Finite Group Databasebeta

← #6493483

Subgroups — #6493483

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic, complemented
4#33067C211yesmaximal, characteristic, complemented
5#32906C212yesmaximal
6#29996C211yesmaximal, characteristic, complemented
7#29922C212yesmaximal
8#29799C211yesmaximal, characteristic, complemented
9#27409C214yesmaximal
10#27209C214yesmaximal, complemented
11#26187C214yesmaximal, complemented
12#25173C214yesmaximal
13#22154C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic
15#21506C212yesmaximal, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#462C2212yes—
18C2×#462C2212yes—
19C2×#462C2214yes—
20C2×#451C2214yes—
21C2×#441C2214yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2647—21no—
26#2629—21no—
27#2507—22no—
28#2497—22no—
29#2446C2214yes—
30#2429C2214yes—
31#2355—22no—
32#2355C2212yes—
33Q8×C4⋊C4C2212yes—
34#2345—22no—
35#2345C2212yes—
36Q8×C22⋊C4C2212yes—
37#2330—21no—
38#2314—21no—
39C4×C4⋊Q8C2212yes—
40#2295—21no—
41#2295—21no—
42#2295C2211yescharacteristic
43#2295C2211yescharacteristic
44#2285—21no—
45#2285C2211yescharacteristic
46C42×Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48C4×#117—21no—
49C4×#117C2211yescharacteristic
50#2096C2214yes—
51#2088C2214yes—
52#2072C2214yes—
53#2066C2214yes—
54#2043C2214yes—
55#2035C2214yes—
56#2035C2214yes—
57#2016C2214yes—
58#1981C2212yes—
59#1981C2214yes—
60#1981C2212yes—
61#1971C2214yes—
62#1948C2212yes—
63#1948C2212yes—
64#1935C2214yes—
65#1927C2214yes—
66#1924C2214yes—
67#1919C2214yes—
68#1917C2214yes—
69#1878C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1872C2214yes—
74#1865C2214yes—
75#1865C2214yes—
76#1842C2211yescharacteristic
77#1831C2211yescharacteristic
78#1803C2212yes—
79#1803C2212yes—
80#1797C2212yes—
81#1767C2212yes—
82#1763C2214yes—
83C23×Q8C2311yescharacteristic
84C22×Q16C2314yes—
85C22×Q16C2314yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2314yes—
88C2×#125—21no—
89C2×C22⋊Q8C2314yes—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8C2311yescharacteristic
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×Q8C2311yescharacteristic
100C2×C4×Q8C2314yes—
101C2×#117—21no—
102C2×#117D411yescharacteristic
103C2×#117D411yescharacteristic
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#462—28no—
109#462—28no—
110#462—48no—
111#451—48no—
112#441—48no—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×C22⋊C8C2314yes—
127C2×#97C2314yes—
128#400—24no—
129#398—24no—
130#398—24no—
131#396—24no—
132#391—22no—
133#391—22no—
134#391C2314yes—
135#391C2312yes—
136#391C2314yes—
137#391C2312yes—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—24no—
144#389—22no—
145#389—22no—
146#389C2314yes—
147#389C2312yes—
148#389C2312yes—
149#388—24no—
150#387—24no—
151#387—22no—
152#387—22no—
153#387—24no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#384—22no—
159#384—21no—
160#384—21no—
161#384—22no—
162#384C2312yes—
163#384C2312yes—
164#382—22no—
165#382—21no—
166#382—21no—
167#380—22no—
168#380—21no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380C2312yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2311yescharacteristic
188C4×C4⋊C4C2311yescharacteristic
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4C2312yes—
194#376—21no—
195#376—21no—
196#376D411yescharacteristic
197#376D411yescharacteristic
198C43—21no—
199C43—21no—
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—21no—
212C22×Q8—24no—
213C22×Q8—42no—
214C22×Q8C2414yes—
215C22×Q8C2411yescharacteristic
216C22×Q8C2411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C2×Q16—28no—
223C2×Q16—28no—
224C2×Q16—28no—
225C2×Q16—28no—
226C2×Q16—44no—
227C2×Q16—44no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—24no—
231C4⋊Q8—24no—
232C4⋊Q8—44no—
233C4⋊Q8—44no—
234#125—44no—
235C22⋊Q8—44no—
236C22⋊Q8—44no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8#4114yes—
252C4×Q8#4114yes—
253#117—24no—
254#117—24no—
255#117—44no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
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—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2411yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C42—28no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2412yes—
313C2×C42C2411yescharacteristic
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316#106—44no—
317#106—44no—
318C4⋊C8—24no—
319C4⋊C8—24no—
320C4⋊C8—44no—
321#103—28no—
322#103—28no—
323#103—28no—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329C22⋊C8—44no—
330#97—44no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—21no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351C24C22×D411yescharacteristic
352C2×Q8—24no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8C2×#4112yes—
369C2×Q8C2×#4112yes—
370C2×Q8H(5,𝔽2)14yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—82no—
391C22×C4#13714yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C2511yescharacteristic
402Q16—416no—
403Q16—88no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4C2×#4112yes—
441C4⋊C4C2×#4112yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—44no—
445C22⋊C4—84no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—21no—
456C42—21no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—42no—
462C42—42no—
463C42—44no—
464C42—44no—
465C42—42no—
466C42—42no—
467C42—82no—
468C42—82no—
469C42—84no—
470C42#13714yes—
471C42C2×#4112yes—
472C42C2×#4112yes—
473C23—22no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23C2×#13712yes—
478C23C23×D411yescharacteristic
479Q8—22no—
480Q8—24no—
481Q8—24no—
482Q8—48no—
483Q8—41no—
484Q8—41no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8#54014yes—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—28no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4#52512yes—
536C2×C4#52512yes—
537C2×C4C2×#13714yes—
538C2×C4C2×#13712yes—
539C2×C4C2×#13712yes—
540C2×C4C2×#13712yes—
541C2×C4C22×#4111yescharacteristic
542C2×C4C22×#4111yescharacteristic
543C2×C4C2×H(5,𝔽2)12yes—
544C2×C4#58512yes—
545C8—44no—
546C8—44no—
547C22—24no—
548C22—22no—
549C22—42no—
550C22—41no—
551C22—41no—
552C22C2×#52511yescharacteristic
553C22#348812yes—
554C22C22×#13711yescharacteristic
555C22C22×#13711yescharacteristic
556C22#358612yes—
557C4—24no—
558C4—22no—
559C4—22no—
560C4—24no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4—84no—
573C4#311314yes—
574C4C2×#54011yescharacteristic
575C4#345511yescharacteristic
576C4#349412yes—
577C2—42no—
578C2#3526111yescharacteristic
579C2C2×#311311yescharacteristic
580C2#6056911yescharacteristic
581C2#6196412yes—
582C2#6196612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33163C2semidirect
4. #33067C2semidirect
6. #29996C2semidirect
8. #29799C2semidirect
10. #27209C2semidirect
11. #26187C2semidirect
13. #22154C2semidirect
15. #21506C2semidirect

Maximal subgroups

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

subgroupindex
#352842
Q8×#1172
#331632
#330672
#329062
#299962
#299222
#297992
#274092
#272092
#261872
#251732
#221542
#219262
#215062