Finite Group Databasebeta

← #7545351

Subgroups — #7545351

Order 512.

1533 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59828C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#32082C214yesmaximal
4#28748C211yesmaximal, characteristic
5#28748C211yesmaximal, characteristic
6#28639C212yesmaximal
7#27749C212yesmaximal
8#26756C214yesmaximal
9#26528C212yesmaximal
10#26320C212yesmaximal
11#25647C212yesmaximal
12#25276C212yesmaximal
13#23214C212yesmaximal
14#23149C211yesmaximal, characteristic
15#22542C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#557C2212yes—
20C2×#487C2212yes—
21C2×#487C2212yes—
22C2×#475C2211yescharacteristic
23C2×#475C2211yescharacteristic
24C2×C8×Q8C2212yes—
25#2868C2211yescharacteristic
26#2859—21no—
27#2726—22no—
28#2582—21no—
29#2568C2212yes—
30#2565—22no—
31#2552C2212yes—
32#2546C2212yes—
33#2546C2212yes—
34#2546C2212yes—
35#2506—22no—
36#2506—22no—
37#2506C2214yes—
38#2453—22no—
39#2446—22no—
40#2365C2212yes—
41#2365C2212yes—
42#2365C2212yes—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2096C2214yes—
50#2072C2214yes—
51#2066C2214yes—
52#2066C2214yes—
53#2043C2214yes—
54#2043C2214yes—
55#2043C2214yes—
56#2043C2214yes—
57#1988C2211yescharacteristic
58#1988C2211yescharacteristic
59#1984C2212yes—
60#1981C2212yes—
61#1981C2214yes—
62#1981C2212yes—
63#1981C2212yes—
64#1981C2212yes—
65#1956C2212yes—
66#1956C2214yes—
67#1956C2212yes—
68#1948C2211yescharacteristic
69#1948C2211yescharacteristic
70#1939C2212yes—
71#1933C2214yes—
72#1933C2214yes—
73#1927C2214yes—
74#1927C2214yes—
75#1919C2212yes—
76#1919C2212yes—
77#1916C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1865C2212yes—
81#1865C2212yes—
82#1803C2211yescharacteristic
83#1803C2214yes—
84#1803C2211yescharacteristic
85C4×#106C2212yes—
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2214yes—
92C4×#103C2212yes—
93#557—28no—
94#557—44no—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—22no—
105C2×#125—22no—
106C2×#125—24no—
107C2×#125—24no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118#487—28no—
119#487—44no—
120#475—24no—
121#475—24no—
122C8×Q8—44no—
123C2×#106C2312yes—
124C2×#106C2312yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#400—24no—
149#398—24no—
150#398—24no—
151#398—24no—
152#398—24no—
153#398—24no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—21no—
158#391—21no—
159#391D412yes—
160#391C2314yes—
161#391C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—21no—
172#389—21no—
173#389D412yes—
174#389D412yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#389C2314yes—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384D411yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2314yes—
191#382—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C43—21no—
208C43C2311yescharacteristic
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8C2×D412yes—
224C22×Q8C2414yes—
225C22×Q8C2411yescharacteristic
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C4⋊Q8—28no—
231C4⋊Q8—22no—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—24no—
236C4⋊Q8—22no—
237C4⋊Q8—24no—
238C4⋊Q8—42no—
239C4⋊Q8—42no—
240C4⋊Q8—44no—
241C4⋊Q8—48no—
242C4⋊Q8SD1618yes—
243C4⋊Q8#4114yes—
244#125—48no—
245#125—416no—
246#125—416no—
247#125—48no—
248C4×Q8—24no—
249C4×Q8—22no—
250C4×Q8—24no—
251C4×Q8—28no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—41no—
255C4×Q8—41no—
256C4×Q8—44no—
257C4×Q8—48no—
258C4×Q8—48no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8#4114yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2414yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318#106—24no—
319#106—24no—
320#106—42no—
321#106—42no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—44no—
327#103—24no—
328#103—24no—
329#103—28no—
330#103—28no—
331#103—42no—
332#103—42no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339C4×C8—24no—
340C4×C8—24no—
341C4×C8—44no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—21no—
367C2×Q8—24no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8C2×SD1618yes—
378C2×Q8C2×#4112yes—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D414yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D414yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2511yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C4⋊C4—24no—
410C4⋊C4—28no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—84no—
449C4⋊C4C2×SD1618yes—
450C4⋊C4C2×SD1618yes—
451C4⋊C4C2×#4114yes—
452C4⋊C4C2×#4112yes—
453C42—24no—
454C42—22no—
455C42—22no—
456C42—24no—
457C42—24no—
458C42—21no—
459C42—21no—
460C42—41no—
461C42—48no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—48no—
466C42—44no—
467C42—44no—
468C42—42no—
469C42—42no—
470C42C2×SD1614yes—
471C42C2×#4112yes—
472C42C2×#4112yes—
473C42H(5,𝔽2)12yes—
474C422−1+412yes—
475C23C23×D411yescharacteristic
476Q8—216no—
477Q8—22no—
478Q8—22no—
479Q8—48no—
480Q8—41no—
481Q8—41no—
482Q8—44no—
483Q8—44no—
484Q8—44no—
485Q8—44no—
486Q8—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4#54612yes—
529C2×C4#54612yes—
530C2×C4#54812yes—
531C2×C4#54812yes—
532C2×C4C22×SD1618yes—
533C2×C4C22×SD1612yes—
534C2×C4C22×SD1612yes—
535C2×C4C22×SD1612yes—
536C2×C4#57812yes—
537C2×C4#57812yes—
538C2×C4C22×#4111yescharacteristic
539C2×C4C22×#4111yescharacteristic
540C2×C4C2×H(5,𝔽2)11yescharacteristic
541C2×C4C2×2−1+411yescharacteristic
542C2×C4#58511yescharacteristic
543C2×C4#58511yescharacteristic
544C8—42no—
545C8—42no—
546C8—42no—
547C8—42no—
548C22C2×#54611yescharacteristic
549C22C2×#54811yescharacteristic
550C22#348711yescharacteristic
551C22C23×SD1611yescharacteristic
552C22C2×#57811yescharacteristic
553C22C2×#57811yescharacteristic
554C22#358611yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—28no—
559C4—44no—
560C4—42no—
561C4—44no—
562C4—42no—
563C4—41no—
564C4—81no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—84no—
569C4—81no—
570C4#329814yes—
571C4#335914yes—
572C4#348011yescharacteristic
573C4#350711yescharacteristic
574C4#351711yescharacteristic
575C4#355411yescharacteristic
576C2#5907811yescharacteristic
577C2C2×#329811yescharacteristic
578C2#6121011yescharacteristic
579C2C2×#335911yescharacteristic
580C2#6134211yescharacteristic
581C2#6188311yescharacteristic
582C2#6193611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊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
#598282
#365602
#320822
#287482
#286392
#277492
#267562
#265282
#263202
#256472
#252762
#232142
#231492
#225422
C4×#4752