Finite Group Databasebeta

← #6971553

Subgroups — #6971553

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36454C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#32744C212yesmaximal, complemented
4#31759C214yesmaximal, complemented
5#31109C214yesmaximal, complemented
6#30205C211yesmaximal, characteristic, complemented
7#30168C212yesmaximal, complemented
8#29645C212yesmaximal, complemented
9#29355C211yesmaximal, characteristic, complemented
10#29213C211yesmaximal, characteristic, complemented
11#26461C212yesmaximal, complemented
12#26443C212yesmaximal, complemented
13#26303C212yesmaximal
14#26233C212yesmaximal
15#24306C212yesmaximal, complemented
16#23838C211yesmaximal, characteristic, complemented
17#22012C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#485C2214yes—
20C2×#479C2214yes—
21#2868C2211yescharacteristic
22#2862C2211yescharacteristic
23#2857C2212yes—
24#2855C2212yes—
25#2575—22no—
26#2552C2212yes—
27#2547C2211yescharacteristic
28#2542—21no—
29#2542C2211yescharacteristic
30#2538—21no—
31#2517—22no—
32#2511—21no—
33#2489C2212yes—
34#2447—21no—
35#2441C2214yes—
36#2429—22no—
37#2423—22no—
38#2421C2212yes—
39#2367—21no—
40#2365—21no—
41#2365C2211yescharacteristic
42#2365C2211yescharacteristic
43#2355—21no—
44#2354C2212yes—
45Q8×C4⋊C4—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C42×D4C2211yescharacteristic
50#2087C2214yes—
51#2085C2212yes—
52#2084C2211yescharacteristic
53#2083C2211yescharacteristic
54#2077C2214yes—
55#2076C2214yes—
56#2072C2214yes—
57#2072C2212yes—
58#2068C2214yes—
59#2059C2212yes—
60#2059C2214yes—
61#2058C2212yes—
62#2058C2211yescharacteristic
63#2057C2212yes—
64#2057C2211yescharacteristic
65#2036C2214yes—
66#2035C2214yes—
67#2026C2214yes—
68#2024C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#1992C2212yes—
72#1990C2212yes—
73#1989C2214yes—
74#1988C2212yes—
75#1981C2212yes—
76#1956C2212yes—
77#1954C2212yes—
78#1948C2212yes—
79#1947C2212yes—
80#1939C2212yes—
81#1935C2214yes—
82#1924C2214yes—
83#1919C2214yes—
84#1873C2214yes—
85#1865C2214yes—
86#1854C2214yes—
87#1847C2212yes—
88#1825C2214yes—
89#1817C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C4×#106C2212yes—
93#1767C2211yescharacteristic
94#1744C2211yescharacteristic
95#533—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125C2311yescharacteristic
103C2×#125C2314yes—
104C2×#124—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—22no—
114C2×C4×Q8C2312yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117C2314yes—
119C22×C42C2312yes—
120#485—48no—
121#479—48no—
122C2×#107C2312yes—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#399—22no—
141#399—24no—
142#398—24no—
143#397—22no—
144#397—22no—
145#397—22no—
146#396—24no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—21no—
152#391—21no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#390—22no—
157#390—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—24no—
163#389C2311yescharacteristic
164#389C2311yescharacteristic
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#388—22no—
169#388—22no—
170#388—24no—
171#387—21no—
172#387—21no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#385—22no—
181#385—22no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—21no—
191#382—22no—
192#382C2311yescharacteristic
193#381—22no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C43C2311yescharacteristic
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#336C2311yescharacteristic
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×D4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C4⋊Q8—82no—
248#125—28no—
249#125—22no—
250#125—41no—
251#125—44no—
252#124—44no—
253C22⋊Q8—24no—
254C22⋊Q8—42no—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257C22⋊Q8—44no—
258C22⋊Q8—84no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×D4—24no—
265C4×D4—42no—
266C4×D4—42no—
267#117—28no—
268#117—44no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—44no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—24no—
330#107—42no—
331#107—44no—
332#106—24no—
333#106—24no—
334#106—44no—
335C4⋊C8—24no—
336C4⋊C8—44no—
337#103—28no—
338#103—44no—
339#103—44no—
340#103—44no—
341C22⋊C8—44no—
342C22⋊C8—44no—
343#97—42no—
344C4×C8—24no—
345#95—21no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24C22×D412yes—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2511yescharacteristic
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4H(5,𝔽2)14yes—
461C4⋊C42−1+412yes—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—82no—
470C22⋊C4—84no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—42no—
481C42—48no—
482C42—44no—
483C42—42no—
484C42—44no—
485C42—42no—
486C42—82no—
487C422−1+414yes—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C23×D411yescharacteristic
496Q8—48no—
497Q8—44no—
498Q8—44no—
499Q8—88no—
500Q8—84no—
501D4—44no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4#53612yes—
555C2×C4#53612yes—
556C2×C4#57512yes—
557C2×C4#57512yes—
558C2×C4#57712yes—
559C2×C4#57812yes—
560C2×C4C2×H(5,𝔽2)12yes—
561C2×C4C2×2−1+411yescharacteristic
562C2×C4C2×2−1+412yes—
563C2×C4#58511yescharacteristic
564C8—42no—
565C8—44no—
566C8—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22C2×#53611yescharacteristic
575C22#348411yescharacteristic
576C22#348911yescharacteristic
577C22C2×#57511yescharacteristic
578C22C2×#57511yescharacteristic
579C22C2×#57711yescharacteristic
580C22C2×#57811yescharacteristic
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—44no—
586C4—42no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—81no—
592C4—81no—
593C4—84no—
594C4—82no—
595C4—81no—
596C4—82no—
597C4—82no—
598C4#324014yes—
599C4#355012yes—
600C4#355412yes—
601C2—82no—
602C2#4248111yescharacteristic
603C2C2×#324011yescharacteristic
604C2#6087811yescharacteristic
605C2#6159511yescharacteristic
606C2#6159611yescharacteristic
607C2#6213311yescharacteristic
608C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32744C2semidirect
4. #31759C2semidirect
5. #31109C2semidirect
6. #30205C2semidirect
7. #30168C2semidirect
8. #29645C2semidirect
9. #29355C2semidirect
10. #29213C2semidirect
11. #26461C2semidirect
12. #26443C2semidirect
15. #24306C2semidirect
16. #23838C2semidirect

Maximal subgroups

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

subgroupindex
#364542
#361342
#327442
#317592
#311092
#302052
#301682
#296452
#293552
#292132
#264612
#264432
#263032
#262332
#243062
#238382
#220122