Finite Group Databasebeta

← #7538118

Subgroups — #7538118

Order 512.

1347 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#32656C214yesmaximal
4#28830C212yesmaximal
5#28689C211yesmaximal, characteristic
6#28532C212yesmaximal
7#28512C211yesmaximal, characteristic
8#26531C212yesmaximal
9#26089C212yesmaximal
10#25456C212yesmaximal
11#25066C214yesmaximal
12#24818C212yesmaximal
13#23247C211yesmaximal, characteristic
14#23096C212yesmaximal
15#22964C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#500C2212yes—
19C2×#475C2212yes—
20C2×#475C2211yescharacteristic
21C2×#475C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2862—21no—
26#2805—22no—
27#2582C2212yes—
28#2581C2214yes—
29#2571C2212yes—
30#2571C2212yes—
31#2570—22no—
32#2568C2212yes—
33#2568C2212yes—
34#2567—21no—
35#2511—22no—
36#2511—22no—
37#2509—22no—
38#2446—22no—
39#2365C2212yes—
40#2365C2212yes—
41#2355—21no—
42#2355C2212yes—
43#2355C2212yes—
44Q8×C4⋊C4C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2097C2214yes—
51#2097C2214yes—
52#2096C2214yes—
53#2085C2214yes—
54#2085C2214yes—
55#2083C2214yes—
56#2068C2214yes—
57#2066C2214yes—
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1984C2214yes—
61#1981C2211yescharacteristic
62#1969C2214yes—
63#1956C2212yes—
64#1949C2214yes—
65#1948C2211yescharacteristic
66#1947C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1941C2214yes—
70#1939C2212yes—
71#1939C2212yes—
72#1928C2214yes—
73#1928C2212yes—
74#1920C2214yes—
75#1919C2212yes—
76#1878C2214yes—
77#1878C2212yes—
78#1865C2212yes—
79#1865C2214yes—
80#1846C2214yes—
81#1844C2212yes—
82#1805C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#106C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2214yes—
92C4×#103C2212yes—
93C22×Q16C2312yes—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2312yes—
108#500—44no—
109#475—28no—
110#475—24no—
111#475—44no—
112#462Q818yescomplemented
113C4×Q16—44no—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#107C2312yes—
117C2×#106C2314yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120C2×#106C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2314yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#400—24no—
138#400—24no—
139#400—24no—
140#400—24no—
141#400—24no—
142#398—24no—
143#395—24no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391D412yes—
153#391C2312yes—
154#391C2314yes—
155#391C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—24no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#387—24no—
170#387—24no—
171#387—24no—
172#387—24no—
173#387—24no—
174#387—24no—
175#384—22no—
176#384—21no—
177#384—21no—
178#384D411yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#382—22no—
184#382—22no—
185#382—22no—
186#382C2314yes—
187#382C2314yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C43—21no—
206C43C2311yescharacteristic
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#336C2312yes—
216#336C2314yes—
217#336C2312yes—
218#336C2314yes—
219C22×Q8—22no—
220C22×Q8—21no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2411yescharacteristic
224C2×Q16—42no—
225C2×Q16—42no—
226C2×Q16C2×Q818yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—22no—
234C4⋊Q8—24no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—44no—
238C4⋊Q8—82no—
239C4⋊Q8C2×Q814yes—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—24no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8—41no—
248C4×Q8—41no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×Q8—84no—
253C4×Q8C2×Q814yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2411yescharacteristic
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2414yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#107—42no—
306#107—44no—
307#107—44no—
308#106—28no—
309#106—24no—
310#106—42no—
311#106—44no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—44no—
315C4⋊C8C2×Q814yes—
316C4⋊C8#4114yes—
317#103—28no—
318#103—24no—
319#103—42no—
320#103—42no—
321#103—42no—
322#103—42no—
323#103—44no—
324#103C2×Q818yes—
325C4×C8—24no—
326C4×C8—24no—
327C4×C8—44no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95C2×D414yes—
347#95C2×D414yes—
348C2×Q8—24no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—42no—
367C2×Q8C22×Q814yes—
368C2×Q8C22×Q812yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2511yescharacteristic
387Q16—48no—
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×C8—22no—
397C2×C8—22no—
398C2×C8C22×Q814yes—
399C2×C8C2×#4114yes—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4C22×Q812yes—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—24no—
444C42—24no—
445C42—21no—
446C42—21no—
447C42—41no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—44no—
452C42—44no—
453C42—42no—
454C42—42no—
455C42—84no—
456C42—84no—
457C42#13514yes—
458C42C22×Q812yes—
459C42C2×#4112yes—
460C42H(5,𝔽2)12yes—
461C422−1+412yes—
462C23C23×D411yescharacteristic
463Q8—22no—
464Q8—24no—
465Q8—22no—
466Q8—24no—
467Q8—41no—
468Q8—42no—
469Q8—41no—
470Q8—42no—
471Q8—84no—
472Q8—84no—
473Q8—88no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4D4×Q812yes—
517C2×C4D4×Q812yes—
518C2×C4#55012yes—
519C2×C4#55012yes—
520C2×C4C2×#13512yes—
521C2×C4C2×#13512yes—
522C2×C4C2×#13512yes—
523C2×C4#57712yes—
524C2×C4#57812yes—
525C2×C4C23×Q811yescharacteristic
526C2×C4C22×#4111yescharacteristic
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C2×C4C2×2−1+411yescharacteristic
529C2×C4#58511yescharacteristic
530C2×C4#58511yescharacteristic
531C8—24no—
532C8—24no—
533C8—42no—
534C8—42no—
535C22C2×D4×Q811yescharacteristic
536C22C2×#55011yescharacteristic
537C22#348511yescharacteristic
538C22C22×#13511yescharacteristic
539C22C2×#57711yescharacteristic
540C22C2×#57811yescharacteristic
541C22#358611yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—44no—
546C4—44no—
547C4—41no—
548C4—42no—
549C4—44no—
550C4—42no—
551C4—82no—
552C4—81no—
553C4—82no—
554C4—84no—
555C4—81no—
556C4—82no—
557C4#338314yes—
558C4#339414yes—
559C4Q8×#4111yescharacteristic
560C4#350111yescharacteristic
561C4#351511yescharacteristic
562C4#354711yescharacteristic
563C2#5905211yescharacteristic
564C2C2×#338311yescharacteristic
565C2#6138311yescharacteristic
566C2C2×#339411yescharacteristic
567C2#6142211yescharacteristic
568C2#6170111yescharacteristic
569C2#6178911yescharacteristic
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
112. #462Q8semidirect

Maximal subgroups

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

subgroupindex
#598702
#365982
#326562
#288302
#286892
#285322
#285122
#265312
#260892
#254562
#250662
#248182
#232472
#230962
#229642
C4×#4752
C4×#4622