Finite Group Databasebeta

← #6854547

Subgroups — #6854547

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36573C211yesmaximal, characteristic, complemented
2#36562C211yesmaximal, characteristic, complemented
3#33296C211yesmaximal, characteristic, complemented
4#31530C214yesmaximal, complemented
5#31526C214yesmaximal, complemented
6#30312C212yesmaximal, complemented
7#29197C212yesmaximal, complemented
8#29149C212yesmaximal, complemented
9#29146C211yesmaximal, characteristic, complemented
10#28521C211yesmaximal, characteristic, complemented
11#24833C212yesmaximal, complemented
12#24829C212yesmaximal, complemented
13#24340C212yesmaximal, complemented
14#24339C212yesmaximal, complemented
15#22272C212yesmaximal
16#21701C211yesmaximal, characteristic
17#21346C211yesmaximal, characteristic, complemented
18C2×#557C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×#496C2211yescharacteristic
21C2×#476C2212yes—
22C2×#474C2212yes—
23C2×#442C2212yes—
24C2×#440C2212yes—
25#2862C2211yescharacteristic
26#2859C2212yes—
27#2859C2211yescharacteristic
28#2859C2212yes—
29#2580—22no—
30#2580C2212yes—
31#2567C2211yescharacteristic
32#2566C2212yes—
33#2565C2212yes—
34#2556—22no—
35#2556—22no—
36#2542C2211yescharacteristic
37#2454—22no—
38#2454—22no—
39#2454C2214yes—
40#2429—22no—
41#2360—22no—
42#2349—22no—
43#2349C2212yes—
44C4×#127C2212yes—
45C4×#127C2211yescharacteristic
46C4×#124C2212yes—
47C4×#124C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2092C2214yes—
50#2084C2214yes—
51#2083C2214yes—
52#2068C2214yes—
53#2065C2214yes—
54#2059C2214yes—
55#2059C2214yes—
56#2026C2212yes—
57#2025C2214yes—
58#2025C2214yes—
59#2025C2212yes—
60#1993C2212yes—
61#1993C2212yes—
62#1991C2214yes—
63#1985C2212yes—
64#1970C2212yes—
65#1968C2212yes—
66#1961C2212yes—
67#1961C2211yescharacteristic
68#1959C2212yes—
69#1959C2211yescharacteristic
70#1940C2212yes—
71#1916C2214yes—
72#1880C2214yes—
73#1879C2214yes—
74#1879C2214yes—
75#1877C2214yes—
76#1865C2214yes—
77#1862C2214yes—
78#1847C2212yes—
79#1847C2211yescharacteristic
80#1836C2212yes—
81#1836C2212yes—
82#1836C2212yes—
83#1805C2212yes—
84#1804C2212yes—
85#1802C2214yes—
86#1802C2214yes—
87#1767C2212yes—
88#1767C2211yescharacteristic
89#1745C2212yes—
90C4×#97C2211yescharacteristic
91C22×SD16C2312yes—
92C22×D8C2312yes—
93#557—44no—
94#554—44no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#125—22no—
100C2×#125—22no—
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×#125C2312yes—
104C2×#124C2311yescharacteristic
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8D412yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2314yes—
119C2×C4×D4C2312yes—
120C22×C4⋊C4—21no—
121#496Q818yescomplemented
122#476—44no—
123#474—44no—
124#442—44no—
125#440—44no—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#97C2311yescharacteristic
141C2×#97C2312yes—
142C2×#97C2312yes—
143C2×#97C2311yescharacteristic
144C2×C4×C8C2312yes—
145#399—24no—
146#399—22no—
147#399—21no—
148#399—21no—
149#398—22no—
150#398—22no—
151#396—22no—
152#396—22no—
153#396—24no—
154#395—21no—
155#395—21no—
156#393—22no—
157#391—22no—
158#391—22no—
159#391C2311yescharacteristic
160#391C2312yes—
161#390—22no—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2312yes—
166#390C2311yescharacteristic
167#390C2312yes—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#385—24no—
185#385—24no—
186#385—24no—
187#385—22no—
188#385—22no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382—22no—
194#382C2312yes—
195#382C2312yes—
196#382C2311yescharacteristic
197#382C2312yes—
198#380—24no—
199#380—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C2×#95—22no—
209C43C2312yes—
210C43C2311yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2314yes—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8C2×D411yescharacteristic
224C22×Q8C2411yescharacteristic
225C22×D4—22no—
226C22×D4C2411yescharacteristic
227C22×D4C2412yes—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—24no—
232C2×SD16—42no—
233C2×SD16—42no—
234C2×SD16C2×Q818yes—
235C2×D8—42no—
236C2×D8—42no—
237C2×D8C2×Q818yes—
238C22×C8C2414yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—44no—
244#127—22no—
245#127C2×Q814yes—
246#125—44no—
247#125—44no—
248#125—44no—
249#125—48no—
250#125—84no—
251#124—22no—
252#124C2×Q814yes—
253C22⋊Q8—48no—
254C22⋊Q8—48no—
255C22⋊Q8—44no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—44no—
265C4×D4—48no—
266C4×D4—44no—
267C4×D4—44no—
268C4×D4—84no—
269C4×D4—84no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—42no—
325#107—44no—
326#106—42no—
327#106—44no—
328C4⋊C8—24no—
329C4⋊C8—44no—
330#103—44no—
331#103—42no—
332#102—44no—
333#102—42no—
334#102—42no—
335#102—42no—
336#102—44no—
337#102—44no—
338#97—24no—
339#97—22no—
340#97—24no—
341#97C2×Q814yes—
342C4×C8—42no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95C2×D411yescharacteristic
356#95C2×D412yes—
357#95C2×D411yescharacteristic
358#95C2×D412yes—
359C24—21no—
360C24—22no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—82no—
370C2×Q8C22×Q812yes—
371C2×D4—28no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4C22×Q812yes—
383C2×D4C22×Q814yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C2511yescharacteristic
410SD16—48no—
411D8—48no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8C22×Q814yes—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C22⋊C4—28no—
453C22⋊C4—48no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—82no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—24no—
464C42—24no—
465C42—22no—
466C42—24no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—44no—
473C42—42no—
474C42C22×Q812yes—
475C42H(5,𝔽2)14yes—
476C422−1+412yes—
477C23—22no—
478C23—24no—
479C23—44no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—44no—
487C23C23×D411yescharacteristic
488Q8—24no—
489Q8—44no—
490Q8—42no—
491Q8—84no—
492Q8—162no—
493D4—24no—
494D4—24no—
495D4—24no—
496D4—416no—
497D4—42no—
498D4—42no—
499D4—42no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4—168no—
538C2×C4—164no—
539C2×C4—164no—
540C2×C4#53512yes—
541C2×C4#53512yes—
542C2×C4D4×Q814yes—
543C2×C4#57514yes—
544C2×C4C23×Q811yescharacteristic
545C2×C4C2×H(5,𝔽2)12yes—
546C2×C4C2×2−1+411yescharacteristic
547C2×C4#58512yes—
548C8—24no—
549C8—42no—
550C8—44no—
551C22—48no—
552C22—44no—
553C22—42no—
554C22—44no—
555C22—84no—
556C22—82no—
557C22—81no—
558C22—82no—
559C22—81no—
560C22—82no—
561C22C2×#53511yescharacteristic
562C22C2×D4×Q811yescharacteristic
563C22#350011yescharacteristic
564C22C2×#57512yes—
565C22#358512yes—
566C4—24no—
567C4—22no—
568C4—22no—
569C4—24no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—84no—
576C4—81no—
577C4—82no—
578C4—81no—
579C4—82no—
580C4—82no—
581C4#350112yes—
582C4#354712yes—
583C2—84no—
584C2—82no—
585C2#3961811yescharacteristic
586C2#6080211yescharacteristic
587C2#6082511yescharacteristic
588C2#6136712yes—
589C2#6257812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#496C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36573C2semidirect
2. #36562C2semidirect
3. #33296C2semidirect
4. #31530C2semidirect
5. #31526C2semidirect
6. #30312C2semidirect
7. #29197C2semidirect
8. #29149C2semidirect
9. #29146C2semidirect
10. #28521C2semidirect
11. #24833C2semidirect
12. #24829C2semidirect
13. #24340C2semidirect
14. #24339C2semidirect
17. #21346C2semidirect
121. #496Q8semidirect

Maximal subgroups

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

subgroupindex
#365732
#365622
#332962
#315302
#315262
#303122
#291972
#291492
#291462
#285212
#248332
#248292
#243402
#243392
#222722
#217012
#213462