Finite Group Databasebeta

← #6469477

Subgroups — #6469477

Order 512.

1686 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic, complemented
2C4×#540C211yesmaximal, characteristic
3#29401C214yesmaximal, complemented
4#29274C212yesmaximal, complemented
5#28201C212yesmaximal, complemented
6#27646C212yesmaximal, complemented
7#27620C212yesmaximal, complemented
8#24298C212yesmaximal, complemented
9#24070C212yesmaximal
10#23844C212yesmaximal, complemented
11#23828C212yesmaximal, complemented
12#21613C212yesmaximal, complemented
13C4×#474C212yesmaximal, complemented
14#21590C214yesmaximal, complemented
15#20786C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#482C2214yes—
19C2×#474C2212yes—
20C2×#474C2212yes—
21#2369—22no—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2335—22no—
26#2333—22no—
27#2331C2212yes—
28#2331C2212yes—
29#2330C2211yescharacteristic
30#2328C2211yescharacteristic
31#2324C2212yes—
32#2319C2212yes—
33#2316C2212yes—
34#2314C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×#127C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×#123—22no—
40C4×#121—22no—
41C4×#121C2212yes—
42#2285C2211yescharacteristic
43#2274C2212yes—
44C42×D4C2212yes—
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2085C2214yes—
48#2083C2214yes—
49#2073C2214yes—
50#2059C2214yes—
51#2057C2214yes—
52#2041C2214yes—
53#2023C2214yes—
54#1988C2212yes—
55#1988C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1925C2214yes—
59#1925C2214yes—
60#1921C2214yes—
61#1921C2214yes—
62#1919C2214yes—
63#1918C2214yes—
64#1874C2214yes—
65#1864C2214yes—
66#1862C2214yes—
67#1852C2214yes—
68#1840C2212yes—
69#1831C2212yes—
70#1825C2214yes—
71#1818C2212yes—
72#1802C2212yes—
73#1802C2212yes—
74#1797C2212yes—
75#1776C2214yes—
76C4×#107C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1762C2214yes—
80C4×#102C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83C23×Q8—21no—
84#540—28no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×#123—22no—
93C2×#121—22no—
94C2×#121C2312yes—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2312yes—
107C22×C42—22no—
108C22×C42C2311yescharacteristic
109#482—216no—
110#474—28no—
111#474—28no—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124C2×C4×C8C2312yes—
125#391—22no—
126#391—22no—
127#391—24no—
128#391—24no—
129#390—22no—
130#390—22no—
131#390C2312yes—
132#390C2312yes—
133#389—24no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389C2312yes—
138#389C2314yes—
139#389C2314yes—
140#389C2312yes—
141#388—22no—
142#388—22no—
143#388—22no—
144#388—22no—
145#388—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—24no—
150#387—24no—
151#386—22no—
152#386—22no—
153#386—22no—
154#386—22no—
155#385—22no—
156#384C2311yescharacteristic
157#384C2312yes—
158#382C2311yescharacteristic
159#382C2312yes—
160#381—22no—
161#380—21no—
162#380—21no—
163#380—24no—
164#380—22no—
165#380C2312yes—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C22⋊C4—24no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186#376C2311yescharacteristic
187#376C2311yescharacteristic
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202#336C2314yes—
203C22×Q8—28no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—84no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4C2412yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—28no—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222C4⋊Q8—24no—
223C4⋊Q8#4118yes—
224#127—24no—
225#125—24no—
226#123—216no—
227#121—216no—
228#121—24no—
229#121#4118yes—
230C4×Q8—84no—
231C4×Q8—84no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—84no—
239C4×D4—84no—
240#117—24no—
241#117—22no—
242#117—22no—
243#117—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—84no—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—28no—
297#107—28no—
298#107—28no—
299#106—28no—
300C4⋊C8—24no—
301C4⋊C8—24no—
302#102—28no—
303#102—28no—
304#102—28no—
305#102—28no—
306C22⋊C8—28no—
307#97—24no—
308C4×C8—24no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326C24—22no—
327C24C22×D411yescharacteristic
328C2×Q8—28no—
329C2×Q8—28no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—84no—
333C2×Q8—816no—
334C2×D4—28no—
335C2×D4—28no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4C2×#4114yes—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—84no—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4114yes—
379C22×C4C2511yescharacteristic
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4C2×#4118yes—
408C4⋊C4C2×#4114yes—
409C22⋊C4—28no—
410C22⋊C4—24no—
411C22⋊C4—28no—
412C22⋊C4—24no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—24no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—42no—
431C42—42no—
432C42—44no—
433C42—44no—
434C42—82no—
435C42—82no—
436C42—84no—
437C42—84no—
438C42C2×#4114yes—
439C422−1+412yes—
440C422−1+412yes—
441C23—22no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23#52512yes—
453C23C23×D411yescharacteristic
454Q8—416no—
455Q8—88no—
456D4—24no—
457D4—24no—
458D4—42no—
459D4—42no—
460D4—88no—
461D4—88no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—88no—
500C2×C4—88no—
501C2×C4—88no—
502C2×C4—88no—
503C2×C4#52512yes—
504C2×C4#52512yes—
505C2×C4#52512yes—
506C2×C4#53614yes—
507C2×C4#57614yes—
508C2×C4C22×#4112yes—
509C2×C4C22×#4112yes—
510C2×C4C2×2−1+411yescharacteristic
511C2×C4C2×2−1+411yescharacteristic
512C8—44no—
513C8—44no—
514C22—22no—
515C22—24no—
516C22—41no—
517C22—41no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22C2×#52511yescharacteristic
527C22C2×#52511yescharacteristic
528C22C2×#53611yescharacteristic
529C22C2×#57612yes—
530C22#358612yes—
531C4—22no—
532C4—22no—
533C4—24no—
534C4—41no—
535C4—48no—
536C4—41no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—42no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4#312214yes—
545C4#348212yes—
546C4#348212yes—
547C2—42no—
548C2—84no—
549C2#3497111yescharacteristic
550C2C2×#312211yescharacteristic
551C2#6056611yescharacteristic
552C2#6059612yes—
553C2#6088812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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
1. #34216C2semidirect
3. #29401C2semidirect
4. #29274C2semidirect
5. #28201C2semidirect
6. #27646C2semidirect
7. #27620C2semidirect
8. #24298C2semidirect
10. #23844C2semidirect
11. #23828C2semidirect
12. #21613C2semidirect
13. C4×#474C2semidirect
14. #21590C2semidirect
15. #20786C2semidirect

Maximal subgroups

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

subgroupindex
#342162
C4×#5402
#294012
#292742
#282012
#276462
#276202
#242982
#240702
#238442
#238282
#216132
C4×#4742
#215902
#207862