Finite Group Databasebeta

← #6498467

Subgroups — #6498467

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C211yesmaximal, characteristic, complemented
2C2×#3215C211yesmaximal, characteristic
3#35338C211yesmaximal, characteristic
4#34225C211yesmaximal, characteristic, complemented
5#33037C212yesmaximal, complemented
6#33035C212yesmaximal, complemented
7#29841C212yesmaximal, complemented
8#29840C212yesmaximal, complemented
9#28659C212yesmaximal, complemented
10#28658C212yesmaximal, complemented
11#28619C212yesmaximal, complemented
12#28619C212yesmaximal, complemented
13#28132C214yesmaximal, complemented
14#28130C214yesmaximal
15#21839C211yesmaximal, characteristic, complemented
16#21808C211yesmaximal, characteristic
17#20811C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×#124C2211yescharacteristic
21#3237—28no—
22#3215—28no—
23C2×#501C2212yescomplemented
24C2×#497C2211yescharacteristic
25C2×#497C2211yescharacteristic
26C2×#491C2212yescomplemented
27C2×#489C2211yescharacteristic
28C2×#489C2211yescharacteristic
29C2×#485C2214yes—
30C2×#477C2212yescomplemented
31C2×#475C2212yescomplemented
32C2×#464C2212yes—
33C2×#464C2212yes—
34C2×#451C2212yes—
35C2×#450C2212yes—
36C2×#432C2211yescharacteristic
37C2×#432C2211yescharacteristic
38#2861C2212yes—
39#2861C2212yes—
40#2640C2211yescharacteristic
41#2640C2211yescharacteristic
42#2640C2211yescharacteristic
43#2640C2211yescharacteristic
44#2592—21no—
45#2592—21no—
46#2592—22no—
47#2561—21no—
48#2561—21no—
49#2561—22no—
50#2370—22no—
51#2362—22no—
52#2356—21no—
53#2351—21no—
54#2347—21no—
55#2346—21no—
56#2323C2214yes—
57#2320C2212yescomplemented
58#2310C2212yes—
59#2309C2212yes—
60C4×C4⋊Q8C2212yescomplemented
61C4×C4⋊Q8C2212yescomplemented
62#2285C2211yescharacteristic
63#2276C2212yescomplemented
64C4×#117C2211yescharacteristic
65#2068C2214yescomplemented
66#2067C2214yes—
67#2066C2214yescomplemented
68#2065C2214yes—
69#2056C2214yescomplemented
70#2056C2214yescomplemented
71#2055C2214yes—
72#2055C2214yes—
73#1992C2212yescomplemented
74#1992C2212yescomplemented
75#1988C2212yescomplemented
76#1987C2212yescomplemented
77#1983C2212yes—
78#1983C2212yes—
79#1959C2212yes—
80#1959C2212yes—
81#1928C2214yescomplemented
82#1927C2214yescomplemented
83#1926C2214yes—
84#1925C2214yes—
85#1880C2214yes—
86#1880C2214yes—
87#1878C2214yescomplemented
88#1878C2214yescomplemented
89#1843C2212yescomplemented
90#1832C2212yescomplemented
91#1825C2214yes—
92#1798C2214yes—
93#1787C2212yes—
94#1786C2212yes—
95#1767C2212yescomplemented
96#1767C2212yescomplemented
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×Q16C2312yes—
100C22×Q16C2312yes—
101C22×SD16C2312yes—
102C22×SD16C2312yes—
103#533—24no—
104#533—24no—
105#533—24no—
106#533D418yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—21no—
112C2×#125C2312yes—
113C2×#124—21no—
114C2×#124—21no—
115C2×#124—24no—
116C2×#124—24no—
117C2×#124C2311yescharacteristic
118C2×#124C2311yescharacteristic
119C2×#124C2311yescharacteristic
120C2×#124C2311yescharacteristic
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—22no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2314yes—
131C2×#117—21no—
132C2×#117—21no—
133C2×#117C2312yes—
134C2×#117C2312yes—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C42C2311yescharacteristic
138#501—28no—
139#497—24no—
140#497—24no—
141#491—28no—
142#489—24no—
143#489—24no—
144#485—216no—
145#477—28no—
146#475—28no—
147#464—28no—
148#464—28no—
149#451—28no—
150#450—28no—
151#432—24no—
152#432—24no—
153C2×#107C2314yes—
154C2×#106C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×#97C2311yescharacteristic
172C2×#97C2311yescharacteristic
173C2×#97C2312yes—
174#397—21no—
175#397—21no—
176#397—22no—
177#397—21no—
178#397—21no—
179#397—22no—
180#393—21no—
181#393—22no—
182#393—21no—
183#393—21no—
184#393—21no—
185#393—22no—
186#391—22no—
187#391—22no—
188#391—22no—
189#391C2314yes—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#390—22no—
195#390—22no—
196#390C2312yes—
197#390C2312yes—
198#390C2312yes—
199#390C2312yes—
200#389—24no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389C2314yes—
205#388—24no—
206#388—24no—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#387—24no—
216#387—24no—
217#384C2312yes—
218#383—22no—
219#382—21no—
220#382—22no—
221#381—22no—
222#381—22no—
223#380—21no—
224#380C2312yes—
225#379—22no—
226#379—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—24no—
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4D411yescharacteristic
235C4×C4⋊C4D412yes—
236C4×C4⋊C4C2314yes—
237C4×C4⋊C4C2312yes—
238C4×C22⋊C4D411yescharacteristic
239C4×C22⋊C4D411yescharacteristic
240C4×C22⋊C4D412yes—
241#376—22no—
242#376C2311yescharacteristic
243#376C2311yescharacteristic
244C43C2312yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251C25—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—44no—
259C22×Q8C2411yescharacteristic
260C22×Q8C2411yescharacteristic
261C22×Q8C2412yes—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—22no—
265C22×D4—44no—
266C22×D4C2411yescharacteristic
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—44no—
275C2×Q16—44no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—44no—
279C2×SD16—44no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8C2×D414yes—
290C4⋊Q8C2×D414yes—
291#125—28no—
292#125—24no—
293#125—24no—
294#124—22no—
295#124—22no—
296#124—22no—
297#124—22no—
298#124—22no—
299#124—22no—
300#124—22no—
301#124—22no—
302#124—44no—
303#124—44no—
304C22⋊Q8—28no—
305C22⋊Q8—24no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C22⋊Q8C2×D418yes—
309C4×Q8—28no—
310C4×Q8—24no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—84no—
316C4×Q8C2×D418yes—
317#117—28no—
318#117—24no—
319#117—24no—
320#117—44no—
321#117—82no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4—44no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—24no—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D414yes—
376C2×C42#4114yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—28no—
382#106—28no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C4⋊C8—44no—
386C4⋊C8—44no—
387#103—28no—
388#103—28no—
389#103—24no—
390#103—24no—
391#103—44no—
392#103—44no—
393#102—24no—
394#102—24no—
395#102—44no—
396#102—44no—
397C22⋊C8—24no—
398C22⋊C8—24no—
399#97—22no—
400#97—22no—
401#97—24no—
402#95—22no—
403#95—24no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95C2×D411yescharacteristic
417#95C2×D412yes—
418#95C2×D411yescharacteristic
419#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D412yes—
422C24—21no—
423C24—21no—
424C24—44no—
425C24—42no—
426C24—81no—
427C24—82no—
428C24—81no—
429C24C22×D411yescharacteristic
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—81no—
447C2×Q8—82no—
448C2×Q8—81no—
449C2×Q8—84no—
450C2×Q8—84no—
451C2×Q8C22×D414yes—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—84no—
461C2×D4—84no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4#12114yes—
476C22×C4#12114yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C2×#4112yes—
488C22×C4C2511yescharacteristic
489Q16—84no—
490Q16—84no—
491SD16—84no—
492SD16—84no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—84no—
522C4⋊C4C22×D414yes—
523C22⋊C4—28no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—28no—
545C42—24no—
546C42—24no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—48no—
552C42—44no—
553C42—82no—
554C42—84no—
555C42#12114yes—
556C42#12114yes—
557C42C22×D412yes—
558C42C22×D412yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C23—21no—
562C23—22no—
563C23—21no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—42no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—164no—
579C23—164no—
580C23C2×#12112yes—
581C23C23×D411yescharacteristic
582Q8—24no—
583Q8—24no—
584Q8—48no—
585Q8—44no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—84no—
591Q8—84no—
592Q8—84no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—84no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4C2×#12112yes—
641C2×C4C2×#12112yes—
642C2×C4C2×#12112yes—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4#58514yes—
648C8—44no—
649C8—42no—
650C8—42no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—168no—
667C22C22×#12111yescharacteristic
668C22#344912yes—
669C22#358511yescharacteristic
670C22#358612yes—
671C22#358611yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—84no—
686C4—81no—
687C4—81no—
688C4#350012yes—
689C4#351012yes—
690C2—42no—
691C2—161no—
692C2—161no—
693C2#3532711yescharacteristic
694C2#6042411yescharacteristic
695C2#6042511yescharacteristic
696C2#6101612yes—
697C2#6101812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. C2×#3237C2semidirect
4. #34225C2semidirect
5. #33037C2semidirect
6. #33035C2semidirect
7. #29841C2semidirect
8. #29840C2semidirect
9. #28659C2semidirect
10. #28658C2semidirect
11. #28619C2semidirect
12. #28619C2semidirect
13. #28132C2semidirect
15. #21839C2semidirect
17. #20811C2semidirect
23. C2×#501C22semidirect
26. C2×#491C22semidirect
30. C2×#477C22semidirect
31. C2×#475C22semidirect
57. #2320C22semidirect
60. C4×C4⋊Q8C22semidirect
61. C4×C4⋊Q8C22semidirect
63. #2276C22semidirect
65. #2068C22semidirect
67. #2066C22semidirect
69. #2056C22semidirect
70. #2056C22semidirect
73. #1992C22semidirect
74. #1992C22semidirect
75. #1988C22semidirect
76. #1987C22semidirect
81. #1928C22semidirect
82. #1927C22semidirect
87. #1878C22semidirect
88. #1878C22semidirect
89. #1843C22semidirect
90. #1832C22semidirect
95. #1767C22semidirect
96. #1767C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
C2×#32152
#353382
#342252
#330372
#330352
#298412
#298402
#286592
#286582
#286192
#281322
#281302
#218392
#218082
#208112