Finite Group Databasebeta

← #355460

Subgroups — #355460

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2929C211yesmaximal, characteristic
2C2×#1896C211yesmaximal, characteristic
3#20967C211yesmaximal, characteristic
4#20385C211yesmaximal, characteristic, complemented
5C22×#336C211yesmaximal, characteristic, complemented
6#10934C212yesmaximal, complemented
7#10887C212yesmaximal, complemented
8#10809C212yesmaximal, complemented
9#10791C212yesmaximal, complemented
10#10723C211yesmaximal, characteristic, complemented
11#10719C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C23×C42C2211yescharacteristic
17#2929—24no—
18#2929C418yescomplemented
19C2×#435—21no—
20C2×#435—21no—
21C2×C8×D4—21no—
22C2×C8×D4—21no—
23C2×#424C2211yescharacteristic
24C22×#104C2212yes—
25C2×#416C2212yes—
26C2×#408C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2212yes—
31#2280C2211yescharacteristic
32#1896C418yescomplemented
33#1893C412yescomplemented
34#1893C412yescomplemented
35#1892C412yescomplemented
36#1892C412yescomplemented
37#1752—21no—
38#1752—21no—
39C8×C22⋊C4—21no—
40C8×C22⋊C4—21no—
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1735C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1729C2212yes—
48C2×#336—24no—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C22×#41C2311yescharacteristic
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C2×#131—22no—
55C2×#131—42no—
56C2×#131—42no—
57C2×#131C2×C414yescomplemented
58C22×M16—21no—
59C22×M16C2311yescharacteristic
60C22×M16C2312yes—
61C22×M16C2311yescharacteristic
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—21no—
65C23×C8—42no—
66C23×C8—42no—
67C23×C8—42no—
68C23×C8—42no—
69C23×C8C2×C412yes—
70C23×C8C2×C412yes—
71C23×C8C2×C414yes—
72C23×C8D411yescharacteristic
73C23×C8D411yescharacteristic
74C23×C8D411yescharacteristic
75C23×C8D411yescharacteristic
76C23×C8D414yes—
77C23×C8C2312yes—
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80#525—22no—
81#525—22no—
82#525C2×C414yes—
83#525C2×C414yes—
84C2×#123—21no—
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×#121C2×C411yescharacteristic
89C2×#120—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×#117C2312yes—
93C2×#117C2311yescharacteristic
94C22×C42—22no—
95C22×C42—21no—
96C22×C42—42no—
97C22×C42—42no—
98C22×C42C2×C414yes—
99C22×C42C2312yes—
100#435—28no—
101#435—28no—
102C8×D4—28no—
103C8×D4—28no—
104#424—22no—
105#424C2×C414yescomplemented
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×#104—42no—
109C2×#104—42no—
110C2×#104C2×C418yes—
111#416—42no—
112#416—42no—
113C2×#103C2×C414yescomplemented
114C2×#102C2×C414yescomplemented
115#408—22no—
116#408C2×C414yescomplemented
117#407—22no—
118#407—22no—
119#407C2×C414yes—
120#407C2×C414yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127#405—22no—
128#405—42no—
129#405—42no—
130#405C2×C414yescomplemented
131C2×#97—21no—
132C2×#97—22no—
133C2×C4×C8—21no—
134C2×C4×C8—22no—
135#381—21no—
136#380C2×C412yes—
137#380C2×C412yes—
138#379—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141#341—42no—
142#341—42no—
143#340C2×C414yescomplemented
144#340C2×C414yescomplemented
145#339—42no—
146#339—42no—
147#339C2×C418yes—
148#336—24no—
149#336—24no—
150#336—22no—
151#336—22no—
152C25#4111yescharacteristic
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41C22×C412yes—
160C2×#41C22×C412yes—
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4111yescharacteristic
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186#131—48no—
187#131—44no—
188#131—44no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—22no—
192C2×M16—24no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—44no—
202C2×M16C22×C412yes—
203C2×M16C22×C414yes—
204C2×M16C22×C412yes—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—22no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—24no—
221C22×C8—48no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—44no—
229C22×C8—44no—
230C22×C8—44no—
231C22×C8—44no—
232C22×C8—44no—
233C22×C8—48no—
234C22×C8C22⋊C414yes—
235C22×C8C22⋊C414yes—
236C22×C8C22⋊C414yes—
237C22×C8C22⋊C414yes—
238C22×C8C22×C414yes—
239C22×C8C22×C412yes—
240C22×C8C22×C412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247#123—24no—
248#123—24no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8C4214yes—
252C22⋊Q8C4214yes—
253#121—22no—
254#121—22no—
255#121C4214yes—
256#121C4214yes—
257#120—24no—
258#120—24no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263#117—21no—
264#117—21no—
265#117—22no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117C22×C412yes—
270#117C22×C412yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C22×C414yes—
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—48no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42C22×C414yes—
297C4⋊C8—24no—
298C4⋊C8—24no—
299#104—48no—
300#103—44no—
301#102—44no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308#97—24no—
309#97—44no—
310C4×C8—24no—
311C4×C8—44no—
312#95—22no—
313#95—22no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—21no—
319C24#11712yes—
320C24C4×D412yes—
321C24C4×D411yescharacteristic
322C24C4×D411yescharacteristic
323C24C2×#4111yescharacteristic
324#41—24no—
325#41—24no—
326#41—24no—
327#41—24no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—42no—
333#41—42no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8C2×C4212yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4C2×C4212yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—48no—
380C22×C4—432no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4C2×C4212yes—
388C22×C4C2×C4212yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4#11712yes—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4C4×D412yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C23×C411yescharacteristic
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+412yes—
409M16—24no—
410M16—24no—
411M16—24no—
412M16—24no—
413M16—48no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—42no—
419M16—42no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—48no—
441C2×C8—48no—
442C2×C8—48no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8C2×C22⋊C412yes—
452C2×C8C2×C22⋊C412yes—
453C2×C8C2×C22⋊C412yes—
454C2×C8C2×C22⋊C412yes—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4C2×C4212yes—
462C4⋊C4C2×C4212yes—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—24no—
477C42—48no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C23—216no—
483C23—216no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C4×C22⋊C412yes—
502C23C4×C22⋊C412yes—
503C23#44314yes—
504C23#44414yes—
505C23C2×#11711yescharacteristic
506C23C2×C4×D411yescharacteristic
507C23C2×C4×D411yescharacteristic
508C23#53812yes—
509C23#53912yes—
510Q8—24no—
511Q8—41no—
512Q8—41no—
513D4—24no—
514D4—44no—
515D4—42no—
516D4—42no—
517D4—48no—
518D4—48no—
519D4—41no—
520D4—41no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—432no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4C4×C22⋊C412yes—
557C2×C4C4×C22⋊C412yes—
558C2×C4C22×C4211yescharacteristic
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4#51812yes—
563C2×C4#51812yes—
564C2×C4#51812yes—
565C2×C4#52012yes—
566C2×C4#52012yes—
567C2×C4#52012yes—
568C2×C4#53411yescharacteristic
569C2×C4#53411yescharacteristic
570C2×C4#53412yes—
571C2×C4#53511yescharacteristic
572C2×C4#53611yescharacteristic
573C2×C4#53712yes—
574C2×C4#53812yes—
575C2×C4#53912yes—
576C8—22no—
577C8—22no—
578C8—22no—
579C8—22no—
580C8—42no—
581C8—42no—
582C8—44no—
583C8—44no—
584C22—28no—
585C22—216no—
586C22—28no—
587C22—24no—
588C22—24no—
589C22—28no—
590C22—28no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—84no—
603C22#176414yes—
604C22#176514yes—
605C22C2×C4×C22⋊C411yescharacteristic
606C22#231812yes—
607C22#232111yescharacteristic
608C22#232211yescharacteristic
609C22#232512yes—
610C22C2×#44312yes—
611C22C2×#44412yes—
612C4—22no—
613C4—22no—
614C4—48no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—84no—
619C4#227512yes—
620C4#230811yescharacteristic
621C4#231111yescharacteristic
622C4#232111yescharacteristic
623C4#232211yescharacteristic
624C4#232512yes—
625C2—21no—
626C2—22no—
627C2—21no—
628C2—24no—
629C2—24no—
630C2—82no—
631C2#1378411yescharacteristic
632C2C2×#176411yescharacteristic
633C2C2×#176511yescharacteristic
634C2#2220414yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20385C2semidirect
5. C22×#336C2semidirect
6. #10934C2semidirect
7. #10887C2semidirect
8. #10809C2semidirect
9. #10791C2semidirect
10. #10723C2semidirect
11. #10719C2semidirect
18. #2929C4semidirect
32. #1896C4semidirect
33. #1893C4semidirect
34. #1893C4semidirect
35. #1892C4semidirect
36. #1892C4semidirect
57. C2×#131C2×C4semidirect
105. #424C2×C4semidirect
113. C2×#103C2×C4semidirect
114. C2×#102C2×C4semidirect
116. #408C2×C4semidirect
130. #405C2×C4semidirect
143. #340C2×C4semidirect
144. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29292
C2×#18962
#209672
#203852
C22×#3362
#109342
#108872
#108092
#107912
#107232
#107192