Finite Group Databasebeta

← #7507700

Subgroups — #7507700

Order 512.

1689 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#35627C211yesmaximal, characteristic, complemented
3#32101C214yesmaximal, complemented
4#32100C214yesmaximal, complemented
5#32099C214yesmaximal, complemented
6#32098C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic
8#23615C211yesmaximal, characteristic, complemented
9#23512C211yesmaximal, characteristic
10#23510C211yesmaximal, characteristic
11#23510C211yesmaximal, characteristic
12#23507C211yesmaximal, characteristic, complemented
13#21262C211yesmaximal, characteristic
14C2×#1960C211yesmaximal, characteristic, complemented
15#21086C211yesmaximal, characteristic
16#21064C211yesmaximal, characteristic, complemented
17#21021C211yesmaximal, 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×#97C2211yescharacteristic
40#2571—22no—
41#2570—22no—
42#2546C2212yes—
43#2546C2212yes—
44#2545C2212yes—
45#2545C2212yes—
46#2509C2214yes—
47#2508C2214yes—
48#2505—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×#382C2211yescharacteristic
60#2274C2211yescharacteristic
61#2097C2212yes—
62#2097C2212yes—
63#2092C2212yes—
64#2092C2212yes—
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#1960C2214yes—
78#1960C2214yes—
79#1936C2212yes—
80#1936C2212yes—
81#1935C2212yes—
82#1935C2212yes—
83#1928C2214yes—
84#1927C2214yes—
85#1917C2214yes—
86#1917C2214yes—
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×#124C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×#123—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—22no—
122#519—42no—
123#519—42no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×#117—22no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#428—42no—
139#428—42no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144#424—42no—
145C2×C4⋊C8C2314yes—
146#419—24no—
147#419—24no—
148#419—24no—
149#419—42no—
150#417—42no—
151#417—42no—
152#416—24no—
153#416—42no—
154C2×#103—42no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103—42no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#97—42no—
175C2×#97C2312yes—
176C2×#97C2312yes—
177#400—24no—
178#400—24no—
179#400—24no—
180#398—24no—
181#398—24no—
182#398—24no—
183#396—24no—
184#396—24no—
185#391—22no—
186#391C2312yes—
187#391C2312yes—
188#391C2311yescharacteristic
189#391C2311yescharacteristic
190#390C2311yescharacteristic
191#390C2311yescharacteristic
192#389—22no—
193#389—24no—
194#389—24no—
195#389—24no—
196#389D412yes—
197#389D412yes—
198#389C2314yes—
199#389C2314yes—
200#389C2314yes—
201#389C2312yes—
202#389C2312yes—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385D414yes—
212#384—21no—
213#384—21no—
214#382—21no—
215#382—21no—
216#382—22no—
217#382—24no—
218#382C2312yes—
219#382C2312yes—
220#381—22no—
221#380—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4C2314yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C2×#95—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#124—24no—
297#124—42no—
298#124—42no—
299#124—42no—
300#123—48no—
301#123—48no—
302C22⋊Q8—48no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C4×Q8—42no—
306C4×Q8—42no—
307C4×Q8—42no—
308C4×Q8—42no—
309C4×Q8—44no—
310C4×Q8—44no—
311#117—22no—
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—48no—
321#117#4114yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—48no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42C2×D412yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—42no—
376#107—42no—
377#106—42no—
378#106—42no—
379C4⋊C8—44no—
380#103—28no—
381#103—24no—
382#103—28no—
383#103—28no—
384#103—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—42no—
389#103—42no—
390#103—42no—
391#103—44no—
392#102—24no—
393#102—42no—
394#102—42no—
395#102—42no—
396#97—42no—
397#97—42no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95C2×D414yes—
411#95C2×D414yes—
412C24—21no—
413C24C22×D411yescharacteristic
414#41—44no—
415#41—44no—
416#41—44no—
417#41—44no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8C2×#4114yes—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
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—82no—
483C22×C4—82no—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C42−1+412yes—
496C22×C4C2511yescharacteristic
497M16—44no—
498M16—44no—
499C2×C8—22no—
500C2×C8—22no—
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—44no—
512C2×C8—44no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4C2×#4114yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C42—24no—
555C42—24no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—44no—
565C42—44no—
566C42—42no—
567C42—42no—
568C42—84no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23#53512yes—
577C23#54612yes—
578C23#54812yes—
579C23C23×D411yescharacteristic
580Q8—24no—
581Q8—24no—
582Q8—42no—
583Q8—48no—
584Q8—42no—
585Q8—42no—
586Q8—41no—
587Q8—42no—
588Q8—41no—
589Q8—42no—
590Q8—41no—
591Q8—42no—
592Q8—41no—
593Q8—84no—
594D4—41no—
595D4—41no—
596D4—41no—
597D4—41no—
598D4—84no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—162no—
640C2×C4—162no—
641C2×C4#53512yes—
642C2×C4#53914yes—
643C2×C4#54014yes—
644C2×C4#54612yes—
645C2×C4#54812yes—
646C2×C4#55314yes—
647C2×C4#56214yes—
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C8—42no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22—24no—
659C22—22no—
660C22—22no—
661C22—24no—
662C22—82no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—82no—
667C22#271114yes—
668C22C2×#53511yescharacteristic
669C22C2×#54611yescharacteristic
670C22C2×#54811yescharacteristic
671C22#358612yes—
672C22#358612yes—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—81no—
681C4—81no—
682C4—82no—
683C4—82no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4#271114yes—
688C4C2×#53911yescharacteristic
689C4C2×#54011yescharacteristic
690C4C2×#55311yescharacteristic
691C4C2×#56211yescharacteristic
692C2—24no—
693C2—161no—
694C2C2×#271111yescharacteristic
695C2#6082811yescharacteristic
696C2#6082811yescharacteristic
697C2#6117612yes—
698C2#6133812yes—
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. #35627C2semidirect
3. #32101C2semidirect
4. #32100C2semidirect
5. #32099C2semidirect
6. #32098C2semidirect
8. #23615C2semidirect
12. #23507C2semidirect
14. C2×#1960C2semidirect
16. #21064C2semidirect
17. #21021C2semidirect
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
#356272
#321012
#321002
#320992
#320982
#237332
#236152
#235122
#235102
#235072
#212622
C2×#19602
#210862
#210642
#210212
#208772