Finite Group Databasebeta

← #6461959

Subgroups — #6461959

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2#34056C211yesmaximal, characteristic
3#29591C214yesmaximal, complemented
4#29590C214yesmaximal, complemented
5#29587C214yesmaximal, complemented
6#29586C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23733C211yesmaximal, characteristic
9#23658C211yesmaximal, characteristic, complemented
10#23630C211yesmaximal, characteristic
11#21159C212yesmaximal, complemented
12#21093C212yesmaximal
13#21034C212yesmaximal, complemented
14#20875C212yesmaximal
15#20365C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#428C2212yes—
18C2×#428C2212yes—
19C2×#417C2212yes—
20C2×#417C2212yes—
21C2×C4×M16C2212yes—
22#2647—21no—
23#2647—21no—
24#2647—21no—
25#2647—21no—
26#2647C2214yes—
27#2647C2214yes—
28C2×#400—22no—
29C2×#398—22no—
30#2320C2212yes—
31#2320C2212yes—
32#2319C2212yes—
33#2319C2212yes—
34#2318—22no—
35#2318—22no—
36#2298—21no—
37#2298—21no—
38#2297C2211yescharacteristic
39#2297C2211yescharacteristic
40#2296C2211yescharacteristic
41#2296C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#384C2211yescharacteristic
45#2283—21no—
46#2283—21no—
47C2×#382C2211yescharacteristic
48C4×#117C2211yescharacteristic
49C2×#376C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2097C2214yes—
53#2092C2214yes—
54#2084C2214yes—
55#2083C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#1944C2214yes—
59#1943C2214yes—
60#1942C2214yes—
61#1941C2214yes—
62#1936C2214yes—
63#1935C2214yes—
64#1934C2214yes—
65#1933C2214yes—
66#1921C2214yes—
67#1920C2214yes—
68#1919C2214yes—
69#1918C2214yes—
70#1865C2214yes—
71#1864C2214yes—
72#1863C2214yes—
73#1862C2214yes—
74#1746C2214yes—
75#1746C2214yes—
76#1736C2212yes—
77#1736C2212yes—
78#1736C2212yes—
79#1736C2212yes—
80#1728C2212yes—
81#1728C2211yescharacteristic
82#1728C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C22×M16C2312yes—
85C22×M16C2312yes—
86C4×#41—24no—
87C4×#41—24no—
88C2×C4×Q8C2311yescharacteristic
89C2×C4×Q8C2311yescharacteristic
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4C2311yescharacteristic
93C2×C4×D4C2311yescharacteristic
94C2×#117—21no—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#428—28no—
108#428—28no—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113#417—28no—
114#417—28no—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C4×M16—28no—
120C2×#97C2314yes—
121C2×C4×C8C2314yes—
122#400—24no—
123#400—24no—
124#400—24no—
125#400—24no—
126#398—24no—
127#398—24no—
128#398—24no—
129#398—24no—
130#391C2312yes—
131#391C2312yes—
132#390C2312yes—
133#390C2312yes—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389C2312yes—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#389C2314yes—
143#389C2314yes—
144#389C2314yes—
145#389C2314yes—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#386—22no—
155#386—22no—
156#385—22no—
157#385—22no—
158#384—21no—
159#384—21no—
160#384C2314yes—
161#383—22no—
162#383—22no—
163#382—21no—
164#382—21no—
165#382—22no—
166#382—22no—
167#382C2314yes—
168#381—22no—
169#381—22no—
170#380—22no—
171#380—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184#376—21no—
185#376D411yescharacteristic
186#376D411yescharacteristic
187#376D411yescharacteristic
188#376D411yescharacteristic
189#376C2312yes—
190#376C2312yes—
191C2×#95—22no—
192C2×#95—22no—
193C2×#95—22no—
194C2×#95D412yes—
195C2×#95D412yes—
196C43—21no—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209C2×#41—22no—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41#4114yes—
213C22×Q8C2411yescharacteristic
214C22×D4—21no—
215C22×D4C2411yescharacteristic
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2411yescharacteristic
226C2×M16—24no—
227C2×M16—24no—
228C2×M16—24no—
229C2×M16—24no—
230C2×M16—24no—
231C2×M16#4118yes—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244#117—28no—
245#117—22no—
246#117—24no—
247#117—24no—
248#117#4114yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—84no—
268C2×C4⋊C4—84no—
269C2×C4⋊C4—84no—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2411yescharacteristic
275C2×C4⋊C4C2411yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C42—28no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2412yes—
306C2×C42C2412yes—
307#107—28no—
308#107—28no—
309#106—28no—
310#106—28no—
311#103—28no—
312#103—28no—
313#102—28no—
314#102—28no—
315#97—28no—
316C4×C8—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341C24—21no—
342C24C22×D411yescharacteristic
343#41—216no—
344#41—48no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8C2×#4112yes—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4C2×#4112yes—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4114yes—
398C22×C42−1+414yes—
399C22×C4C2511yescharacteristic
400M16—216no—
401M16—48no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8C2×#4118yes—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4C2×#4112yes—
432C4⋊C4C2×#4112yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C42—24no—
443C42—24no—
444C42—22no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—24no—
458C42—42no—
459C42—42no—
460C42—82no—
461C42—82no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23#52512yes—
470C23#53212yes—
471C23#53612yes—
472C23C23×D411yescharacteristic
473Q8—24no—
474Q8—42no—
475D4—24no—
476D4—48no—
477D4—42no—
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—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—88no—
525C2×C4—88no—
526C2×C4#52512yes—
527C2×C4#53212yes—
528C2×C4#53612yes—
529C2×C4#55218yes—
530C2×C4#55218yes—
531C2×C4C22×#4111yescharacteristic
532C2×C4C22×#4111yescharacteristic
533C2×C4C22×#4112yes—
534C2×C4C2×2−1+412yes—
535C8—28no—
536C8—44no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22#261114yes—
547C22C2×#52511yescharacteristic
548C22C2×#53211yescharacteristic
549C22C2×#53611yescharacteristic
550C22#358512yes—
551C22#358612yes—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—82no—
564C4—84no—
565C4—84no—
566C4—82no—
567C4#261114yes—
568C4C2×#55212yes—
569C4C2×#55212yes—
570C2—24no—
571C2—82no—
572C2C2×#261111yescharacteristic
573C2#6056811yescharacteristic
574C2#6056911yescharacteristic
575C2#6070312yes—
576C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2647C2semidirect
3. #29591C2semidirect
4. #29590C2semidirect
5. #29587C2semidirect
6. #29586C2semidirect
7. #23746C2semidirect
9. #23658C2semidirect
11. #21159C2semidirect
13. #21034C2semidirect
15. #20365C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
#340562
#295912
#295902
#295872
#295862
#237462
#237332
#236582
#236302
#211592
#210932
#210342
#208752
#203652