemented
77#2041C2214yes—
78#2041C2214yes—
79#2025C2214yescomplemented
80#2023C2214yescomplemented
81#1944C2212yescomplemented
82#1944C2212yescomplemented
83#1943C2212yescomplemented
84#1943C2212yescomplemented
85#1926C2214yescomplemented
86#1926C2214yescomplemented
87#1925C2214yescomplemented
88#1925C2214yescomplemented
89#1917C2214yescomplemented
90#1917C2214yescomplemented
91#1864C2214yescomplemented
92#1862C2214yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×D4—21no—
102C23×D4—21no—
103C23×D4C2311yescharacteristic
104C23×D4C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#544—24no—
110#544—42no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2312yes—
113C2×#127—24no—
114C2×#127—22no—
115C2×#127—42no—
116C2×#127C2314yes—
117C2×#125—21no—
118C2×#125—24no—
119C2×#125—22no—
120C2×#125—42no—
121C2×#125C2312yes—
122C2×#125C2314yes—
123C2×#123—24no—
124C2×#123—24no—
125C2×C22⋊Q8—22no—
126#518—24no—
127#518—42no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—22no—
143C22×C42C2311yescharacteristic
144#490—216no—
145#490—216no—
146#478—28no—
147#478—28no—
148#476—28no—
149#474—28no—
150#428—24no—
151#428—42no—
152#427—24no—
153#427—24no—
154C2×#107C2311yescharacteristic
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2311yescharacteristic
158C2×#107C2312yes—
159C2×#106C2311yescharacteristic
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2311yescharacteristic
163C2×#106C2312yes—
164#424—24no—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167#418—24no—
168#418—24no—
169#418—24no—
170#418—42no—
171C2×#102—24no—
172C2×#102—24no—
173C2×#102—24no—
174C2×#102D418yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2314yes—
178C2×#102C2312yes—
179C2×#102C2314yes—
180C2×#102C2312yes—
181C2×#102C2314yes—
182C2×#102C2314yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#97—24no—
188C2×#97C2312yes—
189C2×#97C2312yes—
190#398—24no—
191#398—24no—
192#398—24no—
193#395—24no—
194#394—24no—
195#394—24no—
196#394—24no—
197#394—24no—
198#391—24no—
199#390—22no—
200#390—21no—
201#390—21no—
202#390—22no—
203#390C2314yes—
204#390C2314yes—
205#390C2312yes—
206#390C2312yes—
207#389—22no—
208#389—24no—
209#389—24no—
210#389D412yes—
211#389D412yes—
212#389C2312yes—
213#389C2312yes—
214#389C2314yes—
215#389C2312yes—
216#389C2312yes—
217#386—24no—
218#386—24no—
219#385—21no—
220#385—21no—
221#385—21no—
222#385—21no—
223#385—22no—
224#385—22no—
225#385D412yes—
226#385D412yes—
227#384—21no—
228#384—24no—
229#384C2312yes—
230#382—21no—
231#382C2312yes—
232#379—22no—
233#379—22no—
234#379—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4C2314yes—
239C4×C4⋊C4C2314yes—
240C4×C22⋊C4—22no—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252C25—21no—
253C25—21no—
254C22×Q8—22no—
255C22×D4—21no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—24no—
259C22×D4—24no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—44no—
271C22×D4—44no—
272C22×D4—84no—
273C22×D4—84no—
274C22×D4C2×D414yes—
275C22×D4C2412yes—
276C22×D4C2412yes—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2411yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8C2×D414yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2411yescharacteristic
300C22×C8C2411yescharacteristic
301C4⋊Q8—24no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304#127—28no—
305#127—44no—
306#127—48no—
307#125—28no—
308#125—24no—
309#125—44no—
310#125—42no—
311#125—48no—
312#125—44no—
313#123—88no—
314#123—88no—
315C22⋊Q8—48no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—42no—
319C4×D4—42no—
320C4×D4—44no—
321C4×D4—44no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117#4114yes—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
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—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
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⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—24no—
388C2×C42C2×D412yes—
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2412yes—
392#107—22no—
393#107—22no—
394#107—24no—
395#107—24no—
396#107—42no—
397#106—22no—
398#106—22no—
399#106—24no—
400#106—24no—
401#106—42no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404#102—28no—
405#102—24no—
406#102—24no—
407#102—24no—
408#102—24no—
409#102—28no—
410#102—28no—
411#102—28no—
412#102—42no—
413#102—42no—
414#102C2×D418yes—
415#102C2×D418yes—
416#97—24no—
417#97—24no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—42no—
431C24—42no—
432C24—42no—
433C24—42no—
434C24—42no—
435C24—82no—
436C24—82no—
437C24—82no—
438C24—82no—
439C24C22×D411yescharacteristic
440C2×Q8—44no—
441C2×Q8—84no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—28no—
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—48no—
457C2×D4—48no—
458C2×D4—48no—
459C2×D4—48no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—84no—
472C2×D4—84no—
473C2×D4—816no—
474C2×D4—816no—
475C2×D4C22×D414yes—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—84no—
493C22×C4—84no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C2×#4112yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C42−1+412yes—
506C22×C42−1+412yes—
507C22×C42−1+412yes—
508C22×C4C2511yescharacteristic
509M16—44no—
510M16—44no—
511C2×C8—28no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—44no—
526C2×C8C22×D412yes—
527C2×C8C22×D412yes—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—84no—
557C4⋊C4—84no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4C22×D414yes—
561C4⋊C4C2×#4114yes—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—24no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C22⋊C4—84no—
573C22⋊C4—84no—
574C22⋊C4—84no—
575C22⋊C4—84no—
576C42—24no—
577C42—24no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—42no—
585C42—42no—
586C42—42no—
587C42—84no—
588C23—22no—
589C23—22no—
590C23—24no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—48no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—84no—
602C23—84no—
603C23—84no—
604C23—84no—
605C23—84no—
606C23—164no—
607C23—164no—
608C23—164no—
609C23—164no—
610C23#53612yes—
611C23D4212yes—
612C23#54812yes—
613C23C23×D411yescharacteristic
614Q8—164no—
615D4—24no—
616D4—24no—
617D4—48no—
618D4—42no—
619D4—42no—
620D4—42no—
621D4—42no—
622D4—42no—
623D4—42no—
624D4—816no—
625D4—816no—
626D4—84no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—164no—
663C2×C4—164no—
664C2×C4#53612yes—
665C2×C4#53614yes—
666C2×C4#54214yes—
667C2×C4D4212yes—
668C2×C4#54714yes—
669C2×C4#54812yes—
670C2×C4#56314yes—
671C2×C4C23×D411yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×2−1+411yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C2×C4C2×2−1+411yescharacteristic
677C8—22no—
678C8—22no—
679C8—42no—
680C8—42no—
681C8—42no—
682C22—22no—
683C22—22no—
684C22—24no—
685C22—24no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—84no—
690C22—84no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—84no—
695C22—84no—
696C22—168no—
697C22—168no—
698C22#267414yes—
699C22C2×#53611yescharacteristic
700C22C2×D4211yescharacteristic
701C22C2×#54811yescharacteristic
702C22#358512yes—
703C22#358512yes—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—42no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—81no—
712C4—81no—
713C4—81no—
714C4—82no—
715C4—81no—
716C4—82no—
717C4#267414yes—
718C4C2×#53611yescharacteristic
719C4C2×#54211yescharacteristic
720C4C2×#54711yescharacteristic
721C4C2×#56311yescharacteristic
722C2—24no—
723C2—84no—
724C2—162no—
725C2C2×#267411yescharacteristic
726C2#6088312yes—
727C2#6105811yescharacteristic
728C2#6105911yescharacteristic
729C2#6133712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
2. C2×#3081C2semidirect
3. #35640C2semidirect
5. #30835C2semidirect
6. #30833C2semidirect
7. #30827C2semidirect
8. #30825C2semidirect
9. #23735C2semidirect
10. #23632C2semidirect
11. #23612C2semidirect
12. #23553C2semidirect
14. #23456C2semidirect
15. #21154C2semidirect
16. #21065C2semidirect
17. #21063C2semidirect
18. #21021C2semidirect
19. #20876C2semidirect
28. C2×#478C22semidirect
29. C2×#478C22semidirect
30. C2×#476C22semidirect
31. C2×#474C22semidirect
48. #2552C22semidirect
50. #2550C22semidirect
67. #2094C22semidirect
68. #2093C22semidirect
69. #2092C22semidirect
70. #2092C22semidirect
71. #2073C22semidirect
72. #2071C22semidirect
73. #2067C22semidirect
74. #2065C22semidirect
75. #2059C22semidirect
76. #2059C22semidirect
79. #2025C22semidirect
80. #2023C22semidirect
81. #1944C22semidirect
82. #1944C22semidirect
83. #1943C22semidirect
84. #1943C22semidirect
85. #1926C22semidirect
86. #1926C22semidirect
87. #1925C22semidirect
88. #1925C22semidirect
89. #1917C22semidirect
90. #1917C22semidirect
91. #1864C22semidirect
92. #1862C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4902
C2×#30812
#356402
#355542
#308352
#308332
#308272
#308252
#237352
#236322
#236122
#235532
#235052
#234562
#211542
#210652
#210632
#210212
#208762