Finite Group Databasebeta

← #6633285

Subgroups — #6633285

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37420C211yesmaximal, characteristic, complemented
2#35679C211yesmaximal, characteristic, complemented
3#32918C212yesmaximal, complemented
4#31648C214yesmaximal, complemented
5#31644C214yesmaximal, complemented
6#30746C214yesmaximal, complemented
7#30738C214yesmaximal, complemented
8#29727C212yesmaximal, complemented
9#29333C212yesmaximal, complemented
10#25724C212yesmaximal, complemented
11#23943C211yesmaximal, characteristic, complemented
12#23942C211yesmaximal, characteristic, complemented
13#23940C211yesmaximal, characteristic, complemented
14#23939C211yesmaximal, characteristic
15#21913C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C2×#519C2211yescharacteristic
20C2×#495C2214yes—
21C2×#464C2214yescomplemented
22C2×#451C2212yes—
23C2×#448C2212yes—
24C2×#450C2212yes—
25C2×#447C2212yes—
26#2850C2212yes—
27#2707—24no—
28#2691—24no—
29#2620C2212yes—
30#2566C2212yescomplemented
31#2565C2212yescomplemented
32#2548C2212yescomplemented
33#2547C2212yescomplemented
34#2482—24no—
35#2448—24no—
36#2448C2214yes—
37#2448C2214yes—
38#2310C2211yescharacteristic
39#2309C2211yescharacteristic
40#2309C2211yescharacteristic
41#2296C2211yescharacteristic
42#2296C2211yescharacteristic
43#2101C2214yescomplemented
44#2071C2214yes—
45#2055C2214yes—
46#2054C2214yes—
47#2042C2214yescomplemented
48#2026C2212yescomplemented
49#2026C2212yescomplemented
50#2025C2212yes—
51#2025C2212yes—
52#2016C2212yescomplemented
53#2015C2214yes—
54#2014C2212yescomplemented
55#2013C2212yes—
56#2012C2214yes—
57#2012C2214yes—
58#2011C2212yes—
59#1958C2214yes—
60#1934C2214yescomplemented
61#1933C2214yescomplemented
62#1916C2214yescomplemented
63#1916C2214yescomplemented
64#1880C2214yes—
65#1880C2214yescomplemented
66#1879C2214yes—
67#1878C2214yescomplemented
68#1878C2214yescomplemented
69#1877C2214yescomplemented
70#1875C2212yes—
71#1875C2212yes—
72#1873C2212yes—
73#1873C2212yes—
74#1818C2211yescharacteristic
75#1818C2211yescharacteristic
76#1803C2214yescomplemented
77#1798C2212yes—
78C4×C22⋊C8C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C22×Q16C2314yes—
82C22×SD16C2314yes—
83C22×SD16C2314yes—
84C22×D8C2314yes—
85#541—44no—
86#540—44no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127C2311yescharacteristic
89C2×#126—22no—
90C2×#125—22no—
91C2×#125C2311yescharacteristic
92C2×#125C2311yescharacteristic
93C2×#124—22no—
94C2×#124—44no—
95C2×#124C2314yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8C2312yes—
100C2×#121—22no—
101C2×#121C2312yes—
102#519—44no—
103C2×C4×Q8—22no—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C22×C22⋊C4—22no—
113C22×C42C2311yescharacteristic
114#495—48no—
115#464—48no—
116#451—44no—
117#450—44no—
118#448—44no—
119#447—44no—
120C2×C4⋊C8C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C4×C8C2314yes—
132#399—24no—
133#399—24no—
134#398—24no—
135#397—24no—
136#397—24no—
137#397—24no—
138#396—24no—
139#396—24no—
140#394—24no—
141#394—24no—
142#391—22no—
143#391C2312yes—
144#391C2312yes—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390—24no—
150#390C2314yes—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#389D414yes—
156#389C2314yes—
157#388—24no—
158#388—24no—
159#388—24no—
160#388—24no—
161#388—24no—
162#387—24no—
163#386—24no—
164#385—22no—
165#385—24no—
166#385D414yes—
167#384C2312yes—
168#382C2312yes—
169#381—22no—
170#380—24no—
171#379—22no—
172#379—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2312yes—
182#336C2312yes—
183#336C2312yes—
184#336C2312yes—
185#336C2314yes—
186C25—21no—
187C22×Q8—24no—
188C22×Q8—21no—
189C22×Q8—24no—
190C22×Q8—42no—
191C22×Q8C2412yes—
192C22×Q8C2412yes—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4C2412yes—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4C2×D412yes—
204C23×C4C2411yescharacteristic
205C2×Q16—28no—
206C2×Q16—44no—
207C2×Q16—44no—
208C2×SD16—28no—
209C2×SD16—44no—
210C2×SD16—44no—
211C2×SD16—44no—
212C2×SD16—44no—
213C2×SD16—44no—
214C2×D8—44no—
215C2×D8—44no—
216C2×D8—44no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—42no—
220#127—42no—
221#126—48no—
222#125—42no—
223#125—42no—
224#125—48no—
225#124—28no—
226#124—44no—
227#124—44no—
228#124—44no—
229#124—48no—
230#123—48no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233#121—24no—
234#121—24no—
235#121—48no—
236C4×Q8—28no—
237C4×Q8—44no—
238C4×Q8—42no—
239C4×Q8—42no—
240C4×Q8—48no—
241C4×D4—42no—
242C4×D4—42no—
243#117—24no—
244#117—42no—
245#117—42no—
246#117—42no—
247#117—48no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
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—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—24no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—42no—
287C2×C42C2×D412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2412yes—
292C4⋊C8—44no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C4×C8—44no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95C2×D414yes—
311#95C2×D414yes—
312C24—22no—
313C24—22no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—84no—
318C24C22×D411yescharacteristic
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—416no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8H(5,𝔽2)14yes—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—88no—
345C2×D4—84no—
346C2×D4—84no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—84no—
362C22×C4C22×D414yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D414yes—
366C22×C4C22×D411yescharacteristic
367C22×C4H(5,𝔽2)14yes—
368C22×C4C2511yescharacteristic
369Q16—88no—
370SD16—88no—
371SD16—88no—
372D8—88no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C22⋊C4—24no—
398C22⋊C4—42no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C42—24no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—42no—
419C42—48no—
420C42—42no—
421C42—42no—
422C42—44no—
423C23—24no—
424C23—21no—
425C23—21no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—84no—
437C23—88no—
438C23—164no—
439C23—164no—
440C23#53512yes—
441C23C23×D411yescharacteristic
442Q8—28no—
443Q8—48no—
444Q8—44no—
445Q8—44no—
446Q8—44no—
447D4—44no—
448D4—44no—
449D4—44no—
450D4—44no—
451D4—84no—
452D4—164no—
453D4—164no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—84no—
483C2×C4—164no—
484C2×C4#53512yes—
485C2×C4C2×H(5,𝔽2)12yes—
486C2×C4C2×H(5,𝔽2)12yes—
487C2×C4#58512yes—
488C8—44no—
489C8—44no—
490C22—22no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—84no—
499C22—84no—
500C22—82no—
501C22—82no—
502C22—84no—
503C22—84no—
504C22—168no—
505C22C2×#53511yescharacteristic
506C22#348411yescharacteristic
507C22#348711yescharacteristic
508C22#358414yes—
509C4—22no—
510C4—44no—
511C4—42no—
512C4—42no—
513C4—44no—
514C4—44no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4#353414yes—
521C2—42no—
522C2—162no—
523C2—162no—
524C2#3671011yescharacteristic
525C2#6081912yes—
526C2#6163712yes—
527C2#6193012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23939C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37420C2semidirect
2. #35679C2semidirect
3. #32918C2semidirect
4. #31648C2semidirect
5. #31644C2semidirect
6. #30746C2semidirect
7. #30738C2semidirect
8. #29727C2semidirect
9. #29333C2semidirect
10. #25724C2semidirect
11. #23943C2semidirect
12. #23942C2semidirect
13. #23940C2semidirect
15. #21913C2semidirect
21. C2×#464C22semidirect
30. #2566C22semidirect
31. #2565C22semidirect
32. #2548C22semidirect
33. #2547C22semidirect
43. #2101C22semidirect
47. #2042C22semidirect
48. #2026C22semidirect
49. #2026C22semidirect
52. #2016C22semidirect
54. #2014C22semidirect
60. #1934C22semidirect
61. #1933C22semidirect
62. #1916C22semidirect
63. #1916C22semidirect
65. #1880C22semidirect
67. #1878C22semidirect
68. #1878C22semidirect
69. #1877C22semidirect
76. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#374202
#356792
#329182
#316482
#316442
#307462
#307382
#297272
#293332
#257242
#239432
#239422
#239402
#239392
#219132