Finite Group Databasebeta

← #7507694

Subgroups — #7507694

Order 512.

1745 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#35555C211yesmaximal, characteristic, complemented
3#32085C214yesmaximal, complemented
4#32084C214yesmaximal, complemented
5#32083C214yesmaximal, complemented
6#32082C214yesmaximal, complemented
7#23734C211yesmaximal, characteristic
8#23615C211yesmaximal, characteristic, complemented
9#23512C211yesmaximal, characteristic
10#23510C211yesmaximal, characteristic
11#23510C211yesmaximal, characteristic
12#23507C211yesmaximal, characteristic, complemented
13#21262C211yesmaximal, characteristic
14C2×#1956C211yesmaximal, characteristic, complemented
15#21083C211yesmaximal, characteristic
16#21064C211yesmaximal, characteristic, complemented
17#21020C211yesmaximal, characteristic, complemented
18#20877C211yesmaximal, characteristic, complemented
19#20877C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2570—22no—
41#2552—22no—
42#2546C2212yes—
43#2546C2212yes—
44#2545C2212yes—
45#2545C2212yes—
46#2507C2214yes—
47#2506C2214yes—
48#2473—24no—
49#2429—22no—
50#2429—22no—
51#2413—24no—
52#2317C2211yescharacteristic
53#2317C2211yescharacteristic
54#2316C2211yescharacteristic
55#2310C2211yescharacteristic
56#2297C2211yescharacteristic
57#2296C2211yescharacteristic
58#2296C2211yescharacteristic
59C2×#384C2211yescharacteristic
60#2274C2211yescharacteristic
61#2096C2212yes—
62#2095C2212yes—
63#2094C2212yes—
64#2093C2212yes—
65#2068C2214yes—
66#2066C2214yes—
67#2043C2214yes—
68#2043C2214yes—
69#2043C2214yes—
70#2043C2214yes—
71#2043C2214yes—
72#2042C2214yes—
73#2042C2214yes—
74#2042C2214yes—
75#1983C2214yes—
76#1981C2214yes—
77#1956C2214yes—
78#1956C2214yes—
79#1934C2212yes—
80#1934C2212yes—
81#1933C2212yes—
82#1933C2212yes—
83#1928C2214yes—
84#1927C2214yes—
85#1916C2214yes—
86#1916C2214yes—
87#1865C2214yes—
88#1865C2214yes—
89#1863C2214yes—
90#1863C2214yes—
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104#536—24no—
105#536—42no—
106#536—42no—
107#536—42no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—42no—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#125—24no—
116C2×#124C2312yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×#123—22no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—22no—
123#519—42no—
124#519—42no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×#117—22no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#428—42no—
141#428—42no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#424—42no—
147C2×C4⋊C8C2314yes—
148#419—24no—
149#419—24no—
150#419—24no—
151#419—42no—
152#417—42no—
153#417—42no—
154#416—24no—
155#416—42no—
156C2×#103—42no—
157C2×#103—42no—
158C2×#103—42no—
159C2×#103—42no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×C4×C8—42no—
177C2×C4×C8C2312yes—
178C2×C4×C8C2312yes—
179#400—24no—
180#398—24no—
181#398—24no—
182#398—24no—
183#398—24no—
184#398—24no—
185#396—24no—
186#396—24no—
187#391—22no—
188#391C2312yes—
189#391C2312yes—
190#391C2311yescharacteristic
191#391C2311yescharacteristic
192#390C2311yescharacteristic
193#390C2311yescharacteristic
194#389—22no—
195#389—24no—
196#389—24no—
197#389—24no—
198#389D412yes—
199#389D412yes—
200#389C2314yes—
201#389C2314yes—
202#389C2314yes—
203#389C2312yes—
204#389C2312yes—
205#388—24no—
206#388—24no—
207#387—24no—
208#387—24no—
209#386—22no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385D414yes—
214#384—21no—
215#384—21no—
216#384—22no—
217#384C2312yes—
218#384C2312yes—
219#382—21no—
220#382—21no—
221#381—22no—
222#380—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4C2314yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8#4114yes—
259C22×Q8C2412yes—
260C22×Q8C2411yescharacteristic
261C22×Q8C2412yes—
262C22×D4—21no—
263C22×D4C2411yescharacteristic
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—42no—
291C4⋊Q8—42no—
292C4⋊Q8—42no—
293C4⋊Q8—42no—
294C4⋊Q8—42no—
295C4⋊Q8—48no—
296#125—88no—
297#124—24no—
298#124—42no—
299#124—42no—
300#124—42no—
301#123—48no—
302#123—48no—
303C22⋊Q8—48no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C4×Q8—42no—
307C4×Q8—42no—
308C4×Q8—42no—
309C4×Q8—42no—
310C4×Q8—44no—
311C4×Q8—44no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—48no—
322#117#4114yes—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
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—48no—
344C2×C4⋊C4—84no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42C2×D412yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—42no—
380#107—42no—
381#106—42no—
382#106—42no—
383C4⋊C8—44no—
384#103—28no—
385#103—24no—
386#103—28no—
387#103—28no—
388#103—44no—
389#103—44no—
390#103—44no—
391#103—44no—
392#103—42no—
393#103—42no—
394#103—42no—
395#103—44no—
396#102—24no—
397#102—42no—
398#102—42no—
399#102—42no—
400C4×C8—42no—
401C4×C8—42no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95C2×D414yes—
413#95C2×D414yes—
414C24—21no—
415C24C22×D411yescharacteristic
416#41—44no—
417#41—44no—
418#41—44no—
419#41—44no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—28no—
423C2×Q8—28no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8C2×#4114yes—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C22×C4C2511yescharacteristic
499M16—44no—
500M16—44no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—44no—
514C2×C8—44no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4C2×#4114yes—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C42—24no—
559C42—24no—
560C42—21no—
561C42—21no—
562C42—22no—
563C42—21no—
564C42—21no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42—42no—
572C42—84no—
573C23—22no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23#53512yes—
581C23#54612yes—
582C23#54812yes—
583C23C23×D411yescharacteristic
584Q8—24no—
585Q8—24no—
586Q8—42no—
587Q8—48no—
588Q8—42no—
589Q8—42no—
590Q8—41no—
591Q8—42no—
592Q8—41no—
593Q8—42no—
594Q8—41no—
595Q8—42no—
596Q8—41no—
597Q8—84no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—84no—
603C2×C4—216no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—162no—
642C2×C4—162no—
643C2×C4#53512yes—
644C2×C4#53914yes—
645C2×C4#54014yes—
646C2×C4#54612yes—
647C2×C4#54812yes—
648C2×C4#55314yes—
649C2×C4#56214yes—
650C2×C4#57714yes—
651C2×C4#57814yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C8—42no—
659C8—42no—
660C8—42no—
661C8—42no—
662C22—24no—
663C22—22no—
664C22—22no—
665C22—24no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—82no—
671C22#271114yes—
672C22C2×#53511yescharacteristic
673C22C2×#54611yescharacteristic
674C22C2×#54811yescharacteristic
675C22C2×#57712yes—
676C22C2×#57812yes—
677C4—24no—
678C4—24no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4—81no—
689C4—82no—
690C4—82no—
691C4#271114yes—
692C4C2×#53911yescharacteristic
693C4C2×#54011yescharacteristic
694C4C2×#55311yescharacteristic
695C4C2×#56211yescharacteristic
696C2—24no—
697C2—161no—
698C2C2×#271111yescharacteristic
699C2#6082011yescharacteristic
700C2#6082111yescharacteristic
701C2#6116812yes—
702C2#6133012yes—
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. #35555C2semidirect
3. #32085C2semidirect
4. #32084C2semidirect
5. #32083C2semidirect
6. #32082C2semidirect
8. #23615C2semidirect
12. #23507C2semidirect
14. C2×#1956C2semidirect
16. #21064C2semidirect
17. #21020C2semidirect
18. #20877C2semidirect
19. #20877C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#355552
#320852
#320842
#320832
#320822
#237342
#236152
#235122
#235102
#235072
#212622
C2×#19562
#210832
#210642
#210202
#208772