Finite Group Databasebeta

← #6506795

Subgroups — #6506795

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic
2#35224C211yesmaximal, characteristic
3#29311C212yesmaximal, complemented
4#29307C212yesmaximal, complemented
5#28776C214yesmaximal, complemented
6#28770C214yesmaximal, complemented
7#27957C212yesmaximal, complemented
8#26068C214yesmaximal, complemented
9#26060C214yesmaximal, complemented
10#25820C212yesmaximal, complemented
11#21941C212yesmaximal
12C2×#1853C212yesmaximal
13#20528C211yesmaximal, characteristic
14C2×#544C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#501C2212yescomplemented
17C2×#498C2212yescomplemented
18C22×#106C2214yes—
19C22×C22⋊C8C2212yes—
20#2848—22no—
21#2841—22no—
22#2659—22no—
23#2646—22no—
24#2629C2212yes—
25#2628C2212yes—
26#2627—22no—
27#2442—22no—
28#2442C2214yes—
29#2442C2214yes—
30#2440—22no—
31#2440C2214yes—
32#2440C2214yes—
33#2437—22no—
34#2317C2211yescharacteristic
35#2317C2211yescharacteristic
36#2316C2211yescharacteristic
37#2280C2211yescharacteristic
38#2280C2211yescharacteristic
39C2×#380C2212yes—
40#2085C2214yescomplemented
41#2085C2214yescomplemented
42#2084C2214yescomplemented
43#2084C2214yescomplemented
44#2059C2214yescomplemented
45#2059C2214yescomplemented
46#2058C2214yescomplemented
47#2058C2214yescomplemented
48#1993C2212yescomplemented
49#1992C2212yescomplemented
50#1991C2212yescomplemented
51#1991C2212yescomplemented
52#1944C2214yescomplemented
53#1943C2214yescomplemented
54#1941C2214yescomplemented
55#1941C2214yescomplemented
56#1854C2214yes—
57#1854C2214yes—
58#1853C2214yes—
59#1853C2212yes—
60#1853C2214yes—
61#1853C2214yes—
62#1853C2214yes—
63#1853C2212yes—
64#1852C2212yes—
65#1852C2212yes—
66#1847C2212yescomplemented
67#1844C2212yescomplemented
68#1808C2214yes—
69#1807C2214yes—
70#1752C2214yes—
71C8×C22⋊C4C2214yes—
72#1750C2212yes—
73#1750C2212yes—
74C2×#340C2214yes—
75C24×C4—21no—
76C24×C4C2311yescharacteristic
77C23×C8C2314yes—
78#544—44no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×#125—21no—
83C2×#125C2312yes—
84C2×#125C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—22no—
101#501—44no—
102#498—44no—
103C2×#107C2314yes—
104C2×#107C2314yes—
105C2×#106—48no—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×C22⋊C8Q8116yescomplemented
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2314yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×C4×C8C2312yes—
119C2×C4×C8C2312yes—
120#400—22no—
121#400—24no—
122#400—24no—
123#399—22no—
124#398—24no—
125#398—24no—
126#398—24no—
127#398—24no—
128#398—24no—
129#396—24no—
130#395—22no—
131#395—24no—
132#395—24no—
133#393—22no—
134#389—22no—
135#389—22no—
136#389C2314yes—
137#389C2314yes—
138#389C2314yes—
139#389C2314yes—
140#385—22no—
141#385—22no—
142#385—24no—
143#385—24no—
144#385—24no—
145#385—24no—
146#384—21no—
147#384C2312yes—
148#384C2312yes—
149#382—21no—
150#382C2312yes—
151#382C2312yes—
152#381—21no—
153#381D412yes—
154#380—22no—
155#380—22no—
156#380—22no—
157#380—24no—
158#380—24no—
159#380D412yes—
160#380D412yes—
161#380C2312yes—
162#380C2314yes—
163#380C2312yes—
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—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—84no—
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×Q818yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C22×C8—44no—
198C22×C8—44no—
199C22×C8C2×Q8116yes—
200C22×C8C2414yes—
201C22×C8C2414yes—
202C4⋊Q8—24no—
203C4⋊Q8—42no—
204C4⋊Q8—42no—
205C4⋊Q8—42no—
206C4⋊Q8—44no—
207#125—24no—
208#125—42no—
209#125—42no—
210#125—42no—
211#125—44no—
212C22⋊Q8—48no—
213C22⋊Q8—88no—
214C22⋊Q8—88no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2411yescharacteristic
250C2×C4⋊C4C2411yescharacteristic
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—84no—
260C2×C22⋊C4C2×D414yes—
261C2×C22⋊C4C2×D414yes—
262C2×C22⋊C4C2×D412yes—
263C2×C22⋊C4C2×D412yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C2412yes—
267C2×C42C2412yes—
268#107—44no—
269#107—44no—
270#106—28no—
271#106—28no—
272#106—44no—
273#106—44no—
274#106—44no—
275#106—44no—
276C22⋊C8C2×Q818yes—
277C22⋊C8C2×Q818yes—
278C22⋊C8C2×Q8116yes—
279#97—42no—
280#97—42no—
281C4×C8—42no—
282C4×C8—42no—
283#95—24no—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—24no—
294#95C2×D414yes—
295#95C2×D414yes—
296C24—24no—
297C24—24no—
298C24—41no—
299C24—41no—
300C24C4⋊Q814yes—
301C24C22×D411yescharacteristic
302C24C22×D412yes—
303C2×Q8—44no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—82no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—416no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—48no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—82no—
330C22×C4—84no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—816no—
337C22×C4C4⋊Q814yes—
338C22×C4C4⋊Q818yes—
339C22×C4C22×D414yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×Q814yes—
345C22×C4C22×Q814yes—
346C22×C4C22×Q814yes—
347C22×C4C2511yescharacteristic
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—416no—
353C2×C8C22×Q818yes—
354C2×C8C22×Q818yes—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—82no—
383C4⋊C4H(5,𝔽2)14yes—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—48no—
392C22⋊C4—44no—
393C22⋊C4—84no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—42no—
399C42—42no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—216no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—22no—
409C23—44no—
410C23—44no—
411C23—48no—
412C23—44no—
413C23C2×C4⋊Q812yes—
414C23C2×C4⋊Q814yes—
415C23#57714yes—
416C23#57714yes—
417C23C23×D411yescharacteristic
418Q8—164no—
419Q8—164no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—84no—
454C2×C4—816no—
455C2×C4—84no—
456C2×C4—164no—
457C2×C4—164no—
458C2×C4C2×C4⋊Q814yes—
459C2×C4C2×C4⋊Q812yes—
460C2×C4C23×Q812yes—
461C2×C4C2×H(5,𝔽2)12yes—
462C2×C4#58512yes—
463C8—28no—
464C8—28no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—22no—
472C22—24no—
473C22—416no—
474C22—42no—
475C22—41no—
476C22—41no—
477C22#316318yes—
478C22C22×C4⋊Q811yescharacteristic
479C22#348411yescharacteristic
480C22#348511yescharacteristic
481C22C2×#57712yes—
482C22C2×#57712yes—
483C4—22no—
484C4—24no—
485C4—44no—
486C4—42no—
487C4—44no—
488C4—42no—
489C4—82no—
490C4—84no—
491C4—82no—
492C4—81no—
493C4—81no—
494C4—84no—
495C4#350114yes—
496C2—28no—
497C2—42no—
498C2#3539111yescharacteristic
499C2C2×#316312yes—
500C2#6163111yescharacteristic
501C2#6163311yescharacteristic
502C2#6176512yes—
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. #29311C2semidirect
4. #29307C2semidirect
5. #28776C2semidirect
6. #28770C2semidirect
7. #27957C2semidirect
8. #26068C2semidirect
9. #26060C2semidirect
10. #25820C2semidirect
16. C2×#501C22semidirect
17. C2×#498C22semidirect
40. #2085C22semidirect
41. #2085C22semidirect
42. #2084C22semidirect
43. #2084C22semidirect
44. #2059C22semidirect
45. #2059C22semidirect
46. #2058C22semidirect
47. #2058C22semidirect
48. #1993C22semidirect
49. #1992C22semidirect
50. #1991C22semidirect
51. #1991C22semidirect
52. #1944C22semidirect
53. #1943C22semidirect
54. #1941C22semidirect
55. #1941C22semidirect
66. #1847C22semidirect
67. #1844C22semidirect
112. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#374772
#352242
#293112
#293072
#287762
#287702
#279572
#260682
#260602
#258202
#219412
C2×#18532
#205282