Finite Group Databasebeta

← #6444632

Subgroups — #6444632

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3083C211yesmaximal, characteristic
2C22×#477C211yesmaximal, characteristic, complemented
3#35576C211yesmaximal, characteristic, complemented
4#34266C211yesmaximal, characteristic
5#26478C214yesmaximal, complemented
6#26474C214yesmaximal, complemented
7#26468C214yesmaximal, complemented
8#26466C214yesmaximal, complemented
9#23613C211yesmaximal, characteristic, complemented
10#23609C211yesmaximal, characteristic
11#23457C211yesmaximal, characteristic, complemented
12#23452C211yesmaximal, characteristic
13#21262C211yesmaximal, characteristic
14#21205C211yesmaximal, characteristic
15#21173C211yesmaximal, characteristic, complemented
16#21147C211yesmaximal, characteristic, complemented
17#20881C211yesmaximal, characteristic
18#20874C211yesmaximal, characteristic, complemented
19#20455C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#3083—28no—
26C2×#479C2212yes—
27C2×#477—28no—
28C2×#477C2214yes—
29C2×#477C2214yes—
30C2×#477C2212yes—
31C2×#478C2212yes—
32C2×#474C2212yes—
33C2×#428C2211yescharacteristic
34C2×#427C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#419C2211yescharacteristic
43C2×#419C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C2×#416C2211yescharacteristic
46C22×#103C2211yescharacteristic
47C22×#103C2211yescharacteristic
48C2×#405C2211yescharacteristic
49#2583—22no—
50#2551—22no—
51#2551C2214yes—
52#2550—22no—
53#2550C2214yes—
54#2434—22no—
55#2367C2214yes—
56#2365C2214yes—
57#2362—22no—
58#2360—22no—
59#2322—22no—
60#2317C2211yescharacteristic
61#2317C2211yescharacteristic
62#2310C2211yescharacteristic
63C4×#123—22no—
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2296C2211yescharacteristic
67#2274C2211yescharacteristic
68#2274C2211yescharacteristic
69#2068C2212yes—
70#2068C2214yes—
71#2067C2212yes—
72#2066C2212yes—
73#2066C2214yes—
74#2065C2212yes—
75#2026C2214yes—
76#2024C2214yes—
77#2024C2214yes—
78#2023C2214yes—
79#1983C2214yes—
80#1981C2214yes—
81#1960C2212yes—
82#1959C2212yes—
83#1958C2212yes—
84#1956C2212yes—
85#1949C2214yes—
86#1947C2214yes—
87#1940C2214yes—
88#1939C2214yes—
89#1865C2214yes—
90#1863C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1776C2214yes—
94C4×#107C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C23×Q8C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110#536—24no—
111#536Q818yescomplemented
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—42no—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2314yes—
119C2×C4⋊Q8C2312yes—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#124C2312yes—
123C2×#123—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126#520—24no—
127#520—42no—
128C2×C4×Q8—22no—
129C2×C4×Q8—24no—
130C2×C4×Q8—42no—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—22no—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C42—21no—
148C22×C42C2311yescharacteristic
149#479—28no—
150#478—28no—
151#477—216no—
152#477—216no—
153#477—28no—
154#474—28no—
155#428—24no—
156#427—24no—
157C2×#107—24no—
158C2×#107D418yes—
159C2×#107C2314yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#107C2312yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166#424—24no—
167#424—24no—
168C2×C4⋊C8—24no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2314yes—
174#419—24no—
175#419—42no—
176#417—24no—
177#417—42no—
178#416—24no—
179#416—24no—
180C2×#103—24no—
181C2×#103—24no—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2311yescharacteristic
188C2×#103C2312yes—
189C2×#103C2314yes—
190C2×#103C2312yes—
191C2×#103C2312yes—
192C2×#103C2312yes—
193C2×#102C2311yescharacteristic
194C2×#102C2312yes—
195C2×#102C2311yescharacteristic
196C2×#102C2312yes—
197C2×#102C2312yes—
198#405—24no—
199C2×#97C2312yes—
200C2×C4×C8C2312yes—
201#398—24no—
202#395—24no—
203#395—24no—
204#395—24no—
205#391—21no—
206#391—21no—
207#391—24no—
208#391C2314yes—
209#391C2312yes—
210#390C2312yes—
211#389—22no—
212#389—24no—
213#389—24no—
214#389—24no—
215#389D412yes—
216#389D412yes—
217#389C2314yes—
218#389C2312yes—
219#389C2312yes—
220#388—24no—
221#388—24no—
222#387—24no—
223#387—24no—
224#386—21no—
225#386—21no—
226#385—21no—
227#385—21no—
228#385—24no—
229#385—22no—
230#385—22no—
231#385D412yes—
232#385D412yes—
233#384—21no—
234#384C2312yes—
235#382—21no—
236#382C2312yes—
237#381—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—24no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4C2314yes—
243C4×C4⋊C4C2314yes—
244C4×C4⋊C4C2314yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—24no—
249#340C2312yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#336C2314yes—
255#336C2314yes—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41C2×Q814yes—
262C22×Q8—21no—
263C22×Q8—24no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—44no—
271C22×Q8—84no—
272C22×Q8C2×Q814yes—
273C22×Q8C2412yes—
274C22×Q8C2411yescharacteristic
275C22×D4—21no—
276C22×D4C2411yescharacteristic
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8C2×D414yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C4⋊Q8—28no—
304C4⋊Q8—24no—
305C4⋊Q8—24no—
306C4⋊Q8—44no—
307C4⋊Q8—42no—
308C4⋊Q8—48no—
309C4⋊Q8—44no—
310C4⋊Q8—88no—
311C4⋊Q8C2×Q818yescomplemented
312#125—24no—
313#125—42no—
314#125—44no—
315#124—24no—
316#124C2×Q818yescomplemented
317#123—216no—
318C22⋊Q8—216no—
319C22⋊Q8—48no—
320C4×Q8—28no—
321C4×Q8—24no—
322C4×Q8—44no—
323C4×Q8—42no—
324C4×Q8—48no—
325C4×D4—24no—
326C4×D4—42no—
327C4×D4—48no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—44no—
336#117C2×Q814yes—
337#117#4114yes—
338#117#4114yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—24no—
390C2×C42—24no—
391C2×C42—24no—
392C2×C42—44no—
393C2×C42C2×D412yes—
394C2×C42C2412yes—
395C2×C42C2412yes—
396C2×C42C2412yes—
397#107—28no—
398#107—24no—
399#107—24no—
400#107C2×D418yes—
401#107C2×D418yes—
402#106—24no—
403#106—24no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—28no—
409#103—24no—
410#103—24no—
411#103—22no—
412#103—22no—
413#103—28no—
414#103—24no—
415#103—24no—
416#103—28no—
417#103—42no—
418#103—42no—
419#103—42no—
420#102—22no—
421#102—22no—
422#102—24no—
423#102—24no—
424#102—42no—
425#97—24no—
426C4×C8—24no—
427#95—22no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95—22no—
432#95—24no—
433#95—22no—
434#95—22no—
435#95—24no—
436#95—24no—
437#95C2×D414yes—
438#95C2×D414yes—
439C24—21no—
440C24C22×D411yescharacteristic
441#41—28no—
442#41—28no—
443#41—44no—
444#41—44no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—28no—
449C2×Q8—28no—
450C2×Q8—22no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—21no—
458C2×Q8—48no—
459C2×Q8—44no—
460C2×Q8—48no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—84no—
466C2×Q8—84no—
467C2×Q8—816no—
468C2×Q8C22×Q814yes—
469C2×Q8C22×Q812yes—
470C2×D4—24no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4C22×Q812yes—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—84no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D414yes—
510C22×C4C22×D412yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×Q812yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C42−1+412yes—
518C22×C4C2511yescharacteristic
519M16—44no—
520M16—44no—
521C2×C8—28no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—44no—
536C2×C8C22×D412yes—
537C2×C8C22×D412yes—
538C4⋊C4—216no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—28no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—22no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—48no—
558C4⋊C4—48no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—42no—
564C4⋊C4—44no—
565C4⋊C4—84no—
566C4⋊C4—84no—
567C4⋊C4C22⋊Q818yes—
568C4⋊C4C22×D414yes—
569C4⋊C4C22×D414yes—
570C4⋊C4C2×#4114yes—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—216no—
574C22⋊C4—44no—
575C22⋊C4—42no—
576C22⋊C4—42no—
577C22⋊C4—44no—
578C22⋊C4—82no—
579C22⋊C4C22⋊Q818yes—
580C42—24no—
581C42—24no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—48no—
587C42—44no—
588C42—42no—
589C42—42no—
590C42—84no—
591C42—84no—
592C42C22×Q814yes—
593C42C2×#4114yes—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—42no—
598C23—41no—
599C23—41no—
600C23—42no—
601C23C2×C22⋊Q812yes—
602C23D4212yes—
603C23#54812yes—
604C23C23×D411yescharacteristic
605Q8—24no—
606Q8—22no—
607Q8—24no—
608Q8—22no—
609Q8—42no—
610Q8—41no—
611Q8—42no—
612Q8—41no—
613Q8—48no—
614Q8—816no—
615Q8—164no—
616D4—22no—
617D4—22no—
618D4—48no—
619D4—41no—
620D4—41no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—164no—
662C2×C4C2×C22⋊Q818yes—
663C2×C4C2×C22⋊Q812yes—
664C2×C4#53214yes—
665C2×C4#54214yes—
666C2×C4D4212yes—
667C2×C4#54812yes—
668C2×C4D4×Q814yes—
669C2×C4#55414yes—
670C2×C4#57814yes—
671C2×C4C23×D411yescharacteristic
672C2×C4C23×Q811yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4C2×2−1+411yescharacteristic
677C8—22no—
678C8—22no—
679C8—42no—
680C8—42no—
681C8—42no—
682C22—24no—
683C22—22no—
684C22—22no—
685C22—24no—
686C22—44no—
687C22—42no—
688C22—81no—
689C22—81no—
690C22—82no—
691C22#245214yes—
692C22C22×C22⋊Q811yescharacteristic
693C22C2×D4211yescharacteristic
694C22C2×#54811yescharacteristic
695C22C2×#57812yes—
696C22#358512yes—
697C4—28no—
698C4—24no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—81no—
708C4—82no—
709C4—81no—
710C4—82no—
711C4#245214yes—
712C4C2×#53211yescharacteristic
713C4C2×#54211yescharacteristic
714C4C2×D4×Q811yescharacteristic
715C4C2×#55411yescharacteristic
716C2—24no—
717C2—82no—
718C2C2×#245211yescharacteristic
719C2#6049812yes—
720C2#6105311yescharacteristic
721C2#6105811yescharacteristic
722C2#6134812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#477C2semidirect
3. #35576C2semidirect
5. #26478C2semidirect
6. #26474C2semidirect
7. #26468C2semidirect
8. #26466C2semidirect
9. #23613C2semidirect
11. #23457C2semidirect
15. #21173C2semidirect
16. #21147C2semidirect
18. #20874C2semidirect
19. #20455C2semidirect
111. #536Q8semidirect
311. C4⋊Q8C2×Q8semidirect
316. #124C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30832
C22×#4772
#355762
#342662
#264782
#264742
#264682
#264662
#236132
#236092
#234572
#234522
#212622
#212052
#211732
#211472
#208812
#208742
#204552