Finite Group Databasebeta

← #366265

Subgroups — #366265

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2#20774C211yesmaximal, characteristic, complemented
3#20387C211yesmaximal, characteristic
4#11376C211yesmaximal, characteristic, complemented
5#11258C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11125C211yesmaximal, characteristic
8#10923C211yesmaximal, characteristic
9#10911C211yesmaximal, characteristic, complemented
10#10909C211yesmaximal, characteristic, complemented
11#10809C211yesmaximal, characteristic, complemented
12#10795C211yesmaximal, characteristic, complemented
13#10791C211yesmaximal, characteristic
14#10791C211yesmaximal, characteristic
15C4×#339C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#2297C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1928C411yescharacteristic, complemented
30#1928C411yescharacteristic, complemented
31#1927C411yescharacteristic, complemented
32#1927C411yescharacteristic, complemented
33#1926C411yescharacteristic, complemented
34#1926C411yescharacteristic, complemented
35#1925C411yescharacteristic, complemented
36#1925C411yescharacteristic, complemented
37#1840—21no—
38#1840—21no—
39#1839—21no—
40#1839—21no—
41#1753—21no—
42#1753—21no—
43#1752—21no—
44C8×C22⋊C4—21no—
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48#1735C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—22no—
71C2×#131—22no—
72C2×#131—41no—
73C2×#131—41no—
74C22×M16—21no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×#108—22no—
93C2×#108—41no—
94C2×#108—41no—
95#427—41no—
96#427—41no—
97C2×#107C2×C411yescharacteristic
98C2×#107C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—41no—
106C2×#104—41no—
107#416—22no—
108#416—22no—
109#416—41no—
110#416—41no—
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119#408—22no—
120#408—22no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123#405—22no—
124#405—41no—
125C4×M16—22no—
126C4×M16—41no—
127C2×#97—21no—
128C2×#97—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131#391C2×C411yescharacteristic
132#391C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#386—21no—
136#386—21no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145#376—21no—
146C43—21no—
147C43—22no—
148C43—41no—
149#341—22no—
150#341—22no—
151#341—41no—
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#339—22no—
157#339—22no—
158#339—22no—
159#339—22no—
160#339—22no—
161#339—41no—
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—22no—
175#336—41no—
176#336—41no—
177#336—41no—
178#336C2×C411yescharacteristic
179#336C2×C411yescharacteristic
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182#336D411yescharacteristic
183#336D411yescharacteristic
184#336D411yescharacteristic
185#336D411yescharacteristic
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41#4112yes—
197C2×#41#4112yes—
198C22×Q8C22×C411yescharacteristic
199C22×D4—21no—
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4#4111yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4C2411yescharacteristic
207#131—44no—
208#131—44no—
209#131—44no—
210#131—44no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—24no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—42no—
233C2×M16—42no—
234C2×M16#4112yes—
235C2×M16#4112yes—
236C2×M16#4112yes—
237C2×M16#4112yes—
238C22×C8—21no—
239C22×C8—24no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—42no—
268C22×C8—42no—
269C22×C8C22×C411yescharacteristic
270C22×C8C22×C411yescharacteristic
271C22×C8C22×C411yescharacteristic
272C22×C8C22×C411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×D411yescharacteristic
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D411yescharacteristic
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—24no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42#4112yes—
334C2×C42#4112yes—
335#108—44no—
336#107—41no—
337#107—41no—
338#106—41no—
339#106—41no—
340#104—44no—
341#104—44no—
342#103—22no—
343#103—22no—
344#103—41no—
345#103—41no—
346#102—22no—
347#102—22no—
348#102—41no—
349#102—41no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352#97—42no—
353#97—42no—
354C4×C8—42no—
355C4×C8—42no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364C24—21no—
365C24C2×#4111yescharacteristic
366#41—22no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8#11712yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4#11712yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C4H(5,𝔽2)11yescharacteristic
452C22×C42−1+411yescharacteristic
453M16—22no—
454M16—22no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—48no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16#12614yes—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—48no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—48no—
558C42—48no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—42no—
570C42—42no—
571C42—81no—
572C42—81no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23C2×C4×D411yescharacteristic
585C23#52911yescharacteristic
586C23#52911yescharacteristic
587C23#55111yescharacteristic
588C23#55211yescharacteristic
589C23#55311yescharacteristic
590C23#55511yescharacteristic
591Q8—22no—
592Q8—22no—
593Q8—41no—
594Q8—41no—
595D4—22no—
596D4—22no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—41no—
602D4—41no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4#44312yes—
657C2×C4#44412yes—
658C2×C4C2×#11711yescharacteristic
659C2×C4C4×#4111yescharacteristic
660C2×C4C4×#4111yescharacteristic
661C2×C4C4×#4111yescharacteristic
662C2×C4#51811yescharacteristic
663C2×C4#52011yescharacteristic
664C2×C4#52511yescharacteristic
665C2×C4#52511yescharacteristic
666C2×C4C2×#12611yescharacteristic
667C2×C4#52911yescharacteristic
668C2×C4#52911yescharacteristic
669C2×C4#52911yescharacteristic
670C2×C4#53811yescharacteristic
671C2×C4#53811yescharacteristic
672C2×C4#54211yescharacteristic
673C2×C4#54211yescharacteristic
674C2×C4#54611yescharacteristic
675C2×C4#54611yescharacteristic
676C2×C4#54711yescharacteristic
677C2×C4#55011yescharacteristic
678C2×C4#55111yescharacteristic
679C2×C4#55111yescharacteristic
680C2×C4#55111yescharacteristic
681C2×C4#55111yescharacteristic
682C2×C4#55211yescharacteristic
683C2×C4#55211yescharacteristic
684C2×C4#55311yescharacteristic
685C2×C4#55311yescharacteristic
686C8—24no—
687C8—41no—
688C8—41no—
689C8—41no—
690C8—41no—
691C8—42no—
692C8—41no—
693C8—41no—
694C8—41no—
695C8—41no—
696C8—81no—
697C8—81no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22#229011yescharacteristic
718C22#235911yescharacteristic
719C22#236311yescharacteristic
720C22#249511yescharacteristic
721C22#251511yescharacteristic
722C22#257211yescharacteristic
723C22#257811yescharacteristic
724C22C2×#44311yescharacteristic
725C22C2×#44411yescharacteristic
726C22#295611yescharacteristic
727C22#295611yescharacteristic
728C4—22no—
729C4—22no—
730C4—44no—
731C4—44no—
732C4—41no—
733C4—42no—
734C4—41no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4—81no—
740C4—81no—
741C4C4×#12611yescharacteristic
742C4#233711yescharacteristic
743C4#234311yescharacteristic
744C4#235311yescharacteristic
745C4#246711yescharacteristic
746C4#246811yescharacteristic
747C4#247911yescharacteristic
748C4#248011yescharacteristic
749C2—21no—
750C2—21no—
751C2—21no—
752C2—21no—
753C2—82no—
754C2#1399011yescharacteristic
755C2#2113711yescharacteristic
756C2#2113911yescharacteristic
757C2#2294811yescharacteristic
758C2#2295011yescharacteristic
759C2#2311311yescharacteristic
760C2#2311411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20774C2semidirect
4. #11376C2semidirect
5. #11258C2semidirect
9. #10911C2semidirect
10. #10909C2semidirect
11. #10809C2semidirect
12. #10795C2semidirect
15. C4×#339C2semidirect
29. #1928C4semidirect
30. #1928C4semidirect
31. #1927C4semidirect
32. #1927C4semidirect
33. #1926C4semidirect
34. #1926C4semidirect
35. #1925C4semidirect
36. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
#207742
#203872
#113762
#112582
#111912
#111252
#109232
#109112
#109092
#108092
#107952
#107912
C4×#3392