Finite Group Databasebeta

← #371082

Subgroups — #371082

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2#20384C211yesmaximal, characteristic
3#20367C211yesmaximal, characteristic, complemented
4#11376C211yesmaximal, characteristic, complemented
5#11317C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11125C211yesmaximal, characteristic
8#10933C211yesmaximal, characteristic
9#10911C211yesmaximal, characteristic, complemented
10#10910C211yesmaximal, characteristic, complemented
11#10909C211yesmaximal, characteristic, complemented
12#10808C211yesmaximal, characteristic, complemented
13#10798C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2297C2211yescharacteristic
28C4×#117C2211yescharacteristic
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#1753—21no—
38#1753—21no—
39#1752—21no—
40#1752—21no—
41#1746—21no—
42#1746—21no—
43#1745—21no—
44#1745—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
66C4×#95C2211yescharacteristic
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×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#108—22no—
91C2×#108—22no—
92C2×#108—41no—
93#427—41no—
94#427—41no—
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#104—22no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—41no—
104C2×#104—41no—
105#416—22no—
106#416—22no—
107#416—41no—
108#416—41no—
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117#408—22no—
118#408—22no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121#405—41no—
122#405—41no—
123C4×M16—22no—
124C4×M16—22no—
125C2×#97—22no—
126C2×#97—22no—
127C2×#97—21no—
128C2×#97—21no—
129C2×#97D411yescharacteristic
130C2×#97D411yescharacteristic
131C2×#97D411yescharacteristic
132C2×#97D411yescharacteristic
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135#391C2×C411yescharacteristic
136#391C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#386—21no—
140#386—21no—
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4C2×C411yescharacteristic
143C4×C4⋊C4C2×C411yescharacteristic
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149#376—21no—
150C2×#95—21no—
151C2×#95—21no—
152C43—21no—
153#341—22no—
154#341—22no—
155#341—41no—
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#339—22no—
161#339—22no—
162#339—41no—
163#339—41no—
164#339—41no—
165#339—41no—
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—21no—
175#336—21no—
176#336—22no—
177#336—22no—
178#336—21no—
179#336—21no—
180#336—41no—
181#336—41no—
182#336C2×C411yescharacteristic
183#336C2×C411yescharacteristic
184#336C2×C411yescharacteristic
185#336C2×C411yescharacteristic
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—24no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
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—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—24no—
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—42no—
266C22×C8—42no—
267C22×C8C22×C411yescharacteristic
268C22×C8C22×C411yescharacteristic
269C22×C8C22×C411yescharacteristic
270C22×C8C22×C411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×D411yescharacteristic
275C22×C8#4112yes—
276C22×C8#4112yes—
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—21no—
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—21no—
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329#108—44no—
330#107—41no—
331#107—41no—
332#106—41no—
333#106—41no—
334#104—44no—
335#104—44no—
336#103—22no—
337#103—22no—
338#103—41no—
339#103—41no—
340#102—22no—
341#102—22no—
342#102—41no—
343#102—41no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346#97—24no—
347#97—22no—
348#97—22no—
349#97—22no—
350#97—22no—
351#97—42no—
352C4×C8—24no—
353C4×C8—42no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—42no—
363#95—42no—
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—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
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—22no—
459M16—22no—
460M16—22no—
461M16—48no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
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—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—48no—
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#12312yes—
522C2×C8#12312yes—
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—81no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—81no—
568C42—81no—
569C42—81no—
570C42—81no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C2×C4×D411yescharacteristic
583C23#52911yescharacteristic
584C23#52911yescharacteristic
585C23#52911yescharacteristic
586C23#52911yescharacteristic
587C23#54011yescharacteristic
588C23#54211yescharacteristic
589Q8—22no—
590Q8—22no—
591Q8—41no—
592Q8—41no—
593D4—22no—
594D4—22no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—41no—
600D4—41no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—41no—
615C2×C4—41no—
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—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4C2×#11711yescharacteristic
657C2×C4C4×#4111yescharacteristic
658C2×C4C4×#4111yescharacteristic
659C2×C4C4×#4111yescharacteristic
660C2×C4#51811yescharacteristic
661C2×C4#52011yescharacteristic
662C2×C4C2×#12311yescharacteristic
663C2×C4#52511yescharacteristic
664C2×C4#52511yescharacteristic
665C2×C4#52511yescharacteristic
666C2×C4#53511yescharacteristic
667C2×C4#53711yescharacteristic
668C2×C4#55111yescharacteristic
669C2×C4#55111yescharacteristic
670C2×C4#55111yescharacteristic
671C2×C4#55111yescharacteristic
672C2×C4#55111yescharacteristic
673C2×C4#55111yescharacteristic
674C2×C4#55111yescharacteristic
675C2×C4#55211yescharacteristic
676C2×C4#55211yescharacteristic
677C2×C4#55211yescharacteristic
678C2×C4#55311yescharacteristic
679C2×C4#55311yescharacteristic
680C2×C4#55311yescharacteristic
681C2×C4#55511yescharacteristic
682C2×C4#55511yescharacteristic
683C2×C4#55611yescharacteristic
684C8—22no—
685C8—22no—
686C8—42no—
687C8—42no—
688C8—41no—
689C8—41no—
690C8—42no—
691C8—41no—
692C8—41no—
693C8—42no—
694C8—41no—
695C8—41no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22#229011yescharacteristic
716C22C4×#12311yescharacteristic
717C22#232411yescharacteristic
718C22#252611yescharacteristic
719C22#252711yescharacteristic
720C22#253011yescharacteristic
721C22#253111yescharacteristic
722C22#295611yescharacteristic
723C22#295611yescharacteristic
724C22#295711yescharacteristic
725C22#295711yescharacteristic
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—42no—
731C4—82no—
732C4—82no—
733C4—81no—
734C4—81no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4#234311yescharacteristic
740C4#235311yescharacteristic
741C4#237211yescharacteristic
742C4#237311yescharacteristic
743C4#248211yescharacteristic
744C4#248511yescharacteristic
745C4#259811yescharacteristic
746C4#260911yescharacteristic
747C2—21no—
748C2—21no—
749C2—21no—
750C2—21no—
751C2—82no—
752C2#1406611yescharacteristic
753C2#2113911yescharacteristic
754C2#2114011yescharacteristic
755C2#2159411yescharacteristic
756C2#2159411yescharacteristic
757C2#2216711yescharacteristic
758C2#2216711yescharacteristic
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 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20367C2semidirect
4. #11376C2semidirect
5. #11317C2semidirect
9. #10911C2semidirect
10. #10910C2semidirect
11. #10909C2semidirect
12. #10808C2semidirect
13. #10798C2semidirect
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
#203842
#203672
#113762
#113172
#111912
#111252
#109332
#109112
#109102
#109092
#108082
#107982
#107912