Finite Group Databasebeta

← #7542161

Subgroups — #7542161

Order 512.

1417 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59828C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32346C214yesmaximal
4#29206C211yesmaximal, characteristic
5#29206C211yesmaximal, characteristic
6#28829C212yesmaximal
7#26860C214yesmaximal
8#26510C212yesmaximal
9#26278C212yesmaximal
10#26088C212yesmaximal
11#25472C212yesmaximal
12#25060C212yesmaximal
13#23149C211yesmaximal, characteristic
14#23062C212yesmaximal
15Q8×#106C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#475C2211yescharacteristic
20C2×#475C2211yescharacteristic
21C2×C8×Q8C2212yes—
22#2868C2211yescharacteristic
23#2859—21no—
24#2726—22no—
25#2582—21no—
26#2582C2212yes—
27#2567C2212yes—
28#2552C2212yes—
29#2552C2212yes—
30#2550—22no—
31#2546C2212yes—
32#2507—22no—
33#2506C2214yes—
34#2453—22no—
35#2451—22no—
36#2446—22no—
37#2369C2212yes—
38#2365C2212yes—
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#2096C2214yes—
49#2088C2214yes—
50#2084C2214yes—
51#2084C2214yes—
52#2066C2214yes—
53#2058C2214yes—
54#2058C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1988C2211yescharacteristic
58#1988C2211yescharacteristic
59#1984C2212yes—
60#1948C2212yes—
61#1948C2212yes—
62#1948C2212yes—
63#1948C2212yes—
64#1948C2214yes—
65#1948C2211yescharacteristic
66#1948C2211yescharacteristic
67#1948C2214yes—
68#1941C2214yes—
69#1939C2212yes—
70#1933C2214yes—
71#1927C2214yes—
72#1920C2214yes—
73#1919C2214yes—
74#1919C2212yes—
75#1919C2212yes—
76#1919C2214yes—
77#1916C2214yes—
78#1865C2212yes—
79#1865C2212yes—
80#1844C2212yes—
81#1844C2212yes—
82#1844C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#106C2214yes—
86C4×#106C2212yes—
87C4×#106C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93#557—48no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—22no—
102C2×#125—24no—
103C2×#125—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113#475—24no—
114#475Q818yescomplemented
115C8×Q8—28no—
116C2×#106C2312yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2312yes—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×C4×C8C2314yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2312yes—
136#400—24no—
137#398—24no—
138#398—24no—
139#398—24no—
140#398—24no—
141#398—24no—
142#395—24no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—24no—
149#391D412yes—
150#391C2314yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389—24no—
161#389—21no—
162#389—21no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#389C2314yes—
170#387—24no—
171#387—24no—
172#387—24no—
173#384—21no—
174#384—21no—
175#384D411yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2314yes—
179#384C2312yes—
180#384C2314yes—
181#384C2312yes—
182#382—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C43—21no—
200C43C2311yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216#336C2314yes—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8C2×D412yes—
221C22×Q8C2411yescharacteristic
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231C4⋊Q8—48no—
232C4⋊Q8—82no—
233C4⋊Q8—82no—
234C4⋊Q8SD1618yes—
235C4⋊Q8C2×Q814yes—
236#125—416no—
237#125—48no—
238#125—84no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—41no—
245C4×Q8—41no—
246C4×Q8—48no—
247C4×Q8—48no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×Q8—84no—
251C4×Q8C2×Q814yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2412yes—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—24no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2414yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#106—24no—
308#106—44no—
309#106—44no—
310#106—42no—
311#106—42no—
312#106—44no—
313#106—44no—
314#106C2×Q818yes—
315C4⋊C8—28no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8C2×Q814yes—
319C4⋊C8#4114yes—
320#103—24no—
321#103—42no—
322#103—42no—
323#103C2×Q818yes—
324C4×C8—28no—
325C4×C8—24no—
326C4×C8—24no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—22no—
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—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—44no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×Q8C2×SD1618yes—
360C2×Q8C22×Q812yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D414yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D414yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2511yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8C22×Q814yes—
390C2×C8C2×#4114yes—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4C2×SD1618yes—
433C4⋊C4C2×SD1618yes—
434C4⋊C4C22×Q814yes—
435C4⋊C4C22×Q812yes—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—24no—
440C42—24no—
441C42—21no—
442C42—21no—
443C42—41no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—48no—
448C42—44no—
449C42—44no—
450C42—42no—
451C42—42no—
452C42—84no—
453C42C2×SD1614yes—
454C42C22×Q812yes—
455C42C2×#4112yes—
456C422−1+412yes—
457C422−1+412yes—
458C23C23×D411yescharacteristic
459Q8—216no—
460Q8—22no—
461Q8—22no—
462Q8—48no—
463Q8—41no—
464Q8—41no—
465Q8—84no—
466Q8—164no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4#54712yes—
509C2×C4#54712yes—
510C2×C4D4×Q812yes—
511C2×C4D4×Q812yes—
512C2×C4C22×SD1618yes—
513C2×C4C22×SD1612yes—
514C2×C4C22×SD1612yes—
515C2×C4C22×SD1612yes—
516C2×C4#57712yes—
517C2×C4#57712yes—
518C2×C4C23×Q811yescharacteristic
519C2×C4C22×#4111yescharacteristic
520C2×C4C2×2−1+411yescharacteristic
521C2×C4C2×2−1+411yescharacteristic
522C2×C4#58511yescharacteristic
523C2×C4#58511yescharacteristic
524C8—24no—
525C8—24no—
526C8—42no—
527C8—42no—
528C22C2×#54711yescharacteristic
529C22C2×D4×Q811yescharacteristic
530C22#348711yescharacteristic
531C22C23×SD1611yescharacteristic
532C22C2×#57711yescharacteristic
533C22C2×#57711yescharacteristic
534C22#358611yescharacteristic
535C4—28no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—44no—
540C4—44no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—82no—
545C4—81no—
546C4—82no—
547C4—82no—
548C4—81no—
549C4—84no—
550C4#333514yes—
551C4Q8×SD1614yes—
552C4Q8×#4111yescharacteristic
553C4#350211yescharacteristic
554C4#351511yescharacteristic
555C4#355511yescharacteristic
556C2#5906711yescharacteristic
557C2C2×#333511yescharacteristic
558C2#6128111yescharacteristic
559C2C2×Q8×SD1611yescharacteristic
560C2#6138311yescharacteristic
561C2#6188311yescharacteristic
562C2#6193311yescharacteristic
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
114. #475Q8semidirect

Maximal subgroups

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

subgroupindex
#598282
#365632
#323462
#292062
#288292
#268602
#265102
#262782
#260882
#254722
#250602
#231492
#230622
Q8×#1062
C4×#4752