Finite Group Databasebeta

← #6800419

Subgroups — #6800419

Order 512.

1329 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#32337C214yesmaximal, complemented
5#31481C214yesmaximal, complemented
6#29967C212yesmaximal
7#28799C212yesmaximal, complemented
8#27969C212yesmaximal
9#27961C211yesmaximal, characteristic
10#27738C212yesmaximal, complemented
11#27653C212yesmaximal
12#27648C211yesmaximal, characteristic
13#26511C212yesmaximal, complemented
14#26113C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#22085C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C2×#484C2212yes—
19C2×#483C2211yescharacteristic
20#2857C2212yes—
21#2857C2211yescharacteristic
22#2586—22no—
23#2571—22no—
24#2571—21no—
25#2571—21no—
26#2571C2212yes—
27#2571C2214yes—
28#2571C2212yes—
29#2567C2211yescharacteristic
30#2551C2211yescharacteristic
31#2549—22no—
32#2507—21no—
33#2507C2212yes—
34#2451C2212yes—
35#2440—21no—
36C2×#400—22no—
37#2369—21no—
38#2369C2211yescharacteristic
39Q8×C4⋊C4C2211yescharacteristic
40#2345—21no—
41C4×C22⋊Q8C2211yescharacteristic
42C4×C22⋊Q8C2212yes—
43C2×#389—21no—
44C2×#389C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2085C2214yes—
57#2083C2212yes—
58#2074C2214yes—
59#2072C2212yes—
60#2058C2214yes—
61#1993C2214yes—
62#1990C2212yes—
63#1987C2212yes—
64#1949C2214yes—
65#1948C2212yes—
66#1947C2212yes—
67#1943C2214yes—
68#1939C2212yes—
69#1935C2214yes—
70#1927C2212yes—
71#1920C2212yes—
72#1920C2214yes—
73#1918C2214yes—
74#1917C2214yes—
75#1872C2211yescharacteristic
76#1865C2214yes—
77#1853C2211yescharacteristic
78#1853C2212yes—
79#1836C2214yes—
80#1834C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1824C2212yes—
84#1823C2212yes—
85#1823C2211yescharacteristic
86#1805C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C2311yescharacteristic
101C2×C4×Q8C2311yescharacteristic
102C2×C4×Q8C2312yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114#484—28no—
115#483—24no—
116C2×#107C2314yes—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#103C2314yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×C22⋊C8C2311yescharacteristic
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2311yescharacteristic
128C2×#97C2314yes—
129C2×C4×C8C2312yes—
130#400—24no—
131#400—24no—
132#400—24no—
133#400—24no—
134#398—22no—
135#398—22no—
136#398—22no—
137#395—22no—
138#395—22no—
139#395—22no—
140#391—21no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391C2312yes—
145#391C2311yescharacteristic
146#389—21no—
147#389—21no—
148#389—22no—
149#389—24no—
150#389—22no—
151#389—21no—
152#389—21no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—21no—
169#387—22no—
170#385—22no—
171#385—21no—
172#384—21no—
173#384—21no—
174#384—21no—
175#384C2312yes—
176#384C2312yes—
177#384C2312yes—
178#384C2311yescharacteristic
179#382—22no—
180#382—22no—
181#382—24no—
182#382—22no—
183#382—22no—
184#382C2314yes—
185#382C2314yes—
186#380—21no—
187#380—22no—
188#380C2312yes—
189#380C2311yescharacteristic
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—22no—
208C2×#95—21no—
209C2×#95—22no—
210C2×#95—22no—
211C2×#95—22no—
212C2×#95D412yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8C2411yescharacteristic
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2411yescharacteristic
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C4⋊Q8—24no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C22⋊Q8—22no—
248C22⋊Q8—82no—
249C22⋊Q8—84no—
250C22⋊Q8#4114yes—
251C4×Q8—24no—
252C4×Q8—44no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—84no—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C42—24no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—28no—
321#106—24no—
322#106—42no—
323#106—44no—
324C4⋊C8—24no—
325C4⋊C8—44no—
326#103—24no—
327#103—28no—
328#103—42no—
329C22⋊C8—22no—
330C22⋊C8—24no—
331C22⋊C8—22no—
332#97—44no—
333C4×C8—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—21no—
343#95—21no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—22no—
351#95—21no—
352#95—22no—
353#95—21no—
354#95—22no—
355#95C2×D412yes—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359C24C22×D411yescharacteristic
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—81no—
366C2×Q8—82no—
367C2×Q8—82no—
368C2×Q8—81no—
369C2×Q8C2×#4112yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4#13714yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4112yes—
412C22×C42−1+412yes—
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—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C4⋊C4—82no—
457C4⋊C4—81no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4C2×#4112yes—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—24no—
471C42—24no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—44no—
476C42—44no—
477C42—42no—
478C42—82no—
479C42—82no—
480C422−1+414yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23#54812yes—
486C23C2×#13712yes—
487C23#57712yes—
488C23C23×D411yescharacteristic
489Q8—24no—
490Q8—42no—
491Q8—162no—
492Q8—164no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
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—21no—
518C2×C4—22no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—88no—
549C2×C4—84no—
550C2×C4#53612yes—
551C2×C4#53612yes—
552C2×C4#54812yes—
553C2×C4C2×#13712yes—
554C2×C4C2×#13712yes—
555C2×C4#57512yes—
556C2×C4C22×#4111yescharacteristic
557C2×C4C2×2−1+412yes—
558C2×C4C2×2−1+411yescharacteristic
559C2×C4#58512yes—
560C8—42no—
561C8—44no—
562C8—42no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22#336914yes—
570C22C2×#53611yescharacteristic
571C22C2×#54811yescharacteristic
572C22#348511yescharacteristic
573C22C22×#13711yescharacteristic
574C22C2×#57511yescharacteristic
575C22C2×#57711yescharacteristic
576C22#358511yescharacteristic
577C4—22no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—42no—
587C4—82no—
588C4—82no—
589C4—81no—
590C4—82no—
591C4—84no—
592C4—81no—
593C4—82no—
594C4#323614yes—
595C4#351512yes—
596C4#356312yes—
597C2—24no—
598C2#3912211yescharacteristic
599C2C2×#323611yescharacteristic
600C2#6087311yescharacteristic
601C2C2×#336911yescharacteristic
602C2#6132411yescharacteristic
603C2#6172111yescharacteristic
604C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32337C2semidirect
5. #31481C2semidirect
7. #28799C2semidirect
10. #27738C2semidirect
13. #26511C2semidirect
14. #26113C2semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#25712
#331482
#323372
#314812
#299672
#287992
#279692
#279612
#277382
#276532
#276482
#265112
#261132
#226292
#220852
C4×#4832