Finite Group Databasebeta

← #7542163

Subgroups — #7542163

Order 512.

1337 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#59871C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32348C214yesmaximal
4#29208C211yesmaximal, characteristic
5#29208C211yesmaximal, characteristic
6#28827C212yesmaximal
7#26862C214yesmaximal
8#26512C212yesmaximal
9#26282C212yesmaximal
10#26092C212yesmaximal
11#25471C212yesmaximal
12#25062C212yesmaximal
13#23149C211yesmaximal, characteristic
14#23063C212yesmaximal
15Q8×#107C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#477C211yesmaximal, characteristic
18C2×#477C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×C8×Q8C2212yes—
21#2868C2211yescharacteristic
22#2859—21no—
23#2806—22no—
24#2584—22no—
25#2582—22no—
26#2582C2212yes—
27#2567C2212yes—
28#2567C2212yes—
29#2552—21no—
30#2552C2212yes—
31#2552C2212yes—
32#2550—22no—
33#2507—22no—
34#2507C2214yes—
35#2451—22no—
36#2369C2212yes—
37#2369C2212yes—
38#2365—21no—
39#2355C2212yes—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C42×Q8C2211yescharacteristic
47#2100C2214yes—
48#2095C2214yes—
49#2087C2214yes—
50#2083C2214yes—
51#2083C2214yes—
52#2068C2214yes—
53#2057C2214yes—
54#2057C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1987C2211yescharacteristic
58#1987C2211yescharacteristic
59#1984C2212yes—
60#1947C2212yes—
61#1947C2212yes—
62#1947C2212yes—
63#1947C2212yes—
64#1947C2214yes—
65#1947C2211yescharacteristic
66#1947C2211yescharacteristic
67#1947C2214yes—
68#1942C2214yes—
69#1939C2212yes—
70#1933C2214yes—
71#1928C2214yes—
72#1921C2214yes—
73#1921C2212yes—
74#1921C2212yes—
75#1921C2214yes—
76#1921C2214yes—
77#1916C2214yes—
78#1863C2212yes—
79#1863C2212yes—
80#1845C2212yes—
81#1845C2212yes—
82#1845C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#107C2214yes—
86C4×#107C2212yes—
87C4×#107C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—22no—
100C2×#125—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109#477—24no—
110#477Q818yescomplemented
111C8×Q8—28no—
112C2×#107C2312yes—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2312yes—
118C2×#107C2312yes—
119C2×#107C2312yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2314yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×C4×C8C2314yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132#398—24no—
133#398—24no—
134#398—24no—
135#398—24no—
136#398—24no—
137#395—24no—
138#395—24no—
139#391—22no—
140#391—22no—
141#391—21no—
142#391—21no—
143#391—24no—
144#391—24no—
145#391D412yes—
146#391C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—24no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—24no—
158#389—21no—
159#389—21no—
160#389—24no—
161#389D412yes—
162#389D412yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#387—24no—
169#387—24no—
170#384—22no—
171#384—21no—
172#384—21no—
173#384D411yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2314yes—
177#384C2312yes—
178#384C2314yes—
179#384C2312yes—
180#382—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2311yescharacteristic
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C43—21no—
199C43C2311yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8C2×D412yes—
220C22×Q8C2411yescharacteristic
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230C4⋊Q8—82no—
231C4⋊Q8—82no—
232C4⋊Q8Q1618yes—
233C4⋊Q8C2×Q814yes—
234#125—84no—
235#125—84no—
236C4×Q8—24no—
237C4×Q8—22no—
238C4×Q8—24no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—41no—
242C4×Q8—41no—
243C4×Q8—82no—
244C4×Q8—82no—
245C4×Q8—84no—
246C4×Q8—84no—
247C4×Q8C2×Q814yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2411yescharacteristic
278C2×C4⋊C4C2411yescharacteristic
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2411yescharacteristic
284C2×C4⋊C4C2412yes—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2414yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—24no—
304#107—44no—
305#107—44no—
306#107—42no—
307#107—42no—
308#107—44no—
309#107—44no—
310#107C2×Q818yes—
311C4⋊C8—28no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8C2×Q814yes—
315C4⋊C8#4114yes—
316#103—24no—
317#103—42no—
318#103—42no—
319#103C2×Q818yes—
320C4×C8—28no—
321C4×C8—24no—
322C4×C8—24no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—24no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95C2×D414yes—
338#95C2×D414yes—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—81no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×Q8C2×Q1618yes—
356C2×Q8C22×Q812yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4C22×D414yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D414yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2511yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8C22×Q814yes—
386C2×C8C2×#4114yes—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—84no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4C2×Q1618yes—
429C4⋊C4C2×Q1618yes—
430C4⋊C4C22×Q814yes—
431C4⋊C4C22×Q812yes—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—24no—
436C42—24no—
437C42—21no—
438C42—21no—
439C42—41no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42—44no—
446C42—42no—
447C42—84no—
448C42—84no—
449C42C2×Q1614yes—
450C42C22×Q812yes—
451C42C2×#4112yes—
452C422−1+412yes—
453C422−1+412yes—
454C23C23×D411yescharacteristic
455Q8—216no—
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—84no—
461Q8—84no—
462Q8—164no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4#54712yes—
505C2×C4#54712yes—
506C2×C4D4×Q812yes—
507C2×C4D4×Q812yes—
508C2×C4C22×Q1612yes—
509C2×C4C22×Q1618yes—
510C2×C4C22×Q1612yes—
511C2×C4C22×Q1612yes—
512C2×C4#57612yes—
513C2×C4#57612yes—
514C2×C4C23×Q811yescharacteristic
515C2×C4C22×#4111yescharacteristic
516C2×C4C2×2−1+411yescharacteristic
517C2×C4C2×2−1+411yescharacteristic
518C2×C4#58511yescharacteristic
519C2×C4#58511yescharacteristic
520C8—24no—
521C8—24no—
522C8—42no—
523C8—42no—
524C22C2×#54711yescharacteristic
525C22C2×D4×Q811yescharacteristic
526C22#348711yescharacteristic
527C22C23×Q1611yescharacteristic
528C22C2×#57611yescharacteristic
529C22C2×#57611yescharacteristic
530C22#358611yescharacteristic
531C4—28no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—44no—
536C4—42no—
537C4—42no—
538C4—44no—
539C4—41no—
540C4—81no—
541C4—82no—
542C4—81no—
543C4—82no—
544C4—82no—
545C4—84no—
546C4#333814yes—
547C4Q8×Q1614yes—
548C4Q8×#4111yescharacteristic
549C4#350211yescharacteristic
550C4#351511yescharacteristic
551C4#355511yescharacteristic
552C2#5906711yescharacteristic
553C2C2×#333811yescharacteristic
554C2#6127811yescharacteristic
555C2C2×Q8×Q1611yescharacteristic
556C2#6138111yescharacteristic
557C2#6188411yescharacteristic
558C2#6193211yescharacteristic
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
110. #477Q8semidirect

Maximal subgroups

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

subgroupindex
#598712
#365632
#323482
#292082
#288272
#268622
#265122
#262822
#260922
#254712
#250622
#231492
#230632
Q8×#1072
C4×#4772