Finite Group Databasebeta

← #6635554

Subgroups — #6635554

Order 512.

1812 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37453C211yesmaximal, characteristic, complemented
2#35540C211yesmaximal, characteristic
3#32914C212yesmaximal, complemented
4#31953C214yesmaximal, complemented
5#31941C214yesmaximal, complemented
6#31002C214yesmaximal, complemented
7#30986C214yesmaximal, complemented
8#29766C212yesmaximal, complemented
9#28320C212yesmaximal, complemented
10#27985C212yesmaximal, complemented
11#24010C212yesmaximal, complemented
12#23919C212yesmaximal
13#21908C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2212yes—
15C22×#121C2211yescharacteristic
16C2×#518C2211yescharacteristic
17C2×#485C2214yes—
18C2×#461C2212yescomplemented
19C2×#461C2212yescomplemented
20C2×#448C2214yes—
21C2×#450C2214yes—
22#2817—22no—
23#2627—22no—
24#2627—22no—
25#2621C2212yes—
26#2621C2212yes—
27#2503C2214yescomplemented
28#2483—22no—
29#2483—22no—
30#2434C2214yescomplemented
31#2422—22no—
32#2422C2214yes—
33#2420—22no—
34#2420C2214yes—
35#2405—22no—
36#2317C2211yescharacteristic
37#2317C2211yescharacteristic
38#2309C2211yescharacteristic
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41#2092C2214yescomplemented
42#2085C2214yescomplemented
43#2074C2214yescomplemented
44#2072C2214yescomplemented
45#2068C2214yescomplemented
46#2066C2214yescomplemented
47#2043C2214yescomplemented
48#2036C2214yescomplemented
49#2035C2214yescomplemented
50#2013C2214yescomplemented
51#2000C2214yes—
52#2000C2214yes—
53#1973C2214yescomplemented
54#1968C2214yescomplemented
55#1960C2212yescomplemented
56#1956C2212yescomplemented
57#1924C2214yescomplemented
58#1923C2214yescomplemented
59#1877C2214yescomplemented
60#1877C2214yescomplemented
61#1874C2214yes—
62#1873C2214yes—
63#1872C2214yes—
64#1853C2212yescomplemented
65#1852C2212yescomplemented
66#1836C2212yescomplemented
67#1835C2212yescomplemented
68#1804C2212yescomplemented
69#1804C2212yescomplemented
70#1798C2212yes—
71#1787C2212yescomplemented
72#1787C2212yescomplemented
73#1779C2212yes—
74#1750C2212yes—
75C23×Q8C2312yes—
76C23×D4—21no—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×SD16C2314yes—
80C22×SD16C2314yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×#123—24no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—48no—
87C2×C22⋊Q8C2314yes—
88C2×C22⋊Q8C2314yes—
89C2×#121—22no—
90C2×#121—44no—
91C2×#121C2314yes—
92#518—44no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—22no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—22no—
107#485—48no—
108#461—44no—
109#461—44no—
110#450—48no—
111#448—48no—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2314yes—
128C2×#97C2312yes—
129C2×C4×C8C2312yes—
130#400—24no—
131#400—24no—
132#398—22no—
133#398—22no—
134#398—24no—
135#397—22no—
136#397—22no—
137#397—24no—
138#397—24no—
139#397—24no—
140#395—24no—
141#395—24no—
142#394—24no—
143#394—24no—
144#391C2314yes—
145#391C2314yes—
146#390—22no—
147#390C2314yes—
148#389—22no—
149#389C2314yes—
150#386—24no—
151#385—22no—
152#385—22no—
153#385—24no—
154#385—24no—
155#385—24no—
156#384—21no—
157#384C2312yes—
158#382—21no—
159#382C2312yes—
160#382C2312yes—
161#382C2312yes—
162#381—21no—
163#381—24no—
164#381—24no—
165#381—24no—
166#381D412yes—
167#380D412yes—
168#380D412yes—
169#380C2312yes—
170#380C2312yes—
171#379—21no—
172#379—22no—
173#379D412yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2314yes—
182C25—21no—
183C25C2×D411yescharacteristic
184C22×Q8—22no—
185C22×Q8—44no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—44no—
189C22×Q8C2414yes—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—21no—
193C22×D4—24no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—44no—
197C22×D4—84no—
198C22×D4C2412yes—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C2×SD16—28no—
213C2×SD16—28no—
214C2×SD16—44no—
215C2×SD16—44no—
216C2×SD16—44no—
217C2×SD16—44no—
218C22×C8C2414yes—
219C22×C8C2414yes—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223C4⋊Q8—42no—
224#123—88no—
225C22⋊Q8—28no—
226C22⋊Q8—28no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231#121—28no—
232#121—44no—
233#121—48no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—44no—
239C4×D4—44no—
240#117—24no—
241#117—41no—
242#117—41no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2411yescharacteristic
263C2×C4⋊C4C2411yescharacteristic
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D414yes—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C2412yes—
286C2×C42C2412yes—
287#107—44no—
288#106—44no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291#103—44no—
292#103—44no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#102—44no—
298#102—44no—
299C22⋊C8—28no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—44no—
303#97—42no—
304C4×C8—42no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95C2×D414yes—
317#95C2×D414yes—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—41no—
323C24—42no—
324C24—41no—
325C24—42no—
326C24—42no—
327C24—82no—
328C24—82no—
329C24C22×D411yescharacteristic
330C24C22×D412yes—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8—82no—
339C2×Q8—82no—
340C2×Q8—88no—
341C2×Q8—88no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—42no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—42no—
354C2×D4—84no—
355C2×D4—816no—
356C2×D4H(5,𝔽2)14yes—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4C22×D412yes—
391C22×C4C22×D414yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4H(5,𝔽2)14yes—
396C22×C4C2511yescharacteristic
397SD16—88no—
398SD16—88no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C22⋊C4—28no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—42no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—82no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—42no—
443C42—42no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—44no—
456C23—48no—
457C23—48no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—82no—
463C23—84no—
464C23—82no—
465C23—84no—
466C23—84no—
467C23—164no—
468C23—164no—
469C23#53512yes—
470C23#57714yes—
471C23C23×D411yescharacteristic
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—88no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—42no—
481D4—42no—
482D4—816no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—88no—
517C2×C4—84no—
518C2×C4—88no—
519C2×C4—164no—
520C2×C4—164no—
521C2×C4#53512yes—
522C2×C4C2×H(5,𝔽2)12yes—
523C2×C4C2×H(5,𝔽2)12yes—
524C2×C4#58511yescharacteristic
525C2×C4#58511yescharacteristic
526C8—44no—
527C8—44no—
528C22—24no—
529C22—22no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22—41no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—88no—
544C22—84no—
545C22—84no—
546C22—168no—
547C22C2×#53511yescharacteristic
548C22#348411yescharacteristic
549C22#348511yescharacteristic
550C22C2×#57712yes—
551C22#358612yes—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—84no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4—81no—
562C4—81no—
563C4#353412yes—
564C4#353412yes—
565C2—44no—
566C2—42no—
567C2—84no—
568C2#3672111yescharacteristic
569C2#6081312yes—
570C2#6163111yescharacteristic
571C2#6164911yescharacteristic
572C2#6175312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
1. #37453C2semidirect
3. #32914C2semidirect
4. #31953C2semidirect
5. #31941C2semidirect
6. #31002C2semidirect
7. #30986C2semidirect
8. #29766C2semidirect
9. #28320C2semidirect
10. #27985C2semidirect
11. #24010C2semidirect
13. #21908C2semidirect
18. C2×#461C22semidirect
19. C2×#461C22semidirect
27. #2503C22semidirect
30. #2434C22semidirect
41. #2092C22semidirect
42. #2085C22semidirect
43. #2074C22semidirect
44. #2072C22semidirect
45. #2068C22semidirect
46. #2066C22semidirect
47. #2043C22semidirect
48. #2036C22semidirect
49. #2035C22semidirect
50. #2013C22semidirect
53. #1973C22semidirect
54. #1968C22semidirect
55. #1960C22semidirect
56. #1956C22semidirect
57. #1924C22semidirect
58. #1923C22semidirect
59. #1877C22semidirect
60. #1877C22semidirect
64. #1853C22semidirect
65. #1852C22semidirect
66. #1836C22semidirect
67. #1835C22semidirect
68. #1804C22semidirect
69. #1804C22semidirect
71. #1787C22semidirect
72. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#374532
#355402
#329142
#319532
#319412
#310022
#309862
#297662
#283202
#279852
#240102
#239192
#219082