Finite Group Databasebeta

← #373193

Subgroups — #373193

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21178C211yesmaximal, characteristic, complemented
2#21098C211yesmaximal, characteristic
3C2×#1938C211yesmaximal, characteristic
4C2×#1861C211yesmaximal, characteristic
5#20852C211yesmaximal, characteristic
6C2×#1773C211yesmaximal, characteristic, complemented
7#20422C211yesmaximal, characteristic
8#11341C211yesmaximal, characteristic, complemented
9#11241C211yesmaximal, characteristic, complemented
10#11104C211yesmaximal, characteristic
11#11080C211yesmaximal, characteristic
12#10898C211yesmaximal, characteristic, complemented
13#10898C211yesmaximal, characteristic, complemented
14#10894C211yesmaximal, characteristic, complemented
15#10845C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C2×#427C2211yescharacteristic
21#2905C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C2×#421C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2211yescharacteristic
29#2879C2211yescharacteristic
30C2×#408C2211yescharacteristic
31#2873C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#1950—24no—
36#1938—22no—
37#1938C418yescomplemented
38#1937—22no—
39#1861—24no—
40#1855—24no—
41#1773—24no—
42#1765—22no—
43#1764—22no—
44C2×#344C2211yescharacteristic
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47C2×#344C2211yescharacteristic
48#1738C2211yescharacteristic
49#1738C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C22×#41C2311yescharacteristic
61#568—22no—
62#568—22no—
63#568—22no—
64#568—41no—
65C2×#131—22no—
66C2×#131—41no—
67C22×M16—22no—
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#430—22no—
83#430—22no—
84#430—22no—
85#430—41no—
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108D414yes—
91#427—22no—
92#427—22no—
93C2×#107—21no—
94C2×#106—21no—
95#424—21no—
96#424—21no—
97#424—22no—
98#424—41no—
99#424C2×C414yes—
100#423—22no—
101#423—22no—
102#423—22no—
103#423—22no—
104#423—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—41no—
107C2×C4⋊C8—41no—
108#421—22no—
109#421—22no—
110#421—22no—
111#421—41no—
112C2×#104—22no—
113C2×#104—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104C2×C414yes—
117#416C2×C414yes—
118#416D412yes—
119#416D412yes—
120C2×#103Q812yes—
121C2×#102Q812yes—
122#408—21no—
123#408—21no—
124#408—22no—
125#408—41no—
126#408C2×C414yes—
127#407—22no—
128C2×C22⋊C8—22no—
129#405—22no—
130#405—22no—
131#405—22no—
132#405—41no—
133#405—41no—
134#405—41no—
135C4×M16—22no—
136C2×#97—22no—
137C2×#97—22no—
138C2×#97—21no—
139C2×#97—21no—
140C2×#97C2×C414yes—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143#344—22no—
144#344—22no—
145#344—22no—
146#344—41no—
147#341—22no—
148#341—22no—
149#341—22no—
150#341—21no—
151#341—21no—
152#341—22no—
153#341—21no—
154#341—21no—
155#341C2×C414yes—
156#339—22no—
157#339—22no—
158#339—22no—
159#339C2×C414yes—
160#339D414yes—
161C2×#41—21no—
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×#41C22×C412yes—
172C2×#41C2×D412yes—
173C22×Q8C2×Q811yescharacteristic
174C22×D4—21no—
175C22×D4C2×Q811yescharacteristic
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C2411yescharacteristic
179#131—28no—
180#131—44no—
181#131—44no—
182#131—44no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
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—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—42no—
224C2×M16—44no—
225C2×M16—44no—
226C2×M16C22×C412yes—
227C2×M16C22×C412yes—
228C2×M16C2×D412yes—
229C2×M16C2×D412yes—
230C2×M16C2×D412yes—
231C2×M16#4112yes—
232C2×M16#4112yes—
233C2×M16#4112yes—
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—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—42no—
251C22×C8C22×C412yes—
252C22×C8C22×C412yes—
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8C2×D412yes—
256C22×C8C2×Q812yes—
257C22×C8#4111yescharacteristic
258C22×C8#4111yescharacteristic
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117C22×C412yes—
268#117C2×D412yes—
269#117C2×D412yes—
270#117#4112yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C2×Q812yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C22×C412yes—
284C2×C42C2×D412yes—
285C2×C42#4112yes—
286#108—28no—
287#108—44no—
288#108—44no—
289#107—24no—
290#106—24no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293C4⋊C8—44no—
294C4⋊C8—44no—
295C4⋊C8—44no—
296#104—28no—
297#104—28no—
298#104—44no—
299#103C4⋊C414yes—
300#103C4⋊C414yes—
301#102C4⋊C414yes—
302#102C4⋊C414yes—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—44no—
306#97—28no—
307#97—42no—
308#97—42no—
309#97—42no—
310C4×C8—42no—
311C4×C8—42no—
312C4×C8—42no—
313C24—21no—
314C242−1+411yescharacteristic
315#41—22no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—22no—
322#41—22no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8C2×C4⋊C412yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4C2×C4⋊C412yes—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—82no—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377M16—22no—
378M16—22no—
379M16—22no—
380M16—22no—
381M16—22no—
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—22no—
390M16—48no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16#12114yes—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—48no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
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×C8C2×C4⋊C412yes—
452C2×C8C2×C4⋊C412yes—
453C2×C8C4×D412yes—
454C2×C8C4×D412yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C2×C4⋊C414yes—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—41no—
475C42—41no—
476C42C4×D414yes—
477C23—22no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—82no—
482C23#52011yescharacteristic
483C23#53911yescharacteristic
484C23#54111yescharacteristic
485C23#54711yescharacteristic
486C23#54711yescharacteristic
487C23D4×Q811yescharacteristic
488C23#55211yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—41no—
492Q8—41no—
493D4—22no—
494D4—22no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—41no—
500D4—41no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4C22×C4⋊C411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C4×#4111yescharacteristic
522C2×C4C4×#4111yescharacteristic
523C2×C4#52011yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4#52511yescharacteristic
526C2×C4#52511yescharacteristic
527C2×C4#52511yescharacteristic
528C2×C4#52911yescharacteristic
529C2×C4#53211yescharacteristic
530C2×C4#53311yescharacteristic
531C2×C4#53311yescharacteristic
532C2×C4#53811yescharacteristic
533C2×C4#54211yescharacteristic
534C2×C4D4211yescharacteristic
535C2×C4D4211yescharacteristic
536C2×C4#54611yescharacteristic
537C2×C4#54611yescharacteristic
538C2×C4#54711yescharacteristic
539C2×C4#54811yescharacteristic
540C2×C4D4×Q811yescharacteristic
541C2×C4#55111yescharacteristic
542C2×C4#55411yescharacteristic
543C2×C4#55411yescharacteristic
544C2×C4#55611yescharacteristic
545C2×C4#55911yescharacteristic
546C8—22no—
547C8—22no—
548C8—24no—
549C8—22no—
550C8—22no—
551C8—42no—
552C8—42no—
553C8—42no—
554C8—42no—
555C8—42no—
556C8—81no—
557C8—81no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—44no—
567C22—161no—
568C22—161no—
569C22#233411yescharacteristic
570C22D4×C4⋊C411yescharacteristic
571C22#235911yescharacteristic
572C22#242911yescharacteristic
573C22#245511yescharacteristic
574C22#247811yescharacteristic
575C22#255611yescharacteristic
576C4—44no—
577C4—42no—
578C4—44no—
579C4—82no—
580C4—82no—
581C4#229611yescharacteristic
582C4C4×#12111yescharacteristic
583C4D4×C4⋊C411yescharacteristic
584C4#236711yescharacteristic
585C4#242411yescharacteristic
586C4#246111yescharacteristic
587C4#246511yescharacteristic
588C4#258911yescharacteristic
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—82no—
596C2#1411211yescharacteristic
597C2#2269512yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21178C2semidirect
6. C2×#1773C2semidirect
8. #11341C2semidirect
9. #11241C2semidirect
12. #10898C2semidirect
13. #10898C2semidirect
14. #10894C2semidirect
15. #10845C2semidirect
37. #1938C4semidirect

Maximal subgroups

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

subgroupindex
#211782
#210982
C2×#19382
C2×#18612
#208522
C2×#17732
#204222
#113412
#112412
#111042
#110802
#108982
#108942
#108452