Finite Group Databasebeta

← #7507720

Subgroups — #7507720

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#34815C211yesmaximal, characteristic, complemented
3#32105C214yesmaximal, complemented
4#32103C214yesmaximal, complemented
5#32089C214yesmaximal, complemented
6#32087C214yesmaximal, complemented
7#23732C211yesmaximal, characteristic, complemented
8#23619C211yesmaximal, characteristic, complemented
9#23507C211yesmaximal, characteristic, complemented
10#23507C211yesmaximal, characteristic, complemented
11#23506C211yesmaximal, characteristic
12#23506C211yesmaximal, characteristic
13#21261C211yesmaximal, characteristic
14#21204C211yesmaximal, characteristic
15#21094C211yesmaximal, characteristic, complemented
16#21061C211yesmaximal, characteristic, complemented
17#21023C211yesmaximal, characteristic, complemented
18#20873C211yesmaximal, characteristic
19#20873C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#419C2211yescharacteristic
34C2×#419C2211yescharacteristic
35C2×#418C2211yescharacteristic
36C2×#418C2211yescharacteristic
37C2×#418C2211yescharacteristic
38C2×#418C2211yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#2547—22no—
41#2547—22no—
42#2547C2214yes—
43#2547C2214yes—
44#2509C2212yescomplemented
45#2508C2212yescomplemented
46#2507C2212yescomplemented
47#2506C2212yescomplemented
48#2505—22no—
49#2473—22no—
50#2402—22no—
51C2×#397—22no—
52C4×#125—22no—
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#391C2211yescharacteristic
58C2×#391C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×#380—22no—
61C4×#117C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63#2097C2212yescomplemented
64#2097C2212yescomplemented
65#2096C2212yescomplemented
66#2095C2212yescomplemented
67#2068C2214yescomplemented
68#2066C2214yescomplemented
69#2043C2214yescomplemented
70#2043C2214yescomplemented
71#2043C2214yescomplemented
72#2043C2214yescomplemented
73#2042C2214yes—
74#2042C2214yes—
75#2042C2214yes—
76#2042C2214yes—
77#1983C2214yes—
78#1983C2214yes—
79#1959C2212yes—
80#1959C2212yes—
81#1958C2212yes—
82#1958C2212yes—
83#1935C2214yescomplemented
84#1933C2214yescomplemented
85#1928C2214yescomplemented
86#1927C2214yescomplemented
87#1917C2214yescomplemented
88#1916C2214yescomplemented
89#1864C2214yes—
90#1864C2214yes—
91#1862C2214yes—
92#1862C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1728C2211yescharacteristic
96#1728C2211yescharacteristic
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C23×Q8C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101C23×D4C2311yescharacteristic
102C24×C4—21no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—42no—
109C2×C4⋊Q8—42no—
110C2×C4⋊Q8C2314yes—
111C2×#125—22no—
112C2×#124—24no—
113C2×#124—24no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124—22no—
117C2×#124—42no—
118C2×#124—42no—
119C2×#124C2312yes—
120C2×#124C2312yes—
121C2×#124C2314yes—
122C2×C4×D4—22no—
123C2×C4×D4—42no—
124C2×C4×D4—42no—
125C2×C4×D4C2314yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4D412yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#428—42no—
141#428—42no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#423—42no—
147C2×C4⋊C8C2314yes—
148#419—24no—
149#419—24no—
150#419—42no—
151#419—42no—
152#419—42no—
153#419—42no—
154#419—42no—
155#419—42no—
156#418—24no—
157#418—24no—
158#418—42no—
159#418—42no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#102C2314yes—
173C2×#102C2314yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C4×M16—42no—
179C2×#97C2312yes—
180C2×C4×C8C2312yes—
181#400—22no—
182#400—22no—
183#400—22no—
184#400—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#398—22no—
189#397—24no—
190#397—24no—
191#397—24no—
192#397—24no—
193#391—21no—
194#391—21no—
195#391—22no—
196#391C2314yes—
197#391C2312yes—
198#391C2312yes—
199#389—22no—
200#389—22no—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#385—21no—
214#385—21no—
215#385—22no—
216#385—22no—
217#385—21no—
218#385—21no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#382—22no—
222#382C2311yescharacteristic
223#382C2311yescharacteristic
224#380—22no—
225#380—24no—
226#380—24no—
227#380—24no—
228#380—24no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4D412yes—
236C4×C4⋊C4D412yes—
237C4×C4⋊C4C2312yes—
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C22⋊C4—22no—
242#376—21no—
243C2×#95—22no—
244C2×#95—22no—
245C2×#95D412yes—
246C43—21no—
247#340C2314yes—
248#340C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252#336C2314yes—
253#336C2314yes—
254C25—21no—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8C2412yes—
270C22×Q8C2412yes—
271C22×D4—21no—
272C22×D4—22no—
273C22×D4—22no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4—22no—
277C22×D4#4114yes—
278C22×D4C2412yes—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—24no—
283C23×C4—44no—
284C23×C4—48no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D412yes—
288C23×C4C2×D412yes—
289C23×C4C2411yescharacteristic
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16—22no—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308C4⋊Q8—48no—
309#125—48no—
310#124—24no—
311#124—24no—
312#124—28no—
313#124—42no—
314#124—42no—
315#124—44no—
316#124—48no—
317#124—44no—
318#124—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—48no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117#4114yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—48no—
369C2×C22⋊C4—48no—
370C2×C22⋊C4—44no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42C2×D412yes—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107—42no—
392#107—42no—
393#106—42no—
394#106—42no—
395C4⋊C8—44no—
396#103—28no—
397#103—28no—
398#103—42no—
399#103—42no—
400#103—42no—
401#103—42no—
402#103—44no—
403#103—44no—
404#103—42no—
405#103—42no—
406#103—42no—
407#103—42no—
408#102—28no—
409#102—28no—
410#102—42no—
411#102—42no—
412#102—42no—
413#102—42no—
414#97—42no—
415C4×C8—42no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95C2×D412yes—
433#95C2×D412yes—
434#95C2×D412yes—
435#95C2×D412yes—
436C24—24no—
437C24—22no—
438C24—44no—
439C24—44no—
440C24—42no—
441C24—42no—
442C24C22×D411yescharacteristic
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—44no—
467C2×Q8—44no—
468C2×Q8—44no—
469C2×D4—28no—
470C2×D4—24no—
471C2×D4—28no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—42no—
478C2×D4—44no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4C2×#4114yes—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—44no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—48no—
512C22×C4—816no—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4H(5,𝔽2)12yes—
523C22×C4H(5,𝔽2)12yes—
524C22×C4H(5,𝔽2)12yes—
525C22×C42−1+412yes—
526C22×C4C2511yescharacteristic
527M16—44no—
528M16—44no—
529M16—44no—
530M16—44no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—22no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—48no—
561C4⋊C4—48no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—42no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4C2×#4114yes—
574C22⋊C4—24no—
575C22⋊C4—48no—
576C22⋊C4—48no—
577C22⋊C4—48no—
578C22⋊C4—48no—
579C22⋊C4—48no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—42no—
583C42—24no—
584C42—24no—
585C42—22no—
586C42—22no—
587C42—24no—
588C42—22no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—44no—
597C42—42no—
598C42—42no—
599C42—42no—
600C42—82no—
601C23—22no—
602C23—22no—
603C23—24no—
604C23—416no—
605C23—44no—
606C23—42no—
607C23—42no—
608C23—48no—
609C23—44no—
610C23—84no—
611C23—84no—
612C23—88no—
613C23#53512yes—
614C23#54612yes—
615C23#54812yes—
616C23C23×D411yescharacteristic
617Q8—42no—
618Q8—42no—
619Q8—42no—
620Q8—42no—
621Q8—42no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—84no—
626Q8—84no—
627D4—24no—
628D4—24no—
629D4—42no—
630D4—48no—
631D4—42no—
632C2×C4—28no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—816no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4#53512yes—
680C2×C4#53914yes—
681C2×C4#54014yes—
682C2×C4#54612yes—
683C2×C4#54812yes—
684C2×C4#55314yes—
685C2×C4#56214yes—
686C2×C4#57514yes—
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×H(5,𝔽2)11yescharacteristic
690C2×C4C2×H(5,𝔽2)11yescharacteristic
691C2×C4C2×H(5,𝔽2)11yescharacteristic
692C2×C4C2×2−1+411yescharacteristic
693C8—42no—
694C8—42no—
695C8—42no—
696C8—42no—
697C22—22no—
698C22—22no—
699C22—24no—
700C22—24no—
701C22—44no—
702C22—48no—
703C22—816no—
704C22—82no—
705C22—82no—
706C22—84no—
707C22#271114yes—
708C22C2×#53511yescharacteristic
709C22C2×#54611yescharacteristic
710C22C2×#54811yescharacteristic
711C22C2×#57512yes—
712C22#358612yes—
713C4—24no—
714C4—42no—
715C4—44no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—81no—
721C4—81no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#271114yes—
729C4C2×#53911yescharacteristic
730C4C2×#54011yescharacteristic
731C4C2×#55311yescharacteristic
732C4C2×#56211yescharacteristic
733C2—24no—
734C2—84no—
735C2C2×#271111yescharacteristic
736C2#6079911yescharacteristic
737C2#6080711yescharacteristic
738C2#6115912yes—
739C2#6132312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. #34815C2semidirect
3. #32105C2semidirect
4. #32103C2semidirect
5. #32089C2semidirect
6. #32087C2semidirect
7. #23732C2semidirect
8. #23619C2semidirect
9. #23507C2semidirect
10. #23507C2semidirect
15. #21094C2semidirect
16. #21061C2semidirect
17. #21023C2semidirect
44. #2509C22semidirect
45. #2508C22semidirect
46. #2507C22semidirect
47. #2506C22semidirect
63. #2097C22semidirect
64. #2097C22semidirect
65. #2096C22semidirect
66. #2095C22semidirect
67. #2068C22semidirect
68. #2066C22semidirect
69. #2043C22semidirect
70. #2043C22semidirect
71. #2043C22semidirect
72. #2043C22semidirect
83. #1935C22semidirect
84. #1933C22semidirect
85. #1928C22semidirect
86. #1927C22semidirect
87. #1917C22semidirect
88. #1916C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#348152
#321052
#321032
#320892
#320872
#237322
#236192
#235072
#235062
#212612
#212042
#210942
#210612
#210232
#208732