Finite Group Databasebeta

← #7186903

Subgroups — #7186903

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#59817C211yesmaximal, characteristic, complemented
3#32246C214yesmaximal, complemented
4#31547C214yesmaximal, complemented
5#29276C211yesmaximal, characteristic, complemented
6#28156C211yesmaximal, characteristic
7#27443C214yesmaximal, complemented
8#27080C214yesmaximal, complemented
9#25933C212yesmaximal, complemented
10#25300C212yesmaximal, complemented
11#25057C212yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22523C212yesmaximal, complemented
14#22242C211yesmaximal, characteristic, complemented
15#22220C211yesmaximal, characteristic, complemented
16C2×D4×Q8—21no—
17C2×D4×Q8C2212yes—
18C2×#486C2211yescharacteristic
19C2×#459C2212yes—
20C2×#446C2212yes—
21#2808—22no—
22#2683—22no—
23#2592—22no—
24#2563C2212yes—
25#2551C2212yes—
26#2548C2212yes—
27#2547—21no—
28#2538—21no—
29#2538C2212yes—
30#2513C2214yes—
31#2485—22no—
32#2452—22no—
33#2452C2214yes—
34#2409—22no—
35#2366C2212yes—
36#2355C2212yes—
37#2354—21no—
38#2354C2212yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#127C2211yescharacteristic
44#2100C2212yes—
45#2099C2212yes—
46#2092C2214yes—
47#2085C2214yes—
48#2071C2212yes—
49#2071C2212yes—
50#2065C2214yes—
51#2058C2212yes—
52#2054C2214yes—
53#2041C2212yes—
54#2023C2212yes—
55#2023C2214yes—
56#2023C2214yes—
57#2011C2212yes—
58#1993C2214yes—
59#1990C2211yescharacteristic
60#1988C2212yes—
61#1987C2214yes—
62#1984C2212yes—
63#1972C2214yes—
64#1957C2211yescharacteristic
65#1948C2212yes—
66#1936C2214yes—
67#1925C2214yes—
68#1925C2214yes—
69#1921C2214yes—
70#1920C2214yes—
71#1917C2214yes—
72#1917C2214yes—
73#1879C2214yes—
74#1879C2214yes—
75#1879C2214yes—
76#1862C2214yes—
77#1845C2211yescharacteristic
78#1831C2211yescharacteristic
79#1804C2214yes—
80#1802C2212yes—
81#1762C2214yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C23×Q8—22no—
86C22×D8C2314yes—
87D4×Q8—28no—
88D4×Q8—44no—
89D4×Q8—48no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8D411yescharacteristic
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#124—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2314yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114#486—24no—
115#459—44no—
116#446—44no—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2314yes—
121C2×#102C2312yes—
122C2×#102C2312yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2312yes—
127C2×#102C2314yes—
128C2×#102C2312yes—
129C2×#97C2314yes—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#397—24no—
133#397—24no—
134#396—24no—
135#396—24no—
136#395—24no—
137#394—24no—
138#393—24no—
139#393—24no—
140#391—22no—
141#391—21no—
142#391—21no—
143#391D412yes—
144#390—22no—
145#390—22no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390C2312yes—
151#390C2312yes—
152#390C2314yes—
153#390C2314yes—
154#390C2314yes—
155#389—22no—
156#389—21no—
157#389—21no—
158#389D412yes—
159#389D412yes—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#388—24no—
164#388—24no—
165#388—24no—
166#387—24no—
167#386—24no—
168#385—24no—
169#384D411yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#382C2314yes—
175#380—22no—
176#380—22no—
177#380—22no—
178#380—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2312yes—
193#340C2312yes—
194#340C2312yes—
195#340C2312yes—
196#340C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201#336C2314yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—48no—
207C22×Q8—48no—
208C22×Q8—84no—
209C22×Q8C2×D412yes—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C2412yes—
215C22×D4C2412yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C2×D8—28no—
220C2×D8—44no—
221C2×D8—44no—
222C22×C8C2414yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C4⋊Q8—28no—
226C4⋊Q8—24no—
227C4⋊Q8—22no—
228C4⋊Q8—42no—
229C4⋊Q8—44no—
230C4⋊Q8—48no—
231C4⋊Q8#4114yes—
232#127—24no—
233#127—22no—
234#127—22no—
235#124—84no—
236C22⋊Q8—416no—
237C22⋊Q8—48no—
238C22⋊Q8—416no—
239C22⋊Q8—48no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—44no—
243C4×Q8—82no—
244C4×D4—28no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—48no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—82no—
253C4×D4—82no—
254#117—84no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
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—24no—
293C2×C22⋊C4—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2414yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#107—44no—
306#106—44no—
307C4⋊C8—24no—
308C4⋊C8—44no—
309#102—24no—
310#102—24no—
311#102—28no—
312#102—42no—
313#102—42no—
314#102—44no—
315#102—44no—
316#102—44no—
317#97—44no—
318C4×C8—22no—
319C4×C8—22no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95C2×D414yes—
329#95C2×D414yes—
330C24—22no—
331C24—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—81no—
342C2×Q8—81no—
343C2×Q8—84no—
344C2×Q8—832no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
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×D42−1+414yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—48no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
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
391D8—88no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—22no—
401C4⋊C4—24no—
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—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—82no—
430C4⋊C4C2×#4114yes—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—22no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—41no—
450C42—41no—
451C42—44no—
452C42—44no—
453C42—84no—
454C42AGL(1,ℤ/8)14yes—
455C42C2×#4112yes—
456C42H(5,𝔽2)12yes—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23C23×D411yescharacteristic
472Q8—48no—
473Q8—48no—
474Q8—816no—
475Q8—84no—
476D4—24no—
477D4—24no—
478D4—48no—
479D4—42no—
480D4—44no—
481D4—42no—
482D4—44no—
483D4—84no—
484D4—84no—
485D4—84no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
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—88no—
523C2×C4—816no—
524C2×C4—88no—
525C2×C4#54612yes—
526C2×C4#54612yes—
527C2×C4C2×AGL(1,ℤ/8)12yes—
528C2×C4C2×AGL(1,ℤ/8)12yes—
529C2×C4C2×AGL(1,ℤ/8)12yes—
530C2×C4#57612yes—
531C2×C4C22×#4111yescharacteristic
532C2×C4C2×H(5,𝔽2)11yescharacteristic
533C2×C4C2×2−1+412yes—
534C2×C4#58512yes—
535C8—42no—
536C8—42no—
537C8—44no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22C2×#54611yescharacteristic
553C22#348611yescharacteristic
554C22#348711yescharacteristic
555C22C22×AGL(1,ℤ/8)11yescharacteristic
556C22C2×#57611yescharacteristic
557C22#358511yescharacteristic
558C22#358611yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—44no—
563C4—41no—
564C4—41no—
565C4—44no—
566C4—44no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4#332514yes—
573C4#351612yes—
574C4#356112yes—
575C2—84no—
576C2—162no—
577C2#4526811yescharacteristic
578C2C2×#332511yescharacteristic
579C2#6122011yescharacteristic
580C2#6182711yescharacteristic
581C2#6184711yescharacteristic
582C2#6189111yescharacteristic
583C2#6191211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28156C2
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. #59863C2semidirect
2. #59817C2semidirect
3. #32246C2semidirect
4. #31547C2semidirect
5. #29276C2semidirect
7. #27443C2semidirect
8. #27080C2semidirect
9. #25933C2semidirect
10. #25300C2semidirect
11. #25057C2semidirect
12. #23815C2semidirect
13. #22523C2semidirect
14. #22242C2semidirect
15. #22220C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#598172
#322462
#315472
#292762
#281562
#274432
#270802
#259332
#253002
#250572
#238152
#225232
#222422
#222202