Finite Group Databasebeta

← #6805696

Subgroups — #6805696

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic, complemented
2Q8×#124C211yesmaximal, characteristic, complemented
3#33298C211yesmaximal, characteristic, complemented
4#32441C214yesmaximal, complemented
5#32437C214yesmaximal, complemented
6#30213C212yesmaximal, complemented
7#29211C212yesmaximal, complemented
8#28727C211yesmaximal, characteristic, complemented
9#28621C212yesmaximal
10#28521C211yesmaximal, characteristic, complemented
11#24697C212yesmaximal, complemented
12#24693C212yesmaximal, complemented
13#24341C212yesmaximal, complemented
14#24338C212yesmaximal, complemented
15#22224C212yesmaximal, complemented
16#21701C211yesmaximal, characteristic
17#21346C211yesmaximal, characteristic, complemented
18C2×Q82C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#496C2211yescharacteristic
21C2×#489C2212yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#442C2212yes—
25C2×#440C2212yes—
26#2862C2211yescharacteristic
27#2862C2212yes—
28#2862C2211yescharacteristic
29#2858C2212yes—
30#2592—22no—
31#2580C2212yes—
32#2568C2211yescharacteristic
33#2568C2212yes—
34#2563C2211yescharacteristic
35#2563C2212yes—
36#2513C2214yes—
37#2492—22no—
38#2492—22no—
39#2492—22no—
40#2448—22no—
41#2447—22no—
42#2367—22no—
43#2357C2212yes—
44Q8×C22⋊C4—22no—
45C4×C4⋊Q8C2212yes—
46C4×#127C2211yescharacteristic
47C4×#124C2211yescharacteristic
48C4×#124C2212yes—
49C42×Q8C2211yescharacteristic
50#2092C2214yes—
51#2074C2212yes—
52#2073C2214yes—
53#2073C2214yes—
54#2073C2212yes—
55#2058C2214yes—
56#2057C2214yes—
57#2055C2214yes—
58#2054C2214yes—
59#2023C2214yes—
60#2023C2214yes—
61#1993C2212yes—
62#1993C2212yes—
63#1993C2212yes—
64#1993C2212yes—
65#1993C2212yes—
66#1985C2212yes—
67#1970C2212yes—
68#1968C2212yes—
69#1961C2211yescharacteristic
70#1961C2212yes—
71#1959C2211yescharacteristic
72#1949C2214yes—
73#1949C2214yes—
74#1940C2212yes—
75#1927C2214yes—
76#1925C2214yes—
77#1916C2214yes—
78#1880C2214yes—
79#1879C2214yes—
80#1864C2214yes—
81#1863C2214yes—
82#1847C2211yescharacteristic
83#1847C2212yes—
84#1835C2214yes—
85#1805C2212yes—
86#1804C2212yes—
87#1767C2212yes—
88#1767C2211yescharacteristic
89#1745C2212yes—
90C4×#97C2211yescharacteristic
91C23×Q8—21no—
92C22×SD16C2312yes—
93C22×D8C2312yes—
94Q82—44no—
95D4×Q8—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#124—22no—
105C2×#124—22no—
106C2×#124C2311yescharacteristic
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D412yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119#496—24no—
120#489—28no—
121#476—44no—
122#474—44no—
123#442—44no—
124#440—44no—
125C2×#107C2312yes—
126C2×#107C2314yes—
127C2×#106C2312yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#97C2312yes—
140C2×#97C2311yescharacteristic
141C2×#97C2312yes—
142C2×#97C2311yescharacteristic
143C2×C4×C8C2312yes—
144#400—21no—
145#400—21no—
146#400—22no—
147#400—22no—
148#399—22no—
149#397—24no—
150#397—24no—
151#393—22no—
152#393—21no—
153#393—21no—
154#393—22no—
155#393—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391C2312yes—
169#391C2311yescharacteristic
170#390—22no—
171#390—22no—
172#390—22no—
173#390C2312yes—
174#390C2312yes—
175#390C2311yescharacteristic
176#390C2314yes—
177#390C2314yes—
178#389—22no—
179#389C2314yes—
180#388—24no—
181#388—24no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—24no—
186#385—24no—
187#384C2311yescharacteristic
188#382C2312yes—
189#382C2312yes—
190#382C2312yes—
191#382C2312yes—
192#382C2312yes—
193#382C2311yescharacteristic
194#382C2312yes—
195#381—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C43C2311yescharacteristic
211C43C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2314yes—
222C22×Q8—22no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—21no—
227C22×Q8—44no—
228C22×Q8—48no—
229C22×Q8C2×D411yescharacteristic
230C22×Q8C2411yescharacteristic
231C22×D4—22no—
232C22×D4C2412yes—
233C22×D4C2411yescharacteristic
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—22no—
237C2×SD16—24no—
238C2×SD16—24no—
239C2×SD16—42no—
240C2×D8—24no—
241C2×D8—24no—
242C2×D8—42no—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—48no—
253C4⋊Q8—44no—
254#127—22no—
255#127—22no—
256#124—22no—
257#124—48no—
258#124—84no—
259#124#4114yes—
260C22⋊Q8—44no—
261C22⋊Q8—48no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—84no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—44no—
272C4×D4—84no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—24no—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2412yes—
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#107—42no—
320#107—44no—
321#106—42no—
322#106—44no—
323C4⋊C8—28no—
324C4⋊C8—24no—
325#103—28no—
326#103—42no—
327#102—28no—
328#102—42no—
329#102—42no—
330#102—42no—
331#102—44no—
332#102—44no—
333#97—24no—
334#97—22no—
335#97—24no—
336#97—22no—
337C4×C8—24no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95C2×D411yescharacteristic
356#95C2×D411yescharacteristic
357#95C2×D412yes—
358#95C2×D412yes—
359C24—21no—
360C24—22no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8—816no—
380C2×Q8C2×#4112yes—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4C2×#4112yes—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—84no—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2511yescharacteristic
417SD16—48no—
418D8—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C82−1+414yes—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—84no—
450C22⋊C4—28no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—48no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—24no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—44no—
472C42—42no—
473C42—84no—
474C42C2×#4112yes—
475C42H(5,𝔽2)14yes—
476C422−1+412yes—
477C23—22no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23C23×D411yescharacteristic
488Q8—28no—
489Q8—24no—
490Q8—42no—
491Q8—88no—
492Q8—84no—
493Q8—88no—
494Q8—162no—
495Q8—164no—
496D4—24no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—88no—
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—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4—88no—
541C2×C4—164no—
542C2×C4#53512yes—
543C2×C4#53512yes—
544C2×C4#54814yes—
545C2×C4#57514yes—
546C2×C4C22×#4111yescharacteristic
547C2×C4C2×H(5,𝔽2)12yes—
548C2×C4C2×2−1+412yes—
549C2×C4C2×2−1+411yescharacteristic
550C8—24no—
551C8—44no—
552C8—42no—
553C22—44no—
554C22—42no—
555C22—82no—
556C22—81no—
557C22—81no—
558C22—84no—
559C22—82no—
560C22—84no—
561C22—82no—
562C22—82no—
563C22C2×#53511yescharacteristic
564C22C2×#54811yescharacteristic
565C22#349911yescharacteristic
566C22C2×#57512yes—
567C22#358512yes—
568C4—22no—
569C4—24no—
570C4—24no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—82no—
578C4—81no—
579C4—84no—
580C4—82no—
581C4—82no—
582C4—81no—
583C4#352012yes—
584C4#355012yes—
585C2—82no—
586C2—162no—
587C2#3916711yescharacteristic
588C2#6079911yescharacteristic
589C2#6082511yescharacteristic
590C2#6132412yes—
591C2#6250812yes—
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. #36609C2semidirect
2. Q8×#124C2semidirect
3. #33298C2semidirect
4. #32441C2semidirect
5. #32437C2semidirect
6. #30213C2semidirect
7. #29211C2semidirect
8. #28727C2semidirect
10. #28521C2semidirect
11. #24697C2semidirect
12. #24693C2semidirect
13. #24341C2semidirect
14. #24338C2semidirect
15. #22224C2semidirect
17. #21346C2semidirect

Maximal subgroups

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

subgroupindex
#366092
Q8×#1242
#332982
#324412
#324372
#302132
#292112
#287272
#286212
#285212
#246972
#246932
#243412
#243382
#222242
#217012
#213462