Finite Group Databasebeta

← #6570160

Subgroups — #6570160

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36509C211yesmaximal, characteristic
2#36209C211yesmaximal, characteristic
3#30318C212yesmaximal, complemented
4#30308C211yesmaximal, characteristic, complemented
5#29129C211yesmaximal, characteristic, complemented
6#28920C214yesmaximal, complemented
7#28723C211yesmaximal, characteristic, complemented
8#28581C212yesmaximal, complemented
9#28234C212yesmaximal
10#28148C212yesmaximal
11#28076C214yesmaximal, complemented
12#27864C212yesmaximal, complemented
13#23801C211yesmaximal, characteristic, complemented
14#23757C212yesmaximal, complemented
15#22809C212yesmaximal, complemented
16#22769C212yesmaximal, complemented
17#20567C211yesmaximal, characteristic
18C2×#491C2212yes—
19C2×#470C2214yes—
20C2×C4×Q16C2212yes—
21C2×C4×Q16C2212yes—
22C2×#436C2212yes—
23#2864C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2855C2212yes—
27#2590—22no—
28#2536C2211yescharacteristic
29#2536C2212yes—
30#2536C2211yescharacteristic
31#2515—22no—
32#2508—22no—
33#2496C2212yes—
34#2495—22no—
35#2489C2214yes—
36#2485—22no—
37#2466—22no—
38#2429C2212yes—
39#2373—22no—
40#2356C2211yescharacteristic
41#2356C2212yes—
42#2351C2211yescharacteristic
43#2346—22no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2285C2212yes—
48C42×D4C2211yescharacteristic
49#2100C2212yes—
50#2095C2212yes—
51#2095C2211yescharacteristic
52#2095C2211yescharacteristic
53#2084C2212yes—
54#2083C2212yes—
55#2076C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#2059C2214yes—
59#2043C2212yes—
60#2036C2214yes—
61#2036C2214yes—
62#2016C2211yescharacteristic
63#2016C2211yescharacteristic
64#2016C2212yes—
65#2001C2214yes—
66#1992C2212yes—
67#1969C2214yes—
68#1935C2214yes—
69#1933C2214yes—
70#1927C2214yes—
71#1927C2214yes—
72#1918C2214yes—
73#1894C2214yes—
74#1894C2214yes—
75#1878C2214yes—
76#1846C2212yes—
77#1835C2212yes—
78#1809C2214yes—
79#1805C2212yes—
80#1805C2212yes—
81#1805C2212yes—
82#1805C2212yes—
83C4×#107C2214yes—
84C8×C4⋊C4C2212yes—
85#1767C2211yescharacteristic
86C4×#103C2214yes—
87#1753C2214yes—
88C4×C22⋊C8C2212yes—
89#1744C2211yescharacteristic
90#1728C2212yes—
91C22×Q16C2312yes—
92C22×Q16C2312yes—
93C22×M16C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#126—22no—
97C2×#125C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2314yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×Q8C2311yescharacteristic
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4D412yes—
106C2×C4×D4C2311yescharacteristic
107C2×#117—22no—
108C22×C42C2312yes—
109#491—44no—
110#470—48no—
111C4×Q16—28no—
112C4×Q16—28no—
113#436—28no—
114C2×#107C2312yes—
115C2×#107C2312yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×#103C2314yes—
120C2×#103C2312yes—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2312yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×C22⋊C8C2314yes—
128C2×#97C2312yes—
129C2×C4×C8C2314yes—
130C2×C4×C8C2312yes—
131#400—22no—
132#400—24no—
133#398—21no—
134#398—21no—
135#398—22no—
136#398—22no—
137#397—22no—
138#397—24no—
139#396—22no—
140#394—21no—
141#394—21no—
142#394—22no—
143#391C2311yescharacteristic
144#391C2311yescharacteristic
145#391C2312yes—
146#391C2312yes—
147#391C2314yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#388—24no—
160#388—24no—
161#388—22no—
162#388—22no—
163#388—22no—
164#387—22no—
165#387—24no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#385—22no—
173#385—22no—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2312yes—
177#383—22no—
178#383—21no—
179#382—22no—
180#382—24no—
181#382—21no—
182#382C2311yescharacteristic
183#382C2312yes—
184#382C2311yescharacteristic
185#381—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380C2312yes—
190#380C2314yes—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205#376—22no—
206#376—21no—
207C43C2311yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2311yescharacteristic
217#336C2311yescharacteristic
218#336C2314yes—
219C22×Q8C2411yescharacteristic
220C22×Q8C2412yes—
221C22×Q8C2411yescharacteristic
222C22×D4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2412yes—
226C2×Q16—24no—
227C2×Q16—24no—
228C2×Q16—24no—
229C2×Q16—42no—
230C2×Q16—42no—
231C2×Q16—42no—
232C2×M16—28no—
233C2×M16—44no—
234C2×M16—44no—
235C22×C8C2414yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C4⋊Q8—22no—
239C4⋊Q8—42no—
240C4⋊Q8—41no—
241#126—48no—
242#125—42no—
243#125—42no—
244C22⋊Q8—24no—
245C22⋊Q8—42no—
246C22⋊Q8—44no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—44no—
254C4×D4—24no—
255C4×D4—44no—
256#117—48no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2411yescharacteristic
279C2×C4⋊C4C2411yescharacteristic
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—44no—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#107—24no—
307#107—24no—
308#107—44no—
309#106—44no—
310C4⋊C8—24no—
311#103—28no—
312#103—24no—
313#103—24no—
314#103—24no—
315#103—24no—
316#103—44no—
317#103—44no—
318#103—44no—
319C22⋊C8—28no—
320#97—42no—
321C4×C8—44no—
322C4×C8#4118yes—
323#95—24no—
324#95—21no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—21no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—21no—
341#95—21no—
342#95—22no—
343#95—22no—
344#95C2×D411yescharacteristic
345#95C2×D411yescharacteristic
346#95C2×D412yes—
347#95C2×D412yes—
348C24C22×D412yes—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8H(5,𝔽2)12yes—
363C2×D4—22no—
364C2×D4—42no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C2511yescharacteristic
392Q16—48no—
393Q16—48no—
394M16—88no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8C2×#4114yes—
403C2×C8C2×#4114yes—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—82no—
440C22⋊C4—84no—
441C42—24no—
442C42—21no—
443C42—24no—
444C42—22no—
445C42—21no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—44no—
451C42—42no—
452C42—48no—
453C42—44no—
454C42—82no—
455C42C2×#4114yes—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23C23×D411yescharacteristic
464Q8—24no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—42no—
472D4—44no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—48no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—84no—
516C2×C4#53212yes—
517C2×C4#53212yes—
518C2×C4#57514yes—
519C2×C4#57814yes—
520C2×C4C22×#4112yes—
521C2×C4C2×H(5,𝔽2)11yescharacteristic
522C2×C4#58511yescharacteristic
523C2×C4#58512yes—
524C8—24no—
525C8—24no—
526C8—44no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22C2×#53211yescharacteristic
535C22#348511yescharacteristic
536C22#348811yescharacteristic
537C22C2×#57512yes—
538C22C2×#57812yes—
539C4—22no—
540C4—24no—
541C4—24no—
542C4—22no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—82no—
548C4—81no—
549C4—81no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4—81no—
554C4—82no—
555C4#349312yes—
556C4#349412yes—
557C2—82no—
558C2#3606211yescharacteristic
559C2#6067611yescharacteristic
560C2#6071511yescharacteristic
561C2#6173812yes—
562C2#6199812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30318C2semidirect
4. #30308C2semidirect
5. #29129C2semidirect
6. #28920C2semidirect
7. #28723C2semidirect
8. #28581C2semidirect
11. #28076C2semidirect
12. #27864C2semidirect
13. #23801C2semidirect
14. #23757C2semidirect
15. #22809C2semidirect
16. #22769C2semidirect

Maximal subgroups

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

subgroupindex
#365092
#362092
#303182
#303082
#291292
#289202
#287232
#285812
#282342
#281482
#280762
#278642
#238012
#237572
#228092
#227692
#205672