Finite Group Databasebeta

← #378850

Subgroups — #378850

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 779 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23479C211yesmaximal, characteristic
2C2×#1938C211yesmaximal, characteristic, complemented
3#21050C211yesmaximal, characteristic, complemented
4C2×#1896C211yesmaximal, characteristic
5#20899C211yesmaximal, characteristic
6C2×#1765C211yesmaximal, characteristic, complemented
7C2×#1749C211yesmaximal, characteristic, complemented
8#11221C211yesmaximal, characteristic, complemented
9#11125C211yesmaximal, characteristic, complemented
10#11097C211yesmaximal, characteristic, complemented
11#10860C211yesmaximal, characteristic, complemented
12#10819C211yesmaximal, characteristic, complemented
13#10809C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic, complemented
15#10719C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2280C2211yescharacteristic
35#2036C412yescomplemented
36#2035C412yescomplemented
37#2034C412yescomplemented
38#2033C412yescomplemented
39#1938—24no—
40#1924—22no—
41#1923—22no—
42#1896—24no—
43#1870—22no—
44#1869—22no—
45#1765—24no—
46#1749—24no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—41no—
71C2×#131—41no—
72C22×M16—21no—
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C23×C8—21no—
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82#525—22no—
83#525—22no—
84#525—41no—
85#525—41no—
86C2×#123—21no—
87C2×C22⋊Q8C2×C412yes—
88C2×#121C2×C412yes—
89C2×#120—21no—
90C2×C4×D4—22no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C42—21no—
95C22×C42—22no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42—41no—
100C22×C42—41no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C2×#108—22no—
104C2×#108—41no—
105C2×#108—41no—
106#427—41no—
107#427—41no—
108C2×#107C2×C412yescomplemented
109C2×#106C2×C412yescomplemented
110#424—22no—
111#424—41no—
112C2×C4⋊C8—22no—
113C2×#104—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—41no—
118C2×#104D414yes—
119#416—22no—
120#416—22no—
121#416—22no—
122#416—22no—
123C2×#103C2×C411yescharacteristic, complemented
124C2×#103C2×C412yescomplemented
125C2×#103C2×C411yescharacteristic, complemented
126C2×#102C2×C411yescharacteristic, complemented
127C2×#102C2×C412yescomplemented
128C2×#102C2×C411yescharacteristic, complemented
129#408—22no—
130#408—41no—
131#407—22no—
132#407—22no—
133#407—41no—
134#407—41no—
135C2×C22⋊C8—22no—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—22no—
138C2×C22⋊C8—22no—
139C2×C22⋊C8—22no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8C2×C412yes—
143C2×C22⋊C8C2×C412yes—
144C2×C22⋊C8D412yes—
145C2×C22⋊C8D412yes—
146#405—22no—
147#405—22no—
148C2×#97—21no—
149C2×#97—22no—
150C2×#97—22no—
151C2×#97—22no—
152C2×#97—41no—
153C2×#97—41no—
154C2×#97D414yescomplemented
155C2×C4×C8—21no—
156#381—21no—
157#380C2×C412yes—
158#380C2×C412yes—
159#379—21no—
160C4×C22⋊C4—22no—
161#341—22no—
162#341—22no—
163#341—22no—
164#340C2×C412yescomplemented
165#340C2×C412yescomplemented
166#339—22no—
167#339—22no—
168#339—22no—
169#339—22no—
170#339—22no—
171#339—41no—
172#336—21no—
173#336—22no—
174#336—21no—
175#336—21no—
176#336—22no—
177#336—21no—
178#336—22no—
179#336—22no—
180C25#4111yescharacteristic
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41C2×D412yes—
192C2×#41#4112yes—
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4—22no—
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C22×C412yes—
218C23×C4C2×D412yes—
219C23×C4#4111yescharacteristic
220C23×C4C2411yescharacteristic
221#131—28no—
222#131—44no—
223#131—44no—
224#131—44no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—42no—
248C2×M16—42no—
249C2×M16—42no—
250C2×M16C2×D412yes—
251C2×M16#4112yes—
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—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—21no—
276C22×C8—42no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8C22×C412yes—
280C22×C8C22×C411yescharacteristic
281C22×C8C22×C411yescharacteristic
282C22×C8C2×D412yes—
283C22×C8C2×D412yes—
284C22×C8C2×D412yes—
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8#4112yes—
288C22×C8#4112yes—
289#123—42no—
290#123—42no—
291C22⋊Q8—24no—
292C22⋊Q8—42no—
293#121—24no—
294#121—42no—
295#120—42no—
296#120—42no—
297C4×D4—44no—
298C4×D4—44no—
299#117—21no—
300#117—21no—
301#117—21no—
302#117—21no—
303#117—21no—
304#117—21no—
305#117—21no—
306#117—21no—
307#117—21no—
308#117—21no—
309#117—21no—
310#117—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C22×C412yes—
313C2×C4⋊C4C22×C411yescharacteristic
314C2×C4⋊C4C22×C411yescharacteristic
315C2×C4⋊C4C22×C412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—44no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42#4112yes—
344C2×C42#4112yes—
345#108—44no—
346#107—42no—
347#106—42no—
348C4⋊C8—44no—
349#104—28no—
350#104—44no—
351#103—24no—
352#103—22no—
353#103—22no—
354#102—24no—
355#102—22no—
356#102—22no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—44no—
362C22⋊C8—44no—
363C22⋊C8—44no—
364#97—44no—
365#97—42no—
366#97—44no—
367C4×C8—42no—
368#95—22no—
369#95—22no—
370C24—21no—
371C24—21no—
372C24—24no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—21no—
378C24—42no—
379C24C4×D412yes—
380C24C2×#4111yescharacteristic
381#41—22no—
382#41—22no—
383#41—22no—
384#41—22no—
385#41—22no—
386#41—22no—
387#41—22no—
388#41—22no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C4×D412yes—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—82no—
417C2×D4C4×D412yes—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4C4×D412yes—
479C22×C4C23×C411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4111yescharacteristic
487C22×C4C2×#4111yescharacteristic
488C22×C4C2×#4111yescharacteristic
489C22×C4H(5,𝔽2)11yescharacteristic
490C22×C4H(5,𝔽2)11yescharacteristic
491C22×C4H(5,𝔽2)11yescharacteristic
492C22×C42−1+411yescharacteristic
493M16—22no—
494M16—22no—
495M16—22no—
496M16—22no—
497M16—48no—
498M16—41no—
499M16—41no—
500M16—41no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505M16—41no—
506M16—41no—
507M16—41no—
508M16—41no—
509M16—41no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—48no—
535C2×C8—41no—
536C2×C8—41no—
537C2×C8—41no—
538C2×C8—41no—
539C2×C8—41no—
540C2×C8—41no—
541C2×C8—41no—
542C2×C8—41no—
543C2×C8—41no—
544C2×C8—41no—
545C2×C8—41no—
546C2×C8—41no—
547C2×C8—41no—
548C2×C8—41no—
549C2×C8—41no—
550C2×C8—41no—
551C2×C8—41no—
552C2×C8—41no—
553C2×C8C4×D412yes—
554C2×C8C4×D412yes—
555C2×C8#12112yes—
556C2×C8#12112yes—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—41no—
573C22⋊C4—41no—
574C22⋊C4—44no—
575C22⋊C4—82no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—48no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42#12114yes—
591C23—22no—
592C23—22no—
593C23—24no—
594C23—24no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—42no—
607C23—42no—
608C23—41no—
609C23—41no—
610C23—41no—
611C23—41no—
612C23—41no—
613C23—42no—
614C23—42no—
615C23—41no—
616C23#44414yes—
617C23C2×C4×D411yescharacteristic
618C23#52511yescharacteristic
619C23#52911yescharacteristic
620C23#54611yescharacteristic
621C23#55111yescharacteristic
622C23#55111yescharacteristic
623C23#55211yescharacteristic
624Q8—22no—
625Q8—22no—
626Q8—41no—
627Q8—41no—
628D4—22no—
629D4—22no—
630D4—42no—
631D4—42no—
632D4—42no—
633D4—42no—
634D4—41no—
635D4—41no—
636D4—88no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—432no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
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—41no—
679C2×C4—41no—
680C2×C4C2×C4×D411yescharacteristic
681C2×C4C2×C4×D411yescharacteristic
682C2×C4C4×#4111yescharacteristic
683C2×C4C4×#4111yescharacteristic
684C2×C4#51811yescharacteristic
685C2×C4#52011yescharacteristic
686C2×C4C2×#12111yescharacteristic
687C2×C4C2×#12111yescharacteristic
688C2×C4#52511yescharacteristic
689C2×C4#52911yescharacteristic
690C2×C4#53211yescharacteristic
691C2×C4#53211yescharacteristic
692C2×C4#53411yescharacteristic
693C2×C4#53711yescharacteristic
694C2×C4#53811yescharacteristic
695C2×C4#53911yescharacteristic
696C2×C4D4211yescharacteristic
697C2×C4#54611yescharacteristic
698C2×C4#54611yescharacteristic
699C2×C4#54611yescharacteristic
700C2×C4#54711yescharacteristic
701C2×C4#54711yescharacteristic
702C2×C4#55111yescharacteristic
703C2×C4#55111yescharacteristic
704C2×C4#55111yescharacteristic
705C2×C4#55211yescharacteristic
706C2×C4#55511yescharacteristic
707C2×C4#55511yescharacteristic
708C8—22no—
709C8—22no—
710C8—22no—
711C8—22no—
712C8—42no—
713C8—42no—
714C8—42no—
715C8—42no—
716C8—81no—
717C8—81no—
718C22—21no—
719C22—21no—
720C22—21no—
721C22—21no—
722C22—22no—
723C22—22no—
724C22—22no—
725C22—22no—
726C22—24no—
727C22—24no—
728C22—24no—
729C22—44no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—42no—
734C22—42no—
735C22—42no—
736C22—42no—
737C22—44no—
738C22—42no—
739C22—81no—
740C22—81no—
741C22—82no—
742C22C4×#12111yescharacteristic
743C22#235811yescharacteristic
744C22#235911yescharacteristic
745C22#253411yescharacteristic
746C22#255311yescharacteristic
747C22#257211yescharacteristic
748C22#260911yescharacteristic
749C22C2×#44412yes—
750C22#295612yes—
751C4—44no—
752C4—44no—
753C4—42no—
754C4—42no—
755C4—81no—
756C4—81no—
757C4—82no—
758C4—82no—
759C4C4×#12111yescharacteristic
760C4#232511yescharacteristic
761C4#235911yescharacteristic
762C4#237211yescharacteristic
763C4#253411yescharacteristic
764C4#253711yescharacteristic
765C4#255311yescharacteristic
766C4#255511yescharacteristic
767C2—21no—
768C2—21no—
769C2—21no—
770C2—21no—
771C2—44no—
772C2—82no—
773C2#1422411yescharacteristic
774C2#2154611yescharacteristic
775C2#2155111yescharacteristic
776C2#2288612yes—
777C2#2295012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1938C2semidirect
3. #21050C2semidirect
6. C2×#1765C2semidirect
7. C2×#1749C2semidirect
8. #11221C2semidirect
9. #11125C2semidirect
10. #11097C2semidirect
11. #10860C2semidirect
12. #10819C2semidirect
13. #10809C2semidirect
14. #10791C2semidirect
15. #10719C2semidirect
35. #2036C4semidirect
36. #2035C4semidirect
37. #2034C4semidirect
38. #2033C4semidirect
108. C2×#107C2×C4semidirect
109. C2×#106C2×C4semidirect
123. C2×#103C2×C4semidirect
124. C2×#103C2×C4semidirect
125. C2×#103C2×C4semidirect
126. C2×#102C2×C4semidirect
127. C2×#102C2×C4semidirect
128. C2×#102C2×C4semidirect
154. C2×#97D4semidirect
164. #340C2×C4semidirect
165. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#234792
C2×#19382
#210502
C2×#18962
#208992
C2×#17652
C2×#17492
#112212
#111252
#110972
#108602
#108192
#108092
#107912
#107192