Finite Group Databasebeta

← #7503486

Subgroups — #7503486

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×#3082C211yesmaximal, characteristic, complemented
3#35672C211yesmaximal, characteristic
4#35151C211yesmaximal, characteristic
5#31109C214yesmaximal, complemented
6#31107C214yesmaximal, complemented
7#31085C214yesmaximal, complemented
8#31083C214yesmaximal, complemented
9#23667C211yesmaximal, characteristic, complemented
10#23661C211yesmaximal, characteristic, complemented
11#23638C211yesmaximal, characteristic
12#23633C211yesmaximal, characteristic, complemented
13#23556C211yesmaximal, characteristic, complemented
14#23481C211yesmaximal, characteristic
15#21088C211yesmaximal, characteristic, complemented
16#21053C211yesmaximal, characteristic
17#20877C211yesmaximal, characteristic, complemented
18#20845C211yesmaximal, characteristic
19#20803C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#485—28no—
25C2×#485C2214yes—
26C2×#485C2214yes—
27#3082—28no—
28C2×#479C2212yescomplemented
29C2×#479C2212yescomplemented
30C2×#477C2212yescomplemented
31C2×#475C2212yescomplemented
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C2×#407C2211yescharacteristic
44C22×C22⋊C8C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2855—22no—
47#2626—22no—
48#2484—22no—
49#2482—22no—
50#2441—22no—
51#2441C2214yes—
52#2439—22no—
53#2439C2214yes—
54#2423—22no—
55#2423C2214yes—
56#2421—22no—
57#2421C2214yes—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2317C2211yescharacteristic
62#2316C2211yescharacteristic
63#2310C2211yescharacteristic
64#2280C2211yescharacteristic
65#2280C2211yescharacteristic
66#2280C2211yescharacteristic
67#2087C2214yescomplemented
68#2086C2214yescomplemented
69#2085C2214yescomplemented
70#2084C2214yescomplemented
71#2077C2212yes—
72#2077C2212yes—
73#2076C2212yes—
74#2075C2212yes—
75#2074C2214yescomplemented
76#2072C2214yescomplemented
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2036C2214yes—
80#2035C2214yes—
81#1935C2214yescomplemented
82#1933C2214yescomplemented
83#1924C2214yes—
84#1923C2214yes—
85#1865C2214yescomplemented
86#1863C2214yescomplemented
87#1854C2214yes—
88#1854C2214yes—
89#1847C2212yescomplemented
90#1847C2212yescomplemented
91#1846C2212yescomplemented
92#1845C2212yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C23×Q8C2311yescharacteristic
102C24×C4C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—24no—
108#544—42no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#125C2312yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—42no—
117C2×C22⋊Q8C2314yes—
118#519—24no—
119#519—42no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×#117—22no—
125C2×#117—22no—
126C2×#117—24no—
127C2×#117—42no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2314yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C42—21no—
142#485—216no—
143#485—216no—
144#479—28no—
145#479—28no—
146#477—28no—
147#475—28no—
148#428—24no—
149#428—42no—
150#428—42no—
151#427—24no—
152C2×#107—24no—
153C2×#107C2311yescharacteristic
154C2×#107C2312yes—
155C2×#107C2311yescharacteristic
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#106—24no—
161C2×#106C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2311yescharacteristic
164C2×#106C2314yes—
165C2×#106C2312yes—
166C2×#106C2312yes—
167#424—24no—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170#419—24no—
171#419—24no—
172C2×#103—24no—
173C2×#103D418yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180#407—24no—
181C2×C22⋊C8—24no—
182C2×C22⋊C8C2312yes—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2314yes—
185#405—24no—
186C2×#97C2312yes—
187C2×C4×C8C2312yes—
188#400—24no—
189#399—24no—
190#399—24no—
191#398—24no—
192#397—24no—
193#396—24no—
194#396—24no—
195#393—24no—
196#391—22no—
197#391—24no—
198#391C2314yes—
199#389—22no—
200#389—22no—
201#389—22no—
202#389—24no—
203#389C2314yes—
204#389C2314yes—
205#389C2314yes—
206#388—24no—
207#388—24no—
208#387—24no—
209#387—24no—
210#387—24no—
211#387—24no—
212#385—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#384—21no—
217#384—21no—
218#384C2312yes—
219#384C2312yes—
220#382—21no—
221#382—21no—
222#382C2312yes—
223#382C2312yes—
224#381—21no—
225#381—21no—
226#381—22no—
227#381D412yes—
228#380—22no—
229#380D412yes—
230#380D412yes—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234#380C2312yes—
235#379—21no—
236#379—21no—
237#379D412yes—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2312yes—
242#340C2314yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C25C2×D411yescharacteristic
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8C2×D414yes—
261C22×Q8C2412yes—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D412yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16#4114yes—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8C2×D414yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2411yescharacteristic
291C22×C8C2411yescharacteristic
292C22×C8C2412yes—
293C4⋊Q8—24no—
294C4⋊Q8—42no—
295C4⋊Q8—44no—
296#125—24no—
297#125—42no—
298#125—44no—
299C22⋊Q8—28no—
300C22⋊Q8—48no—
301C22⋊Q8—44no—
302C22⋊Q8—48no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—42no—
306C4×Q8—42no—
307C4×Q8—44no—
308C4×Q8—44no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—28no—
322#117—48no—
323#117—44no—
324#117—48no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—24no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—44no—
382C2×C42C2412yes—
383C2×C42C2412yes—
384#107—24no—
385#107—24no—
386#107—22no—
387#107—22no—
388#107—24no—
389#107—42no—
390#107—42no—
391#106—22no—
392#106—22no—
393#106—28no—
394#106—24no—
395#106—42no—
396#106—42no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399#103—28no—
400#103—24no—
401#103—24no—
402#103—28no—
403#103C2×D418yes—
404#103C2×D418yes—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—28no—
408#97—24no—
409C4×C8—24no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95C2×D414yes—
422#95C2×D414yes—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—42no—
427C24—42no—
428C24C22×D412yes—
429C24C22×D411yescharacteristic
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—28no—
433C2×Q8—28no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—84no—
444C2×Q8C22×D414yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492M16—28no—
493M16—44no—
494C2×C8—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—44no—
508C2×C8C22×D412yes—
509C2×C8C22×D412yes—
510C2×C8C2×#4114yes—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4C22×D414yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—48no—
565C42—44no—
566C42—44no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42—42no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—84no—
584C23—84no—
585C23#53612yes—
586C23D4212yes—
587C23#54712yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—48no—
592Q8—42no—
593Q8—42no—
594Q8—164no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4#53614yes—
639C2×C4#53612yes—
640C2×C4#53914yes—
641C2×C4D4212yes—
642C2×C4#54712yes—
643C2×C4#54714yes—
644C2×C4#56314yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—24no—
652C8—22no—
653C8—22no—
654C8—42no—
655C8—42no—
656C22—24no—
657C22—22no—
658C22—22no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—88no—
668C22#268014yes—
669C22C2×#53611yescharacteristic
670C22C2×D4211yescharacteristic
671C22C2×#54711yescharacteristic
672C22#358412yes—
673C22#358612yes—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4#268014yes—
689C4C2×#53611yescharacteristic
690C4C2×#53911yescharacteristic
691C4C2×#54711yescharacteristic
692C4C2×#56311yescharacteristic
693C2—24no—
694C2—82no—
695C2C2×#268011yescharacteristic
696C2#6087612yes—
697C2#6105411yescharacteristic
698C2#6106011yescharacteristic
699C2#6127512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3082C2semidirect
5. #31109C2semidirect
6. #31107C2semidirect
7. #31085C2semidirect
8. #31083C2semidirect
9. #23667C2semidirect
10. #23661C2semidirect
12. #23633C2semidirect
13. #23556C2semidirect
15. #21088C2semidirect
17. #20877C2semidirect
19. #20803C2semidirect
28. C2×#479C22semidirect
29. C2×#479C22semidirect
30. C2×#477C22semidirect
31. C2×#475C22semidirect
67. #2087C22semidirect
68. #2086C22semidirect
69. #2085C22semidirect
70. #2084C22semidirect
75. #2074C22semidirect
76. #2072C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
81. #1935C22semidirect
82. #1933C22semidirect
85. #1865C22semidirect
86. #1863C22semidirect
89. #1847C22semidirect
90. #1847C22semidirect
91. #1846C22semidirect
92. #1845C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#30822
#356722
#351512
#311092
#311072
#310852
#310832
#236672
#236612
#236382
#236332
#235562
#234812
#210882
#210532
#208772
#208452
#208032