Finite Group Databasebeta

← #6484563

Subgroups — #6484563

Order 512.

1664 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#37642C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#32964C212yesmaximal, complemented
4#32962C212yesmaximal
5#29960C212yesmaximal, complemented
6#29959C212yesmaximal
7#26545C214yesmaximal, complemented
8#26543C214yesmaximal, complemented
9#26478C214yesmaximal, complemented
10#26476C214yesmaximal, complemented
11#21956C211yesmaximal, characteristic, complemented
12#21808C212yesmaximal
13#21364C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C2×#519C2211yescharacteristic
16C2×#518C2211yescharacteristic
17C2×#477C2212yescomplemented
18C2×#475C2212yescomplemented
19C2×#476C2212yes—
20C2×#474C2212yes—
21C2×#432C2212yes—
22#2844—21no—
23#2844—21no—
24#2628—24no—
25#2628C2212yes—
26#2628C2212yes—
27#2571—21no—
28#2571C2214yescomplemented
29#2551—21no—
30#2551C2214yescomplemented
31#2367C2214yes—
32#2367C2214yes—
33#2342—24no—
34#2326—21no—
35#2326—21no—
36#2310C2211yescharacteristic
37#2309C2211yescharacteristic
38C4×#124—21no—
39C4×#124—21no—
40#2285C2212yes—
41#2285C2211yescharacteristic
42C4×#117C2211yescharacteristic
43#2097C2214yescomplemented
44#2092C2214yes—
45#2068C2214yescomplemented
46#2067C2214yes—
47#2066C2214yescomplemented
48#2065C2214yes—
49#2024C2214yescomplemented
50#2023C2214yes—
51#1983C2214yes—
52#1983C2214yes—
53#1959C2214yes—
54#1959C2214yes—
55#1948C2212yescomplemented
56#1948C2212yescomplemented
57#1947C2212yescomplemented
58#1947C2212yescomplemented
59#1940C2214yescomplemented
60#1940C2214yescomplemented
61#1865C2214yescomplemented
62#1864C2214yes—
63#1863C2214yescomplemented
64#1862C2214yes—
65#1854C2214yes—
66#1854C2214yes—
67#1805C2212yescomplemented
68#1805C2212yescomplemented
69#1804C2212yes—
70#1804C2212yes—
71#1798C2212yes—
72#1798C2212yes—
73#1787C2214yes—
74#1786C2214yes—
75#1776C2214yescomplemented
76#1776C2214yescomplemented
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×#124—21no—
83C2×#124—21no—
84C2×#124—24no—
85C2×#124Q818yescomplemented
86C2×#124C2314yes—
87#519—42no—
88#519—42no—
89#518—42no—
90#518—42no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2312yes—
98C2×C4×D4C2312yes—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C42C2311yescharacteristic
108#477—44no—
109#476—44no—
110#475—44no—
111#474—44no—
112#432—28no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2314yes—
133#400—22no—
134#400—22no—
135#400—24no—
136#400—24no—
137#399—22no—
138#399—22no—
139#395—22no—
140#395—22no—
141#395—24no—
142#395—24no—
143#393—22no—
144#393—22no—
145#391—21no—
146#391—21no—
147#391C2314yes—
148#390—21no—
149#390—21no—
150#390C2314yes—
151#389—24no—
152#389—24no—
153#389C2314yes—
154#389C2314yes—
155#388—24no—
156#388—24no—
157#388—24no—
158#388—24no—
159#385—24no—
160#385—24no—
161#384D411yescharacteristic
162#384D411yescharacteristic
163#384C2312yes—
164#384C2312yes—
165#382—24no—
166#382—22no—
167#382D411yescharacteristic
168#382D411yescharacteristic
169#382C2312yes—
170#382C2312yes—
171#381—22no—
172#380—22no—
173#380D414yes—
174#380C2314yes—
175#379—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—24no—
183C4×C22⋊C4—24no—
184#376—21no—
185C43—21no—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—41no—
198C22×Q8—41no—
199C22×Q8—41no—
200C22×Q8—41no—
201C22×Q8C2×Q814yes—
202C22×Q8C2412yes—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4—41no—
209C22×D4C2×Q814yes—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4C2×D412yes—
215C23×C4C2411yescharacteristic
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—88no—
223#124—28no—
224#124—28no—
225#124—28no—
226#124C2×Q8116yes—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—44no—
232C4×Q8—44no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—44no—
238C4×D4—44no—
239#117—22no—
240#117—22no—
241#117—41no—
242#117—41no—
243#117—41no—
244#117—41no—
245#117—48no—
246#117—41no—
247#117—41no—
248#117—84no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C4⋊C4C2411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×Q814yes—
293C2×C42#4114yes—
294C2×C42C2412yes—
295C2×C42C2412yes—
296C2×C42C2412yes—
297#107—28no—
298#107—44no—
299#106—28no—
300#106—44no—
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—42no—
304C4⋊C8—42no—
305C4⋊C8—42no—
306C4⋊C8—42no—
307#103—44no—
308#103—44no—
309#103—44no—
310#103—44no—
311#102—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—28no—
316#97—28no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C24—24no—
333C24—42no—
334C24—42no—
335C24—82no—
336C24—82no—
337C24C22×D411yescharacteristic
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—82no—
351C2×Q8—82no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×Q8C22×Q814yes—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4C22×Q814yes—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4C22⋊Q818yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×Q812yes—
396C22×C4C2×#4112yes—
397C22×C4C2511yescharacteristic
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—44no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4H(5,𝔽2)12yes—
429C4⋊C4H(5,𝔽2)12yes—
430C22⋊C4—24no—
431C22⋊C4—216no—
432C22⋊C4—216no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—44no—
447C42—44no—
448C42—42no—
449C42—48no—
450C42—82no—
451C42—82no—
452C42C22⋊Q814yes—
453C42C22⋊Q814yes—
454C42C22×Q814yes—
455C42C2×#4114yes—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—48no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—88no—
467C23—162no—
468C23—162no—
469C23—164no—
470C23C2×C22⋊Q812yes—
471C23C23×D411yescharacteristic
472Q8—24no—
473Q8—24no—
474Q8—48no—
475Q8—42no—
476Q8—42no—
477Q8—164no—
478Q8—164no—
479D4—24no—
480D4—24no—
481D4—48no—
482D4—42no—
483D4—42no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—164no—
521C2×C4—164no—
522C2×C4C2×C22⋊Q812yes—
523C2×C4C2×C22⋊Q814yes—
524C2×C4C2×C22⋊Q814yes—
525C2×C4C23×Q811yescharacteristic
526C2×C4C22×#4111yescharacteristic
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C2×C4C2×H(5,𝔽2)11yescharacteristic
529C2×C4#58511yescharacteristic
530C2×C4#58511yescharacteristic
531C8—44no—
532C8—44no—
533C22—24no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—48no—
539C22—44no—
540C22—82no—
541C22—82no—
542C22—84no—
543C22—168no—
544C22C22×C22⋊Q811yescharacteristic
545C22#348411yescharacteristic
546C22#348711yescharacteristic
547C22#358512yes—
548C22#358612yes—
549C4—24no—
550C4—24no—
551C4—22no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—84no—
560C4—84no—
561C4—82no—
562C4#346011yescharacteristic
563C4#346011yescharacteristic
564C4#350111yescharacteristic
565C4#350111yescharacteristic
566C2—42no—
567C2—84no—
568C2#3517411yescharacteristic
569C2#6049312yes—
570C2#6167311yescharacteristic
571C2#6167411yescharacteristic
572C2#6194412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37642C2semidirect
3. #32964C2semidirect
5. #29960C2semidirect
7. #26545C2semidirect
8. #26543C2semidirect
9. #26478C2semidirect
10. #26476C2semidirect
11. #21956C2semidirect
13. #21364C2semidirect
17. C2×#477C22semidirect
18. C2×#475C22semidirect
28. #2571C22semidirect
30. #2551C22semidirect
43. #2097C22semidirect
45. #2068C22semidirect
47. #2066C22semidirect
49. #2024C22semidirect
55. #1948C22semidirect
56. #1948C22semidirect
57. #1947C22semidirect
58. #1947C22semidirect
59. #1940C22semidirect
60. #1940C22semidirect
61. #1865C22semidirect
63. #1863C22semidirect
67. #1805C22semidirect
68. #1805C22semidirect
75. #1776C22semidirect
76. #1776C22semidirect
85. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#342112
#329642
#329622
#299602
#299592
#265452
#265432
#264782
#264762
#219562
#218082
#213642