Finite Group Databasebeta

← #7187071

Subgroups — #7187071

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#59821C211yesmaximal, characteristic, complemented
3#32189C214yesmaximal, complemented
4#31654C214yesmaximal, complemented
5#29254C211yesmaximal, characteristic, complemented
6#28160C211yesmaximal, characteristic, complemented
7#27571C214yesmaximal, complemented
8#27206C214yesmaximal, complemented
9#25633C212yesmaximal
10#25303C212yesmaximal, complemented
11#24996C212yesmaximal, complemented
12#23816C211yesmaximal, characteristic, complemented
13#22546C212yesmaximal, complemented
14#22269C211yesmaximal, characteristic
15#22242C211yesmaximal, characteristic, complemented
16C2×#559C2212yes—
17C2×#556—21no—
18C2×#493C2211yescharacteristic
19C2×#460C2212yes—
20C2×#446C2212yes—
21#2727—22no—
22#2724—22no—
23#2582C2212yes—
24#2548C2212yes—
25#2547—21no—
26#2547C2212yes—
27#2542C2212yes—
28#2538—21no—
29#2512C2214yes—
30#2486—22no—
31#2448—22no—
32#2447—22no—
33#2447C2214yes—
34#2415—22no—
35#2354C2212yes—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2212yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#127C2211yescharacteristic
44#2100C2212yes—
45#2100C2214yes—
46#2093C2212yes—
47#2072C2212yes—
48#2072C2214yes—
49#2071C2212yes—
50#2067C2214yes—
51#2055C2214yes—
52#2043C2212yes—
53#2024C2212yes—
54#2023C2212yes—
55#2013C2212yes—
56#2012C2214yes—
57#2012C2214yes—
58#1984C2212yes—
59#1983C2214yes—
60#1982C2212yes—
61#1973C2214yes—
62#1959C2214yes—
63#1957C2211yescharacteristic
64#1957C2211yescharacteristic
65#1947C2212yes—
66#1936C2214yes—
67#1928C2214yes—
68#1921C2214yes—
69#1917C2214yes—
70#1917C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1879C2214yes—
74#1879C2214yes—
75#1877C2214yes—
76#1862C2214yes—
77#1834C2211yescharacteristic
78#1831C2211yescharacteristic
79#1803C2212yes—
80#1802C2214yes—
81#1762C2214yes—
82C4×#103C2212yes—
83C4×#103C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86C22×D8C2314yes—
87#559—28no—
88#559—44no—
89#556—48no—
90C2×C4⋊Q8D411yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—24no—
96C2×#124—22no—
97C2×#124—22no—
98C2×#124C2314yes—
99C2×#123—24no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2314yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C22×C4⋊C4—22no—
116#493—24no—
117#460—44no—
118#446—44no—
119C2×#107C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#97C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#399—24no—
136#398—24no—
137#397—24no—
138#397—24no—
139#397—24no—
140#397—24no—
141#396—24no—
142#394—24no—
143#391—22no—
144#391—21no—
145#391D412yes—
146#391C2312yes—
147#391C2314yes—
148#390—22no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390C2312yes—
154#390C2312yes—
155#390C2314yes—
156#390C2314yes—
157#389—22no—
158#389—24no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2314yes—
167#388—24no—
168#388—24no—
169#385—24no—
170#385—24no—
171#384D411yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2314yes—
177#381—24no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2312yes—
192#340C2314yes—
193#340C2312yes—
194#340C2314yes—
195#340C2312yes—
196#340C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202#336C2314yes—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C2×SD16—44no—
216C2×SD16—44no—
217C2×SD16—44no—
218C2×D8—28no—
219C2×D8—44no—
220C2×D8—44no—
221C22×C8C2412yes—
222C22×C8C2414yes—
223C22×C8C2412yes—
224C4⋊Q8—24no—
225C4⋊Q8—42no—
226C4⋊Q8—42no—
227#127—22no—
228#127—22no—
229#127—22no—
230#127#4114yes—
231#126—416no—
232#124—28no—
233#124—44no—
234#124—48no—
235#124—48no—
236#123—416no—
237C22⋊Q8—48no—
238C22⋊Q8—84no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—48no—
244C4×Q8—44no—
245C4×D4—28no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—48no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—82no—
253C4×D4—82no—
254#117—48no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—84no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2414yes—
306C2×C42C2411yescharacteristic
307#107—44no—
308C4⋊C8—24no—
309C4⋊C8—44no—
310#103—42no—
311#103—44no—
312#103—42no—
313#103—42no—
314#103—42no—
315#102—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319#102—44no—
320#97—44no—
321C4×C8—22no—
322C4×C8—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24—22no—
334C24—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—81no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4C2×#4114yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D414yes—
385C22×C4C22×D414yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2511yescharacteristic
391SD16—88no—
392D8—88no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—84no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—84no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—84no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—84no—
445C42—22no—
446C42—24no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—41no—
451C42—41no—
452C42—48no—
453C42—44no—
454C42—44no—
455C42AGL(1,ℤ/8)14yes—
456C42C2×#4112yes—
457C42H(5,𝔽2)12yes—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C23×D411yescharacteristic
473Q8—48no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—84no—
479D4—28no—
480D4—48no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—84no—
485D4—84no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—88no—
524C2×C4—88no—
525C2×C4—168no—
526C2×C4#54612yes—
527C2×C4#54612yes—
528C2×C4C2×AGL(1,ℤ/8)12yes—
529C2×C4C2×AGL(1,ℤ/8)12yes—
530C2×C4C2×AGL(1,ℤ/8)12yes—
531C2×C4#57812yes—
532C2×C4C22×#4111yescharacteristic
533C2×C4C2×H(5,𝔽2)11yescharacteristic
534C2×C4#58512yes—
535C2×C4#58512yes—
536C8—42no—
537C8—42no—
538C8—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22C2×#54611yescharacteristic
554C22#348711yescharacteristic
555C22#348711yescharacteristic
556C22C22×AGL(1,ℤ/8)11yescharacteristic
557C22C2×#57811yescharacteristic
558C22#358511yescharacteristic
559C22#358511yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—44no—
564C4—41no—
565C4—41no—
566C4—44no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4#333014yes—
574C4#352112yes—
575C4#356212yes—
576C2—84no—
577C2—162no—
578C2#4526911yescharacteristic
579C2C2×#333011yescharacteristic
580C2#6121811yescharacteristic
581C2#6189011yescharacteristic
582C2#6189111yescharacteristic
583C2#6191711yescharacteristic
584C2#6191711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
2. #59821C2semidirect
3. #32189C2semidirect
4. #31654C2semidirect
5. #29254C2semidirect
6. #28160C2semidirect
7. #27571C2semidirect
8. #27206C2semidirect
10. #25303C2semidirect
11. #24996C2semidirect
12. #23816C2semidirect
13. #22546C2semidirect
15. #22242C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598212
#321892
#316542
#292542
#281602
#275712
#272062
#256332
#253032
#249962
#238162
#225462
#222692
#222422