Finite Group Databasebeta

← #6495160

Subgroups — #6495160

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35343C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#33299C211yesmaximal, characteristic, complemented
4#33294C211yesmaximal, characteristic, complemented
5#33237C211yesmaximal, characteristic, complemented
6#33230C211yesmaximal, characteristic, complemented
7#30217C212yesmaximal, complemented
8#30216C212yesmaximal, complemented
9#29153C212yesmaximal, complemented
10#29152C212yesmaximal, complemented
11#28730C214yesmaximal, complemented
12#27986C212yesmaximal
13#27985C212yesmaximal, complemented
14#25916C214yesmaximal, complemented
15#21868C212yesmaximal, complemented
16#21867C212yesmaximal, complemented
17#20467C211yesmaximal, characteristic, complemented
18C2×#544C2212yes—
19C2×#529C2212yes—
20C2×#489C2214yes—
21C2×#488C2214yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2861C2212yes—
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2858C2211yescharacteristic
30#2660C2212yes—
31#2660C2212yes—
32#2621C2211yescharacteristic
33#2620C2212yes—
34#2619C2211yescharacteristic
35#2602—22no—
36#2602—22no—
37#2601—22no—
38#2601—22no—
39#2552—22no—
40#2551—22no—
41#2440—22no—
42#2440—22no—
43#2328C2211yescharacteristic
44#2327C2211yescharacteristic
45#2326C2212yes—
46#2317C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2074C2214yes—
51#2073C2214yes—
52#2072C2214yes—
53#2071C2214yes—
54#2026C2214yes—
55#2025C2214yes—
56#2024C2214yes—
57#2023C2214yes—
58#1992C2212yes—
59#1992C2212yes—
60#1992C2212yes—
61#1992C2212yes—
62#1992C2214yes—
63#1990C2214yes—
64#1961C2212yes—
65#1960C2212yes—
66#1959C2212yes—
67#1959C2212yes—
68#1959C2212yes—
69#1959C2212yes—
70#1957C2212yes—
71#1956C2212yes—
72#1875C2214yes—
73#1874C2214yes—
74#1873C2214yes—
75#1872C2214yes—
76#1836C2212yes—
77#1836C2212yes—
78#1835C2212yes—
79#1835C2212yes—
80#1805C2214yes—
81#1804C2214yes—
82#1803C2214yes—
83#1802C2214yes—
84#1768C2212yes—
85#1768C2212yes—
86#1750C2212yes—
87#1750C2212yes—
88#1745C2212yes—
89C4×#97C2211yescharacteristic
90C4×#97C2211yescharacteristic
91#544—48no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×#127C2311yescharacteristic
99#529—216no—
100C2×#126—22no—
101C2×#125—22no—
102C2×#125C2312yes—
103C2×#125C2312yes—
104C2×#125C2312yes—
105C2×#124C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122#489—48no—
123#488—48no—
124#432—28no—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2312yes—
136C2×#97C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#398—22no—
142#398—24no—
143#398—24no—
144#397—22no—
145#396—22no—
146#396—22no—
147#395—24no—
148#395—24no—
149#394—22no—
150#394—22no—
151#394—22no—
152#392—22no—
153#391C2312yes—
154#391C2312yes—
155#390—22no—
156#390—22no—
157#390C2312yes—
158#390C2312yes—
159#389—24no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#386—22no—
175#386—22no—
176#385—24no—
177#385—22no—
178#385—22no—
179#384—22no—
180#384—22no—
181#384—22no—
182#384C2312yes—
183#384C2312yes—
184#384C2312yes—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#382C2312yes—
191#382C2312yes—
192#382C2312yes—
193#382C2311yescharacteristic
194#382C2311yescharacteristic
195#380—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201#376C2311yescharacteristic
202#376C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2314yes—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4C2412yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—28no—
228C4⋊Q8—24no—
229C4⋊Q8—44no—
230C4⋊Q8—48no—
231C4⋊Q8—84no—
232C4⋊Q8—84no—
233#127—24no—
234#126—48no—
235#125—28no—
236#125—28no—
237#125—44no—
238#125—48no—
239#124—28no—
240#124—28no—
241#123—48no—
242C22⋊Q8—28no—
243C22⋊Q8—48no—
244C22⋊Q8—48no—
245#121—28no—
246C4×Q8—28no—
247C4×D4—28no—
248C4×D4—48no—
249#117—24no—
250#117—24no—
251#117—44no—
252#117—48no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—84no—
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4C2×D411yescharacteristic
298C2×C22⋊C4C2×D411yescharacteristic
299C2×C22⋊C4C2×D412yes—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42#4118yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C4⋊C8—28no—
315#103—44no—
316#103—44no—
317#103—44no—
318#103—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323C22⋊C8—28no—
324#97—24no—
325#97—24no—
326#97—24no—
327#97—24no—
328C4×C8—24no—
329C4×C8—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95C2×D412yes—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343C24—22no—
344C24C22×D411yescharacteristic
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—82no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—84no—
377C22×C4#12418yes—
378C22×C4C22×D412yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C2×#4114yes—
388C22×C4C2511yescharacteristic
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C42—22no—
441C42—22no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—44no—
447C42—44no—
448C42#12418yes—
449C42C2×#4114yes—
450C42C2×#4114yes—
451C42H(5,𝔽2)14yes—
452C422−1+414yes—
453C23—22no—
454C23—22no—
455C23—21no—
456C23—21no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×#12412yes—
465C23C23×D411yescharacteristic
466Q8—48no—
467Q8—164no—
468Q8—164no—
469D4—48no—
470D4—88no—
471D4—164no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—88no—
509C2×C4—164no—
510C2×C4C2×#12414yes—
511C2×C4C2×#12412yes—
512C2×C4C2×#12414yes—
513C2×C4#53512yes—
514C2×C4#53512yes—
515C2×C4#53614yes—
516C2×C4C22×#4112yes—
517C2×C4C2×H(5,𝔽2)12yes—
518C2×C4C2×2−1+412yes—
519C8—44no—
520C8—44no—
521C22—24no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22C22×#12411yescharacteristic
534C22C2×#53511yescharacteristic
535C22C2×#53611yescharacteristic
536C22#358512yes—
537C22#358612yes—
538C4—22no—
539C4—22no—
540C4—24no—
541C4—24no—
542C4—24no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4#351114yes—
555C2—42no—
556C2—162no—
557C2#3528911yescharacteristic
558C2#6062112yes—
559C2#6083011yescharacteristic
560C2#6083111yescharacteristic
561C2#6088512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35275C2semidirect
3. #33299C2semidirect
4. #33294C2semidirect
5. #33237C2semidirect
6. #33230C2semidirect
7. #30217C2semidirect
8. #30216C2semidirect
9. #29153C2semidirect
10. #29152C2semidirect
11. #28730C2semidirect
13. #27985C2semidirect
14. #25916C2semidirect
15. #21868C2semidirect
16. #21867C2semidirect
17. #20467C2semidirect

Maximal subgroups

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

subgroupindex
#353432
#352752
#332992
#332942
#332372
#332302
#302172
#302162
#291532
#291522
#287302
#279862
#279852
#259162
#218682
#218672
#204672