Finite Group Databasebeta

← #7597144

Subgroups — #7597144

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic
2#59832C211yesmaximal, characteristic
3#31881C214yesmaximal
4#29287C211yesmaximal, characteristic
5#28954C214yesmaximal
6#28797C212yesmaximal
7#28284C211yesmaximal, characteristic
8#26527C212yesmaximal
9#26440C212yesmaximal
10#25841C212yesmaximal
11#25480C212yesmaximal
12#25063C212yesmaximal
13#23222C212yesmaximal
14#22525C212yesmaximal
15C4×#500C211yesmaximal, characteristic
16#21712C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#500C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#477C2212yes—
22C2×#446C2212yes—
23#2805—22no—
24#2734—22no—
25#2582C2212yes—
26#2581—22no—
27#2571C2212yes—
28#2568—21no—
29#2567C2212yes—
30#2551—21no—
31#2551C2212yes—
32#2546—22no—
33#2511—22no—
34#2509C2214yes—
35#2451—22no—
36#2451C2214yes—
37#2365—21no—
38#2365C2212yes—
39#2365C2212yes—
40#2355C2212yes—
41Q8×C4⋊C4—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47#2097C2214yes—
48#2088C2214yes—
49#2086C2214yes—
50#2085C2214yes—
51#2068C2214yes—
52#2066C2214yes—
53#2057C2214yes—
54#2057C2214yes—
55#1993C2214yes—
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1988C2212yes—
60#1988C2214yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1944C2214yes—
69#1942C2212yes—
70#1942C2212yes—
71#1939C2212yes—
72#1939C2212yes—
73#1935C2214yes—
74#1933C2212yes—
75#1933C2212yes—
76#1928C2214yes—
77#1918C2214yes—
78#1863C2214yes—
79#1863C2214yes—
80#1845C2211yescharacteristic
81#1842C2211yescharacteristic
82#1834C2211yescharacteristic
83#1805C2214yes—
84#1803C2212yes—
85C4×#107C2212yes—
86C4×#107C2212yes—
87#1770C2212yes—
88#1770C2212yes—
89#1763C2214yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93Q82—28no—
94Q82—44no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112#500—24no—
113#487Q818yescomplemented
114#477—44no—
115#446—44no—
116C2×#107C2312yes—
117C2×#107C2312yes—
118C2×#107C2312yes—
119C2×#107C2314yes—
120C2×#107C2312yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2312yes—
126C2×#103C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#97C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#400—24no—
140#400—24no—
141#400—24no—
142#398—24no—
143#398—24no—
144#395—24no—
145#395—24no—
146#395—24no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—24no—
155#391D412yes—
156#391C2312yes—
157#391C2314yes—
158#389—22no—
159#389—22no—
160#389—24no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389D412yes—
169#389D412yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#387—24no—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#384—22no—
182#384D411yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—22no—
188#382C2314yes—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C43C2311yescharacteristic
203#340C2312yes—
204#340C2312yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#336C2314yes—
216#336C2314yes—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×C8C2414yes—
224C22×C8C2411yescharacteristic
225C22×C8C2411yescharacteristic
226C22×C8C2411yescharacteristic
227C22×C8C2411yescharacteristic
228C4⋊Q8—28no—
229C4⋊Q8—24no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—44no—
236C4⋊Q8—48no—
237C4⋊Q8—416no—
238C4⋊Q8—84no—
239C4⋊Q8C2×Q814yes—
240C4⋊Q8C2×Q814yes—
241C4⋊Q8#4114yes—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—44no—
248C4×Q8—48no—
249C4×Q8—44no—
250C4×Q8—82no—
251C4×Q8—82no—
252C4×Q8—84no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2414yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#107—24no—
303#107—24no—
304#107—24no—
305#107—42no—
306#107—44no—
307#106—44no—
308C4⋊C8—24no—
309C4⋊C8—24no—
310C4⋊C8—42no—
311C4⋊C8—44no—
312#103—24no—
313#103—44no—
314#103—42no—
315#103—44no—
316#103C2×Q818yes—
317#103C2×Q818yes—
318#97—44no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—41no—
324C4×C8C2×Q814yes—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95C2×D414yes—
342#95C2×D414yes—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—48no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—81no—
358C2×Q8—81no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×Q8—84no—
362C2×Q8—84no—
363C2×Q8C22×Q814yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4C22×D414yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D414yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2511yescharacteristic
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—24no—
389C2×C8—21no—
390C2×C8—24no—
391C2×C8—21no—
392C2×C8—24no—
393C2×C8C22×Q812yes—
394C2×C8C22×Q812yes—
395C2×C82−1+412yes—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—84no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4C22×Q814yes—
437C4⋊C4C2×#4114yes—
438C42—22no—
439C42—22no—
440C42—24no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—48no—
446C42—44no—
447C42—44no—
448C42—44no—
449C42—44no—
450C42—84no—
451C42#13714yes—
452C42C22×Q812yes—
453C42C2×#4112yes—
454C42H(5,𝔽2)12yes—
455C422−1+412yes—
456C23C23×D411yescharacteristic
457Q8—24no—
458Q8—24no—
459Q8—48no—
460Q8—48no—
461Q8—42no—
462Q8—42no—
463Q8—84no—
464Q8—84no—
465Q8—168no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4#54612yes—
506C2×C4#54612yes—
507C2×C4D4×Q812yes—
508C2×C4D4×Q812yes—
509C2×C4C2×#13712yes—
510C2×C4C2×#13712yes—
511C2×C4C2×#13712yes—
512C2×C4#57612yes—
513C2×C4#57812yes—
514C2×C4C23×Q811yescharacteristic
515C2×C4C22×#4111yescharacteristic
516C2×C4C2×H(5,𝔽2)11yescharacteristic
517C2×C4C2×2−1+411yescharacteristic
518C2×C4C2×2−1+411yescharacteristic
519C2×C4#58511yescharacteristic
520C8—22no—
521C8—22no—
522C8—22no—
523C8—41no—
524C8—44no—
525C22C2×#54611yescharacteristic
526C22C2×D4×Q811yescharacteristic
527C22#349711yescharacteristic
528C22C22×#13711yescharacteristic
529C22C2×#57611yescharacteristic
530C22C2×#57811yescharacteristic
531C22#358411yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—44no—
536C4—44no—
537C4—44no—
538C4—41no—
539C4—44no—
540C4—44no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4#332814yes—
547C4#338514yes—
548C4Q8×#4111yescharacteristic
549C4#350311yescharacteristic
550C4#352311yescharacteristic
551C4#355411yescharacteristic
552C2#5941311yescharacteristic
553C2C2×#332811yescharacteristic
554C2#6121111yescharacteristic
555C2C2×#338511yescharacteristic
556C2#6137211yescharacteristic
557C2#6236611yescharacteristic
558C2#6237511yescharacteristic
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

Complements

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

NH ≅ G/Ntype
113. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#598702
#598322
#318812
#292872
#289542
#287972
#282842
#265272
#264402
#258412
#254802
#250632
#232222
#225252
C4×#5002
#217122
C4×#4872