Finite Group Databasebeta

← #6810394

Subgroups — #6810394

Order 512.

2103 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3218C212yesmaximal, complemented
2#36117C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic, complemented
4#32903C211yesmaximal, characteristic
5#30894C214yesmaximal, complemented
6#30760C214yesmaximal, complemented
7#29624C212yesmaximal, complemented
8#28473C212yesmaximal, complemented
9#27777C212yesmaximal, complemented
10#25718C211yesmaximal, characteristic
11#25368C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#24509C212yesmaximal, complemented
14#24212C211yesmaximal, characteristic, complemented
15#23937C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#529C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#3218—216no—
23C2×#497C2214yes—
24C2×#492C2212yes—
25C2×#491C2214yes—
26C2×#462C2214yes—
27C2×#462C2212yes—
28C2×#460C2212yes—
29C2×#461C2214yes—
30C2×#451C2212yes—
31C2×#451C2211yescharacteristic
32C2×#448C2212yes—
33C2×#448C2212yes—
34C2×#448C2212yes—
35C2×C4×Q16C2212yes—
36C2×#432C2212yes—
37#2856C2211yescharacteristic
38#2854C2212yes—
39#2552C2211yescharacteristic
40#2548C2211yescharacteristic
41#2546—21no—
42#2546—22no—
43#2546—21no—
44#2546C2212yes—
45#2546C2214yes—
46#2546C2212yes—
47#2512—21no—
48#2501—21no—
49#2479—22no—
50#2454C2212yes—
51#2448C2212yes—
52#2425—22no—
53C2×#398—22no—
54#2366C2211yescharacteristic
55#2361—21no—
56#2354—21no—
57Q8×C4⋊C4C2211yescharacteristic
58C4×C22⋊Q8C2211yescharacteristic
59C2×#391C2212yes—
60C2×#389—21no—
61C2×#389—21no—
62C2×C4×C4⋊C4C2211yescharacteristic
63#2068C2214yes—
64#2055C2214yes—
65#2043C2214yes—
66#2043C2212yes—
67#2043C2214yes—
68#2026C2214yes—
69#2016C2212yes—
70#2015C2214yes—
71#2013C2214yes—
72#2013C2214yes—
73#2013C2214yes—
74#1981C2212yes—
75#1973C2214yes—
76#1956C2212yes—
77#1939C2212yes—
78#1935C2214yes—
79#1917C2214yes—
80#1880C2214yes—
81#1878C2212yes—
82#1878C2214yes—
83#1877C2214yes—
84#1874C2212yes—
85#1872C2211yescharacteristic
86#1863C2212yes—
87#1852C2211yescharacteristic
88#1817C2212yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1787C2212yes—
92#1787C2211yescharacteristic
93#1770C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2312yes—
97C23×Q8C2311yescharacteristic
98C22×Q16C2314yes—
99C22×Q16C2312yes—
100C22×SD16C2314yes—
101C22×SD16C2314yes—
102C22×SD16C2314yes—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8—44no—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2314yes—
113C2×C4⋊Q8C2312yes—
114C2×#127C2311yescharacteristic
115#529—28no—
116#529—44no—
117C2×#126—22no—
118C2×#125C2312yes—
119C2×#124C2312yes—
120C2×#123—22no—
121C2×#123—22no—
122C2×C22⋊Q8C2311yescharacteristic
123C2×#121—21no—
124C2×#121—21no—
125C2×#121C2312yes—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—44no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—22no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×#117—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4D411yescharacteristic
143C22×C4⋊C4D411yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C42—21no—
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148#497—216no—
149#492—44no—
150#491—216no—
151#462—216no—
152#462—44no—
153#461—216no—
154#460—44no—
155#451—28no—
156#451D418yescomplemented
157#448—28no—
158#448—28no—
159#448—44no—
160C4×Q16—44no—
161#432—28no—
162C2×#107C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2314yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#102C2314yes—
176C2×C22⋊C8C2312yes—
177C2×C22⋊C8C2311yescharacteristic
178C2×C22⋊C8C2311yescharacteristic
179C2×#97C2314yes—
180C2×C4×C8C2312yes—
181#399—22no—
182#399—22no—
183#398—24no—
184#398—24no—
185#398—24no—
186#398—24no—
187#398—22no—
188#397—22no—
189#396—22no—
190#394—22no—
191#391—22no—
192#391—22no—
193#391C2314yes—
194#391C2311yescharacteristic
195#391C2314yes—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390C2312yes—
202#390C2312yes—
203#390C2312yes—
204#389—21no—
205#389—22no—
206#389—22no—
207#389—24no—
208#389—22no—
209#389—24no—
210#389—22no—
211#389—22no—
212#389C2312yes—
213#389C2312yes—
214#389C2311yescharacteristic
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#387—21no—
220#386—22no—
221#386—22no—
222#385—21no—
223#385—22no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#380—22no—
228#380C2311yescharacteristic
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4D411yescharacteristic
232C4×C4⋊C4D412yes—
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C4⋊C4C2311yescharacteristic
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2311yescharacteristic
238C4×C22⋊C4—21no—
239C2×#95—22no—
240C2×#95—22no—
241C2×#95D412yes—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#336C2312yes—
246#336C2314yes—
247#336C2314yes—
248#336C2312yes—
249C22×Q8—21no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8—42no—
255C22×Q8—42no—
256C22×Q8—44no—
257C22×Q8—44no—
258C22×Q8C2×D414yes—
259C22×Q8C2414yes—
260C22×Q8C2411yescharacteristic
261C22×Q8C2412yes—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D412yes—
273C23×C4C2×D411yescharacteristic
274C23×C4C2411yescharacteristic
275C2×Q16—28no—
276C2×Q16—42no—
277C2×Q16—42no—
278C2×Q16—44no—
279C2×Q16—44no—
280C2×Q16C2×D418yes—
281C2×SD16—28no—
282C2×SD16—28no—
283C2×SD16—44no—
284C2×SD16—44no—
285C2×SD16—44no—
286C2×SD16—44no—
287C2×SD16—44no—
288C2×SD16—44no—
289C2×SD16—44no—
290C22×C8C2412yes—
291C22×C8C2414yes—
292C22×C8C2412yes—
293C4⋊Q8—28no—
294C4⋊Q8—28no—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—48no—
301C4⋊Q8—48no—
302C4⋊Q8—44no—
303C4⋊Q8—44no—
304#127—24no—
305#126—48no—
306#125—24no—
307#125—42no—
308#124—24no—
309#124—42no—
310#123—48no—
311#123—84no—
312C22⋊Q8—22no—
313C22⋊Q8C2×D414yes—
314#121—24no—
315#121—24no—
316#121—44no—
317#121—82no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×D4—24no—
328C4×D4—42no—
329C4×D4—48no—
330C4×D4—44no—
331#117—48no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
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—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—48no—
357C2×C4⋊C4—48no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D412yes—
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2411yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C42—24no—
385C2×C42—24no—
386C2×C42—24no—
387C2×C42—22no—
388C2×C42—21no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—21no—
392C2×C42—44no—
393C2×C42C2×D412yes—
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D412yes—
397C2×C42C2412yes—
398C2×C42C2411yescharacteristic
399C2×C42C2412yes—
400C2×C42C2411yescharacteristic
401#107—42no—
402C4⋊C8—28no—
403C4⋊C8—24no—
404#103—28no—
405#103—28no—
406#103—28no—
407#103—42no—
408#103—44no—
409#103—42no—
410#103—42no—
411#103—44no—
412#103—44no—
413#103C2×D418yes—
414#102—44no—
415C22⋊C8—24no—
416C22⋊C8—22no—
417C22⋊C8C2×D414yes—
418#97—28no—
419C4×C8—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—21no—
425#95—21no—
426#95—22no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—22no—
431#95C2×D412yes—
432#95C2×D412yes—
433#95C2×D412yes—
434#95C2×D412yes—
435C24—22no—
436C24C22×D411yescharacteristic
437C2×Q8—24no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—48no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—48no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—88no—
460C2×Q8—88no—
461C2×Q8C22×D414yes—
462C2×Q8C22×D412yes—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—81no—
475C2×D4—81no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—42no—
497C22×C4—41no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—42no—
507C22×C4—44no—
508C22×C4—82no—
509C22×C4#13714yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4H(5,𝔽2)12yes—
521C22×C4C2511yescharacteristic
522Q16—48no—
523Q16—88no—
524SD16—88no—
525SD16—88no—
526SD16—88no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—24no—
533C2×C8—22no—
534C2×C8—24no—
535C2×C8C22×D414yes—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—48no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—48no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—81no—
566C4⋊C4—81no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4#13714yes—
570C4⋊C4C22×D412yes—
571C22⋊C4—24no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—42no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C42—22no—
584C42—22no—
585C42—24no—
586C42—24no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—42no—
596C42—42no—
597C42H(5,𝔽2)14yes—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23D4212yes—
610C23C2×#13712yes—
611C23#57712yes—
612C23C23×D411yescharacteristic
613Q8—24no—
614Q8—24no—
615Q8—24no—
616Q8—48no—
617Q8—42no—
618Q8—42no—
619Q8—42no—
620Q8—44no—
621Q8—44no—
622Q8—44no—
623Q8—44no—
624Q8—88no—
625D4—44no—
626D4—44no—
627D4—88no—
628D4—84no—
629D4—162no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—48no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—48no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—41no—
674C2×C4—44no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—88no—
682C2×C4—84no—
683C2×C4—88no—
684C2×C4#53512yes—
685C2×C4#53512yes—
686C2×C4D4212yes—
687C2×C4C2×#13712yes—
688C2×C4C2×#13714yes—
689C2×C4C2×#13712yes—
690C2×C4C2×#13712yes—
691C2×C4C23×D411yescharacteristic
692C2×C4C2×H(5,𝔽2)12yes—
693C2×C4C2×H(5,𝔽2)11yescharacteristic
694C2×C4#58512yes—
695C8—24no—
696C8—44no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22#327814yes—
712C22C2×#53511yescharacteristic
713C22C2×D4211yescharacteristic
714C22#348711yescharacteristic
715C22C22×#13711yescharacteristic
716C22C22×#13711yescharacteristic
717C22C2×#57711yescharacteristic
718C22#358611yescharacteristic
719C4—22no—
720C4—22no—
721C4—24no—
722C4—22no—
723C4—42no—
724C4—42no—
725C4—41no—
726C4—42no—
727C4—42no—
728C4—44no—
729C4—42no—
730C4—82no—
731C4—81no—
732C4—82no—
733C4—84no—
734C4—81no—
735C4#321814yescomplemented
736C4#349612yes—
737C4#354212yes—
738C2—24no—
739C2—162no—
740C2#3933911yescharacteristic
741C2C2×#321811yescharacteristic
742C2#6081311yescharacteristic
743C2C2×#327811yescharacteristic
744C2D4×#13711yescharacteristic
745C2#6189311yescharacteristic
746C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3218C2semidirect
3. C2×#2546C2semidirect
5. #30894C2semidirect
6. #30760C2semidirect
7. #29624C2semidirect
8. #28473C2semidirect
9. #27777C2semidirect
11. #25368C2semidirect
12. #24732C2semidirect
13. #24509C2semidirect
14. #24212C2semidirect
15. #23937C2semidirect
16. #22603C2semidirect
17. C4×#451C2semidirect
156. #451D4semidirect
735. C4#3218semidirect

Maximal subgroups

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

subgroupindex
C2×#32182
#361172
C2×#25462
#329032
#308942
#307602
#296242
#284732
#277772
#257182
#253682
#247322
#245092
#242122
#239372
#226032
C4×#4512