Finite Group Databasebeta

← #371268

Subgroups — #371268

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21086C211yesmaximal, characteristic
2#21024C211yesmaximal, characteristic, complemented
3#20838C211yesmaximal, characteristic
4C2×#1771C211yesmaximal, characteristic, complemented
5C2×#1765C211yesmaximal, characteristic
6#11291C211yesmaximal, characteristic, complemented
7#11226C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11075C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10882C211yesmaximal, characteristic, complemented
12#10823C211yesmaximal, characteristic, complemented
13#10819C211yesmaximal, characteristic, complemented
14#10809C211yesmaximal, characteristic, complemented
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2297C2211yescharacteristic
29#2296C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1936C412yescomplemented
32#1936C412yescomplemented
33#1935C412yescomplemented
34#1935C412yescomplemented
35#1917—22no—
36#1916—22no—
37#1851—22no—
38#1850—22no—
39#1771—24no—
40#1765—24no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1728C2211yescharacteristic
58#1728C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C2×#131—22no—
63C2×#131—22no—
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—41no—
67C2×#131—41no—
68C22×M16—21no—
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8—21no—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#424—22no—
86#424—41no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8D412yes—
91C2×C4⋊C8D412yes—
92C2×#104—22no—
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—22no—
96C2×#104—22no—
97C2×#104—41no—
98#416—22no—
99#416—22no—
100#416—22no—
101#416—22no—
102C2×#103C2×C412yes—
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#102C2×C412yes—
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108#408—22no—
109#408—41no—
110C2×C22⋊C8—22no—
111#405—22no—
112#405—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—21no—
116C2×#97—41no—
117C2×#97—41no—
118C2×#97C2×C412yes—
119C2×#97C2×C412yes—
120C2×#97D414yes—
121C2×C4×C8—22no—
122C2×C4×C8—21no—
123C2×C4×C8—41no—
124C2×C4×C8—41no—
125#391C2×C412yes—
126#390C2×C412yes—
127#389C2×C412yes—
128#389C2×C412yes—
129#386—21no—
130#386—21no—
131#385—21no—
132#385—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C22⋊C4—22no—
137C2×#95—21no—
138#341—22no—
139#341—22no—
140#341—22no—
141#341—22no—
142#340C2×C412yes—
143#340C2×C412yes—
144#340C2×C412yes—
145#340C2×C412yes—
146#339—22no—
147#339—22no—
148#339—22no—
149#339—41no—
150#339—41no—
151#339—41no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—22no—
156#336—22no—
157#336—21no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41#4112yes—
173C2×#41#4112yes—
174C22×Q8C22×C411yescharacteristic
175C22×D4—21no—
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4C2411yescharacteristic
183#131—28no—
184#131—44no—
185#131—44no—
186#131—44no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
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—42no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16#4112yes—
213C2×M16#4112yes—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
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—42no—
239C22×C8—42no—
240C22×C8—42no—
241C22×C8C22×C412yes—
242C22×C8C22×C411yescharacteristic
243C22×C8C22×C411yescharacteristic
244C22×C8C2×D412yes—
245C22×C8C2×D412yes—
246C22×C8C2×D412yes—
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8#4112yes—
250C22×C8#4112yes—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×C412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42#4112yes—
297C4⋊C8—24no—
298C4⋊C8—24no—
299C4⋊C8—44no—
300#104—28no—
301#104—44no—
302#103—24no—
303#103—22no—
304#103—22no—
305#102—24no—
306#102—22no—
307#102—22no—
308C22⋊C8—44no—
309#97—24no—
310#97—24no—
311#97—44no—
312#97—42no—
313C4×C8—44no—
314C4×C8—42no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320C24—21no—
321C24C2×#4111yescharacteristic
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8#11712yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4#11712yes—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399M16—22no—
400M16—22no—
401M16—22no—
402M16—22no—
403M16—48no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—22no—
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—48no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
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×C8C4×D412yes—
460C2×C8C4×D412yes—
461C2×C8#12112yes—
462C2×C8#12112yes—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—82no—
474C22⋊C4—44no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—82no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23C2×C4×D411yescharacteristic
508C23#52511yescharacteristic
509C23#52911yescharacteristic
510C23#55211yescharacteristic
511C23#55311yescharacteristic
512C23#55611yescharacteristic
513C23#55911yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518D4—22no—
519D4—22no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—41no—
525D4—41no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4C2×#11711yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C4×#4111yescharacteristic
566C2×C4C4×#4111yescharacteristic
567C2×C4#52011yescharacteristic
568C2×C4#52011yescharacteristic
569C2×C4C2×#12111yescharacteristic
570C2×C4#52511yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52911yescharacteristic
573C2×C4#52911yescharacteristic
574C2×C4#53211yescharacteristic
575C2×C4#53411yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53911yescharacteristic
578C2×C4#53911yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54611yescharacteristic
581C2×C4#54811yescharacteristic
582C2×C4#54811yescharacteristic
583C2×C4#55111yescharacteristic
584C2×C4#55111yescharacteristic
585C2×C4#55111yescharacteristic
586C2×C4#55211yescharacteristic
587C2×C4#55211yescharacteristic
588C2×C4#55211yescharacteristic
589C2×C4#55311yescharacteristic
590C2×C4#55611yescharacteristic
591C8—22no—
592C8—22no—
593C8—22no—
594C8—22no—
595C8—42no—
596C8—42no—
597C8—42no—
598C8—42no—
599C8—81no—
600C8—81no—
601C22—22no—
602C22—22no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—44no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22C4×#12111yescharacteristic
615C22#235211yescharacteristic
616C22#236711yescharacteristic
617C22#257511yescharacteristic
618C22#257711yescharacteristic
619C22#259911yescharacteristic
620C22#260011yescharacteristic
621C22#295512yes—
622C22#295612yes—
623C4—44no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4—81no—
630C4—81no—
631C4—82no—
632C4#229011yescharacteristic
633C4#232211yescharacteristic
634C4#234311yescharacteristic
635C4#237311yescharacteristic
636C4#253411yescharacteristic
637C4#253611yescharacteristic
638C4#255811yescharacteristic
639C4#256011yescharacteristic
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—82no—
645C2#1407011yescharacteristic
646C2#2155011yescharacteristic
647C2#2155111yescharacteristic
648C2#2278412yes—
649C2#2323612yes—
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×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21024C2semidirect
4. C2×#1771C2semidirect
6. #11291C2semidirect
7. #11226C2semidirect
11. #10882C2semidirect
12. #10823C2semidirect
13. #10819C2semidirect
14. #10809C2semidirect
31. #1936C4semidirect
32. #1936C4semidirect
33. #1935C4semidirect
34. #1935C4semidirect

Maximal subgroups

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

subgroupindex
#210862
#210242
#208382
C2×#17712
C2×#17652
#112912
#112262
#111912
#110752
#109292
#108822
#108232
#108192
#108092
#107912