Finite Group Databasebeta

← #7569977

Subgroups — #7569977

Order 512.

1425 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#32148C214yesmaximal
4#29218C211yesmaximal, characteristic
5#28799C212yesmaximal
6#28687C211yesmaximal, characteristic
7#28684C212yesmaximal
8#27314C214yesmaximal
9#26442C212yesmaximal
10#26277C212yesmaximal
11#26113C212yesmaximal
12#25837C212yesmaximal
13#23153C211yesmaximal, characteristic
14#22971C212yesmaximal
15#22525C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#498C2212yes—
20C2×#477C2211yescharacteristic
21C2×#475C2211yescharacteristic
22C2×#446C2212yes—
23#2868—21no—
24#2868C2211yescharacteristic
25#2809—22no—
26#2571C2212yes—
27#2568—22no—
28#2567—22no—
29#2567C2212yes—
30#2551—22no—
31#2551—22no—
32#2551C2212yes—
33#2551C2212yes—
34#2546—21no—
35#2546C2212yes—
36#2508—22no—
37#2508C2214yes—
38#2369C2212yes—
39#2355—21no—
40#2355C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48#2101C2214yes—
49#2100C2214yes—
50#2088C2214yes—
51#2088C2214yes—
52#2085C2214yes—
53#2084C2214yes—
54#2058C2214yes—
55#2058C2214yes—
56#1993C2214yes—
57#1990C2212yes—
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1988C2214yes—
61#1987C2212yes—
62#1987C2212yes—
63#1987C2211yescharacteristic
64#1984C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1947C2211yescharacteristic
69#1943C2214yes—
70#1943C2214yes—
71#1939C2212yes—
72#1921C2212yes—
73#1919C2214yes—
74#1919C2212yes—
75#1918C2214yes—
76#1917C2214yes—
77#1865C2214yes—
78#1865C2212yes—
79#1865C2214yes—
80#1863C2212yes—
81#1834C2212yes—
82#1805C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#107C2212yes—
86C4×#106C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93Q82—28no—
94Q82—44no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8D411yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115#498—44no—
116#477Q818yescomplemented
117#475—24no—
118#446—44no—
119C2×#107C2312yes—
120C2×#107C2312yes—
121C2×#106C2312yes—
122C2×#106C2312yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#97C2314yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137#400—24no—
138#400—24no—
139#398—24no—
140#398—24no—
141#395—24no—
142#395—24no—
143#395—24no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—21no—
148#391—21no—
149#391—24no—
150#391D412yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—24no—
158#389—24no—
159#389D412yes—
160#389D412yes—
161#389C2312yes—
162#389C2312yes—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#387—24no—
168#387—24no—
169#384—22no—
170#384—22no—
171#384—22no—
172#384—21no—
173#384—21no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2312yes—
178#384C2312yes—
179#382—22no—
180#382—24no—
181#382C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C43—21no—
200C43C2311yescharacteristic
201#340C2312yes—
202#340C2312yes—
203#340C2314yes—
204#340C2314yes—
205#340C2312yes—
206#340C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8C2×D412yes—
224C22×Q8C2411yescharacteristic
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C4⋊Q8—28no—
230C4⋊Q8—24no—
231C4⋊Q8—24no—
232C4⋊Q8—22no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—44no—
237C4⋊Q8—48no—
238C4⋊Q8—416no—
239C4⋊Q8—82no—
240C4⋊Q8—84no—
241C4⋊Q8—82no—
242C4⋊Q8—84no—
243C4⋊Q8—84no—
244C4⋊Q8C2×Q814yes—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—24no—
248C4×Q8—22no—
249C4×Q8—24no—
250C4×Q8—22no—
251C4×Q8—28no—
252C4×Q8—44no—
253C4×Q8—48no—
254C4×Q8—44no—
255C4×Q8—82no—
256C4×Q8C2×Q814yes—
257C4×Q8#4114yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2411yescharacteristic
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—24no—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2414yes—
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312#107—24no—
313#107C2×Q818yes—
314#106—24no—
315#106—24no—
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—24no—
325#103—24no—
326#103C2×Q818yes—
327#97—44no—
328C4×C8—24no—
329C4×C8—24no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—22no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95C2×D414yes—
345#95C2×D414yes—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—48no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—81no—
359C2×Q8—81no—
360C2×Q8—82no—
361C2×Q8—82no—
362C2×Q8—84no—
363C2×Q8—84no—
364C2×Q8C22×Q812yes—
365C2×Q8C2×#4112yes—
366C2×Q82−1+412yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4C22×D414yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D414yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2511yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8C22×Q814yes—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—28no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4C22×Q814yes—
438C4⋊C4C22×Q812yes—
439C4⋊C4C2×#4112yes—
440C4⋊C42−1+412yes—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—22no—
445C42—22no—
446C42—24no—
447C42—24no—
448C42—21no—
449C42—21no—
450C42—41no—
451C42—48no—
452C42—42no—
453C42—44no—
454C42—42no—
455C42—44no—
456C42—42no—
457C42—84no—
458C42#13714yes—
459C42C22×Q812yes—
460C42C2×#4112yes—
461C42H(5,𝔽2)12yes—
462C422−1+412yes—
463C23C23×D411yescharacteristic
464Q8—22no—
465Q8—22no—
466Q8—22no—
467Q8—48no—
468Q8—84no—
469Q8—84no—
470Q8—164no—
471Q8—168no—
472Q8#55714yes—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4#54612yes—
516C2×C4#54612yes—
517C2×C4D4×Q812yes—
518C2×C4D4×Q812yes—
519C2×C4C2×#13712yes—
520C2×C4C2×#13712yes—
521C2×C4C2×#13712yes—
522C2×C4#57712yes—
523C2×C4#57712yes—
524C2×C4C23×Q811yescharacteristic
525C2×C4C22×#4111yescharacteristic
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4C2×2−1+411yescharacteristic
528C2×C4C2×2−1+411yescharacteristic
529C2×C4C2×2−1+411yescharacteristic
530C8—24no—
531C8—42no—
532C8—42no—
533C8—42no—
534C22C2×#54611yescharacteristic
535C22C2×D4×Q811yescharacteristic
536C22#348611yescharacteristic
537C22C22×#13711yescharacteristic
538C22C2×#57711yescharacteristic
539C22C2×#57711yescharacteristic
540C22#358511yescharacteristic
541C4—24no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—44no—
546C4—42no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—44no—
551C4—44no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4#332914yes—
557C4#338714yes—
558C4C2×#55711yescharacteristic
559C4#350311yescharacteristic
560C4#350911yescharacteristic
561C4#355211yescharacteristic
562C2#5914411yescharacteristic
563C2C2×#332911yescharacteristic
564C2#6120611yescharacteristic
565C2C2×#338711yescharacteristic
566C2#6138211yescharacteristic
567C2#6183011yescharacteristic
568C2#6187111yescharacteristic
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. #477Q8semidirect

Maximal subgroups

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

subgroupindex
#598692
Q8×C4⋊Q82
#321482
#292182
#287992
#286872
#286842
#273142
#264422
#262772
#261132
#258372
#231532
#229712
#225252
C4×#4772
C4×#4752