Finite Group Databasebeta

← #6519216

Subgroups — #6519216

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#34953C211yesmaximal, characteristic
3#29141C212yesmaximal, complemented
4#29123C212yesmaximal, complemented
5#29123C212yesmaximal, complemented
6#28717C212yesmaximal, complemented
7#27815C212yesmaximal, complemented
8#27769C212yesmaximal, complemented
9#27749C212yesmaximal, complemented
10#27729C212yesmaximal, complemented
11#27697C214yesmaximal
12#27258C214yesmaximal
13#21885C211yesmaximal, characteristic
14#21860C211yesmaximal, characteristic
15#20734C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic
17#20543C211yesmaximal, characteristic
18C2×#544—21no—
19C2×#533C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2214yes—
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2212yes—
25#2868C2212yes—
26#2859C2212yes—
27#2660—22no—
28#2584—21no—
29#2584—21no—
30#2583—22no—
31#2567C2211yescharacteristic
32#2567C2211yescharacteristic
33#2552—21no—
34#2552C2212yes—
35#2551—21no—
36#2551C2212yes—
37#2546C2211yescharacteristic
38#2546C2211yescharacteristic
39#2502C2212yes—
40#2440—22no—
41#2440C2214yes—
42#2434C2212yes—
43#2399—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47#2296—21no—
48#2296—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2100C2214yes—
54#2072C2214yes—
55#2072C2214yes—
56#2043C2214yes—
57#2043C2214yes—
58#2024C2214yes—
59#2024C2214yes—
60#1990C2214yes—
61#1990C2212yes—
62#1987C2212yes—
63#1981C2212yes—
64#1956C2212yes—
65#1956C2214yes—
66#1933C2214yes—
67#1933C2214yes—
68#1933C2214yes—
69#1933C2214yes—
70#1892C2214yes—
71#1872C2214yes—
72#1834C2214yes—
73#1824C2212yes—
74#1803C2211yescharacteristic
75#1803C2212yes—
76#1803C2212yes—
77#1803C2212yes—
78#1803C2212yes—
79#1803C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1803C2211yescharacteristic
86#1787C2212yes—
87C8×C4⋊C4C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2212yes—
90C4×C4⋊C8C2212yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C8×C22⋊C4C2214yes—
94C2×#340C2212yes—
95C23×Q8C2311yescharacteristic
96C23×C8C2312yes—
97#544—48no—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×#125—22no—
109C2×#125—21no—
110C2×#125—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×#117—21no—
125C2×#117—21no—
126C2×#117D411yescharacteristic
127C2×#117D411yescharacteristic
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#487—48no—
136#431—24no—
137C2×C4⋊C8—24no—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×#103—44no—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156C2×C4×C8C2314yes—
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—24no—
162#398—24no—
163#395—22no—
164#395—22no—
165#395—22no—
166#395—22no—
167#395—24no—
168#395—24no—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#385—24no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#384—22no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384C2312yes—
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2314yes—
207#382—22no—
208#382—21no—
209#380—21no—
210#380C2312yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376D412yes—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2314yes—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—41no—
245C22×Q8—41no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×Q8C2411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C22×C8—24no—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—42no—
270C4⋊Q8—42no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273C4⋊Q8—84no—
274#125—48no—
275#125—44no—
276#125—48no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—44no—
280C22⋊Q8—416no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294#117—44no—
295#117—44no—
296#117SD1618yes—
297#117Q1618yes—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—82no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—44no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42#4114yes—
365C2×C42#4114yes—
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374#103—28no—
375#103—24no—
376#103—24no—
377#103—44no—
378#103—42no—
379#103—44no—
380#103—44no—
381#103—42no—
382#103—44no—
383#103—44no—
384C22⋊C8—24no—
385C4×C8—24no—
386C4×C8—44no—
387C4×C8—44no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24C22×D411yescharacteristic
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—28no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—48no—
430C2×Q8—81no—
431C2×Q8—81no—
432C2×Q8—84no—
433C2×Q8—84no—
434C2×Q8H(5,𝔽2)12yes—
435C2×Q82−1+412yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4#12414yes—
461C22×C4#12414yes—
462C22×C4C2×SD1614yes—
463C22×C4C2×Q1614yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2511yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—84no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C2×SD1614yes—
531C4⋊C4C2×SD1614yes—
532C4⋊C4C2×Q1614yes—
533C4⋊C4C2×Q1614yes—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4C2×SD1618yes—
537C22⋊C4C2×Q1618yes—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—44no—
549C42—44no—
550C42#12414yes—
551C42#12414yes—
552C42C2×SD1618yes—
553C42C2×Q1618yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23C2×#12412yes—
563C23C22×SD1612yes—
564C23C22×Q1612yes—
565C23C23×D411yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—48no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—84no—
580Q8—168no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—84no—
629C2×C4—164no—
630C2×C4#487116yescomplemented
631C2×C4C2×#12412yes—
632C2×C4C2×#12412yes—
633C2×C4C2×#12412yes—
634C2×C4C2×#12412yes—
635C2×C4C2×#12412yes—
636C2×C4C22×SD1614yes—
637C2×C4C22×SD1614yes—
638C2×C4C22×SD1612yes—
639C2×C4C22×SD1612yes—
640C2×C4C22×Q1614yes—
641C2×C4C22×Q1614yes—
642C2×C4C22×Q1612yes—
643C2×C4C22×Q1612yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C2×C4#58511yescharacteristic
649C2×C4#58511yescharacteristic
650C8—42no—
651C8—42no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22C2×#48714yes—
660C22C22×#12411yescharacteristic
661C22#348811yescharacteristic
662C22#348911yescharacteristic
663C22C23×SD1611yescharacteristic
664C22C23×Q1611yescharacteristic
665C22#358511yescharacteristic
666C22#358611yescharacteristic
667C4—24no—
668C4—24no—
669C4—24no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4C2×#48714yes—
684C4C2×#48714yes—
685C4#349211yescharacteristic
686C4#349311yescharacteristic
687C4#349411yescharacteristic
688C4#349511yescharacteristic
689C2—24no—
690C2#3554211yescharacteristic
691C2C22×#48711yescharacteristic
692C2#6062111yescharacteristic
693C2#6195111yescharacteristic
694C2#6195211yescharacteristic
695C2#6209911yescharacteristic
696C2#6210011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29141C2semidirect
4. #29123C2semidirect
5. #29123C2semidirect
6. #28717C2semidirect
7. #27815C2semidirect
8. #27769C2semidirect
9. #27749C2semidirect
10. #27729C2semidirect
630. C2×C4#487semidirect

Maximal subgroups

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

subgroupindex
#352752
#349532
#291412
#291232
#287172
#278152
#277692
#277492
#277292
#276972
#272582
#218852
#218602
#207342
C2×#18032
#205432