Finite Group Databasebeta

← #7533933

Subgroups — #7533933

Order 512.

1435 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59832C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#31545C214yesmaximal
4#29206C211yesmaximal, characteristic
5#29141C211yesmaximal, characteristic
6#28797C212yesmaximal
7#28274C212yesmaximal
8#27017C214yesmaximal
9#26527C212yesmaximal
10#26022C212yesmaximal
11#25061C212yesmaximal
12#24333C212yesmaximal
13#23153C211yesmaximal, characteristic
14#23102C212yesmaximal
15#22525C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#477C2211yescharacteristic
20C2×#462C2211yescharacteristic
21C2×#446C2212yes—
22C2×C4×Q16C2212yes—
23#2868C2211yescharacteristic
24#2862—21no—
25#2734—22no—
26#2582C2212yes—
27#2582C2212yes—
28#2571C2212yes—
29#2570—22no—
30#2568C2212yes—
31#2551—21no—
32#2551C2212yes—
33#2546—22no—
34#2511—22no—
35#2451—22no—
36#2451C2214yes—
37#2446—22no—
38#2365—21no—
39#2365C2212yes—
40#2365C2212yes—
41#2355C2212yes—
42#2355C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48#2097C2214yes—
49#2085C2214yes—
50#2066C2214yes—
51#2066C2214yes—
52#2058C2214yes—
53#2057C2214yes—
54#2024C2214yes—
55#2024C2214yes—
56#1993C2214yes—
57#1990C2212yes—
58#1988C2214yes—
59#1987C2211yescharacteristic
60#1984C2212yes—
61#1981C2211yescharacteristic
62#1971C2214yes—
63#1956C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1947C2211yescharacteristic
67#1947C2211yescharacteristic
68#1944C2214yes—
69#1939C2212yes—
70#1928C2212yes—
71#1927C2214yes—
72#1921C2212yes—
73#1919C2214yes—
74#1917C2214yes—
75#1878C2214yes—
76#1878C2212yes—
77#1863C2214yes—
78#1863C2212yes—
79#1844C2212yes—
80#1834C2212yes—
81#1805C2214yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2211yescharacteristic
86C4×#107C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C22×Q16C2312yes—
94Q82—28no—
95Q82—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8D411yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115#477—24no—
116#462Q818yescomplemented
117#446—44no—
118C4×Q16—44no—
119C2×#107C2312yes—
120C2×#107C2312yes—
121C2×#107C2312yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#400—24no—
141#400—24no—
142#398—24no—
143#398—24no—
144#398—24no—
145#395—24no—
146#395—24no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—21no—
154#391—24no—
155#391D412yes—
156#391C2312yes—
157#391C2314yes—
158#391C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389D412yes—
168#389D412yes—
169#389C2312yes—
170#389C2314yes—
171#389C2314yes—
172#387—24no—
173#387—24no—
174#387—24no—
175#387—24no—
176#387—24no—
177#384—21no—
178#384—21no—
179#384D411yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#382—22no—
185#382—22no—
186#382C2314yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C43—21no—
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#336C2314yes—
216#336C2312yes—
217#336C2314yes—
218#336C2312yes—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—24no—
222C22×Q8C2×D412yes—
223C22×Q8C2412yes—
224C22×Q8C2411yescharacteristic
225C2×Q16—42no—
226C2×Q16—42no—
227C2×Q16C2×Q818yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C4⋊Q8—28no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238C4⋊Q8—44no—
239C4⋊Q8—48no—
240C4⋊Q8—416no—
241C4⋊Q8—44no—
242C4⋊Q8—84no—
243C4⋊Q8—82no—
244C4⋊Q8C2×Q814yes—
245C4⋊Q8#4114yes—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—22no—
249C4×Q8—28no—
250C4×Q8—24no—
251C4×Q8—22no—
252C4×Q8—24no—
253C4×Q8—22no—
254C4×Q8—41no—
255C4×Q8—41no—
256C4×Q8—44no—
257C4×Q8—48no—
258C4×Q8—82no—
259C4×Q8—84no—
260C4×Q8C2×Q814yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2412yes—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—24no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2414yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—24no—
314#107—42no—
315#107—42no—
316#106—44no—
317#106—44no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—44no—
322C4⋊C8C2×Q814yes—
323#103—24no—
324#103—42no—
325#103—42no—
326#103—44no—
327#103—44no—
328#103—42no—
329#103—42no—
330#103C2×Q818yes—
331#97—44no—
332C4×C8—24no—
333C4×C8—24no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—24no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95C2×D414yes—
351#95C2×D414yes—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—81no—
371C2×Q8—81no—
372C2×Q8—84no—
373C2×Q8—84no—
374C2×Q8C22×Q814yes—
375C2×Q8C22×Q812yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4C22×D414yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D414yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2511yescharacteristic
394Q16—48no—
395C2×C8—22no—
396C2×C8—22no—
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×C8C22×Q814yes—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—84no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—84no—
446C4⋊C4C22×Q812yes—
447C4⋊C4C2×#4114yes—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—24no—
452C42—24no—
453C42—21no—
454C42—21no—
455C42—41no—
456C42—48no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—44no—
461C42—42no—
462C42—44no—
463C42—42no—
464C42—84no—
465C42#13714yes—
466C42C22×Q812yes—
467C42C2×#4112yes—
468C42H(5,𝔽2)12yes—
469C422−1+412yes—
470C23C23×D411yescharacteristic
471Q8—22no—
472Q8—24no—
473Q8—22no—
474Q8—24no—
475Q8—48no—
476Q8—41no—
477Q8—42no—
478Q8—41no—
479Q8—42no—
480Q8—84no—
481Q8—88no—
482Q8—168no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4#54612yes—
526C2×C4#54612yes—
527C2×C4D4×Q812yes—
528C2×C4D4×Q812yes—
529C2×C4C2×#13712yes—
530C2×C4C2×#13712yes—
531C2×C4C2×#13712yes—
532C2×C4#57712yes—
533C2×C4#57812yes—
534C2×C4C23×Q811yescharacteristic
535C2×C4C22×#4111yescharacteristic
536C2×C4C2×H(5,𝔽2)11yescharacteristic
537C2×C4C2×2−1+411yescharacteristic
538C2×C4#58511yescharacteristic
539C2×C4#58511yescharacteristic
540C8—24no—
541C8—42no—
542C8—42no—
543C8—42no—
544C22C2×#54611yescharacteristic
545C22C2×D4×Q811yescharacteristic
546C22#348711yescharacteristic
547C22C22×#13711yescharacteristic
548C22C2×#57711yescharacteristic
549C22C2×#57811yescharacteristic
550C22#358511yescharacteristic
551C4—22no—
552C4—22no—
553C4—22no—
554C4—44no—
555C4—44no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—44no—
560C4—41no—
561C4—81no—
562C4—82no—
563C4—84no—
564C4—81no—
565C4—82no—
566C4#332814yes—
567C4#338714yes—
568C4Q8×#4111yescharacteristic
569C4#350111yescharacteristic
570C4#351811yescharacteristic
571C4#354711yescharacteristic
572C2#5903611yescharacteristic
573C2C2×#332811yescharacteristic
574C2#6120011yescharacteristic
575C2C2×#338711yescharacteristic
576C2#6138411yescharacteristic
577C2#6189211yescharacteristic
578C2#6193411yescharacteristic
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
116. #462Q8semidirect

Maximal subgroups

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

subgroupindex
#598322
#365982
#315452
#292062
#291412
#287972
#282742
#270172
#265272
#260222
#250612
#243332
#231532
#231022
#225252
C4×#4772
C4×#4622