Finite Group Databasebeta

← #6506806

Subgroups — #6506806

Order 512.

1548 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic
2#35151C211yesmaximal, characteristic
3#29310C212yesmaximal, complemented
4#29307C212yesmaximal, complemented
5#28786C214yesmaximal, complemented
6#28785C214yesmaximal, complemented
7#27958C212yesmaximal, complemented
8#26077C214yesmaximal, complemented
9#26072C214yesmaximal, complemented
10#25820C212yesmaximal, complemented
11#21944C212yesmaximal
12#20846C212yesmaximal
13#20529C211yesmaximal, characteristic
14C2×#544C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#501C2212yescomplemented
17C2×#498C2212yescomplemented
18C2×#428C2214yes—
19C2×#407C2212yes—
20#2855—22no—
21#2841—22no—
22#2659—22no—
23#2646—22no—
24#2629C2212yes—
25#2628C2212yes—
26#2626—22no—
27#2442C2214yes—
28#2441—22no—
29#2441C2214yes—
30#2440C2214yes—
31#2439—22no—
32#2439C2214yes—
33#2437—22no—
34#2317C2211yescharacteristic
35#2317C2211yescharacteristic
36#2316C2211yescharacteristic
37#2280C2211yescharacteristic
38#2280C2211yescharacteristic
39C2×#380C2212yes—
40#2085C2214yescomplemented
41#2085C2214yescomplemented
42#2084C2214yescomplemented
43#2083C2214yescomplemented
44#2059C2214yescomplemented
45#2059C2214yescomplemented
46#2058C2214yescomplemented
47#2057C2214yescomplemented
48#1993C2212yescomplemented
49#1992C2212yescomplemented
50#1991C2212yescomplemented
51#1991C2212yescomplemented
52#1944C2214yescomplemented
53#1944C2214yescomplemented
54#1942C2214yescomplemented
55#1941C2214yescomplemented
56#1854C2214yes—
57#1854C2214yes—
58#1854C2214yes—
59#1854C2214yes—
60#1853C2214yes—
61#1853C2212yes—
62#1853C2212yes—
63#1852C2214yes—
64#1852C2212yes—
65#1852C2212yes—
66#1847C2212yescomplemented
67#1845C2212yescomplemented
68#1809C2214yes—
69#1809C2214yes—
70#1753C2214yes—
71#1753C2214yes—
72#1750C2212yes—
73#1750C2212yes—
74#1736C2214yes—
75C24×C4C2311yescharacteristic
76C22×M16C2314yes—
77#544—44no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×#125—21no—
82C2×#125C2312yes—
83C2×#125C2312yes—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—24no—
86C2×#117—22no—
87C2×#117—44no—
88C2×#117C2314yes—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—22no—
99C22×C42—21no—
100#501—44no—
101#498—44no—
102#428—48no—
103C2×#107C2314yes—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111#407Q8116yescomplemented
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×#97C2312yes—
116C2×#97C2312yes—
117C2×C4×C8C2312yes—
118C2×C4×C8C2312yes—
119#400—22no—
120#400—24no—
121#399—22no—
122#399—24no—
123#398—24no—
124#398—24no—
125#398—24no—
126#396—24no—
127#396—24no—
128#396—24no—
129#395—22no—
130#395—24no—
131#393—22no—
132#393—24no—
133#391—24no—
134#389—22no—
135#389—22no—
136#389—24no—
137#389C2314yes—
138#389C2314yes—
139#389C2314yes—
140#389C2314yes—
141#387—24no—
142#387—24no—
143#385—22no—
144#385—22no—
145#385—24no—
146#385—24no—
147#384—21no—
148#384C2312yes—
149#384C2312yes—
150#382—21no—
151#382C2312yes—
152#382C2312yes—
153#381—21no—
154#381D412yes—
155#380—22no—
156#380—22no—
157#380—22no—
158#380—24no—
159#380D412yes—
160#380D412yes—
161#380C2312yes—
162#380C2312yes—
163#380C2314yes—
164#380C2314yes—
165#379—21no—
166#379D412yes—
167C4×C22⋊C4—22no—
168C4×C22⋊C4C2314yes—
169C2×#95—22no—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#336C2314yes—
179#336C2314yes—
180C25C2×D411yescharacteristic
181C22×Q8—22no—
182C22×Q8—22no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×Q818yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16C2×Q8116yes—
197C22×C8C2414yes—
198C22×C8C2414yes—
199C4⋊Q8—24no—
200C4⋊Q8—42no—
201C4⋊Q8—42no—
202C4⋊Q8—42no—
203C4⋊Q8—44no—
204#125—24no—
205#125—42no—
206#125—42no—
207#125—42no—
208#125—44no—
209C22⋊Q8—48no—
210C22⋊Q8—88no—
211#117—28no—
212#117—42no—
213#117—42no—
214#117—44no—
215#117—42no—
216#117—42no—
217#117—48no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2411yescharacteristic
244C2×C4⋊C4C2411yescharacteristic
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—84no—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D414yes—
259C2×C22⋊C4C2×D414yes—
260C2×C42—24no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—44no—
264C2×C42C2412yes—
265C2×C42C2412yes—
266#107—44no—
267#107—44no—
268#107—44no—
269#107—44no—
270#106—28no—
271#106—28no—
272#106—44no—
273#106—44no—
274C22⋊C8C2×Q8116yes—
275C22⋊C8C2×Q818yes—
276C22⋊C8C2×Q818yes—
277#97—42no—
278#97—42no—
279C4×C8—42no—
280C4×C8—42no—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95C2×D414yes—
293#95C2×D414yes—
294C24—41no—
295C24—42no—
296C24—41no—
297C24—42no—
298C24C4⋊Q814yes—
299C24C22×D412yes—
300C24C22×D411yescharacteristic
301C2×Q8—44no—
302C2×Q8—84no—
303C2×Q8—82no—
304C2×Q8—84no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—416no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4C4⋊Q818yes—
332C22×C4C4⋊Q814yes—
333C22×C4C22×D412yes—
334C22×C4C22×D414yes—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q814yes—
339C22×C4C22×Q814yes—
340C22×C4C22×Q814yes—
341C22×C4C2511yescharacteristic
342M16—416no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8C22×Q818yes—
348C2×C8C22×Q818yes—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—82no—
376C4⋊C4H(5,𝔽2)14yes—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—84no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—48no—
393C42—42no—
394C42—42no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—21no—
402C23—22no—
403C23—21no—
404C23—22no—
405C23—44no—
406C23—48no—
407C23—84no—
408C23—82no—
409C23—82no—
410C23C2×C4⋊Q812yes—
411C23C2×C4⋊Q814yes—
412C23C23×D411yescharacteristic
413Q8—164no—
414Q8—164no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—164no—
451C2×C4—164no—
452C2×C4C2×C4⋊Q814yes—
453C2×C4C2×C4⋊Q812yes—
454C2×C4C23×Q812yes—
455C2×C4C2×H(5,𝔽2)12yes—
456C2×C4#58512yes—
457C8—28no—
458C8—28no—
459C22—24no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—41no—
468C22—42no—
469C22—41no—
470C22—42no—
471C22—88no—
472C22C22×C4⋊Q811yescharacteristic
473C22#348411yescharacteristic
474C22#348511yescharacteristic
475C22#358512yes—
476C22#358612yes—
477C4—24no—
478C4—22no—
479C4—44no—
480C4—42no—
481C4—44no—
482C4—42no—
483C4—82no—
484C4—84no—
485C4—81no—
486C4—81no—
487C4—82no—
488C4—84no—
489C4#350114yes—
490C2—44no—
491C2—42no—
492C2#3539111yescharacteristic
493C2#6066412yes—
494C2#6164911yescharacteristic
495C2#6165111yescharacteristic
496C2#6177612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29310C2semidirect
4. #29307C2semidirect
5. #28786C2semidirect
6. #28785C2semidirect
7. #27958C2semidirect
8. #26077C2semidirect
9. #26072C2semidirect
10. #25820C2semidirect
16. C2×#501C22semidirect
17. C2×#498C22semidirect
40. #2085C22semidirect
41. #2085C22semidirect
42. #2084C22semidirect
43. #2083C22semidirect
44. #2059C22semidirect
45. #2059C22semidirect
46. #2058C22semidirect
47. #2057C22semidirect
48. #1993C22semidirect
49. #1992C22semidirect
50. #1991C22semidirect
51. #1991C22semidirect
52. #1944C22semidirect
53. #1944C22semidirect
54. #1942C22semidirect
55. #1941C22semidirect
66. #1847C22semidirect
67. #1845C22semidirect
111. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#374772
#351512
#293102
#293072
#287862
#287852
#279582
#260772
#260722
#258202
#219442
#208462
#205292