Finite Group Databasebeta

← #7193902

Subgroups — #7193902

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#32080C214yesmaximal, complemented
4#31574C214yesmaximal, complemented
5#29258C211yesmaximal, characteristic, complemented
6#28202C211yesmaximal, characteristic, complemented
7#27296C214yesmaximal, complemented
8#26692C214yesmaximal, complemented
9#25649C212yesmaximal, complemented
10#25470C212yesmaximal, complemented
11#25058C212yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22540C212yesmaximal, complemented
14#22238C211yesmaximal, characteristic, complemented
15#22219C211yesmaximal, characteristic, complemented
16C2×#550C2212yes—
17C2×#547—21no—
18C2×#486C2211yescharacteristic
19C2×#476C2212yes—
20C2×C8×Q8C2212yes—
21#2806—22no—
22#2682—22no—
23#2584—22no—
24#2582—22no—
25#2567C2212yes—
26#2552—21no—
27#2552C2212yes—
28#2541C2212yes—
29#2538C2212yes—
30#2537—21no—
31#2507C2214yes—
32#2445C2214yes—
33#2416—22no—
34#2407—22no—
35#2369C2212yes—
36#2365—21no—
37Q8×C4⋊C4C2212yes—
38D4×C4⋊C4C2212yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#127C2211yescharacteristic
44#2099C2212yes—
45#2094C2212yes—
46#2094C2214yes—
47#2088C2214yes—
48#2084C2212yes—
49#2084C2214yes—
50#2071C2212yes—
51#2065C2214yes—
52#2041C2212yes—
53#2041C2212yes—
54#2041C2214yes—
55#2041C2214yes—
56#2023C2212yes—
57#2023C2212yes—
58#1990C2211yescharacteristic
59#1984C2212yes—
60#1982C2212yes—
61#1982C2214yes—
62#1957C2211yescharacteristic
63#1957C2214yes—
64#1948C2212yes—
65#1941C2214yes—
66#1934C2214yes—
67#1926C2214yes—
68#1926C2214yes—
69#1926C2214yes—
70#1919C2214yes—
71#1916C2214yes—
72#1916C2214yes—
73#1864C2214yes—
74#1864C2214yes—
75#1864C2214yes—
76#1864C2214yes—
77#1844C2211yescharacteristic
78#1831C2211yescharacteristic
79#1802C2212yes—
80#1802C2214yes—
81C4×#102C2212yes—
82C4×#102C2212yes—
83C4×#102C2214yes—
84#1744C2211yescharacteristic
85C22×#41—22no—
86#550—28no—
87#550—44no—
88#547—48no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2314yes—
93C2×#127C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#125—22no—
96C2×#123—24no—
97C2×C22⋊Q8—22no—
98C2×#121—22no—
99C2×#121—24no—
100C2×#121—22no—
101C2×#121—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
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—
113C22×C4⋊C4—22no—
114#486—24no—
115#476—44no—
116C8×Q8—44no—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2312yes—
125C2×#102C2312yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2312yes—
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2314yes—
135#398—24no—
136#398—24no—
137#398—24no—
138#396—24no—
139#395—24no—
140#394—24no—
141#394—24no—
142#392—24no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—24no—
147#391D412yes—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390C2312yes—
153#390C2312yes—
154#390C2314yes—
155#389—22no—
156#389—24no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—24no—
162#389D412yes—
163#389D412yes—
164#389C2312yes—
165#389C2312yes—
166#389C2314yes—
167#389C2314yes—
168#389C2314yes—
169#387—24no—
170#386—24no—
171#385—24no—
172#384—22no—
173#384D411yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2314yes—
179#380—22no—
180#380—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2312yes—
194#340C2312yes—
195#340C2314yes—
196#340C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203C2×#41—48no—
204C2×#41—48no—
205C2×#41—84no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8C2×D412yes—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—24no—
215C22×D4C2412yes—
216C22×D4C2412yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C22×C8C2412yes—
221C22×C8C2414yes—
222C22×C8C2412yes—
223C4⋊Q8—24no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8SD1618yes—
227#127—28no—
228#127—22no—
229#127—42no—
230#127—44no—
231#127#4114yes—
232#125—84no—
233#123—416no—
234C22⋊Q8—48no—
235#121—48no—
236#121—416no—
237#121—48no—
238#121—84no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—82no—
242C4×Q8—82no—
243C4×Q8—84no—
244C4×D4—28no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—82no—
252C4×D4—82no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2414yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#106—44no—
304#106—44no—
305#106—44no—
306C4⋊C8—24no—
307C4⋊C8—44no—
308#102—28no—
309#102—28no—
310#102—24no—
311#102—24no—
312#102—44no—
313#102—42no—
314#102—44no—
315#102—44no—
316#102—42no—
317#102—44no—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—44no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—22no—
325#95—24no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C24—22no—
333#41—832no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—42no—
338C2×Q8—81no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×Q8—81no—
342C2×Q8C2×SD1618yes—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—48no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—48no—
362C2×D4—42no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4C2×#4114yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—84no—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D414yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2511yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—84no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—84no—
429C4⋊C4—82no—
430C4⋊C4—81no—
431C4⋊C4—82no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4C2×SD1618yes—
435C4⋊C4C2×SD1618yes—
436C4⋊C42−1+414yes—
437C22⋊C4—48no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—84no—
446C42—22no—
447C42—24no—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—41no—
452C42—41no—
453C42—44no—
454C42—44no—
455C42—84no—
456C42C2×SD1614yes—
457C42C2×#4112yes—
458C42H(5,𝔽2)12yes—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C23×D411yescharacteristic
474Q8—216no—
475Q8—84no—
476Q8—84no—
477D4—28no—
478D4—42no—
479D4—42no—
480D4—48no—
481D4—44no—
482D4—42no—
483D4—42no—
484D4—84no—
485D4—816no—
486D4—84no—
487D4—84no—
488D4—168no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
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—42no—
524C2×C4—816no—
525C2×C4—88no—
526C2×C4—88no—
527C2×C4#54612yes—
528C2×C4#54612yes—
529C2×C4C22×SD1612yes—
530C2×C4C22×SD1612yes—
531C2×C4C22×SD1618yes—
532C2×C4C22×SD1612yes—
533C2×C4#57612yes—
534C2×C4C22×#4111yescharacteristic
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4C2×2−1+412yes—
537C2×C4#58512yes—
538C8—42no—
539C8—42no—
540C8—44no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22C2×#54611yescharacteristic
556C22#348611yescharacteristic
557C22#348711yescharacteristic
558C22C23×SD1611yescharacteristic
559C22C2×#57611yescharacteristic
560C22#358511yescharacteristic
561C22#358511yescharacteristic
562C4—22no—
563C4—22no—
564C4—28no—
565C4—22no—
566C4—44no—
567C4—41no—
568C4—44no—
569C4—41no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4#329514yes—
576C4#351512yes—
577C4#355412yes—
578C2—84no—
579C2—162no—
580C2#4533511yescharacteristic
581C2C2×#329511yescharacteristic
582C2#6121811yescharacteristic
583C2#6182011yescharacteristic
584C2#6184711yescharacteristic
585C2#6188111yescharacteristic
586C2#6191011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
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. #59871C2semidirect
3. #32080C2semidirect
4. #31574C2semidirect
5. #29258C2semidirect
6. #28202C2semidirect
7. #27296C2semidirect
8. #26692C2semidirect
9. #25649C2semidirect
10. #25470C2semidirect
11. #25058C2semidirect
12. #23815C2semidirect
13. #22540C2semidirect
14. #22238C2semidirect
15. #22219C2semidirect

Maximal subgroups

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

subgroupindex
#598712
#598152
#320802
#315742
#292582
#282022
#272962
#266922
#256492
#254702
#250582
#238152
#225402
#222382
#222192