Finite Group Databasebeta

← #355849

Subgroups — #355849

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 785 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1974C211yesmaximal, characteristic
2#20924C211yesmaximal, characteristic
3C2×#1876C211yesmaximal, characteristic
4#20384C211yesmaximal, characteristic, complemented
5#20375C211yesmaximal, characteristic, complemented
6#10934C211yesmaximal, characteristic, complemented
7#10933C211yesmaximal, characteristic, complemented
8#10930C211yesmaximal, characteristic, complemented
9#10930C211yesmaximal, characteristic, complemented
10#10911C211yesmaximal, characteristic, complemented
11#10910C211yesmaximal, characteristic, complemented
12#10818C211yesmaximal, characteristic, complemented
13#10791C211yesmaximal, characteristic, complemented
14#10790C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×#131C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#408C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#2297C2211yescharacteristic
35#1974C418yescomplemented
36#1880C412yescomplemented
37#1879C412yescomplemented
38#1878C412yescomplemented
39#1877C412yescomplemented
40#1876C418yescomplemented
41#1753—21no—
42#1753—21no—
43#1752—21no—
44#1752—21no—
45#1750—21no—
46#1750—21no—
47#1749—21no—
48#1749—21no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#341C2211yescharacteristic
52#1735C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C4×#95C2211yescharacteristic
65C22×#41C2311yescharacteristic
66C22×#41C2311yescharacteristic
67C2×#135—41no—
68C2×#135—41no—
69C2×#135—41no—
70C2×#135—41no—
71C2×#135C2×C414yes—
72C2×#135C2×C414yes—
73C22×Q16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×SD16C2×C411yescharacteristic
76C22×D8C2×C411yescharacteristic
77C2×#131—22no—
78C2×#131—41no—
79C2×#131—41no—
80C2×#131—41no—
81C2×#131—41no—
82C2×#131C2×C414yescomplemented
83C22×M16—21no—
84C22×M16C2311yescharacteristic
85C22×M16C2311yescharacteristic
86C22×M16C2311yescharacteristic
87C22×M16C2311yescharacteristic
88C23×C8—21no—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C4×#41—41no—
94C4×#41—41no—
95C4×#41—41no—
96C4×#41—41no—
97C2×C4×Q8C2×C412yes—
98C2×C4×D4—22no—
99C2×C4×D4C2×C412yes—
100C2×#117—21no—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C2×#108—41no—
108C2×#108—41no—
109C2×#108C2×C414yescomplemented
110C2×#104—41no—
111C2×#104—41no—
112C2×#104—41no—
113C2×#104—41no—
114C2×#104—41no—
115C2×#104—41no—
116C2×#104—41no—
117C2×#104—41no—
118C2×#104C2×C414yes—
119C2×#104C2×C414yes—
120C2×#104C2×C414yescomplemented
121C2×#104C2×C414yescomplemented
122#416—41no—
123#416—41no—
124#416—41no—
125#416—41no—
126C2×#103C2×C412yes—
127C2×#103C2×C412yes—
128C2×#102C2×C412yes—
129C2×#102C2×C412yes—
130#408—22no—
131#408—22no—
132#408—22no—
133#408C2×C414yes—
134#408C2×C414yes—
135#408C2×C414yescomplemented
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142#405—41no—
143#405—41no—
144C4×M16—41no—
145C4×M16—41no—
146C2×#97—21no—
147C2×#97—22no—
148C2×#97—21no—
149C2×#97—41no—
150C2×#97—41no—
151C2×#97—41no—
152C2×#97—41no—
153C2×#97D411yescharacteristic
154C2×#97D411yescharacteristic
155C2×#97D411yescharacteristic
156C2×#97D411yescharacteristic
157C2×C4×C8—21no—
158C2×C4×C8—21no—
159C2×C4×C8—41no—
160C2×C4×C8—41no—
161C2×C4×C8C2×C414yescomplemented
162#391C2×C412yes—
163#390C2×C412yes—
164#386—22no—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—21no—
167C2×#95—22no—
168#341—41no—
169#341—41no—
170#341C2×C414yescomplemented
171#340C2×C412yes—
172#340C2×C412yes—
173#339—41no—
174#339—41no—
175#339—41no—
176#339—41no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—22no—
184#336—21no—
185#336—21no—
186#336C2×C412yes—
187#336C2×C412yes—
188#336C2×C414yescomplemented
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
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×#41C22×C412yes—
210C2×#41C22×C412yes—
211C2×#41C22×C412yes—
212C2×#41C22×C412yes—
213C22×Q8C22×C411yescharacteristic
214C22×Q8C22×C411yescharacteristic
215C22×D4—21no—
216C22×D4—21no—
217C22×D4C22×C411yescharacteristic
218C22×D4C22×C411yescharacteristic
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4#4112yes—
225C23×C4C2411yescharacteristic
226#135—44no—
227#135—44no—
228#135—44no—
229#135—44no—
230C2×Q16—41no—
231C2×Q16—41no—
232C2×Q16C4214yes—
233C2×SD16—41no—
234C2×SD16—41no—
235C2×SD16—41no—
236C2×SD16—41no—
237C2×SD16C4214yes—
238C2×SD16C4214yes—
239C2×D8—41no—
240C2×D8—41no—
241C2×D8C4214yes—
242#131—44no—
243#131—44no—
244#131—44no—
245#131—44no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—24no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
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—21no—
264C2×M16—21no—
265C2×M16—21no—
266C2×M16—21no—
267C2×M16—42no—
268C2×M16—42no—
269C2×M16C22×C412yes—
270C2×M16C22×C412yes—
271C2×M16C22×C412yes—
272C2×M16C22×C412yes—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—21no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—21no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
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—24no—
296C22×C8—42no—
297C22×C8—42no—
298C22×C8C22×C412yes—
299C22×C8C22×C411yescharacteristic
300C22×C8C22×C411yescharacteristic
301C22×C8C22×C412yes—
302C22×C8C22×C412yes—
303C22×C8C22×C412yes—
304C22×C8C22×C412yes—
305C22×C8C2×D411yescharacteristic
306C22×C8C2×D411yescharacteristic
307C22×C8C2×D411yescharacteristic
308C22×C8C2×D411yescharacteristic
309C22×C8#4112yes—
310C22×C8#4112yes—
311C4×Q8—42no—
312C4×Q8—42no—
313C4×D4—42no—
314C4×D4—42no—
315C4×D4—44no—
316C4×D4—44no—
317#117—21no—
318#117—21no—
319#117—21no—
320#117—21no—
321#117—21no—
322#117—21no—
323#117—21no—
324#117—21no—
325#117—42no—
326#117—42no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4C22×C412yes—
329C2×C4⋊C4C22×C412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42C22×C412yes—
355C2×C42C22×C412yes—
356C2×C42C22×C412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359#108—44no—
360#104—44no—
361#104—44no—
362#104—44no—
363#104—44no—
364#103—42no—
365#103—42no—
366#102—42no—
367#102—42no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374#97—41no—
375#97—41no—
376#97—41no—
377#97—41no—
378#97—42no—
379#97—44no—
380#97—42no—
381C4×C8—44no—
382C4×C8—42no—
383C4×C8—42no—
384#95—22no—
385#95—22no—
386#95—44no—
387C24—21no—
388C24—21no—
389C24C2×#4111yescharacteristic
390#41—22no—
391#41—22no—
392#41—22no—
393#41—22no—
394#41—22no—
395#41—22no—
396#41—22no—
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—41no—
407#41—41no—
408#41—41no—
409#41—41no—
410#41—41no—
411#41—41no—
412#41—41no—
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—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8C2×C4212yes—
433C2×Q8C2×C4212yes—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4C2×C4212yes—
457C2×D4C2×C4212yes—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C4×D412yes—
492C22×C4C4×D412yes—
493C22×C4C23×C411yescharacteristic
494C22×C4C23×C411yescharacteristic
495C22×C4C23×C411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4111yescharacteristic
498C22×C4C2×#4111yescharacteristic
499C22×C4H(5,𝔽2)11yescharacteristic
500C22×C4H(5,𝔽2)11yescharacteristic
501C22×C4H(5,𝔽2)11yescharacteristic
502C22×C4H(5,𝔽2)11yescharacteristic
503C22×C4H(5,𝔽2)11yescharacteristic
504C22×C4H(5,𝔽2)11yescharacteristic
505C22×C4H(5,𝔽2)11yescharacteristic
506C22×C42−1+411yescharacteristic
507Q16—44no—
508SD16—44no—
509SD16—44no—
510D8—44no—
511M16—22no—
512M16—22no—
513M16—22no—
514M16—22no—
515M16—22no—
516M16—22no—
517M16—22no—
518M16—22no—
519M16—48no—
520M16—41no—
521M16—41no—
522M16—41no—
523M16—41no—
524M16—41no—
525M16—41no—
526M16—41no—
527M16—41no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—22no—
556C2×C8—22no—
557C2×C8—22no—
558C2×C8—22no—
559C2×C8—48no—
560C2×C8—41no—
561C2×C8—41no—
562C2×C8—41no—
563C2×C8—41no—
564C2×C8—41no—
565C2×C8—41no—
566C2×C8—41no—
567C2×C8—41no—
568C2×C8—41no—
569C2×C8—41no—
570C2×C8—41no—
571C2×C8—41no—
572C2×C8C2×C4212yes—
573C2×C8C2×C4212yes—
574C2×C8C4×D412yes—
575C2×C8C4×D412yes—
576C2×C8C4×D412yes—
577C2×C8C4×D412yes—
578C4⋊C4—22no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—22no—
582C4⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—41no—
585C22⋊C4—41no—
586C22⋊C4—41no—
587C22⋊C4—41no—
588C22⋊C4—82no—
589C22⋊C4—82no—
590C42—22no—
591C42—22no—
592C42—21no—
593C42—21no—
594C42—21no—
595C42—21no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—44no—
601C42—41no—
602C42—41no—
603C42—41no—
604C42—41no—
605C42—41no—
606C42—41no—
607C42—41no—
608C42—41no—
609C23—22no—
610C23—22no—
611C23—22no—
612C23—22no—
613C23—41no—
614C23—41no—
615C23—41no—
616C23—41no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C2×C4×D411yescharacteristic
622C23C4×#4111yescharacteristic
623C23C4×#4111yescharacteristic
624C23#53811yescharacteristic
625C23#53811yescharacteristic
626C23#54011yescharacteristic
627C23#54211yescharacteristic
628Q8—22no—
629Q8—22no—
630Q8—22no—
631Q8—22no—
632Q8—41no—
633Q8—41no—
634Q8—41no—
635Q8—41no—
636D4—22no—
637D4—22no—
638D4—22no—
639D4—22no—
640D4—42no—
641D4—42no—
642D4—42no—
643D4—42no—
644D4—42no—
645D4—42no—
646D4—42no—
647D4—42no—
648D4—41no—
649D4—41no—
650D4—41no—
651D4—41no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
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—82no—
687C2×C4—82no—
688C2×C4—88no—
689C2×C4—82no—
690C2×C4—82no—
691C2×C4C22×C4211yescharacteristic
692C2×C4C2×C4×D411yescharacteristic
693C2×C4C2×C4×D411yescharacteristic
694C2×C4C4×#4111yescharacteristic
695C2×C4#51811yescharacteristic
696C2×C4#51811yescharacteristic
697C2×C4#51811yescharacteristic
698C2×C4#51811yescharacteristic
699C2×C4#51811yescharacteristic
700C2×C4#51811yescharacteristic
701C2×C4#51811yescharacteristic
702C2×C4#51811yescharacteristic
703C2×C4#51811yescharacteristic
704C2×C4#52011yescharacteristic
705C2×C4#52011yescharacteristic
706C2×C4#52011yescharacteristic
707C2×C4#53411yescharacteristic
708C2×C4#53511yescharacteristic
709C2×C4#53511yescharacteristic
710C2×C4#53711yescharacteristic
711C2×C4#53811yescharacteristic
712C2×C4#53811yescharacteristic
713C2×C4#53811yescharacteristic
714C2×C4#53911yescharacteristic
715C2×C4#54011yescharacteristic
716C2×C4#54011yescharacteristic
717C2×C4#54011yescharacteristic
718C2×C4#54211yescharacteristic
719C8—22no—
720C8—22no—
721C8—22no—
722C8—22no—
723C8—22no—
724C8—22no—
725C8—42no—
726C8—42no—
727C8—42no—
728C8—42no—
729C8—42no—
730C22—21no—
731C22—21no—
732C22—21no—
733C22—21no—
734C22—22no—
735C22—22no—
736C22—22no—
737C22—22no—
738C22—42no—
739C22—42no—
740C22—42no—
741C22—42no—
742C22—42no—
743C22—42no—
744C22—82no—
745C22—82no—
746C22—82no—
747C22—82no—
748C22C42×D411yescharacteristic
749C22#232111yescharacteristic
750C22#232411yescharacteristic
751C22#233311yescharacteristic
752C22#233311yescharacteristic
753C22#233511yescharacteristic
754C22#233711yescharacteristic
755C22#295612yes—
756C22#295712yes—
757C4—44no—
758C4—42no—
759C4—42no—
760C4—42no—
761C4—42no—
762C4—82no—
763C4—82no—
764C4—82no—
765C4#227311yescharacteristic
766C4#227511yescharacteristic
767C4#232111yescharacteristic
768C4#232411yescharacteristic
769C4#232511yescharacteristic
770C4#232811yescharacteristic
771C4#233311yescharacteristic
772C4#233511yescharacteristic
773C2—21no—
774C2—21no—
775C2—21no—
776C2—21no—
777C2—82no—
778C2—82no—
779C2#1379111yescharacteristic
780C2#2051211yescharacteristic
781C2#2051411yescharacteristic
782C2#2206212yes—
783C2#2216712yes—
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 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20384C2semidirect
5. #20375C2semidirect
6. #10934C2semidirect
7. #10933C2semidirect
8. #10930C2semidirect
9. #10930C2semidirect
10. #10911C2semidirect
11. #10910C2semidirect
12. #10818C2semidirect
13. #10791C2semidirect
14. #10790C2semidirect
15. #10696C2semidirect
35. #1974C4semidirect
36. #1880C4semidirect
37. #1879C4semidirect
38. #1878C4semidirect
39. #1877C4semidirect
40. #1876C4semidirect
82. C2×#131C2×C4semidirect
109. C2×#108C2×C4semidirect
120. C2×#104C2×C4semidirect
121. C2×#104C2×C4semidirect
135. #408C2×C4semidirect
161. C2×C4×C8C2×C4semidirect
170. #341C2×C4semidirect
188. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19742
#209242
C2×#18762
#203842
#203752
#109342
#109332
#109302
#109112
#109102
#108182
#107912
#107902
#106962