Finite Group Databasebeta

← #7503440

Subgroups — #7503440

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3099C211yesmaximal, characteristic
2C2×#3084C211yesmaximal, characteristic, complemented
3#35620C211yesmaximal, characteristic, complemented
4#35618C211yesmaximal, characteristic
5#31093C214yesmaximal, complemented
6#31090C214yesmaximal, complemented
7#31088C214yesmaximal, complemented
8#31087C214yesmaximal, complemented
9#23668C211yesmaximal, characteristic, complemented
10#23659C211yesmaximal, characteristic, complemented
11#23640C211yesmaximal, characteristic, complemented
12#23631C211yesmaximal, characteristic, complemented
13#23555C211yesmaximal, characteristic, complemented
14#23488C211yesmaximal, characteristic
15#21083C211yesmaximal, characteristic, complemented
16#21046C211yesmaximal, characteristic, complemented
17#20881C211yesmaximal, characteristic, complemented
18#20847C211yesmaximal, characteristic, complemented
19C2×#1846C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#537C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#520C2211yescharacteristic
24#3099—28no—
25C2×#481C2212yes—
26C2×#485C2212yes—
27C2×#482C2212yes—
28C2×#483C2212yes—
29#3084—28no—
30C2×#479C2212yes—
31C2×#475C2212yes—
32C2×#478C2212yes—
33C2×#476C2212yes—
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C2×#424C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#417C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C2×#408C2211yescharacteristic
46C2×#408C2211yescharacteristic
47C22×C4×C8C2211yescharacteristic
48#2568—21no—
49#2566—21no—
50#2565—21no—
51#2563—21no—
52#2504—21no—
53#2503—21no—
54#2500—21no—
55#2499—21no—
56#2450—24no—
57#2444—24no—
58#2442C2214yes—
59#2439C2214yes—
60#2421C2214yes—
61#2420C2214yes—
62#2317C2211yescharacteristic
63#2317C2211yescharacteristic
64#2311C2211yescharacteristic
65#2311C2211yescharacteristic
66#2297C2211yescharacteristic
67#2297C2211yescharacteristic
68#2296C2211yescharacteristic
69C2×#382C2211yescharacteristic
70C2×#382C2211yescharacteristic
71#2088C2214yes—
72#2088C2214yes—
73#2085C2214yes—
74#2085C2214yes—
75#2077C2214yes—
76#2077C2214yes—
77#2074C2214yes—
78#2073C2214yes—
79#2059C2214yes—
80#2057C2214yes—
81#2036C2214yes—
82#2033C2214yes—
83#1934C2212yes—
84#1934C2212yes—
85#1933C2212yes—
86#1933C2212yes—
87#1923C2214yes—
88#1923C2214yes—
89#1863C2214yes—
90#1862C2214yes—
91#1854C2214yes—
92#1852C2214yes—
93#1846C2214yes—
94#1846C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1736C2211yescharacteristic
99#1735C2211yescharacteristic
100#1735C2211yescharacteristic
101#1729C2211yescharacteristic
102#1728C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C22×#41C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#544—24no—
110#544—42no—
111#537—24no—
112#537—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8C2312yes—
120C2×#121—22no—
121C2×#121—21no—
122C2×#121C2312yes—
123#520—24no—
124#520—24no—
125#520—42no—
126#520—42no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4—22no—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2312yes—
145C22×C42C2311yescharacteristic
146#485—28no—
147#483—28no—
148#482—28no—
149#481—28no—
150#479—28no—
151#478—28no—
152#476—28no—
153#475—28no—
154#428—24no—
155#428—24no—
156C2×#107—42no—
157C2×#107—42no—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2314yes—
161C2×#107C2314yes—
162C2×#106—24no—
163C2×#106—24no—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170#424—24no—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173#417—24no—
174#417—24no—
175#416—24no—
176#416D418yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2311yescharacteristic
181C2×#103C2311yescharacteristic
182C2×#102C2312yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2311yescharacteristic
186C2×#102C2311yescharacteristic
187#408—24no—
188#408—24no—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2314yes—
192C2×C4×C8—24no—
193C2×C4×C8C2312yes—
194C2×C4×C8C2312yes—
195#400—24no—
196#400—24no—
197#398—24no—
198#397—24no—
199#397—24no—
200#396—24no—
201#394—24no—
202#393—24no—
203#391—21no—
204#391—21no—
205#391—24no—
206#391D411yescharacteristic
207#391D411yescharacteristic
208#391C2312yes—
209#390—21no—
210#390—21no—
211#390D411yescharacteristic
212#390D411yescharacteristic
213#390C2312yes—
214#389C2314yes—
215#389C2314yes—
216#389C2312yes—
217#389C2312yes—
218#388—24no—
219#387—24no—
220#387—24no—
221#386—22no—
222#386—22no—
223#386—22no—
224#386—24no—
225#386D414yes—
226#385—22no—
227#385—24no—
228#385—22no—
229#385—22no—
230#384—21no—
231#384—21no—
232#382—21no—
233#382—21no—
234#382—21no—
235#382—21no—
236#382—21no—
237#382—21no—
238#382C2314yes—
239#382C2314yes—
240#381—24no—
241#380—24no—
242#380C2314yes—
243#380C2314yes—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C22⋊C4—21no—
247C4×C22⋊C4—21no—
248C4×C22⋊C4C2314yes—
249C2×#95—22no—
250C2×#95—22no—
251#340C2312yes—
252#340C2314yes—
253#340C2312yes—
254#340C2314yes—
255#340C2314yes—
256#340C2312yes—
257#340C2312yes—
258#340C2314yes—
259#336C2314yes—
260#336C2314yes—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—22no—
270C2×#41—22no—
271C2×#41—22no—
272C2×#41C2×D414yes—
273C22×Q8—22no—
274C22×Q8C2×D411yescharacteristic
275C22×Q8C2411yescharacteristic
276C22×D4—21no—
277C22×D4—22no—
278C22×D4C2×D412yes—
279C22×D4C2×D411yescharacteristic
280C22×D4C2411yescharacteristic
281C23×C4—21no—
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—22no—
285C23×C4—21no—
286C23×C4—21no—
287C23×C4—22no—
288C23×C4C2×D412yes—
289C23×C4C2412yes—
290C23×C4C2411yescharacteristic
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8C2×D414yes—
302C22×C8#4114yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2411yescharacteristic
306C22×C8C2411yescharacteristic
307C22×C8C2412yes—
308C4⋊Q8—24no—
309C4⋊Q8—42no—
310C4⋊Q8—44no—
311#125—24no—
312#125—42no—
313#125—44no—
314C22⋊Q8—24no—
315C22⋊Q8—48no—
316C22⋊Q8—42no—
317C22⋊Q8—44no—
318#121—24no—
319#121—42no—
320#121—44no—
321#121—48no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×D4—24no—
326C4×D4—42no—
327C4×D4—44no—
328C4×D4—48no—
329C4×D4—48no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—24no—
338#117—42no—
339#117—44no—
340#117C2×D414yes—
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—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2411yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4C2×D412yes—
388C2×C22⋊C4C2×D414yes—
389C2×C22⋊C4C2×D414yes—
390C2×C42—21no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2412yes—
400C2×C42C2412yes—
401#107—24no—
402#107—24no—
403#107—44no—
404#107—44no—
405#106—24no—
406#106—24no—
407#106—24no—
408#106—24no—
409#106—24no—
410#106—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103C2×D414yes—
417#103C2×D414yes—
418#102—24no—
419#102—24no—
420#102—24no—
421#102C2×D414yes—
422#102C2×D414yes—
423C22⋊C8—24no—
424C22⋊C8—24no—
425C22⋊C8—28no—
426C4×C8—24no—
427C4×C8—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—24no—
437#95—24no—
438#95—24no—
439#95—24no—
440#95C2×D414yes—
441#95C2×D414yes—
442C24—21no—
443C24C22×D412yes—
444C24C22×D411yescharacteristic
445#41—28no—
446#41—28no—
447#41—44no—
448#41—44no—
449#41—44no—
450#41—44no—
451#41—44no—
452#41—44no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—44no—
462C2×Q8—84no—
463C2×Q8C22×D412yes—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—44no—
479C2×D4—84no—
480C2×D4C22×D412yes—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—82no—
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×D411yescharacteristic
520C22×C4C22×D412yes—
521C22×C4C2×#4112yes—
522C22×C4H(5,𝔽2)12yes—
523C22×C42−1+412yes—
524C22×C42−1+412yes—
525C22×C42−1+412yes—
526C22×C4C2511yescharacteristic
527M16—44no—
528M16—44no—
529C2×C8—28no—
530C2×C8—28no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8C22×D412yes—
544C2×C8C22×D412yes—
545C2×C8C2×#4114yes—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
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—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—84no—
573C4⋊C4—84no—
574C4⋊C4—82no—
575C4⋊C4—82no—
576C4⋊C4C22×D412yes—
577C4⋊C4C22×D412yes—
578C22⋊C4—24no—
579C22⋊C4—24no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—42no—
587C22⋊C4—42no—
588C22⋊C4—82no—
589C22⋊C4—82no—
590C22⋊C4—84no—
591C22⋊C4—84no—
592C42—24no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—44no—
600C42—44no—
601C42—42no—
602C42—42no—
603C23—22no—
604C23—22no—
605C23—24no—
606C23—24no—
607C23—22no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—42no—
612C23—42no—
613C23—41no—
614C23—41no—
615C23#53612yes—
616C23D4212yes—
617C23#54712yes—
618C23C23×D411yescharacteristic
619Q8—22no—
620Q8—22no—
621Q8—44no—
622Q8—41no—
623Q8—41no—
624Q8—164no—
625D4—22no—
626D4—22no—
627D4—48no—
628D4—48no—
629D4—44no—
630D4—41no—
631D4—41no—
632D4—164no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—48no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—48no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—164no—
674C2×C4—164no—
675C2×C4—162no—
676C2×C4—162no—
677C2×C4—162no—
678C2×C4—162no—
679C2×C4#53614yes—
680C2×C4#53612yes—
681C2×C4#53914yes—
682C2×C4D4212yes—
683C2×C4#54712yes—
684C2×C4#54714yes—
685C2×C4#56314yes—
686C2×C4C23×D411yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C2×C4C2×2−1+411yescharacteristic
691C2×C4C2×2−1+411yescharacteristic
692C8—24no—
693C8—22no—
694C8—22no—
695C8—42no—
696C8—42no—
697C22—24no—
698C22—22no—
699C22—22no—
700C22—24no—
701C22—44no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—44no—
709C22—82no—
710C22—82no—
711C22#268014yes—
712C22C2×#53611yescharacteristic
713C22C2×D4211yescharacteristic
714C22C2×#54711yescharacteristic
715C22#358412yes—
716C22#358412yes—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4#268014yes—
732C4C2×#53611yescharacteristic
733C4C2×#53911yescharacteristic
734C4C2×#54711yescharacteristic
735C4C2×#56311yescharacteristic
736C2—24no—
737C2—82no—
738C2—82no—
739C2C2×#268011yescharacteristic
740C2#6088212yes—
741C2#6108811yescharacteristic
742C2#6108811yescharacteristic
743C2#6126112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3084C2semidirect
3. #35620C2semidirect
5. #31093C2semidirect
6. #31090C2semidirect
7. #31088C2semidirect
8. #31087C2semidirect
9. #23668C2semidirect
10. #23659C2semidirect
11. #23640C2semidirect
12. #23631C2semidirect
13. #23555C2semidirect
15. #21083C2semidirect
16. #21046C2semidirect
17. #20881C2semidirect
18. #20847C2semidirect
19. C2×#1846C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30992
C2×#30842
#356202
#356182
#310932
#310902
#310882
#310872
#236682
#236592
#236402
#236312
#235552
#234882
#210832
#210462
#208812
#208472
C2×#18462