Finite Group Databasebeta

← #359194

Subgroups — #359194

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2961C211yesmaximal, characteristic, complemented
2#23451C211yesmaximal, characteristic
3C2×#1765C211yesmaximal, characteristic
4C2×#1764C211yesmaximal, characteristic
5#20384C211yesmaximal, characteristic
6#11386C211yesmaximal, characteristic, complemented
7#11291C211yesmaximal, characteristic, complemented
8#11291C211yesmaximal, characteristic, complemented
9#11269C211yesmaximal, characteristic, complemented
10#11068C211yesmaximal, characteristic
11#11068C211yesmaximal, characteristic
12#11053C211yesmaximal, characteristic
13#10707C211yesmaximal, characteristic, complemented
14#10706C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#2961—24no—
19#2961—24no—
20C2×#446—22no—
21C2×#446—22no—
22C2×#424C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C22×#104C2211yescharacteristic
27C2×#416C2211yescharacteristic
28C2×#416C2211yescharacteristic
29C2×#408C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2297C2211yescharacteristic
37#2026C412yescomplemented
38#2025C412yescomplemented
39#2024C412yescomplemented
40#2023C412yescomplemented
41#1765—24no—
42#1764—24no—
43#1753—22no—
44#1752—22no—
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1728C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—22no—
69C2×#131Q814yescomplemented
70C22×M16—21no—
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8—21no—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80#533—22no—
81#533—22no—
82#533—22no—
83#533—22no—
84C2×C4⋊Q8C2×C412yes—
85C2×#125C2×C412yes—
86C2×C22⋊Q8—22no—
87C2×C4×Q8—22no—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#446—216no—
95#446—216no—
96#424—22no—
97#424—22no—
98#424—22no—
99#424—22no—
100#423—22no—
101#423—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111#416—22no—
112#416—22no—
113#416—22no—
114#416—22no—
115C2×#103C2×C412yes—
116C2×#103C2×C412yes—
117C2×#102C2×C412yes—
118C2×#102C2×C412yes—
119#408—22no—
120#408—22no—
121C2×C22⋊C8—22no—
122#405—22no—
123#405—22no—
124#405—22no—
125#405—22no—
126#405D414yes—
127#405D414yes—
128C2×#97—22no—
129C2×#97—22no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—22no—
133C2×#97—21no—
134C2×#97—22no—
135C2×#97—22no—
136C2×#97D412yes—
137C2×#97D412yes—
138C2×C4×C8—21no—
139C2×C4×C8—21no—
140C2×C4×C8—22no—
141C2×C4×C8—21no—
142C2×C4×C8—22no—
143C2×C4×C8—22no—
144#391C2×C411yescharacteristic
145#391C2×C411yescharacteristic
146#390C2×C411yescharacteristic
147#390C2×C411yescharacteristic
148#386—22no—
149C4×C22⋊C4—22no—
150C2×#95—22no—
151#341—22no—
152#341—22no—
153#341—22no—
154#341—22no—
155#340C2×C412yes—
156#340C2×C412yes—
157#340C2×C412yes—
158#340C2×C412yes—
159#339—22no—
160#339—22no—
161#339—22no—
162#339—22no—
163#339—22no—
164#339—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336C2×C412yes—
170#336C2×C412yes—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41C2×Q812yes—
182C2×#41#4112yes—
183C22×Q8—22no—
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192#131—28no—
193#131—28no—
194#131—44no—
195#131—44no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—24no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—42no—
218C2×M16—42no—
219C2×M16C2×D412yes—
220C2×M16C2×D412yes—
221C2×M16C2×Q812yes—
222C2×M16#4112yes—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—24no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8C22×C412yes—
249C22×C8C22×C412yes—
250C22×C8C2×D412yes—
251C22×C8C2×D412yes—
252C22×C8C2×D412yes—
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8C2×Q812yes—
256C22×C8#4112yes—
257C22×C8#4112yes—
258C22×C8#4112yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261#125—24no—
262#125—24no—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C4×Q8—28no—
266C4×Q8—28no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117C2×D412yes—
276#117C2×D412yes—
277#117#4112yes—
278#117#4112yes—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4C22×C412yes—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C2×D412yes—
310C2×C42#4112yes—
311C2×C42#4112yes—
312C4⋊C8—28no—
313C4⋊C8—28no—
314C4⋊C8—28no—
315#104—28no—
316#104—28no—
317#103—24no—
318#103—24no—
319#102—24no—
320#102—24no—
321C22⋊C8—28no—
322#97—24no—
323#97—24no—
324#97—24no—
325#97—24no—
326#97—44no—
327#97—42no—
328C4×C8—24no—
329C4×C8—24no—
330C4×C8—44no—
331C4×C8—42no—
332#95—22no—
333#95—22no—
334#95—44no—
335C24—21no—
336C24C2×#4111yescharacteristic
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353C2×Q8—28no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—82no—
360C2×Q8—82no—
361C2×Q8C4×Q812yes—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4C4×Q812yes—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—48no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—48no—
466C2×C8#12012yes—
467C2×C8#12012yes—
468C2×C8C22⋊Q812yes—
469C2×C8C22⋊Q812yes—
470C2×C8C22⋊Q814yes—
471C2×C8C22⋊Q814yes—
472C2×C8#12412yes—
473C2×C8#12414yes—
474C2×C8#12412yes—
475C2×C8#12414yes—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4C4×D414yes—
485C4⋊C4C4×D414yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23C2×C4×D411yescharacteristic
522C23#52511yescharacteristic
523C23#52511yescharacteristic
524C23#52911yescharacteristic
525C23#52911yescharacteristic
526C23#52911yescharacteristic
527C23#53311yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532Q8—88no—
533D4—22no—
534D4—22no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—41no—
540D4—41no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×C4×Q811yescharacteristic
579C2×C4C4×#4111yescharacteristic
580C2×C4C4×#4111yescharacteristic
581C2×C4C4×#4111yescharacteristic
582C2×C4C2×#12011yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4#52511yescharacteristic
588C2×C4#52511yescharacteristic
589C2×C4#52511yescharacteristic
590C2×C4C2×#12411yescharacteristic
591C2×C4C2×#12411yescharacteristic
592C2×C4C2×#12411yescharacteristic
593C2×C4#52911yescharacteristic
594C2×C4#52911yescharacteristic
595C2×C4#52911yescharacteristic
596C2×C4#52911yescharacteristic
597C2×C4#52911yescharacteristic
598C2×C4#52911yescharacteristic
599C2×C4#53211yescharacteristic
600C2×C4#53211yescharacteristic
601C2×C4#53211yescharacteristic
602C2×C4#53311yescharacteristic
603C2×C4#53311yescharacteristic
604C8—42no—
605C8—42no—
606C8—42no—
607C8—42no—
608C8—42no—
609C8—42no—
610C8—42no—
611C8#39314yes—
612C8#39314yes—
613C22—22no—
614C22—22no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—44no—
623C22—81no—
624C22—81no—
625C22—82no—
626C22C4×#12011yescharacteristic
627C22C4×C22⋊Q811yescharacteristic
628C22C4×#12411yescharacteristic
629C22#239611yescharacteristic
630C22#239611yescharacteristic
631C22#239811yescharacteristic
632C22#239811yescharacteristic
633C22#295612yes—
634C22#295712yes—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—82no—
642C4—81no—
643C4—81no—
644C4C4×C22⋊Q811yescharacteristic
645C4C4×C22⋊Q811yescharacteristic
646C4C4×#12411yescharacteristic
647C4C4×#12411yescharacteristic
648C4C2×#39311yescharacteristic
649C4#238811yescharacteristic
650C4#239611yescharacteristic
651C4#239811yescharacteristic
652C2—21no—
653C2—21no—
654C2—21no—
655C2—21no—
656C2—82no—
657C2C4×#39311yescharacteristic
658C2#2146812yes—
659C2#2162912yes—
660C2#2165911yescharacteristic
661C2#2166211yescharacteristic
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
1. C2×#2961C2semidirect
6. #11386C2semidirect
7. #11291C2semidirect
8. #11291C2semidirect
9. #11269C2semidirect
13. #10707C2semidirect
14. #10706C2semidirect
15. #10696C2semidirect
37. #2026C4semidirect
38. #2025C4semidirect
39. #2024C4semidirect
40. #2023C4semidirect
69. C2×#131Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29612
#234512
C2×#17652
C2×#17642
#203842
#113862
#112912
#112692
#110682
#110532
#107072
#107062
#106962