Finite Group Databasebeta

← #6797945

Subgroups — #6797945

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#31932C218yesmaximal, complemented
5#29843C212yesmaximal, complemented
6#28716C211yesmaximal, characteristic, complemented
7#28512C211yesmaximal, characteristic, complemented
8#27948C212yesmaximal, complemented
9#25598C212yesmaximal, complemented
10#24813C214yesmaximal, complemented
11#24120C214yesmaximal, complemented
12#21860C212yesmaximal, complemented
13C4×#492C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#547—21no—
16C2×#547—21no—
17C2×#547C2212yes—
18C2×#533C2212yes—
19C2×#492C2211yescharacteristic
20C2×#486C2212yes—
21C2×#475C2214yes—
22C2×C4×SD16C2214yes—
23C2×#431C2212yes—
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2856C2212yes—
28#2552—21no—
29#2542—21no—
30#2542—21no—
31#2542—21no—
32#2542C2212yes—
33#2538C2212yes—
34#2501—22no—
35#2501—22no—
36#2501C2214yes—
37#2441—22no—
38#2429—22no—
39#2423C2214yes—
40#2365—21no—
41#2354—21no—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C2×#384C2212yes—
46C42×D4C2211yescharacteristic
47#2096C2214yes—
48#2084C2218yes—
49#2071C2214yes—
50#2065C2218yes—
51#2034C2218yes—
52#2000C2218yes—
53#1990C2212yes—
54#1990C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1968C2214yes—
58#1957C2211yescharacteristic
59#1956C2212yes—
60#1956C2211yescharacteristic
61#1954C2212yes—
62#1939C2212yes—
63#1923C2214yes—
64#1880C2218yes—
65#1872C2218yes—
66#1853C2218yes—
67#1844C2211yescharacteristic
68#1844C2212yes—
69#1834C2214yes—
70#1803C2218yes—
71#1802C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×#41—22no—
76C22×SD16C2314yes—
77#547—48no—
78#547—48no—
79#547—48no—
80#533—28no—
81#533—44no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#127C2311yescharacteristic
90C2×#125—22no—
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—24no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8C2314yes—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121C2314yes—
103C2×C4×Q8—22no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2314yes—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4D411yescharacteristic
109C2×C4×D4D411yescharacteristic
110C2×C4×D4C2312yes—
111C2×C4×D4C2311yescharacteristic
112C2×#117—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2314yes—
116C22×C42C2312yes—
117#492—24no—
118#486—44no—
119#475—48no—
120C4×SD16—48no—
121#431—44no—
122C2×#106C2318yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2318yes—
127C2×#103C2314yes—
128C2×#102C2318yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2312yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2312yes—
137#399—24no—
138#399—24no—
139#399—24no—
140#399—24no—
141#398—24no—
142#396—24no—
143#391—22no—
144#391C2312yes—
145#390—22no—
146#390—21no—
147#390—21no—
148#390—22no—
149#390—22no—
150#390C2312yes—
151#390C2314yes—
152#389—22no—
153#389—21no—
154#389—21no—
155#389—22no—
156#389—24no—
157#389C2314yes—
158#388—24no—
159#387—24no—
160#384—22no—
161#384—21no—
162#384—21no—
163#384—21no—
164#384—21no—
165#384—22no—
166#384—22no—
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#384C2312yes—
170#384C2314yes—
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#380—22no—
175#380C2314yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4C2311yescharacteristic
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184C4×C22⋊C4C2312yes—
185C43C2311yescharacteristic
186#340C2318yes—
187#340C2318yes—
188#340C2318yes—
189#336C2312yes—
190#336C2314yes—
191C2×#41—48no—
192C2×#41—48no—
193C2×#41—48no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8C2412yes—
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4C2×D411yescharacteristic
201C22×D4C2412yes—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2412yes—
207C2×SD16—28no—
208C2×SD16—28no—
209C2×SD16—44no—
210C22×C8C2414yes—
211C22×C8C2412yes—
212C22×C8C2412yes—
213C4⋊Q8—24no—
214C4⋊Q8—22no—
215C4⋊Q8—24no—
216C4⋊Q8—28no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C4⋊Q8—44no—
220C4⋊Q8—82no—
221C4⋊Q8#4114yescomplemented
222#127—22no—
223#127—22no—
224#125—48no—
225#123—416no—
226#123—416no—
227#123—416no—
228C22⋊Q8—28no—
229C22⋊Q8—44no—
230C22⋊Q8—48no—
231C22⋊Q8—48no—
232C22⋊Q8—48no—
233C22⋊Q8—48no—
234C22⋊Q8—84no—
235#121—48no—
236#121—48no—
237#121—48no—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—44no—
241C4×Q8—48no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—28no—
245C4×D4—44no—
246C4×D4—44no—
247C4×D4SD1618yescomplemented
248#117—48no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D412yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2412yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#106—44no—
308#106—48no—
309C4⋊C8—24no—
310C4⋊C8—44no—
311#103—44no—
312#103—48no—
313#102—44no—
314#102—48no—
315C22⋊C8—28no—
316C22⋊C8—44no—
317C4×C8—24no—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—24no—
321C4×C8—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329C24—22no—
330C24C22×D412yes—
331#41—832no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8C2×#4114yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4C2×SD1614yes—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4C2×SD1618yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2511yescharacteristic
389SD16—416no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C82−1+414yes—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—84no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4C2×SD1614yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C22⋊C4—44no—
439C22⋊C4—84no—
440C22⋊C4C2×SD1618yes—
441C42—24no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—48no—
449C42—44no—
450C42C2×SD1614yes—
451C42AGL(1,ℤ/8)14yes—
452C42C2×#4112yes—
453C42H(5,𝔽2)14yes—
454C42H(5,𝔽2)12yes—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23C22×SD1614yes—
469C23C23×D411yescharacteristic
470Q8—28no—
471Q8—44no—
472Q8—88no—
473Q8—88no—
474D4—28no—
475D4—44no—
476D4—44no—
477D4—816no—
478D4—84no—
479D4—84no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—216no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—816no—
514C2×C4—88no—
515C2×C4#53512yes—
516C2×C4#53512yes—
517C2×C4#54612yes—
518C2×C4#54612yes—
519C2×C4C22×SD1612yes—
520C2×C4C22×SD1614yes—
521C2×C4C22×SD1612yes—
522C2×C4C22×SD1612yes—
523C2×C4C2×AGL(1,ℤ/8)12yes—
524C2×C4C2×AGL(1,ℤ/8)12yes—
525C2×C4C2×AGL(1,ℤ/8)12yes—
526C2×C4#57712yes—
527C2×C4#57812yes—
528C2×C4C22×#4111yescharacteristic
529C2×C4C2×H(5,𝔽2)12yes—
530C2×C4C2×H(5,𝔽2)11yescharacteristic
531C2×C4C2×2−1+412yes—
532C8—24no—
533C8—42no—
534C8—44no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C2×#53511yescharacteristic
550C22C2×#54611yescharacteristic
551C22#349911yescharacteristic
552C22C23×SD1611yescharacteristic
553C22C22×AGL(1,ℤ/8)11yescharacteristic
554C22C2×#57711yescharacteristic
555C22C2×#57811yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—24no—
560C4—42no—
561C4—44no—
562C4—42no—
563C4—82no—
564C4—84no—
565C4—82no—
566C4—84no—
567C4#320814yes—
568C4#329814yes—
569C4#333214yes—
570C4#352012yes—
571C4#354812yes—
572C2—44no—
573C2—162no—
574C2#3895111yescharacteristic
575C2C2×#320811yescharacteristic
576C2#6082411yescharacteristic
577C2C2×#329811yescharacteristic
578C2C2×#333211yescharacteristic
579C2#6246911yescharacteristic
580C2#6247911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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
2. #36134C2semidirect
3. #33289C2semidirect
4. #31932C2semidirect
5. #29843C2semidirect
6. #28716C2semidirect
7. #28512C2semidirect
8. #27948C2semidirect
9. #25598C2semidirect
10. #24813C2semidirect
11. #24120C2semidirect
12. #21860C2semidirect
13. C4×#492C2semidirect
14. #21264C2semidirect
221. C4⋊Q8#41semidirect
247. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361792
#361342
#332892
#319322
#298432
#287162
#285122
#279482
#255982
#248132
#241202
#218602
C4×#4922
#212642