Finite Group Databasebeta

← #6506600

Subgroups — #6506600

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37386C211yesmaximal, characteristic
2#35532C211yesmaximal, characteristic
3#29344C212yesmaximal, complemented
4#29332C212yesmaximal, complemented
5#27972C212yesmaximal, complemented
6#27706C214yesmaximal, complemented
7#27700C214yesmaximal, complemented
8#27283C214yesmaximal, complemented
9#27258C214yesmaximal, complemented
10#25914C212yesmaximal, complemented
11#21771C212yesmaximal
12#20913C212yesmaximal
13#20530C211yesmaximal, characteristic
14C2×#544C2211yescharacteristic
15C2×#536C2212yes—
16C2×#520C2211yescharacteristic
17C2×#491C2212yes—
18C2×#487C2212yes—
19C2×#419C2212yes—
20C22×#103C2212yes—
21C2×#408C2212yes—
22#2857C2212yes—
23#2734—21no—
24#2734—21no—
25#2724—21no—
26#2724—21no—
27#2717—24no—
28#2621C2212yes—
29#2567—21no—
30#2566—21no—
31#2565—21no—
32#2542—21no—
33#2499C2214yes—
34#2449—24no—
35#2441C2214yes—
36#2440C2214yes—
37#2434C2214yes—
38#2317C2211yescharacteristic
39#2311C2211yescharacteristic
40#2297C2211yescharacteristic
41#2296C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2101C2214yes—
44#2099C2212yes—
45#2099C2212yes—
46#2072C2214yes—
47#2072C2214yes—
48#2043C2212yes—
49#2043C2214yes—
50#2043C2212yes—
51#2026C2214yes—
52#2024C2214yes—
53#1993C2212yes—
54#1991C2212yes—
55#1956C2214yes—
56#1935C2214yes—
57#1933C2214yes—
58#1933C2214yes—
59#1933C2214yes—
60#1894C2214yes—
61#1892C2214yes—
62#1873C2214yes—
63#1873C2214yes—
64#1873C2214yes—
65#1872C2214yes—
66#1872C2214yes—
67#1872C2214yes—
68#1834C2214yes—
69#1825C2214yes—
70#1787C2214yes—
71#1753C2214yes—
72C8×C22⋊C4C2214yes—
73#1750C2212yes—
74C4×C22⋊C8C2212yes—
75#1736C2212yes—
76#1735C2212yes—
77C22×#41C2311yescharacteristic
78C23×Q8C2312yes—
79C22×M16C2312yes—
80C23×C8C2312yes—
81#544—42no—
82#544—42no—
83#536—28no—
84#536—28no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#125—21no—
91C2×#125C2312yes—
92C2×#124—22no—
93C2×#123—22no—
94C2×#123—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99#520—42no—
100#520—42no—
101C2×C4×Q8—21no—
102C2×C4×D4—21no—
103C2×C4×D4—22no—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2312yes—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#491—44no—
115#487—44no—
116#419—44no—
117C2×#103—44no—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126#408—28no—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×C4×C8C2314yes—
131C2×C4×C8C2312yes—
132#399—24no—
133#398—22no—
134#398—22no—
135#398—24no—
136#397—22no—
137#397—22no—
138#397—24no—
139#397—24no—
140#396—24no—
141#395—24no—
142#395—24no—
143#395—24no—
144#391—21no—
145#391D411yescharacteristic
146#391D411yescharacteristic
147#391C2314yes—
148#391C2314yes—
149#390—21no—
150#390D411yescharacteristic
151#390D411yescharacteristic
152#389—22no—
153#389—22no—
154#389—24no—
155#389C2314yes—
156#389C2312yes—
157#389C2312yes—
158#387—24no—
159#387—24no—
160#387—24no—
161#386—22no—
162#386D414yes—
163#385—22no—
164#385—24no—
165#385—22no—
166#385—24no—
167#385—24no—
168#384—21no—
169#384—21no—
170#384—21no—
171#384C2314yes—
172#384C2312yes—
173#382—21no—
174#382C2312yes—
175#380—22no—
176#380—22no—
177#380C2314yes—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C22⋊C4C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192C2×#41—41no—
193C2×#41—41no—
194C2×#41—42no—
195C2×#41#4118yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8C2×D411yescharacteristic
205C22×Q8C2414yes—
206C22×D4C2×D411yescharacteristic
207C22×D4C2×D412yes—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4C2×D412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C2×M16—24no—
216C2×M16—42no—
217C2×M16—42no—
218C22×C8—24no—
219C22×C8—42no—
220C22×C8—42no—
221C22×C8C2414yes—
222C22×C8C2414yes—
223C4⋊Q8—24no—
224C4⋊Q8—42no—
225C4⋊Q8—44no—
226C4⋊Q8—44no—
227C4⋊Q8—42no—
228C4⋊Q8—42no—
229C4⋊Q8—48no—
230#125—24no—
231#125—42no—
232#125—44no—
233#124—48no—
234#123—48no—
235#123—48no—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C22⋊Q8—48no—
241C22⋊Q8—88no—
242C4×Q8—44no—
243C4×D4—44no—
244C4×D4—48no—
245#117—22no—
246#117—22no—
247#117—24no—
248#117—24no—
249#117—24no—
250#117—41no—
251#117—41no—
252#117—42no—
253#117—42no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—84no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2411yescharacteristic
287C2×C4⋊C4C2411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D412yes—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—42no—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2412yes—
306C2×C42C2412yes—
307#103—28no—
308#103—28no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314#103—44no—
315C22⋊C8—28no—
316C22⋊C8—28no—
317#97—42no—
318C4×C8—44no—
319C4×C8—42no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24C22×D412yes—
333C24C22×D411yescharacteristic
334#41—48no—
335#41—48no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—82no—
349C2×Q8—82no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8#12414yes—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4#12414yes—
359C2×D4#12418yes—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—84no—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4114yes—
394C22×C4C2×#4114yes—
395C22×C4C2511yescharacteristic
396M16—84no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—84no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4H(5,𝔽2)12yes—
435C4⋊C4H(5,𝔽2)12yes—
436C22⋊C4—24no—
437C22⋊C4—28no—
438C22⋊C4—28no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—84no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—41no—
451C42—41no—
452C42—44no—
453C42—42no—
454C42—44no—
455C23—21no—
456C23—22no—
457C23—21no—
458C23—22no—
459C23—24no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23C2×#12412yes—
464C23C2×#12414yes—
465C23C23×D411yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—88no—
472Q8—164no—
473D4—44no—
474D4—48no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—88no—
512C2×C4—164no—
513C2×C4—162no—
514C2×C4—162no—
515C2×C4C2×#12414yes—
516C2×C4C2×#12412yes—
517C2×C4#57714yes—
518C2×C4#57814yes—
519C2×C4C22×#4112yes—
520C2×C4C2×H(5,𝔽2)11yescharacteristic
521C2×C4C2×H(5,𝔽2)11yescharacteristic
522C2×C4#58512yes—
523C8—44no—
524C8—44no—
525C22—24no—
526C22—22no—
527C22—28no—
528C22—24no—
529C22—42no—
530C22—44no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—42no—
535C22C22×#12411yescharacteristic
536C22#348411yescharacteristic
537C22#348511yescharacteristic
538C22C2×#57712yes—
539C22C2×#57812yes—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—24no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—84no—
549C4—82no—
550C4—84no—
551C4—82no—
552C4—82no—
553C4#350512yes—
554C4#350512yes—
555C2—44no—
556C2—42no—
557C2#3537711yescharacteristic
558C2#6062312yes—
559C2#6166211yescharacteristic
560C2#6166511yescharacteristic
561C2#6176712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29344C2semidirect
4. #29332C2semidirect
5. #27972C2semidirect
6. #27706C2semidirect
7. #27700C2semidirect
8. #27283C2semidirect
9. #27258C2semidirect
10. #25914C2semidirect

Maximal subgroups

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

subgroupindex
#373862
#355322
#293442
#293322
#279722
#277062
#277002
#272832
#272582
#259142
#217712
#209132
#205302