Finite Group Databasebeta

← #7538117

Subgroups — #7538117

Order 512.

1383 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32650C214yesmaximal
4#28827C212yesmaximal
5#28689C211yesmaximal, characteristic
6#28531C212yesmaximal
7#28511C211yesmaximal, characteristic
8#26532C212yesmaximal
9#26092C212yesmaximal
10#25455C212yesmaximal
11#25062C214yesmaximal
12#24817C212yesmaximal
13#23247C211yesmaximal, characteristic
14#23096C212yesmaximal
15#22965C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#500C2212yes—
19C2×#477C2212yes—
20C2×#477C2211yescharacteristic
21C2×#477C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2859—21no—
26#2806—22no—
27#2584—22no—
28#2582C2212yes—
29#2582C2214yes—
30#2567C2212yes—
31#2567C2212yes—
32#2552—21no—
33#2552C2212yes—
34#2552C2212yes—
35#2550—22no—
36#2507—22no—
37#2507—22no—
38#2451—22no—
39#2369—21no—
40#2369C2212yes—
41#2365C2212yes—
42#2355C2212yes—
43Q8×C4⋊C4C2212yes—
44Q8×C4⋊C4C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2095C2214yes—
51#2095C2214yes—
52#2095C2214yes—
53#2083C2214yes—
54#2083C2214yes—
55#2083C2214yes—
56#2068C2214yes—
57#2068C2214yes—
58#1990C2212yes—
59#1987C2211yescharacteristic
60#1984C2214yes—
61#1981C2211yescharacteristic
62#1969C2214yes—
63#1956C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2211yescharacteristic
67#1947C2214yes—
68#1947C2211yescharacteristic
69#1942C2214yes—
70#1939C2212yes—
71#1939C2212yes—
72#1928C2214yes—
73#1928C2212yes—
74#1921C2212yes—
75#1921C2214yes—
76#1878C2214yes—
77#1878C2212yes—
78#1863C2212yes—
79#1863C2214yes—
80#1845C2212yes—
81#1845C2214yes—
82#1803C2211yescharacteristic
83#1803C2214yes—
84#1803C2211yescharacteristic
85C4×#107C2212yes—
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×#125—22no—
101C2×#125—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2314yes—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110#500—44no—
111#477—28no—
112#477—24no—
113#477—44no—
114#462Q818yescomplemented
115C4×Q16—44no—
116C2×#107C2314yes—
117C2×#107C2312yes—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#107C2312yes—
121C2×#107C2314yes—
122C2×#107C2312yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2314yes—
139#398—24no—
140#398—24no—
141#398—24no—
142#398—24no—
143#398—24no—
144#395—24no—
145#395—24no—
146#391—22no—
147#391—24no—
148#391—21no—
149#391—24no—
150#391D412yes—
151#391C2312yes—
152#391C2314yes—
153#391C2314yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—24no—
170#389D412yes—
171#389D412yes—
172#389C2312yes—
173#389C2314yes—
174#389C2314yes—
175#387—24no—
176#387—24no—
177#384—22no—
178#384—21no—
179#384—21no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2314yes—
184#384C2314yes—
185#384C2312yes—
186#384C2312yes—
187#382—22no—
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#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#336C2314yes—
216#336C2312yes—
217#336C2314yes—
218#336C2312yes—
219C22×Q8—21no—
220C22×Q8—22no—
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—22no—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—82no—
237C4⋊Q8—82no—
238C4⋊Q8Q1618yescomplemented
239C4⋊Q8C2×Q814yes—
240#125—84no—
241#125—84no—
242C4×Q8—22no—
243C4×Q8—24no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—24no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—41no—
250C4×Q8—41no—
251C4×Q8—44no—
252C4×Q8—44no—
253C4×Q8—82no—
254C4×Q8—84no—
255C4×Q8C2×Q814yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2414yes—
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—28no—
312#107—24no—
313#107—42no—
314#107—42no—
315#107—44no—
316#107—44no—
317#107—44no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—44no—
321C4⋊C8C2×Q814yes—
322C4⋊C8#4114yes—
323#103—28no—
324#103—24no—
325#103—42no—
326#103—42no—
327#103—42no—
328#103—42no—
329#103—44no—
330#103C2×Q818yes—
331C4×C8—24no—
332C4×C8—24no—
333C4×C8—44no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95C2×D414yes—
349#95C2×D414yes—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—44no—
364C2×Q8—81no—
365C2×Q8—81no—
366C2×Q8—82no—
367C2×Q8—82no—
368C2×Q8C2×Q1618yes—
369C2×Q8C22×Q814yes—
370C2×Q8C22×Q812yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4C22×D414yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2511yescharacteristic
389Q16—48no—
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×C8—22no—
399C2×C8—22no—
400C2×C8C22×Q814yes—
401C2×C8C2×#4114yes—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—21no—
408C4⋊C4—24no—
409C4⋊C4—22no—
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—44no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—81no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—81no—
436C4⋊C4—82no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4C2×Q1618yes—
444C4⋊C4C2×Q1618yes—
445C4⋊C4C22×Q812yes—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—24no—
451C42—21no—
452C42—21no—
453C42—41no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—44no—
458C42—44no—
459C42—42no—
460C42—42no—
461C42—84no—
462C42—84no—
463C42C2×Q1614yes—
464C42C22×Q812yes—
465C42C2×#4112yes—
466C42H(5,𝔽2)12yes—
467C422−1+412yes—
468C23C23×D411yescharacteristic
469Q8—22no—
470Q8—24no—
471Q8—22no—
472Q8—24no—
473Q8—48no—
474Q8—41no—
475Q8—42no—
476Q8—41no—
477Q8—42no—
478Q8—84no—
479Q8—164no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4D4×Q812yes—
522C2×C4D4×Q812yes—
523C2×C4#55012yes—
524C2×C4#55012yes—
525C2×C4C22×Q1612yes—
526C2×C4C22×Q1618yes—
527C2×C4C22×Q1612yes—
528C2×C4C22×Q1612yes—
529C2×C4#57612yes—
530C2×C4#57812yes—
531C2×C4C23×Q811yescharacteristic
532C2×C4C22×#4111yescharacteristic
533C2×C4C2×H(5,𝔽2)11yescharacteristic
534C2×C4C2×2−1+411yescharacteristic
535C2×C4#58511yescharacteristic
536C2×C4#58511yescharacteristic
537C8—24no—
538C8—24no—
539C8—42no—
540C8—42no—
541C22C2×D4×Q811yescharacteristic
542C22C2×#55011yescharacteristic
543C22#348511yescharacteristic
544C22C23×Q1611yescharacteristic
545C22C2×#57611yescharacteristic
546C22C2×#57811yescharacteristic
547C22#358511yescharacteristic
548C4—28no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—44no—
554C4—42no—
555C4—44no—
556C4—42no—
557C4—81no—
558C4—82no—
559C4—81no—
560C4—82no—
561C4—82no—
562C4—84no—
563C4Q8×Q1614yes—
564C4#339514yes—
565C4Q8×#4111yescharacteristic
566C4#350111yescharacteristic
567C4#351511yescharacteristic
568C4#354711yescharacteristic
569C2#5905211yescharacteristic
570C2C2×Q8×Q1611yescharacteristic
571C2#6138411yescharacteristic
572C2C2×#339511yescharacteristic
573C2#6141911yescharacteristic
574C2#6169811yescharacteristic
575C2#6179111yescharacteristic
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. #462Q8semidirect
238. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598712
#365632
#326502
#288272
#286892
#285312
#285112
#265322
#260922
#254552
#250622
#248172
#232472
#230962
#229652
C4×#4772
C4×#4622