Finite Group Databasebeta

← #6631337

Subgroups — #6631337

Order 512.

1628 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35333C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#32756C214yesmaximal, complemented
4#32754C214yesmaximal, complemented
5#32747C214yesmaximal, complemented
6#32746C214yesmaximal, complemented
7#23671C212yesmaximal, complemented
8#23656C211yesmaximal, characteristic, complemented
9#23641C212yesmaximal, complemented
10#23628C211yesmaximal, characteristic, complemented
11#23612C212yesmaximal, complemented
12#23553C211yesmaximal, characteristic, complemented
13#23483C212yesmaximal
14C2×#2025C211yesmaximal, characteristic, complemented
15#20630C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#428C2212yes—
20C2×#427C2212yes—
21C2×#424C2212yes—
22C2×#418C2212yes—
23C22×#102C2212yes—
24C2×#408C2212yes—
25#2854C2214yes—
26#2851C2212yescomplemented
27#2851C2212yescomplemented
28#2629C2214yescomplemented
29#2624C2214yes—
30#2579—24no—
31#2566—22no—
32#2565—22no—
33#2448—22no—
34#2445—22no—
35#2418—24no—
36#2317C2212yes—
37#2316C2211yescharacteristic
38#2311C2212yes—
39#2311C2211yescharacteristic
40#2309C2211yescharacteristic
41#2296C2211yescharacteristic
42C2×#390C2211yescharacteristic
43#2285C2211yescharacteristic
44#2088C2214yescomplemented
45#2088C2214yescomplemented
46#2087C2214yescomplemented
47#2086C2214yescomplemented
48#2085C2212yescomplemented
49#2085C2212yescomplemented
50#2084C2212yescomplemented
51#2083C2212yescomplemented
52#2077C2214yescomplemented
53#2077C2214yescomplemented
54#2076C2214yescomplemented
55#2075C2214yescomplemented
56#2073C2212yes—
57#2073C2212yes—
58#2071C2214yes—
59#2067C2214yes—
60#2067C2214yes—
61#2065C2214yes—
62#2065C2214yes—
63#2059C2214yescomplemented
64#2059C2214yescomplemented
65#2034C2214yes—
66#2034C2214yes—
67#2033C2214yes—
68#2033C2214yes—
69#2025C2214yes—
70#2025C2212yes—
71#2025C2212yes—
72#1798C2212yescomplemented
73#1798C2212yescomplemented
74#1797C2214yescomplemented
75#1736C2212yes—
76#1736C2212yes—
77#1735C2212yes—
78C2×#340C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C22×#41C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×M16C2312yes—
84C23×C8C2312yes—
85#535—44no—
86C2×#127—21no—
87C2×#125—22no—
88C2×#125—44no—
89C2×#125C2312yes—
90C2×#125C2312yes—
91C2×#124—21no—
92C2×#123—24no—
93C2×#121C2314yes—
94C2×#120—22no—
95C4×#41—44no—
96C2×C4×Q8D412yes—
97C2×C4×D4D412yes—
98C2×C4×D4D414yes—
99C2×#117C2311yescharacteristic
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2312yes—
104C22×C4⋊C4—22no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—22no—
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110#428—44no—
111#427—44no—
112C2×#107C2314yes—
113C2×#107C2312yes—
114C2×#107C2312yes—
115C2×#106C2314yes—
116C2×#106C2312yes—
117C2×#106C2312yes—
118#424—44no—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2314yes—
121#418—44no—
122C2×#102—44no—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127#408—44no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130#399—24no—
131#398—24no—
132#398—24no—
133#397—24no—
134#396—24no—
135#396—24no—
136#394—24no—
137#394—24no—
138#391—22no—
139#391—21no—
140#390—22no—
141#390—21no—
142#390—22no—
143#390—22no—
144#390C2314yes—
145#390C2312yes—
146#390C2312yes—
147#389—22no—
148#389C2312yes—
149#389C2312yes—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#388—24no—
154#388—24no—
155#387—24no—
156#387—24no—
157#386—24no—
158#386—22no—
159#386—24no—
160#385—24no—
161#385—24no—
162#385—22no—
163#385—24no—
164#384—22no—
165#384C2312yes—
166#382—22no—
167#382C2311yescharacteristic
168#382C2311yescharacteristic
169#380—22no—
170#380—22no—
171#380C2314yes—
172#380C2314yes—
173#380C2314yes—
174#379—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2312yes—
184#340C2312yes—
185#340C2312yes—
186#340C2312yes—
187#336C2312yes—
188#336C2312yes—
189#336C2312yes—
190#336C2311yescharacteristic
191#336C2311yescharacteristic
192C25—21no—
193C2×#41—24no—
194C2×#41—22no—
195C2×#41—24no—
196C2×#41—22no—
197C22×Q8C2×D411yescharacteristic
198C22×D4—24no—
199C22×D4—21no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4C2×D412yes—
205C22×D4C2×D411yescharacteristic
206C22×D4C2412yes—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4C2×D412yes—
210C23×C4C2×D414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C2×M16—24no—
214C2×M16—24no—
215C2×M16—24no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8C2412yes—
220C22×C8C2412yes—
221C22×C8C2414yes—
222#127—44no—
223#125—44no—
224#125—44no—
225#125—48no—
226#124—44no—
227#123—88no—
228#121—44no—
229#121—44no—
230#120—48no—
231C4×Q8—44no—
232C4×D4—44no—
233C4×D4—48no—
234#117—24no—
235#117—24no—
236#117—24no—
237#117—22no—
238#117—22no—
239#117—24no—
240#117—44no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—84no—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D414yes—
278C2×C42—24no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—44no—
283C2×C42C2×D412yes—
284C2×C42C2×D414yes—
285C2×C42C2411yescharacteristic
286C2×C42C2412yes—
287C2×C42C2411yescharacteristic
288#107—42no—
289#107—42no—
290#107—44no—
291#106—42no—
292#106—42no—
293#106—44no—
294C4⋊C8—44no—
295C4⋊C8—44no—
296#102—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300C22⋊C8—44no—
301C22⋊C8—44no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312C24—22no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—82no—
317C24—82no—
318C24C22×D412yes—
319C24C22×D411yescharacteristic
320#41—48no—
321#41—48no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—48no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—84no—
342C2×D4—84no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D414yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4H(5,𝔽2)14yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+414yes—
372C22×C42−1+412yes—
373C22×C4C2511yescharacteristic
374M16—48no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—48no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—84no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C22⋊C4—24no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—24no—
423C42—24no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42—42no—
428C42—44no—
429C42—42no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—88no—
443C23—84no—
444C23—164no—
445C23—162no—
446C23—162no—
447C23#53412yes—
448C23#53612yes—
449C23#53712yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452D4—48no—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—84no—
457D4—168no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—88no—
491C2×C4—88no—
492C2×C4—162no—
493C2×C4#53412yes—
494C2×C4#53612yes—
495C2×C4#53712yes—
496C2×C4#56218yes—
497C2×C4#56318yes—
498C2×C4C2×H(5,𝔽2)11yescharacteristic
499C2×C4C2×H(5,𝔽2)12yes—
500C2×C4C2×2−1+412yes—
501C2×C4C2×2−1+411yescharacteristic
502C8—42no—
503C8—42no—
504C8—44no—
505C22—24no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—44no—
510C22—42no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22—84no—
515C22—82no—
516C22—84no—
517C22—82no—
518C22—82no—
519C22—168no—
520C22#281414yes—
521C22C2×#53411yescharacteristic
522C22C2×#53611yescharacteristic
523C22C2×#53711yescharacteristic
524C22#358412yes—
525C22#358512yes—
526C4—42no—
527C4—44no—
528C4—42no—
529C4—42no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4#281414yes—
537C4C2×#56212yes—
538C4C2×#56312yes—
539C2—24no—
540C2—82no—
541C2—162no—
542C2C2×#281411yescharacteristic
543C2#6076012yes—
544C2#6087911yescharacteristic
545C2#6088311yescharacteristic
546C2#6093312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35333C2semidirect
3. #32756C2semidirect
4. #32754C2semidirect
5. #32747C2semidirect
6. #32746C2semidirect
7. #23671C2semidirect
8. #23656C2semidirect
9. #23641C2semidirect
10. #23628C2semidirect
11. #23612C2semidirect
12. #23553C2semidirect
14. C2×#2025C2semidirect
15. #20630C2semidirect
26. #2851C22semidirect
27. #2851C22semidirect
28. #2629C22semidirect
44. #2088C22semidirect
45. #2088C22semidirect
46. #2087C22semidirect
47. #2086C22semidirect
48. #2085C22semidirect
49. #2085C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2075C22semidirect
63. #2059C22semidirect
64. #2059C22semidirect
72. #1798C22semidirect
73. #1798C22semidirect
74. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#353332
#352542
#327562
#327542
#327472
#327462
#236712
#236562
#236412
#236282
#236122
#235532
#234832
C2×#20252
#206302