Finite Group Databasebeta

← #6490789

Subgroups — #6490789

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3#33115C214yesmaximal, complemented
4#30074C212yesmaximal, complemented
5#30072C212yesmaximal, complemented
6#29269C212yesmaximal, complemented
7#29268C212yesmaximal, complemented
8#29213C212yesmaximal, complemented
9#28757C212yesmaximal, complemented
10#27917C211yesmaximal, characteristic
11#27916C211yesmaximal, characteristic, complemented
12#25913C211yesmaximal, characteristic
13#25912C211yesmaximal, characteristic, complemented
14#24075C214yesmaximal, complemented
15#22226C212yesmaximal, complemented
16#22225C212yesmaximal, complemented
17#20783C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C22×#125C2211yescharacteristic
20C2×#491C2212yes—
21C2×#490C2212yes—
22C2×#485C2214yes—
23C2×#482C2214yes—
24#2868C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2640C2212yes—
29#2620C2212yes—
30#2512—22no—
31#2491—22no—
32#2452—22no—
33#2422—22no—
34#2370—22no—
35#2370—22no—
36#2345—22no—
37#2345—22no—
38#2330C2211yescharacteristic
39#2330C2211yescharacteristic
40#2330C2211yescharacteristic
41#2330C2211yescharacteristic
42#2326C2212yes—
43#2316C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125C2212yes—
47C4×#125C2212yes—
48#2285C2211yescharacteristic
49#2274C2212yes—
50C4×#117C2211yescharacteristic
51#2096C2212yes—
52#2095C2212yes—
53#2094C2212yes—
54#2093C2212yes—
55#2058C2214yes—
56#2057C2214yes—
57#2042C2214yes—
58#2042C2214yes—
59#2026C2212yes—
60#2026C2212yes—
61#2025C2212yes—
62#2025C2212yes—
63#1992C2212yes—
64#1992C2212yes—
65#1983C2214yes—
66#1983C2214yes—
67#1961C2212yes—
68#1960C2212yes—
69#1948C2214yes—
70#1947C2214yes—
71#1928C2214yes—
72#1927C2214yes—
73#1926C2214yes—
74#1925C2214yes—
75#1874C2214yes—
76#1872C2214yes—
77#1865C2214yes—
78#1864C2214yes—
79#1863C2214yes—
80#1862C2214yes—
81#1847C2212yes—
82#1847C2212yes—
83#1840C2212yes—
84#1832C2212yes—
85#1818C2212yes—
86#1798C2212yes—
87#1767C2212yes—
88#1749C2211yescharacteristic
89#1749C2211yescharacteristic
90#1745C2212yes—
91#542—48no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2312yes—
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#125—28no—
98C2×#125—22no—
99C2×#125C2311yescharacteristic
100C2×#125C2311yescharacteristic
101C2×#125C2312yes—
102C2×#124—22no—
103C2×#124C2312yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C4⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119#491—28no—
120#490—28no—
121#485—48no—
122#482—48no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140#397—24no—
141#396—24no—
142#394—24no—
143#393—24no—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390C2312yes—
151#390C2312yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#388—24no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#385—24no—
174#384—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2312yes—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#382C2312yes—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203#376C2311yescharacteristic
204#376C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4C2412yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—24no—
228C4⋊Q8—44no—
229C4⋊Q8—84no—
230#127—24no—
231#126—48no—
232#125—216no—
233#125—24no—
234#125—24no—
235#125—24no—
236#125#4118yes—
237#124—28no—
238#124—44no—
239#124—48no—
240#123—48no—
241C22⋊Q8—28no—
242C22⋊Q8—48no—
243#121—28no—
244#121—84no—
245C4×Q8—44no—
246C4×D4—44no—
247C4×D4—84no—
248#117—24no—
249#117—24no—
250#117—28no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4C2×D411yescharacteristic
291C2×C22⋊C4C2×D411yescharacteristic
292C2×C22⋊C4C2×D412yes—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—44no—
312#107—44no—
313#106—44no—
314#106—44no—
315C4⋊C8—28no—
316#103—28no—
317#103—28no—
318#103—44no—
319#103—44no—
320#102—28no—
321#102—28no—
322#102—44no—
323#102—44no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326#97—24no—
327#97—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343#95C2×D412yes—
344C24—22no—
345C24C22×D411yescharacteristic
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—44no—
350C2×Q8—84no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—84no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—84no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4H(5,𝔽2)14yes—
386C22×C4C2511yescharacteristic
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—84no—
420C4⋊C4C2×#4114yes—
421C4⋊C4C2×#4114yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—24no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—44no—
444C42—44no—
445C42—82no—
446C42—82no—
447C42—82no—
448C42—82no—
449C42C2×#4114yes—
450C422−1+414yes—
451C23—22no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23#53512yes—
461C23C23×D411yescharacteristic
462Q8—48no—
463Q8—164no—
464D4—48no—
465D4—88no—
466D4—164no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—28no—
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—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4—88no—
509C2×C4—164no—
510C2×C4#52514yes—
511C2×C4#53512yes—
512C2×C4#53614yes—
513C2×C4C22×#4112yes—
514C2×C4C2×H(5,𝔽2)12yes—
515C2×C4C2×2−1+412yes—
516C8—44no—
517C8—44no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—84no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—84no—
528C22C2×#52511yescharacteristic
529C22C2×#53511yescharacteristic
530C22C2×#53611yescharacteristic
531C22#358512yes—
532C22#358612yes—
533C4—24no—
534C4—22no—
535C4—24no—
536C4—42no—
537C4—44no—
538C4—44no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4#352214yes—
550C2—42no—
551C2—162no—
552C2#3524911yescharacteristic
553C2#6056712yes—
554C2#6083011yescharacteristic
555C2#6083111yescharacteristic
556C2#6088612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. #34217C2semidirect
3. #33115C2semidirect
4. #30074C2semidirect
5. #30072C2semidirect
6. #29269C2semidirect
7. #29268C2semidirect
8. #29213C2semidirect
9. #28757C2semidirect
11. #27916C2semidirect
13. #25912C2semidirect
14. #24075C2semidirect
15. #22226C2semidirect
16. #22225C2semidirect
17. #20783C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#342172
#331152
#300742
#300722
#292692
#292682
#292132
#287572
#279172
#279162
#259132
#259122
#240752
#222262
#222252
#207832