Finite Group Databasebeta

← #6485301

Subgroups — #6485301

Order 512.

1519 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#35319C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic
3#33041C214yesmaximal, complemented
4#30116C212yesmaximal
5#30109C212yesmaximal
6#29371C212yesmaximal, complemented
7#29370C212yesmaximal, complemented
8#29221C211yesmaximal, characteristic, complemented
9#28755C211yesmaximal, characteristic, complemented
10#28518C211yesmaximal, characteristic, complemented
11#28440C211yesmaximal, characteristic, complemented
12#27902C212yesmaximal
13#25609C212yesmaximal
14#24165C214yesmaximal
15#22277C212yesmaximal, complemented
16#22231C212yesmaximal, complemented
17#20781C211yesmaximal, characteristic
18C2×#541C2211yescharacteristic
19C2×#541C2212yes—
20C2×#494C2212yes—
21C2×#491C2212yes—
22C2×#479C2214yes—
23C2×#462C2214yes—
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2642C2212yes—
29#2621C2212yes—
30#2506—22no—
31#2506—22no—
32#2496—22no—
33#2496—22no—
34#2373—22no—
35#2373—22no—
36#2371—22no—
37#2370—22no—
38#2330C2211yescharacteristic
39#2327C2211yescharacteristic
40#2327C2212yes—
41#2323C2212yes—
42#2314C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#125C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2212yes—
49#2276C2212yes—
50C4×#117C2211yescharacteristic
51#2100C2212yes—
52#2099C2212yes—
53#2096C2212yes—
54#2095C2212yes—
55#2043C2212yes—
56#2043C2212yes—
57#2036C2214yes—
58#2035C2214yes—
59#2016C2212yes—
60#2016C2212yes—
61#2001C2214yes—
62#2001C2214yes—
63#1991C2212yes—
64#1989C2214yes—
65#1981C2214yes—
66#1960C2212yes—
67#1960C2212yes—
68#1956C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1927C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1865C2214yes—
80#1863C2214yes—
81#1847C2212yes—
82#1839C2212yes—
83#1835C2212yes—
84#1832C2212yes—
85#1817C2212yes—
86#1798C2212yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2211yescharacteristic
89#1750C2212yes—
90#1746C2212yes—
91C22×Q16C2314yes—
92C22×Q16C2314yes—
93#541—24no—
94#541—24no—
95#541—48no—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#126—22no—
103C2×#126—22no—
104C2×#125—22no—
105C2×#125—22no—
106C2×#125—22no—
107C2×#125C2312yes—
108C2×#125C2311yescharacteristic
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#494—28no—
121#491—28no—
122#479—48no—
123#462—48no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#400—24no—
140#400—24no—
141#398—24no—
142#397—24no—
143#391C2312yes—
144#391C2312yes—
145#391C2312yes—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—24no—
172#387—24no—
173#387—22no—
174#385—22no—
175#384—22no—
176#384—22no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#382C2311yescharacteristic
181#382C2312yes—
182#380—22no—
183#380—22no—
184#380C2312yes—
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⋊C4—22no—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202#376—22no—
203#376—22no—
204#376C2311yescharacteristic
205#376C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217C22×Q8C2412yes—
218C22×Q8C2412yes—
219C22×Q8C2412yes—
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C2×Q16—28no—
223C2×Q16—28no—
224C2×Q16—44no—
225C2×Q16—44no—
226C2×Q16—44no—
227C2×Q16—44no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—22no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—28no—
234C4⋊Q8—42no—
235C4⋊Q8—42no—
236C4⋊Q8#4114yes—
237#126—216no—
238#126—48no—
239#126—48no—
240#125—216no—
241#125—22no—
242#125—28no—
243#125—48no—
244#125—48no—
245#125#4114yes—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—44no—
249C22⋊Q8—44no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—28no—
253C4×Q8—44no—
254C4×Q8—44no—
255C4×Q8—44no—
256#117—22no—
257#117—22no—
258#117—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—22no—
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⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C2×D411yescharacteristic
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C22⋊C4C2×D412yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316#107—44no—
317#106—44no—
318C4⋊C8—24no—
319C4⋊C8—24no—
320#103—28no—
321#103—28no—
322#103—44no—
323#103—44no—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328C22⋊C8—44no—
329#97—24no—
330C4×C8—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
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—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8C2×#4114yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C2511yescharacteristic
389Q16—88no—
390Q16—88no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4C2×#4114yes—
428C22⋊C4—28no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—82no—
435C42—28no—
436C42—24no—
437C42—24no—
438C42—21no—
439C42—21no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—44no—
447C42—44no—
448C42—44no—
449C42—44no—
450C42—82no—
451C42—82no—
452C42—82no—
453C42C2×#4112yes—
454C42C2×#4112yes—
455C42H(5,𝔽2)14yes—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23C23×D411yescharacteristic
462Q8—28no—
463Q8—44no—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
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—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—84no—
512C2×C4#52514yes—
513C2×C4#53512yes—
514C2×C4#53512yes—
515C2×C4#57814yes—
516C2×C4C22×#4111yescharacteristic
517C2×C4C22×#4111yescharacteristic
518C2×C4C2×H(5,𝔽2)12yes—
519C2×C4#58512yes—
520C8—44no—
521C8—44no—
522C22—42no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22C2×#52511yescharacteristic
528C22C2×#53511yescharacteristic
529C22#344911yescharacteristic
530C22C2×#57812yes—
531C22#358612yes—
532C4—24no—
533C4—22no—
534C4—22no—
535C4—24no—
536C4—42no—
537C4—41no—
538C4—44no—
539C4—41no—
540C4—41no—
541C4—42no—
542C4—44no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4#351412yes—
550C4#351712yes—
551C2—81no—
552C2#3519011yescharacteristic
553C2#6059412yes—
554C2#6082111yescharacteristic
555C2#6083111yescharacteristic
556C2#6098312yes—
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. #33041C2semidirect
6. #29371C2semidirect
7. #29370C2semidirect
8. #29221C2semidirect
9. #28755C2semidirect
10. #28518C2semidirect
11. #28440C2semidirect
15. #22277C2semidirect
16. #22231C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#342812
#330412
#301162
#301092
#293712
#293702
#292212
#287552
#285182
#284402
#279022
#256092
#241652
#222772
#222312
#207812