Finite Group Databasebeta

← #7186861

Subgroups — #7186861

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic, complemented
3#32241C214yesmaximal, complemented
4#31527C214yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#28156C211yesmaximal, characteristic
7#27427C214yesmaximal, complemented
8#27065C214yesmaximal, complemented
9#25930C212yesmaximal, complemented
10#25292C212yesmaximal, complemented
11#25059C212yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22519C212yesmaximal, complemented
14#22238C211yesmaximal, characteristic, complemented
15#22219C211yesmaximal, characteristic, complemented
16C2×#554C2212yes—
17C2×#486C2211yescharacteristic
18C2×#459C2212yes—
19C2×C8×Q8C2212yes—
20#2808—22no—
21#2732—22no—
22#2592—22no—
23#2563C2212yes—
24#2552C2212yes—
25#2548C2212yes—
26#2547—21no—
27#2542—21no—
28#2538—22no—
29#2538C2212yes—
30#2513C2214yes—
31#2485—22no—
32#2454C2214yes—
33#2417—22no—
34#2366C2212yes—
35#2355C2212yes—
36#2354—21no—
37#2354C2212yes—
38D4×C4⋊C4—21no—
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#2067C2214yes—
51#2057C2212yes—
52#2054C2214yes—
53#2041C2212yes—
54#2025C2214yes—
55#2025C2214yes—
56#2023C2212yes—
57#2011C2212yes—
58#1991C2214yes—
59#1990C2211yescharacteristic
60#1989C2214yes—
61#1987C2212yes—
62#1984C2212yes—
63#1970C2214yes—
64#1957C2211yescharacteristic
65#1947C2212yes—
66#1934C2214yes—
67#1926C2214yes—
68#1926C2214yes—
69#1921C2214yes—
70#1920C2214yes—
71#1916C2214yes—
72#1916C2214yes—
73#1879C2214yes—
74#1879C2214yes—
75#1879C2214yes—
76#1864C2214yes—
77#1844C2211yescharacteristic
78#1831C2211yescharacteristic
79#1804C2214yes—
80#1802C2212yes—
81C4×#102C2212yes—
82C4×#102C2212yes—
83C4×#102C2214yes—
84#1744C2211yescharacteristic
85C22×D8C2314yes—
86#554—28no—
87#554—44no—
88C2×C4⋊Q8D411yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#125C2314yes—
94C2×#124—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2314yes—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C22×C4⋊C4—22no—
111#486—24no—
112#459—44no—
113C8×Q8—44no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#102C2312yes—
119C2×#102C2312yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2312yes—
124C2×#102C2314yes—
125C2×#102C2312yes—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2314yes—
129#399—24no—
130#398—24no—
131#397—24no—
132#397—24no—
133#396—24no—
134#396—24no—
135#394—24no—
136#393—24no—
137#391—22no—
138#391—21no—
139#391—21no—
140#391D412yes—
141#390—22no—
142#390—22no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390—21no—
147#390C2312yes—
148#390C2312yes—
149#390C2314yes—
150#390C2314yes—
151#390C2314yes—
152#389—22no—
153#389—21no—
154#389—21no—
155#389D412yes—
156#389D412yes—
157#389C2312yes—
158#389C2312yes—
159#389C2314yes—
160#388—24no—
161#388—24no—
162#388—24no—
163#387—24no—
164#386—24no—
165#385—24no—
166#384—22no—
167#384D411yescharacteristic
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#382C2314yes—
173#380—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—24no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C43C2311yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2312yes—
192#340C2312yes—
193#340C2312yes—
194#340C2312yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201#336C2312yes—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C2412yes—
210C22×D4C2412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C2×D8—28no—
215C2×D8—44no—
216C2×D8—44no—
217C22×C8C2414yes—
218C22×C8C2412yes—
219C22×C8C2412yes—
220C4⋊Q8—24no—
221C4⋊Q8—22no—
222C4⋊Q8—42no—
223C4⋊Q8#4114yes—
224#127—24no—
225#127—22no—
226#127—22no—
227#125—28no—
228#125—44no—
229#124—84no—
230C22⋊Q8—48no—
231C22⋊Q8—416no—
232C22⋊Q8—84no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—82no—
236C4×Q8—82no—
237C4×D4—28no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4—82no—
245C4×D4—84no—
246C4×D4—82no—
247#117—84no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2414yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300#107—44no—
301#106—44no—
302C4⋊C8—24no—
303C4⋊C8—44no—
304#102—24no—
305#102—24no—
306#102—28no—
307#102—42no—
308#102—42no—
309#102—44no—
310#102—44no—
311#102—44no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—44no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—22no—
326C24—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—42no—
332C2×Q8—81no—
333C2×Q8—81no—
334C2×Q8—81no—
335C2×Q8—81no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—81no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D42−1+414yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D414yes—
375C22×C4C22×D414yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2511yescharacteristic
382D8—88no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—82no—
419C4⋊C4—81no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4C2×#4114yes—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4—82no—
431C22⋊C4—84no—
432C22⋊C4—82no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C42—22no—
437C42—24no—
438C42—24no—
439C42—22no—
440C42—22no—
441C42—41no—
442C42—41no—
443C42—44no—
444C42—44no—
445C42—84no—
446C42#13514yes—
447C42C2×#4112yes—
448C42H(5,𝔽2)12yes—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C23×D411yescharacteristic
464Q8—48no—
465Q8—84no—
466Q8—84no—
467D4—24no—
468D4—24no—
469D4—48no—
470D4—42no—
471D4—44no—
472D4—42no—
473D4—44no—
474D4—84no—
475D4—84no—
476D4—84no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—88no—
514C2×C4—88no—
515C2×C4—168no—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4C2×#13512yes—
519C2×C4C2×#13512yes—
520C2×C4C2×#13512yes—
521C2×C4#57612yes—
522C2×C4C22×#4111yescharacteristic
523C2×C4C2×H(5,𝔽2)11yescharacteristic
524C2×C4C2×2−1+412yes—
525C2×C4#58512yes—
526C8—42no—
527C8—42no—
528C8—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22C2×#54611yescharacteristic
544C22#348611yescharacteristic
545C22#348711yescharacteristic
546C22C22×#13511yescharacteristic
547C22C2×#57611yescharacteristic
548C22#358511yescharacteristic
549C22#358511yescharacteristic
550C4—22no—
551C4—22no—
552C4—22no—
553C4—44no—
554C4—44no—
555C4—41no—
556C4—44no—
557C4—41no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4#330514yes—
564C4#351612yes—
565C4#356112yes—
566C2—84no—
567C2—162no—
568C2#4526811yescharacteristic
569C2C2×#330511yescharacteristic
570C2#6121811yescharacteristic
571C2#6181911yescharacteristic
572C2#6184711yescharacteristic
573C2#6188511yescharacteristic
574C2#6191011yescharacteristic
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. #59823C2semidirect
3. #32241C2semidirect
4. #31527C2semidirect
5. #29277C2semidirect
7. #27427C2semidirect
8. #27065C2semidirect
9. #25930C2semidirect
10. #25292C2semidirect
11. #25059C2semidirect
12. #23815C2semidirect
13. #22519C2semidirect
14. #22238C2semidirect
15. #22219C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#598232
#322412
#315272
#292772
#281562
#274272
#270652
#259302
#252922
#250592
#238152
#225192
#222382
#222192