Finite Group Databasebeta

← #7157051

Subgroups — #7157051

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59924C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#32568C214yesmaximal, complemented
4#30813C214yesmaximal, complemented
5#29258C211yesmaximal, characteristic, complemented
6#28202C211yesmaximal, characteristic, complemented
7#27767C212yesmaximal, complemented
8#26692C214yesmaximal, complemented
9#26044C212yesmaximal, complemented
10#25649C214yesmaximal, complemented
11#25274C212yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22963C212yesmaximal, complemented
14#22307C211yesmaximal, characteristic, complemented
15#22219C211yesmaximal, characteristic, complemented
16C2×#550C2212yes—
17C2×#547—21no—
18C2×#486C2211yescharacteristic
19C2×#486C2214yes—
20C2×#476C2212yes—
21C2×#476C2214yes—
22#2811—22no—
23#2682—22no—
24#2567—22no—
25#2567—22no—
26#2567C2214yes—
27#2551—21no—
28#2551C2212yes—
29#2551C2212yes—
30#2541C2212yes—
31#2538C2212yes—
32#2537—21no—
33#2445C2214yes—
34#2416—22no—
35#2407—22no—
36#2369—21no—
37#2369C2212yes—
38#2369C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2099C2212yes—
46#2094C2212yes—
47#2088C2214yes—
48#2084C2212yes—
49#2071C2212yes—
50#2065C2214yes—
51#2058C2214yes—
52#2041C2212yes—
53#2041C2212yes—
54#2041C2214yes—
55#2041C2214yes—
56#2023C2212yes—
57#2023C2214yes—
58#2023C2212yes—
59#1990C2211yescharacteristic
60#1984C2214yes—
61#1982C2214yes—
62#1982C2212yes—
63#1957C2211yescharacteristic
64#1948C2212yes—
65#1941C2214yes—
66#1939C2212yes—
67#1934C2214yes—
68#1926C2214yes—
69#1926C2214yes—
70#1926C2214yes—
71#1919C2214yes—
72#1916C2214yes—
73#1864C2214yes—
74#1864C2214yes—
75#1864C2214yes—
76#1864C2214yes—
77#1844C2211yescharacteristic
78#1831C2211yescharacteristic
79#1802C2212yes—
80C8×C4⋊C4C2212yes—
81C4×#102C2212yes—
82C4×#102C2212yes—
83C4×#102C2214yes—
84#1744C2211yescharacteristic
85C22×#41—22no—
86#550—48no—
87#547—48no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8D411yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2314yes—
94C2×#127C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#127C2314yes—
97C2×#123—24no—
98C2×C22⋊Q8—22no—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121—22no—
102C2×#121—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#486—48no—
116#476—28no—
117#476—48no—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2312yes—
126C2×#102C2312yes—
127C2×#102C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2314yes—
136#396—24no—
137#395—24no—
138#395—24no—
139#395—24no—
140#395—24no—
141#394—24no—
142#394—24no—
143#392—24no—
144#391—24no—
145#391—21no—
146#391—21no—
147#391—24no—
148#391—24no—
149#391D412yes—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#390C2314yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#389C2314yes—
170#386—24no—
171#385—24no—
172#384—24no—
173#384D411yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#380—22no—
179#380—22no—
180C4×C4⋊C4—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#340C2314yes—
191#340C2312yes—
192#340C2312yes—
193#340C2312yes—
194#340C2314yes—
195#340C2312yes—
196#340C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2314yes—
203C2×#41—48no—
204C2×#41—48no—
205C2×#41—84no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—24no—
209C22×Q8C2×D412yes—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—24no—
216C22×D4C2412yes—
217C22×D4C2412yes—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2414yes—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—44no—
227C4⋊Q8—44no—
228C4⋊Q8—84no—
229C4⋊Q8—84no—
230C4⋊Q8SD1618yes—
231C4⋊Q8#4114yescomplemented
232#127—24no—
233#127—28no—
234#127—22no—
235#127—22no—
236#127—44no—
237#123—416no—
238C22⋊Q8—48no—
239#121—48no—
240#121—416no—
241#121—48no—
242#121—84no—
243C4×Q8—44no—
244C4×Q8—82no—
245C4×Q8—82no—
246C4×D4—28no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—82no—
254C4×D4—82no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—48no—
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×C4⋊C4C2412yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2414yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#106—28no—
305#106—44no—
306#106—44no—
307C4⋊C8—24no—
308C4⋊C8—44no—
309#102—24no—
310#102—28no—
311#102—24no—
312#102—28no—
313#102—28no—
314#102—24no—
315#102—24no—
316#102—28no—
317#102—44no—
318#102—44no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8—44no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24—22no—
333C24—22no—
334#41—832no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×Q8—81no—
342C2×Q8—81no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8C2×SD1618yes—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—48no—
366C2×D4—42no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4H(5,𝔽2)14yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—48no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—84no—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D414yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2511yescharacteristic
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—82no—
431C4⋊C4—81no—
432C4⋊C4—82no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4C2×SD1618yes—
437C4⋊C4C2×SD1618yes—
438C4⋊C4C2×#4114yes—
439C4⋊C42−1+414yes—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—84no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42—41no—
456C42—44no—
457C42—44no—
458C42—84no—
459C42C2×SD1614yes—
460C42C2×#4112yes—
461C422−1+412yes—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C23×D411yescharacteristic
477Q8—48no—
478Q8—84no—
479Q8—84no—
480Q8—168no—
481D4—24no—
482D4—24no—
483D4—48no—
484D4—42no—
485D4—44no—
486D4—42no—
487D4—44no—
488D4—816no—
489D4—84no—
490D4—84no—
491D4—84no—
492D4—168no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
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—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—816no—
529C2×C4—88no—
530C2×C4—88no—
531C2×C4#54812yes—
532C2×C4#54812yes—
533C2×C4C22×SD1612yes—
534C2×C4C22×SD1612yes—
535C2×C4C22×SD1618yes—
536C2×C4C22×SD1612yes—
537C2×C4#57612yes—
538C2×C4C22×#4111yescharacteristic
539C2×C4C2×H(5,𝔽2)12yes—
540C2×C4C2×2−1+412yes—
541C2×C4C2×2−1+411yescharacteristic
542C8—42no—
543C8—44no—
544C8—42no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22C2×#54811yescharacteristic
560C22#348411yescharacteristic
561C22#348611yescharacteristic
562C22C23×SD1611yescharacteristic
563C22C2×#57611yescharacteristic
564C22#358511yescharacteristic
565C22#358511yescharacteristic
566C4—22no—
567C4—28no—
568C4—22no—
569C4—22no—
570C4—44no—
571C4—41no—
572C4—44no—
573C4—41no—
574C4—44no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4#336214yes—
580C4#352212yes—
581C4#354412yes—
582C2—84no—
583C2—162no—
584C2#4493711yescharacteristic
585C2C2×#336211yescharacteristic
586C2#6133211yescharacteristic
587C2#6156011yescharacteristic
588C2#6167911yescharacteristic
589C2#6182011yescharacteristic
590C2#6184711yescharacteristic
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. #59924C2semidirect
3. #32568C2semidirect
4. #30813C2semidirect
5. #29258C2semidirect
6. #28202C2semidirect
7. #27767C2semidirect
8. #26692C2semidirect
9. #26044C2semidirect
10. #25649C2semidirect
11. #25274C2semidirect
12. #23815C2semidirect
13. #22963C2semidirect
14. #22307C2semidirect
15. #22219C2semidirect
231. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#599242
#598152
#325682
#308132
#292582
#282022
#277672
#266922
#260442
#256492
#252742
#238152
#229632
#223072
#222192