Finite Group Databasebeta

← #7502030

Subgroups — #7502030

Order 512.

1914 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
1C2×#3136C211yesmaximal, characteristic
2C2×#3083C211yesmaximal, characteristic, complemented
3#35531C211yesmaximal, characteristic
4#35330C211yesmaximal, characteristic, complemented
5#30830C214yesmaximal, complemented
6#30825C214yesmaximal, complemented
7#30820C214yesmaximal, complemented
8#30807C214yesmaximal, complemented
9C2×#2092C211yesmaximal, characteristic, complemented
10#23631C211yesmaximal, characteristic, complemented
11#23612C211yesmaximal, characteristic, complemented
12#23556C211yesmaximal, characteristic, complemented
13#23509C211yesmaximal, characteristic
14#23456C211yesmaximal, characteristic, complemented
15#21156C211yesmaximal, characteristic, complemented
16#21060C211yesmaximal, characteristic, complemented
17#21058C211yesmaximal, characteristic, complemented
18#21025C211yesmaximal, characteristic, complemented
19#20882C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#520C2211yescharacteristic
24#3136—28no—
25C2×#490C2212yes—
26C2×#489C2212yes—
27C2×#488C2212yes—
28C2×#486C2212yes—
29#3083—28no—
30C2×#479C2212yescomplemented
31C2×#477C2212yescomplemented
32C2×#478C2212yescomplemented
33C2×#474C2212yescomplemented
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2862—21no—
49#2859—21no—
50#2660—21no—
51#2660—21no—
52#2550C2214yescomplemented
53#2550C2214yescomplemented
54#2549—24no—
55#2504—21no—
56#2501—21no—
57#2500—21no—
58#2500—21no—
59#2448C2214yes—
60#2445C2214yes—
61#2418—24no—
62#2317C2211yescharacteristic
63#2317C2211yescharacteristic
64#2317C2211yescharacteristic
65#2309C2211yescharacteristic
66#2297C2211yescharacteristic
67C2×#390C2211yescharacteristic
68#2285C2211yescharacteristic
69C2×#382C2211yescharacteristic
70C4×#117C2211yescharacteristic
71#2092C2214yes—
72#2092C2214yes—
73#2074C2214yescomplemented
74#2073C2214yescomplemented
75#2067C2214yes—
76#2065C2214yes—
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2042C2214yes—
80#2041C2214yes—
81#2025C2214yes—
82#2023C2214yes—
83#1944C2214yescomplemented
84#1942C2214yescomplemented
85#1928C2212yescomplemented
86#1927C2212yescomplemented
87#1926C2214yes—
88#1926C2212yescomplemented
89#1925C2214yes—
90#1925C2212yescomplemented
91#1917C2214yescomplemented
92#1916C2214yescomplemented
93#1865C2214yescomplemented
94#1864C2214yescomplemented
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101#1728C2211yescharacteristic
102C2×#336C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C23×D4C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#544—24no—
110#544—24no—
111#544—42no—
112#544—42no—
113#535—24no—
114#535—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×#127C2312yes—
120C2×#125—21no—
121C2×#125—21no—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×#124C2312yes—
125C2×#123—24no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—22no—
128C2×#121—21no—
129C2×#121—21no—
130C2×#120—22no—
131#520—24no—
132#520—42no—
133C2×C4×Q8C2312yes—
134C2×C4×D4—22no—
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C22⋊C4—22no—
147C22×C42C2311yescharacteristic
148#490—28no—
149#489—28no—
150#488—28no—
151#486—28no—
152#479—28no—
153#478—28no—
154#477—28no—
155#474—28no—
156#428—24no—
157#427—24no—
158C2×#107—24no—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#106—42no—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2314yes—
167#424—24no—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170#418—24no—
171#418—24no—
172#417—24no—
173#417—24no—
174#416—24no—
175#416D418yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#102—24no—
182C2×#102—42no—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2314yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2314yes—
192C2×#102C2311yescharacteristic
193C2×#102C2311yescharacteristic
194#405—24no—
195C2×#97C2312yes—
196C2×C4×C8C2312yes—
197#399—24no—
198#398—24no—
199#398—24no—
200#397—24no—
201#395—24no—
202#395—24no—
203#394—24no—
204#394—24no—
205#391—24no—
206#391—24no—
207#391C2312yes—
208#390—21no—
209#390—21no—
210#390—21no—
211#390—21no—
212#390C2314yes—
213#390C2312yes—
214#390C2314yes—
215#389—24no—
216#389—24no—
217#389C2314yes—
218#389C2314yes—
219#389C2314yes—
220#388—24no—
221#388—24no—
222#386—22no—
223#386—24no—
224#385—22no—
225#385—22no—
226#385—22no—
227#384—21no—
228#384—21no—
229#384—21no—
230#384D411yescharacteristic
231#384D411yescharacteristic
232#382—21no—
233#382—21no—
234#382—21no—
235#382—21no—
236#382—21no—
237#382D411yescharacteristic
238#382D411yescharacteristic
239#382C2314yes—
240#380D414yes—
241#379—22no—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2314yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—21no—
247C4×C22⋊C4—21no—
248#376—21no—
249C2×#95—22no—
250C43—21no—
251#340C2312yes—
252#340C2314yes—
253#340C2312yes—
254#340C2314yes—
255#340C2314yes—
256#336C2314yes—
257#336C2314yes—
258#336C2312yes—
259#336C2312yes—
260#336C2314yes—
261C25—21no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41C2×D414yes—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8C2411yescharacteristic
271C22×D4—21no—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4—22no—
277C22×D4—22no—
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—22no—
281C22×D4C2412yes—
282C22×D4C2411yescharacteristic
283C23×C4—21no—
284C23×C4—21no—
285C23×C4—21no—
286C23×C4—21no—
287C23×C4—22no—
288C23×C4C2×D412yes—
289C23×C4C2411yescharacteristic
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8C2×D414yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2411yescharacteristic
306C22×C8C2411yescharacteristic
307C4⋊Q8—24no—
308C4⋊Q8—24no—
309C4⋊Q8—42no—
310C4⋊Q8—42no—
311C4⋊Q8—44no—
312C4⋊Q8—44no—
313#127—24no—
314#127—42no—
315#125—24no—
316#125—24no—
317#125—42no—
318#125—42no—
319#125—44no—
320#125—44no—
321#124—24no—
322#124—42no—
323#123—88no—
324C22⋊Q8—48no—
325C22⋊Q8—48no—
326#121—44no—
327#121—44no—
328#120—48no—
329C4×Q8—24no—
330C4×Q8—42no—
331C4×D4—24no—
332C4×D4—42no—
333C4×D4—48no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—44no—
343#117C2×D414yes—
344#117#4114yes—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D414yes—
370C2×C4⋊C4C2×D414yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2411yescharacteristic
376C2×C4⋊C4C2411yescharacteristic
377C2×C4⋊C4C2412yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—84no—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D412yes—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2412yes—
404C2×C42C2412yes—
405C2×C42C2412yes—
406#107—24no—
407#107—24no—
408#107—24no—
409#107—24no—
410#106—24no—
411#106—24no—
412#106—44no—
413C4⋊C8—24no—
414C4⋊C8—24no—
415#103—24no—
416#103—24no—
417#103—24no—
418#103C2×D414yes—
419#103C2×D414yes—
420#102—24no—
421#102—24no—
422#102—24no—
423#102—24no—
424#102—24no—
425#102—24no—
426#102—24no—
427#102—28no—
428#102—44no—
429#102C2×D414yes—
430#102C2×D414yes—
431#97—24no—
432C4×C8—24no—
433#95—24no—
434#95—24no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—24no—
440#95—24no—
441#95—24no—
442#95C2×D414yes—
443#95C2×D414yes—
444C24—21no—
445C24—22no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—82no—
450C24—82no—
451C24C22×D411yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—84no—
464C2×Q8—84no—
465C2×Q8C22×D412yes—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—44no—
479C2×D4—44no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—44no—
483C2×D4—42no—
484C2×D4—44no—
485C2×D4—42no—
486C2×D4—44no—
487C2×D4—42no—
488C2×D4—44no—
489C2×D4—84no—
490C2×D4—84no—
491C2×D4C22×D412yes—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—84no—
511C22×C4—82no—
512C22×C4—82no—
513C22×C4—82no—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C2×#4112yes—
523C22×C4H(5,𝔽2)12yes—
524C22×C42−1+412yes—
525C22×C42−1+412yes—
526C22×C42−1+412yes—
527C22×C4C2511yescharacteristic
528M16—44no—
529M16—44no—
530C2×C8—28no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—44no—
545C2×C8C22×D412yes—
546C2×C8C22×D412yes—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—84no—
575C4⋊C4—84no—
576C4⋊C4—84no—
577C4⋊C4C22×D412yes—
578C4⋊C4C22×D412yes—
579C4⋊C4C2×#4114yes—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—24no—
583C22⋊C4—24no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—44no—
587C22⋊C4—48no—
588C22⋊C4—44no—
589C22⋊C4—42no—
590C22⋊C4—82no—
591C22⋊C4—84no—
592C22⋊C4—84no—
593C22⋊C4—84no—
594C42—22no—
595C42—22no—
596C42—24no—
597C42—24no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—44no—
605C42—42no—
606C42—82no—
607C42—82no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—42no—
612C23—42no—
613C23—41no—
614C23—41no—
615C23—44no—
616C23—44no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—88no—
621C23—84no—
622C23—162no—
623C23—162no—
624C23—164no—
625C23#53612yes—
626C23D4212yes—
627C23#54812yes—
628C23C23×D411yescharacteristic
629Q8—22no—
630Q8—22no—
631Q8—41no—
632Q8—41no—
633Q8—164no—
634Q8—164no—
635D4—22no—
636D4—22no—
637D4—48no—
638D4—42no—
639D4—41no—
640D4—42no—
641D4—42no—
642D4—41no—
643D4—42no—
644D4—84no—
645D4—164no—
646D4—164no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—24no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—48no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—84no—
682C2×C4—164no—
683C2×C4—162no—
684C2×C4—162no—
685C2×C4—162no—
686C2×C4#53612yes—
687C2×C4#53614yes—
688C2×C4#54214yes—
689C2×C4D4212yes—
690C2×C4#54714yes—
691C2×C4#54812yes—
692C2×C4#56314yes—
693C2×C4C23×D411yescharacteristic
694C2×C4C22×#4111yescharacteristic
695C2×C4C2×H(5,𝔽2)11yescharacteristic
696C2×C4C2×2−1+411yescharacteristic
697C2×C4C2×2−1+411yescharacteristic
698C2×C4C2×2−1+411yescharacteristic
699C8—22no—
700C8—22no—
701C8—42no—
702C8—42no—
703C8—42no—
704C22—22no—
705C22—22no—
706C22—24no—
707C22—24no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—84no—
717C22—84no—
718C22—168no—
719C22#267414yes—
720C22C2×#53611yescharacteristic
721C22C2×D4211yescharacteristic
722C22C2×#54811yescharacteristic
723C22#358412yes—
724C22#358512yes—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—42no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4#267414yes—
740C4C2×#53611yescharacteristic
741C4C2×#54211yescharacteristic
742C4C2×#54711yescharacteristic
743C4C2×#56311yescharacteristic
744C2—24no—
745C2—82no—
746C2—162no—
747C2C2×#267411yescharacteristic
748C2#6087512yes—
749C2#6108211yescharacteristic
750C2#6109011yescharacteristic
751C2#6134912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3083C2semidirect
4. #35330C2semidirect
5. #30830C2semidirect
6. #30825C2semidirect
7. #30820C2semidirect
8. #30807C2semidirect
9. C2×#2092C2semidirect
10. #23631C2semidirect
11. #23612C2semidirect
12. #23556C2semidirect
14. #23456C2semidirect
15. #21156C2semidirect
16. #21060C2semidirect
17. #21058C2semidirect
18. #21025C2semidirect
19. #20882C2semidirect
30. C2×#479C22semidirect
31. C2×#477C22semidirect
32. C2×#478C22semidirect
33. C2×#474C22semidirect
52. #2550C22semidirect
53. #2550C22semidirect
73. #2074C22semidirect
74. #2073C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
83. #1944C22semidirect
84. #1942C22semidirect
85. #1928C22semidirect
86. #1927C22semidirect
88. #1926C22semidirect
90. #1925C22semidirect
91. #1917C22semidirect
92. #1916C22semidirect
93. #1865C22semidirect
94. #1864C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31362
C2×#30832
#355312
#353302
#308302
#308252
#308202
#308072
C2×#20922
#236312
#236122
#235562
#235092
#234562
#211562
#210602
#210582
#210252
#208822