Finite Group Databasebeta

← #357200

Subgroups — #357200

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 789 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21228C211yesmaximal, characteristic
2C2×#1876C211yesmaximal, characteristic
3C2×#1876C211yesmaximal, characteristic
4C2×#1765C211yesmaximal, characteristic, complemented
5C2×#1764C211yesmaximal, characteristic, complemented
6#20387C211yesmaximal, characteristic, complemented
7#20387C211yesmaximal, characteristic, complemented
8#10933C211yesmaximal, characteristic, complemented
9#10933C211yesmaximal, characteristic, complemented
10#10930C211yesmaximal, characteristic, complemented
11#10929C211yesmaximal, characteristic, complemented
12#10909C211yesmaximal, characteristic, complemented
13#10909C211yesmaximal, characteristic, complemented
14#10705C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×#131C2211yescharacteristic
18C22×#131C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2297C2211yescharacteristic
36#2297C2211yescharacteristic
37#1973C412yescomplemented
38#1973C412yescomplemented
39#1972C412yescomplemented
40#1971C412yescomplemented
41#1876C418yescomplemented
42#1876C418yescomplemented
43#1765—24no—
44#1764—24no—
45#1753—21no—
46#1753—21no—
47#1753—21no—
48#1753—21no—
49#1752—21no—
50#1752—21no—
51C8×C22⋊C4—21no—
52C8×C22⋊C4—21no—
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55C2×#341C2211yescharacteristic
56C2×#341C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C22×#41C2311yescharacteristic
69C2×#135—41no—
70C2×#135—41no—
71C2×#135—41no—
72C2×#135—41no—
73C2×#135C2×C414yes—
74C2×#135C2×C414yes—
75C22×Q16C2×C411yescharacteristic
76C22×SD16C2×C411yescharacteristic
77C22×SD16C2×C411yescharacteristic
78C22×D8C2×C411yescharacteristic
79C2×#131—22no—
80C2×#131—22no—
81C2×#131—22no—
82C2×#131—22no—
83C2×#131—22no—
84C2×#131—22no—
85C2×#131—41no—
86C2×#131—41no—
87C2×#131—41no—
88C2×#131—41no—
89C2×#131C2×C414yes—
90C2×#131C2×C414yes—
91C22×M16—21no—
92C22×M16—21no—
93C22×M16C2311yescharacteristic
94C22×M16C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8—21no—
98C23×C8—21no—
99C23×C8C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#427—41no—
109#427—41no—
110C2×#107C2×C412yes—
111C2×#106C2×C412yes—
112C2×#104—22no—
113C2×#104—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—41no—
117C2×#104—41no—
118C2×#104—41no—
119C2×#104—41no—
120C2×#104C2×C414yes—
121C2×#104C2×C414yes—
122C2×#104C2×C414yes—
123C2×#104C2×C414yes—
124#416—22no—
125#416—22no—
126#416—22no—
127#416—22no—
128C2×#103C2×C412yes—
129C2×#103C2×C412yes—
130C2×#102C2×C412yes—
131C2×#102C2×C412yes—
132#408—22no—
133#408—22no—
134#408C2×C414yes—
135#408C2×C414yes—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140#405—22no—
141#405—22no—
142#405—41no—
143#405—41no—
144C4×M16—22no—
145C4×M16—22no—
146C4×M16—41no—
147C4×M16—41no—
148C2×#97—22no—
149C2×#97—21no—
150C2×#97—21no—
151C2×#97—22no—
152C2×#97—21no—
153C2×#97—21no—
154C2×#97—41no—
155C2×#97—41no—
156C2×#97—41no—
157C2×#97C2×C412yes—
158C2×#97C2×C412yes—
159C2×C4×C8—22no—
160C2×C4×C8—21no—
161C2×C4×C8—21no—
162C2×C4×C8—22no—
163C2×C4×C8—21no—
164C2×C4×C8—21no—
165C2×C4×C8—41no—
166#391C2×C412yes—
167#391C2×C412yes—
168#390C2×C412yes—
169#390C2×C412yes—
170#386—22no—
171#386—22no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C2×#95—22no—
177#341—22no—
178#341C2×C414yes—
179#341C2×C414yes—
180#341D414yes—
181#339—22no—
182#339—22no—
183#339—41no—
184#339—41no—
185#336—21no—
186#336—21no—
187#336—21no—
188#336—21no—
189#336C2×C414yes—
190#336C2×C414yes—
191#336D411yescharacteristic
192#336D411yescharacteristic
193#336D411yescharacteristic
194#336D411yescharacteristic
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41—21no—
206C2×#41—21no—
207C2×#41—21no—
208C2×#41—21no—
209C2×#41—21no—
210C2×#41—21no—
211C2×#41—21no—
212C2×#41—21no—
213C2×#41—21no—
214C2×#41—21no—
215C2×#41C22×C412yes—
216C2×#41C22×C412yes—
217C2×#41C22×C412yes—
218C2×#41C22×C412yes—
219C22×Q8C22×C411yescharacteristic
220C22×Q8C22×C411yescharacteristic
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C22×C411yescharacteristic
224C22×D4C22×C411yescharacteristic
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4#4112yes—
230C23×C4C2411yescharacteristic
231#135—44no—
232#135—44no—
233#135—44no—
234#135—44no—
235C2×Q16—41no—
236C2×Q16—41no—
237C2×Q16C4214yes—
238C2×SD16—41no—
239C2×SD16—41no—
240C2×SD16—41no—
241C2×SD16—41no—
242C2×SD16C4214yes—
243C2×SD16C4214yes—
244C2×D8—41no—
245C2×D8—41no—
246C2×D8C4214yes—
247#131—28no—
248#131—28no—
249#131—44no—
250#131—44no—
251#131—44no—
252#131—44no—
253#131—44no—
254#131—44no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—21no—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—21no—
270C2×M16—21no—
271C2×M16—21no—
272C2×M16—21no—
273C2×M16—42no—
274C2×M16—42no—
275C2×M16—42no—
276C2×M16—42no—
277C2×M16C22×C412yes—
278C2×M16C22×C412yes—
279C2×M16C22×C412yes—
280C2×M16C22×C412yes—
281C2×M16C2×D412yes—
282C2×M16C2×D412yes—
283C2×M16#4112yes—
284C2×M16#4112yes—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—21no—
293C22×C8—21no—
294C22×C8—21no—
295C22×C8—21no—
296C22×C8—21no—
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—21no—
300C22×C8—21no—
301C22×C8—21no—
302C22×C8—21no—
303C22×C8—21no—
304C22×C8—21no—
305C22×C8—24no—
306C22×C8—24no—
307C22×C8—42no—
308C22×C8—42no—
309C22×C8—42no—
310C22×C8—42no—
311C22×C8C22×C411yescharacteristic
312C22×C8C22×C411yescharacteristic
313C22×C8C22×C412yes—
314C22×C8C22×C412yes—
315C22×C8C22×C412yes—
316C22×C8C22×C412yes—
317C22×C8C22×C412yes—
318C22×C8C2×D412yes—
319C22×C8C2×D411yescharacteristic
320C22×C8C2×D411yescharacteristic
321C22×C8C2×D411yescharacteristic
322C22×C8C2×D411yescharacteristic
323C22×C8#4112yes—
324C22×C8#4112yes—
325C22×C8#4112yes—
326#117—21no—
327#117—21no—
328#117—21no—
329#117—21no—
330#117—21no—
331#117—21no—
332#117—21no—
333#117—21no—
334C2×C4⋊C4C22×C412yes—
335C2×C4⋊C4C22×C412yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42C22×C412yes—
359C2×C42C22×C412yes—
360C2×C42C22×C412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363#107—42no—
364#106—42no—
365#104—28no—
366#104—28no—
367#104—44no—
368#104—44no—
369#103—24no—
370#103—24no—
371#102—24no—
372#102—24no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377#97—42no—
378#97—42no—
379#97—42no—
380#97—42no—
381#97—44no—
382#97—42no—
383#97—42no—
384C4×C8—42no—
385C4×C8—42no—
386C4×C8—44no—
387C4×C8—42no—
388C4×C8—42no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—44no—
394C24—21no—
395C24—21no—
396C24C2×#4111yescharacteristic
397#41—22no—
398#41—22no—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—22no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—22no—
412#41—22no—
413#41—41no—
414#41—41no—
415#41—41no—
416#41—41no—
417#41—41no—
418#41—41no—
419#41—41no—
420#41—41no—
421#41—41no—
422#41—41no—
423#41—41no—
424#41—41no—
425#41—41no—
426#41—41no—
427#41—41no—
428#41—41no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×C4212yes—
440C2×Q8C2×C4212yes—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4C2×C4212yes—
464C2×D4C2×C4212yes—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4C4×D412yes—
496C22×C4C4×D412yes—
497C22×C4C23×C411yescharacteristic
498C22×C4C23×C411yescharacteristic
499C22×C4C23×C411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C2×#4111yescharacteristic
503C22×C4C2×#4111yescharacteristic
504C22×C4C2×#4111yescharacteristic
505C22×C4C2×#4111yescharacteristic
506C22×C4C2×#4111yescharacteristic
507C22×C4H(5,𝔽2)11yescharacteristic
508C22×C4H(5,𝔽2)11yescharacteristic
509C22×C4H(5,𝔽2)11yescharacteristic
510C22×C4H(5,𝔽2)11yescharacteristic
511Q16—44no—
512SD16—44no—
513SD16—44no—
514D8—44no—
515M16—22no—
516M16—22no—
517M16—22no—
518M16—22no—
519M16—22no—
520M16—22no—
521M16—22no—
522M16—22no—
523M16—22no—
524M16—22no—
525M16—48no—
526M16—48no—
527M16—41no—
528M16—41no—
529M16—41no—
530M16—41no—
531M16C4×D414yes—
532M16C4×D414yes—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8—22no—
556C2×C8—22no—
557C2×C8—22no—
558C2×C8—22no—
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—21no—
565C2×C8—22no—
566C2×C8—22no—
567C2×C8—22no—
568C2×C8—22no—
569C2×C8—22no—
570C2×C8—22no—
571C2×C8—48no—
572C2×C8—48no—
573C2×C8—41no—
574C2×C8—41no—
575C2×C8—41no—
576C2×C8—41no—
577C2×C8C2×C4212yes—
578C2×C8C2×C4212yes—
579C2×C8C4×D412yes—
580C2×C8C4×D412yes—
581C2×C8C4×D412yes—
582C2×C8C4×D412yes—
583C2×C8C4×D414yes—
584C4⋊C4—22no—
585C4⋊C4—22no—
586C4⋊C4—22no—
587C4⋊C4—22no—
588C22⋊C4—44no—
589C22⋊C4—44no—
590C22⋊C4—41no—
591C22⋊C4—41no—
592C22⋊C4—41no—
593C22⋊C4—41no—
594C22⋊C4—82no—
595C22⋊C4—82no—
596C42—22no—
597C42—22no—
598C42—21no—
599C42—21no—
600C42—21no—
601C42—21no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—41no—
607C42—41no—
608C42—41no—
609C42—41no—
610C42—41no—
611C42—41no—
612C42—41no—
613C42—41no—
614C23—22no—
615C23—22no—
616C23—22no—
617C23—22no—
618C23—41no—
619C23—41no—
620C23—41no—
621C23—41no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23C2×C4×D411yescharacteristic
627C23C4×#4111yescharacteristic
628C23C4×#4111yescharacteristic
629C23#55011yescharacteristic
630C23#55311yescharacteristic
631C23#55311yescharacteristic
632C23#55911yescharacteristic
633Q8—22no—
634Q8—22no—
635Q8—22no—
636Q8—22no—
637Q8—41no—
638Q8—41no—
639Q8—41no—
640Q8—41no—
641D4—22no—
642D4—22no—
643D4—22no—
644D4—22no—
645D4—42no—
646D4—42no—
647D4—42no—
648D4—42no—
649D4—42no—
650D4—42no—
651D4—42no—
652D4—42no—
653D4—41no—
654D4—41no—
655D4—41no—
656D4—41no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—82no—
692C2×C4—82no—
693C2×C4—82no—
694C2×C4—82no—
695C2×C4C22×C4211yescharacteristic
696C2×C4C2×C4×D411yescharacteristic
697C2×C4C2×C4×D411yescharacteristic
698C2×C4C2×C4×D411yescharacteristic
699C2×C4C2×C4×D411yescharacteristic
700C2×C4C2×C4×D411yescharacteristic
701C2×C4C4×#4111yescharacteristic
702C2×C4C4×#4111yescharacteristic
703C2×C4C4×#4111yescharacteristic
704C2×C4C4×#4111yescharacteristic
705C2×C4#51811yescharacteristic
706C2×C4#51811yescharacteristic
707C2×C4#51811yescharacteristic
708C2×C4#51811yescharacteristic
709C2×C4#51811yescharacteristic
710C2×C4#51811yescharacteristic
711C2×C4D4211yescharacteristic
712C2×C4#54611yescharacteristic
713C2×C4#54611yescharacteristic
714C2×C4#54611yescharacteristic
715C2×C4#54611yescharacteristic
716C2×C4#55011yescharacteristic
717C2×C4#55111yescharacteristic
718C2×C4#55111yescharacteristic
719C2×C4#55311yescharacteristic
720C2×C4#55311yescharacteristic
721C2×C4#55511yescharacteristic
722C2×C4#55511yescharacteristic
723C8—22no—
724C8—22no—
725C8—22no—
726C8—22no—
727C8—24no—
728C8—24no—
729C8—24no—
730C8—22no—
731C8—22no—
732C8—42no—
733C8—42no—
734C22—21no—
735C22—21no—
736C22—21no—
737C22—21no—
738C22—22no—
739C22—22no—
740C22—22no—
741C22—22no—
742C22—42no—
743C22—42no—
744C22—42no—
745C22—42no—
746C22—42no—
747C22—42no—
748C22—82no—
749C22—82no—
750C22—82no—
751C22—82no—
752C22C42×D411yescharacteristic
753C22#236311yescharacteristic
754C22#236311yescharacteristic
755C22#236411yescharacteristic
756C22#236611yescharacteristic
757C22#237411yescharacteristic
758C22#237411yescharacteristic
759C22#295512yes—
760C22#295612yes—
761C4—44no—
762C4—42no—
763C4—42no—
764C4—42no—
765C4—42no—
766C4—82no—
767C4—82no—
768C4—82no—
769C4C42×D411yescharacteristic
770C4#227311yescharacteristic
771C4#235811yescharacteristic
772C4#235811yescharacteristic
773C4#236311yescharacteristic
774C4#236311yescharacteristic
775C4#236411yescharacteristic
776C4#237211yescharacteristic
777C2—21no—
778C2—21no—
779C2—21no—
780C2—21no—
781C2—82no—
782C2—82no—
783C2#1382411yescharacteristic
784C2#2051011yescharacteristic
785C2#2051211yescharacteristic
786C2#2314812yes—
787C2#2318112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#135C22
LCSG ▷ C2×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
4. C2×#1765C2semidirect
5. C2×#1764C2semidirect
6. #20387C2semidirect
7. #20387C2semidirect
8. #10933C2semidirect
9. #10933C2semidirect
10. #10930C2semidirect
11. #10929C2semidirect
12. #10909C2semidirect
13. #10909C2semidirect
14. #10705C2semidirect
15. #10696C2semidirect
37. #1973C4semidirect
38. #1973C4semidirect
39. #1972C4semidirect
40. #1971C4semidirect
41. #1876C4semidirect
42. #1876C4semidirect

Maximal subgroups

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

subgroupindex
#212282
C2×#18762
C2×#17652
C2×#17642
#203872
#109332
#109302
#109292
#109092
#107052
#106962