Finite Group Databasebeta

← #6627614

Subgroups — #6627614

Order 512.

1473 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×Q82C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#33033C212yesmaximal
5#29355C212yesmaximal
6#29272C212yesmaximal
7#29219C211yesmaximal, characteristic
8#28284C211yesmaximal, characteristic
9#27569C214yesmaximal
10#27149C214yesmaximal
11#24994C212yesmaximal
12#24333C212yesmaximal
13#23069C212yesmaximal
14#22825C211yesmaximal, characteristic
15#22763C212yesmaximal
16#21615C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#477C2212yes—
20C2×#475C2212yes—
21C2×#462C2211yescharacteristic
22C2×C4×Q16C2212yes—
23#2868C2211yescharacteristic
24#2862C2212yes—
25#2582C2212yes—
26#2571—21no—
27#2570—22no—
28#2568—21no—
29#2511C2214yes—
30#2446—22no—
31#2369—22no—
32#2369—21no—
33#2369C2212yes—
34#2369C2211yescharacteristic
35#2365—21no—
36#2355—21no—
37#2355C2212yes—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2211yescharacteristic
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8—22no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C42×Q8—21no—
49C42×Q8C2212yes—
50C42×Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52#2099C2214yes—
53#2099C2214yes—
54#2072C2214yes—
55#2068C2214yes—
56#2057C2214yes—
57#2024C2214yes—
58#1990C2212yes—
59#1988C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1981C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1947C2211yescharacteristic
68#1947C2212yes—
69#1935C2214yes—
70#1935C2214yes—
71#1928C2212yes—
72#1927C2214yes—
73#1927C2214yes—
74#1920C2214yes—
75#1918C2214yes—
76#1918C2214yes—
77#1917C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1878C2212yes—
81#1863C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1831C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96Q82—24no—
97Q82—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120#477—28no—
121#475—28no—
122#462—24no—
123C4×Q16—28no—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#400—24no—
142#400—24no—
143#398—24no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—21no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2314yes—
155#391C2314yes—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#387—24no—
174#387—24no—
175#387—24no—
176#384—21no—
177#384—22no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382—22no—
187#382—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C43—22no—
218C43—21no—
219C43C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C2×Q16—24no—
241C2×Q16—24no—
242C2×Q16—24no—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C4⋊Q8—216no—
247C4⋊Q8—28no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258C4×Q8—28no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—42no—
272C4×Q8—41no—
273C4×Q8—44no—
274C4×Q8—84no—
275C4×Q8—82no—
276C4×Q8—84no—
277C4×Q8#4114yes—
278C4×Q8#4114yes—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—24no—
344#107—28no—
345#106—28no—
346C4⋊C8—22no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349C4⋊C8—22no—
350C4⋊C8—42no—
351#103—24no—
352#103—28no—
353#103—24no—
354#103—24no—
355#103—28no—
356#103—42no—
357#103—44no—
358#97—44no—
359C4×C8—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—81no—
397C2×Q8—82no—
398C2×Q8—81no—
399C2×Q8—82no—
400C2×Q8#13714yes—
401C2×Q8C2×#4112yes—
402C2×Q8C2×#4112yes—
403C2×Q82−1+412yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2511yescharacteristic
424Q16—48no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4C2×#4112yes—
477C4⋊C4C2×#4114yes—
478C4⋊C4C2×#4112yes—
479C42—28no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—21no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—41no—
494C42—42no—
495C42—42no—
496C42—44no—
497C42—44no—
498C42—44no—
499C42—42no—
500C42—42no—
501C42—44no—
502C42—81no—
503C42—82no—
504C42—84no—
505C42#13714yes—
506C42C2×#4112yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C422−1+412yes—
510C23C23×D411yescharacteristic
511Q8—28no—
512Q8—22no—
513Q8—22no—
514Q8—22no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—84no—
520Q8—88no—
521Q8—88no—
522Q8#55614yes—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4#52512yes—
571C2×C4#52512yes—
572C2×C4#54812yes—
573C2×C4#54812yes—
574C2×C4C2×#13714yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4C2×#13712yes—
579C2×C4#57612yes—
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C2×C4#58511yescharacteristic
586C8—42no—
587C8—44no—
588C8—42no—
589C22C2×#52511yescharacteristic
590C22C2×#54811yescharacteristic
591C22#348911yescharacteristic
592C22C22×#13711yescharacteristic
593C22C22×#13711yescharacteristic
594C22C2×#57611yescharacteristic
595C22#358611yescharacteristic
596C4—24no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—24no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—84no—
609C4—82no—
610C4—81no—
611C4—82no—
612C4—82no—
613C4#311314yes—
614C4#337514yes—
615C4C2×#55611yescharacteristic
616C4#348011yescharacteristic
617C4#350711yescharacteristic
618C4#352311yescharacteristic
619C2#3657911yescharacteristic
620C2C2×#311311yescharacteristic
621C2#6059611yescharacteristic
622C2C2×#337511yescharacteristic
623C2#6132211yescharacteristic
624C2#6211311yescharacteristic
625C2#6214911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#365982
C4×Q822
#331962
#330332
#293552
#292722
#292192
#282842
#275692
#271492
#249942
#243332
#230692
#228252
#227632
#216152
C4×#4622