Finite Group Databasebeta

← #7503445

Subgroups — #7503445

Order 512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3099C211yesmaximal, characteristic
2C2×#3083C211yesmaximal, characteristic, complemented
3#35618C211yesmaximal, characteristic
4#35532C211yesmaximal, characteristic, complemented
5#31101C214yesmaximal, complemented
6#31098C214yesmaximal, complemented
7#31096C214yesmaximal, complemented
8#31095C214yesmaximal, complemented
9#23668C211yesmaximal, characteristic, complemented
10#23659C211yesmaximal, characteristic, complemented
11#23639C211yesmaximal, characteristic, complemented
12#23630C211yesmaximal, characteristic, complemented
13#23556C211yesmaximal, characteristic, complemented
14#23489C211yesmaximal, characteristic
15#21086C211yesmaximal, characteristic, complemented
16#21047C211yesmaximal, characteristic, complemented
17#20881C211yesmaximal, characteristic, complemented
18#20847C211yesmaximal, characteristic, complemented
19C2×#1847C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#537C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#520C2211yescharacteristic
24#3099—28no—
25C2×#481C2212yes—
26C2×#485C2212yes—
27C2×#482C2212yes—
28C2×#483C2212yes—
29#3083—28no—
30C2×#479C2212yes—
31C2×#477C2212yes—
32C2×#478C2212yes—
33C2×#474C2212yes—
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×#97C2211yescharacteristic
48#2567—21no—
49#2566—21no—
50#2565—21no—
51#2542—21no—
52#2504—21no—
53#2503—21no—
54#2500—21no—
55#2499—21no—
56#2449—24no—
57#2444—24no—
58#2441C2214yes—
59#2440C2214yes—
60#2421C2214yes—
61#2420C2214yes—
62#2317C2211yescharacteristic
63#2317C2211yescharacteristic
64#2311C2211yescharacteristic
65#2311C2211yescharacteristic
66#2297C2211yescharacteristic
67#2297C2211yescharacteristic
68#2296C2211yescharacteristic
69C2×#384C2211yescharacteristic
70C2×#382C2211yescharacteristic
71#2088C2214yes—
72#2088C2214yes—
73#2085C2214yes—
74#2085C2214yes—
75#2076C2214yes—
76#2075C2214yes—
77#2072C2214yes—
78#2071C2214yes—
79#2059C2214yes—
80#2058C2214yes—
81#2035C2214yes—
82#2034C2214yes—
83#1936C2212yes—
84#1936C2212yes—
85#1935C2212yes—
86#1935C2212yes—
87#1924C2214yes—
88#1924C2214yes—
89#1863C2214yes—
90#1862C2214yes—
91#1854C2214yes—
92#1852C2214yes—
93#1847C2214yes—
94#1847C2214yes—
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⋊Q8—24no—
120C2×C22⋊Q8C2312yes—
121C2×#121—22no—
122C2×#121—21no—
123C2×#121C2312yes—
124#520—24no—
125#520—24no—
126#520—42no—
127#520—42no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×D4—21no—
131C2×C4×D4—22no—
132C2×C4×D4—22no—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2312yes—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—22no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C42C2311yescharacteristic
148#485—28no—
149#483—28no—
150#482—28no—
151#481—28no—
152#479—28no—
153#478—28no—
154#477—28no—
155#474—28no—
156#428—24no—
157#428—24no—
158C2×#107—24no—
159C2×#107—42no—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#107C2312yes—
164C2×#107C2314yes—
165C2×#106—24no—
166C2×#106—42no—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2314yes—
172#424—24no—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175#417—24no—
176#417—24no—
177#416—24no—
178#416D418yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2311yescharacteristic
183C2×#103C2311yescharacteristic
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2311yescharacteristic
188C2×#102C2311yescharacteristic
189#408—24no—
190#408—24no—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×#97—24no—
195C2×#97C2312yes—
196C2×#97C2312yes—
197#400—24no—
198#399—24no—
199#398—24no—
200#397—24no—
201#397—24no—
202#396—24no—
203#395—24no—
204#394—24no—
205#391—21no—
206#391—21no—
207#391D411yescharacteristic
208#391D411yescharacteristic
209#391C2312yes—
210#390—21no—
211#390—21no—
212#390D411yescharacteristic
213#390D411yescharacteristic
214#390C2312yes—
215#389—24no—
216#389C2314yes—
217#389C2314yes—
218#389C2312yes—
219#389C2312yes—
220#388—24no—
221#387—24no—
222#387—24no—
223#386—22no—
224#386—22no—
225#386—22no—
226#386—24no—
227#386D414yes—
228#385—22no—
229#385—24no—
230#385—22no—
231#385—22no—
232#384—21no—
233#384—21no—
234#384—21no—
235#384—21no—
236#384C2314yes—
237#382—21no—
238#382—21no—
239#382—21no—
240#382—21no—
241#382C2314yes—
242#381—24no—
243#380C2314yes—
244#380C2314yes—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C22⋊C4—21no—
248C4×C22⋊C4—21no—
249C4×C22⋊C4C2314yes—
250C2×#95—22no—
251#340C2312yes—
252#340C2314yes—
253#340C2312yes—
254#340C2314yes—
255#340C2312yes—
256#340C2314yes—
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—
318C22⋊Q8—88no—
319#121—24no—
320#121—42no—
321#121—44no—
322#121—48no—
323C4×Q8—24no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×D4—24no—
327C4×D4—42no—
328C4×D4—44no—
329C4×D4—48no—
330C4×D4—48no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—24no—
339#117—42no—
340#117—44no—
341#117C2×D414yes—
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—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—42no—
368C2×C4⋊C4—44no—
369C2×C4⋊C4—84no—
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2411yescharacteristic
379C2×C4⋊C4C2411yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D414yes—
393C2×C22⋊C4C2×D414yes—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2412yes—
404C2×C42C2412yes—
405#107—24no—
406#107—24no—
407#107—24no—
408#107—24no—
409#107—44no—
410#106—24no—
411#106—24no—
412#106—24no—
413#106—24no—
414#106—44no—
415C4⋊C8—24no—
416C4⋊C8—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103C2×D414yes—
421#103C2×D414yes—
422#102—24no—
423#102—24no—
424#102—24no—
425#102C2×D414yes—
426#102C2×D414yes—
427C22⋊C8—24no—
428C22⋊C8—24no—
429C22⋊C8—28no—
430#97—24no—
431#97—24no—
432#95—24no—
433#95—24no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—24no—
439#95—24no—
440#95—24no—
441#95—24no—
442#95C2×D414yes—
443#95C2×D414yes—
444C24—21no—
445C24C22×D412yes—
446C24C22×D411yescharacteristic
447#41—28no—
448#41—28no—
449#41—44no—
450#41—44no—
451#41—44no—
452#41—44no—
453#41—44no—
454#41—44no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—44no—
464C2×Q8—84no—
465C2×Q8C22×D412yes—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—84no—
482C2×D4C22×D412yes—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
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—22no—
499C22×C4—22no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—42no—
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—84no—
514C22×C4—82no—
515C22×C4—82no—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D412yes—
523C22×C4C2×#4112yes—
524C22×C4H(5,𝔽2)12yes—
525C22×C42−1+412yes—
526C22×C42−1+412yes—
527C22×C42−1+412yes—
528C22×C4C2511yescharacteristic
529M16—44no—
530M16—44no—
531C2×C8—28no—
532C2×C8—28no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8C22×D412yes—
546C2×C8C22×D412yes—
547C2×C8C2×#4114yes—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
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—22no—
563C4⋊C4—22no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—48no—
572C4⋊C4—44no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—84no—
578C4⋊C4—84no—
579C4⋊C4—84no—
580C4⋊C4C22×D412yes—
581C4⋊C4C22×D412yes—
582C22⋊C4—24no—
583C22⋊C4—24no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—44no—
587C22⋊C4—44no—
588C22⋊C4—44no—
589C22⋊C4—44no—
590C22⋊C4—42no—
591C22⋊C4—42no—
592C22⋊C4—82no—
593C22⋊C4—82no—
594C22⋊C4—84no—
595C22⋊C4—84no—
596C42—24no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—44no—
604C42—44no—
605C42—42no—
606C42—42no—
607C23—22no—
608C23—22no—
609C23—24no—
610C23—24no—
611C23—22no—
612C23—22no—
613C23—22no—
614C23—22no—
615C23—42no—
616C23—42no—
617C23—41no—
618C23—41no—
619C23#53612yes—
620C23D4212yes—
621C23#54712yes—
622C23C23×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—44no—
626Q8—41no—
627Q8—41no—
628Q8—164no—
629D4—22no—
630D4—22no—
631D4—48no—
632D4—48no—
633D4—44no—
634D4—41no—
635D4—41no—
636D4—164no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—48no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—48no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—164no—
678C2×C4—164no—
679C2×C4—162no—
680C2×C4—162no—
681C2×C4—162no—
682C2×C4—162no—
683C2×C4#53614yes—
684C2×C4#53612yes—
685C2×C4#53914yes—
686C2×C4D4212yes—
687C2×C4#54712yes—
688C2×C4#54714yes—
689C2×C4#56314yes—
690C2×C4C23×D411yescharacteristic
691C2×C4C22×#4111yescharacteristic
692C2×C4C2×H(5,𝔽2)11yescharacteristic
693C2×C4C2×2−1+411yescharacteristic
694C2×C4C2×2−1+411yescharacteristic
695C2×C4C2×2−1+411yescharacteristic
696C8—24no—
697C8—22no—
698C8—22no—
699C8—42no—
700C8—42no—
701C22—24no—
702C22—22no—
703C22—22no—
704C22—24no—
705C22—44no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—44no—
713C22—82no—
714C22—82no—
715C22#268014yes—
716C22C2×#53611yescharacteristic
717C22C2×D4211yescharacteristic
718C22C2×#54711yescharacteristic
719C22#358512yes—
720C22#358612yes—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4#268014yes—
736C4C2×#53611yescharacteristic
737C4C2×#53911yescharacteristic
738C4C2×#54711yescharacteristic
739C4C2×#56311yescharacteristic
740C2—24no—
741C2—82no—
742C2—82no—
743C2C2×#268011yescharacteristic
744C2#6088712yes—
745C2#6108211yescharacteristic
746C2#6108311yescharacteristic
747C2#6126612yes—
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×#3083C2semidirect
4. #35532C2semidirect
5. #31101C2semidirect
6. #31098C2semidirect
7. #31096C2semidirect
8. #31095C2semidirect
9. #23668C2semidirect
10. #23659C2semidirect
11. #23639C2semidirect
12. #23630C2semidirect
13. #23556C2semidirect
15. #21086C2semidirect
16. #21047C2semidirect
17. #20881C2semidirect
18. #20847C2semidirect
19. C2×#1847C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30992
C2×#30832
#356182
#355322
#311012
#310982
#310962
#310952
#236682
#236592
#236392
#236302
#235562
#234892
#210862
#210472
#208812
#208472
C2×#18472