Finite Group Databasebeta

← #6797944

Subgroups — #6797944

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31934C218yesmaximal, complemented
5#29845C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28509C211yesmaximal, characteristic, complemented
8#27948C212yesmaximal, complemented
9#25597C212yesmaximal, complemented
10#24815C214yesmaximal, complemented
11#24118C214yesmaximal, complemented
12#21859C212yesmaximal, complemented
13C4×#492C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#554C2212yes—
16C2×#547—21no—
17C2×#532C2212yes—
18C2×#492C2211yescharacteristic
19C2×#487C2212yes—
20C2×#476C2214yes—
21C2×C4×SD16C2214yes—
22C2×#431C2212yes—
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2567C2212yes—
28#2552—21no—
29#2552—21no—
30#2547—21no—
31#2537—21no—
32#2537C2212yes—
33#2502C2214yes—
34#2440—22no—
35#2440—22no—
36#2432—22no—
37#2422—22no—
38#2422C2214yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C2×#384C2212yes—
46C42×D4C2211yescharacteristic
47#2094C2214yes—
48#2084C2218yes—
49#2072C2214yes—
50#2066C2218yes—
51#2035C2218yes—
52#2000C2218yes—
53#1990C2212yes—
54#1990C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1968C2214yes—
58#1957C2212yes—
59#1957C2211yescharacteristic
60#1956C2211yescharacteristic
61#1954C2212yes—
62#1939C2212yes—
63#1923C2214yes—
64#1877C2218yes—
65#1874C2218yes—
66#1853C2218yes—
67#1844C2211yescharacteristic
68#1844C2212yes—
69#1834C2214yes—
70#1803C2214yes—
71#1802C2218yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×#41—22no—
76C22×SD16C2314yes—
77#554—48no—
78#547—48no—
79#532—28no—
80#532—44no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#125—22no—
90C2×#124—22no—
91C2×#123—24no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2314yes—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121C2314yes—
98C2×C4×Q8C2312yes—
99C2×C4×D4—22no—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—22no—
103C2×C4×D4D411yescharacteristic
104C2×C4×D4D411yescharacteristic
105C2×C4×D4C2312yes—
106C2×C4×D4C2314yes—
107C2×C4×D4C2311yescharacteristic
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4C2314yes—
112C22×C42C2312yes—
113#492—24no—
114#487—44no—
115#476—48no—
116C4×SD16—48no—
117#431—44no—
118C2×#106C2318yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#103C2318yes—
123C2×#103C2314yes—
124C2×#102C2314yes—
125C2×#102C2318yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C4×C8C2312yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2312yes—
133#398—24no—
134#398—24no—
135#397—24no—
136#395—24no—
137#394—24no—
138#394—24no—
139#391—22no—
140#391C2312yes—
141#391C2314yes—
142#390—21no—
143#390—21no—
144#390C2312yes—
145#389—22no—
146#389—22no—
147#389—21no—
148#389—21no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#385—24no—
154#385—24no—
155#385—24no—
156#385—24no—
157#384—21no—
158#384—21no—
159#384—21no—
160#384—21no—
161#384—22no—
162#384—22no—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2314yes—
166#384C2312yes—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380—22no—
175#380C2314yes—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4C2311yescharacteristic
181C4×C22⋊C4D412yes—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4C2312yes—
184C43C2311yescharacteristic
185#340C2318yes—
186#340C2318yes—
187#340C2318yes—
188#336C2312yes—
189#336C2314yes—
190C2×#41—48no—
191C2×#41—48no—
192C2×#41—84no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8C2412yes—
196C22×D4—24no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4C2×D411yescharacteristic
200C22×D4C2412yes—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
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—22no—
218C4⋊Q8—44no—
219C4⋊Q8—82no—
220C4⋊Q8—82no—
221C4⋊Q8—82no—
222#127—22no—
223#127#4114yes—
224#125—48no—
225#124—48no—
226#123—416no—
227C22⋊Q8—416no—
228C22⋊Q8—48no—
229C22⋊Q8—48no—
230#121—28no—
231#121—44no—
232#121—48no—
233#121—48no—
234C4×Q8—28no—
235C4×D4—28no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—28no—
239C4×D4—44no—
240C4×D4—48no—
241C4×D4—48no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4SD1618yescomplemented
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—84no—
270C2×C4⋊C4—84no—
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2412yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#106—44no—
302#106—48no—
303C4⋊C8—24no—
304C4⋊C8—44no—
305#103—44no—
306#103—48no—
307#102—44no—
308#102—48no—
309C22⋊C8—28no—
310C22⋊C8—44no—
311C4×C8—24no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—24no—
315C4×C8—24no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325C24—22no—
326C24C22×D412yes—
327#41—832no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—84no—
352C2×D4C2×SD1614yes—
353C2×D4C2×#4114yes—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4C2×SD1618yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2511yescharacteristic
388SD16—416no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C82−1+414yes—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4C2×SD1614yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4C2×SD1618yes—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—44no—
446C42C2×SD1614yes—
447C42#13714yes—
448C42C2×#4112yes—
449C42H(5,𝔽2)14yes—
450C42H(5,𝔽2)12yes—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C22×SD1614yes—
465C23C23×D411yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—88no—
469Q8—164no—
470D4—28no—
471D4—28no—
472D4—48no—
473D4—44no—
474D4—48no—
475D4—816no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—216no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—816no—
510C2×C4—88no—
511C2×C4—168no—
512C2×C4#53512yes—
513C2×C4#53512yes—
514C2×C4#54612yes—
515C2×C4#54612yes—
516C2×C4C22×SD1612yes—
517C2×C4C22×SD1614yes—
518C2×C4C22×SD1612yes—
519C2×C4C22×SD1612yes—
520C2×C4C2×#13712yes—
521C2×C4C2×#13712yes—
522C2×C4C2×#13712yes—
523C2×C4#57612yes—
524C2×C4#57712yes—
525C2×C4C22×#4111yescharacteristic
526C2×C4C2×H(5,𝔽2)12yes—
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C2×C4C2×2−1+412yes—
529C8—24no—
530C8—42no—
531C8—44no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C2×#53511yescharacteristic
547C22C2×#54611yescharacteristic
548C22#349911yescharacteristic
549C22C23×SD1611yescharacteristic
550C22C22×#13711yescharacteristic
551C22C2×#57611yescharacteristic
552C22C2×#57711yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—24no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—84no—
563C4—84no—
564C4#320614yes—
565C4#329514yes—
566C4#333114yes—
567C4#352012yes—
568C4#354812yes—
569C2—44no—
570C2—84no—
571C2#3895111yescharacteristic
572C2C2×#320611yescharacteristic
573C2#6082211yescharacteristic
574C2C2×#329511yescharacteristic
575C2C2×#333111yescharacteristic
576C2#6246911yescharacteristic
577C2#6248011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
1. #36180C2semidirect
3. #33289C2semidirect
4. #31934C2semidirect
5. #29845C2semidirect
6. #28717C2semidirect
7. #28509C2semidirect
8. #27948C2semidirect
9. #25597C2semidirect
10. #24815C2semidirect
11. #24118C2semidirect
12. #21859C2semidirect
13. C4×#492C2semidirect
14. #21264C2semidirect
244. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361802
#361272
#332892
#319342
#298452
#287172
#285092
#279482
#255972
#248152
#241182
#218592
C4×#4922
#212642