Finite Group Databasebeta

← #6436884

Subgroups — #6436884

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#30003C214yesmaximal, complemented
4#29982C212yesmaximal, complemented
5#29981C212yesmaximal, complemented
6#29977C212yesmaximal, complemented
7#29970C212yesmaximal, complemented
8#29959C212yesmaximal
9#29952C212yesmaximal
10#23786C214yesmaximal, complemented
11#23780C212yesmaximal, complemented
12#23775C212yesmaximal, complemented
13#21369C212yesmaximal, complemented
14#20813C211yesmaximal, characteristic, complemented
15#20744C212yesmaximal
16C2×#518C2211yescharacteristic
17C2×#433C2212yes—
18#2868C2211yescharacteristic, complemented
19#2862C2211yescharacteristic, complemented
20#2857C2212yes—
21#2647C2212yescomplemented
22#2628C2212yes—
23#2511—22no—
24#2492—22no—
25#2451—22no—
26#2423—22no—
27#2357—22no—
28#2354—22no—
29#2342—22no—
30#2342—22no—
31#2331C2212yescomplemented
32#2319C2212yes—
33#2319C2212yes—
34#2317C2211yescharacteristic
35#2314C2212yes—
36#2309C2211yescharacteristic
37C4×C4⋊Q8C2212yescomplemented
38#2296C2211yescharacteristic
39#2293C2212yes—
40#2293C2212yes—
41C2×#390C2211yescharacteristic
42#2274C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2101C2214yescomplemented
45#2100C2212yescomplemented
46#2100C2212yescomplemented
47#2094C2212yes—
48#2094C2212yes—
49#2092C2214yes—
50#2088C2214yescomplemented
51#2088C2214yescomplemented
52#2088C2214yescomplemented
53#2086C2214yescomplemented
54#2067C2214yes—
55#2065C2214yes—
56#2065C2214yes—
57#2065C2214yes—
58#2058C2212yescomplemented
59#2058C2214yescomplemented
60#2057C2212yescomplemented
61#2023C2212yes—
62#2023C2212yes—
63#2023C2214yes—
64#1949C2214yescomplemented
65#1948C2214yescomplemented
66#1854C2214yes—
67#1853C2214yes—
68#1843C2212yescomplemented
69#1831C2212yescomplemented
70#1825C2214yes—
71#1804C2214yes—
72#1802C2214yes—
73#1798C2212yes—
74#1797C2212yes—
75#1786C2214yes—
76#1776C2214yescomplemented
77C4×#106C2214yescomplemented
78#1768C2212yescomplemented
79#1767C2211yescharacteristic, complemented
80C4×C4⋊C8C2211yescharacteristic, complemented
81#1762C2214yes—
82C4×#102C2214yes—
83C23×D4C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C22⋊Q8—22no—
88#518—24no—
89#518Q818yescomplemented
90C2×C4×Q8—22no—
91C2×C4×D4—22no—
92C2×C4×D4C2312yes—
93C2×C4×D4C2312yes—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—22no—
102C22×C22⋊C4—22no—
103C22×C42C2311yescharacteristic
104#433—28no—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×C4⋊C8C2314yes—
110C2×C4⋊C8C2314yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2312yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×C22⋊C8C2314yes—
118C2×#97C2312yes—
119C2×C4×C8C2312yes—
120#400—22no—
121#400—24no—
122#399—22no—
123#398—24no—
124#395—22no—
125#393—22no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391—24no—
131#390—21no—
132#390—21no—
133#390—22no—
134#390C2312yes—
135#390C2312yes—
136#390C2312yes—
137#390C2312yes—
138#389—22no—
139#389—22no—
140#389C2314yes—
141#389C2312yes—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#389C2314yes—
146#388—24no—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#387—24no—
156#387—24no—
157#385—22no—
158#385—21no—
159#385—21no—
160#384—21no—
161#384—22no—
162#384C2312yes—
163#384C2312yes—
164#382—21no—
165#382C2312yes—
166#382C2312yes—
167#381—22no—
168#381—22no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380C2314yes—
173#379—21no—
174#379—21no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4D412yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4D412yes—
188#376C2311yescharacteristic
189#376C2311yescharacteristic
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2312yes—
201#336C2312yes—
202#336C2314yes—
203C25—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4C2×Q814yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—21no—
215C23×C4C2×D412yes—
216C23×C4C2411yescharacteristic
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—44no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—84no—
223C22⋊Q8—84no—
224C4×Q8—84no—
225C4×D4—216no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4C2×Q818yes—
229C4×D4C2×Q818yes—
230#117—22no—
231#117—42no—
232#117—42no—
233#117—84no—
234#117—84no—
235#117C2×Q814yes—
236#117C2×Q814yes—
237#117#4114yes—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—84no—
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D412yes—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—42no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#106—44no—
296#106—44no—
297#106—44no—
298C4⋊C8—28no—
299C4⋊C8—44no—
300C4⋊C8—42no—
301C4⋊C8—42no—
302#102—44no—
303#102—44no—
304#102—44no—
305#102—44no—
306C22⋊C8—28no—
307#97—24no—
308C4×C8—24no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326C24—22no—
327C24—24no—
328C24—42no—
329C24—42no—
330C24—82no—
331C24—82no—
332C24C22×D411yescharacteristic
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—84no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—48no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4C22×Q814yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×Q812yes—
375C22×C4C2×#4112yes—
376C22×C4C2511yescharacteristic
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4C22⋊Q818yes—
405C4⋊C4C22×Q814yes—
406C22⋊C4—24no—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—84no—
421C22⋊C4C22⋊Q818yes—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—82no—
434C42—82no—
435C42—84no—
436C42C22×Q812yes—
437C42C22×Q812yes—
438C42C2×#4112yes—
439C42C2×#4112yes—
440C23—22no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—24no—
445C23—48no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—88no—
453C23—164no—
454C23—162no—
455C23—162no—
456C23C2×C22⋊Q812yes—
457C23C23×D411yescharacteristic
458Q8—88no—
459Q8—164no—
460D4—28no—
461D4—48no—
462D4—44no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—84no—
500C2×C4—88no—
501C2×C4—88no—
502C2×C4—162no—
503C2×C4C2×C22⋊Q812yes—
504C2×C4C2×C22⋊Q814yes—
505C2×C4C2×C22⋊Q814yes—
506C2×C4#57714yes—
507C2×C4C23×Q811yescharacteristic
508C2×C4C22×#4111yescharacteristic
509C2×C4#58512yes—
510C2×C4#58512yes—
511C8—44no—
512C8—44no—
513C22—22no—
514C22—24no—
515C22—42no—
516C22—41no—
517C22—41no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—168no—
526C22C22×C22⋊Q811yescharacteristic
527C22#344911yescharacteristic
528C22#344911yescharacteristic
529C22C2×#57712yes—
530C22#358512yes—
531C4—24no—
532C4—22no—
533C4—24no—
534C4—24no—
535C4—42no—
536C4—44no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4—84no—
545C4#345512yes—
546C4#345812yes—
547C2—42no—
548C2—84no—
549C2#3463411yescharacteristic
550C2#6049512yes—
551C2#6098012yes—
552C2#6101211yescharacteristic
553C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35283C2semidirect
3. #30003C2semidirect
4. #29982C2semidirect
5. #29981C2semidirect
6. #29977C2semidirect
7. #29970C2semidirect
10. #23786C2semidirect
11. #23780C2semidirect
12. #23775C2semidirect
13. #21369C2semidirect
14. #20813C2semidirect
18. #2868C22semidirect
19. #2862C22semidirect
21. #2647C22semidirect
31. #2331C22semidirect
37. C4×C4⋊Q8C22semidirect
44. #2101C22semidirect
45. #2100C22semidirect
46. #2100C22semidirect
50. #2088C22semidirect
51. #2088C22semidirect
52. #2088C22semidirect
53. #2086C22semidirect
58. #2058C22semidirect
59. #2058C22semidirect
60. #2057C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
68. #1843C22semidirect
69. #1831C22semidirect
76. #1776C22semidirect
77. C4×#106C22semidirect
78. #1768C22semidirect
79. #1767C22semidirect
80. C4×C4⋊C8C22semidirect
89. #518Q8semidirect

Maximal subgroups

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

subgroupindex
#352832
#342552
#300032
#299822
#299812
#299772
#299702
#299592
#299522
#237862
#237802
#237752
#213692
#208132
#207442