Finite Group Databasebeta

← #7545521

Subgroups — #7545521

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#59871C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32548C214yesmaximal
4#29218C211yesmaximal, characteristic
5#28717C211yesmaximal, characteristic
6#27729C212yesmaximal
7#27729C212yesmaximal
8#26512C212yesmaximal
9#26112C212yesmaximal
10#25324C212yesmaximal
11#24996C214yesmaximal
12#24682C212yesmaximal
13#23247C211yesmaximal, characteristic
14#23217C212yesmaximal
15#22771C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#487C2212yes—
19C2×#477C2211yescharacteristic
20C2×#462C2211yescharacteristic
21C2×C4×Q16C2212yes—
22#2868C2211yescharacteristic
23#2859—21no—
24#2806—22no—
25#2584—22no—
26#2582C2212yes—
27#2582C2214yes—
28#2567C2212yes—
29#2567C2212yes—
30#2552—21no—
31#2552C2212yes—
32#2552C2212yes—
33#2550—22no—
34#2507—22no—
35#2507—22no—
36#2451—22no—
37#2369—21no—
38#2369C2212yes—
39#2365C2212yes—
40#2355C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48#2100C2214yes—
49#2100C2214yes—
50#2100C2214yes—
51#2087C2214yes—
52#2072C2214yes—
53#2072C2214yes—
54#2072C2214yes—
55#2056C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1987C2212yes—
59#1987C2212yes—
60#1987C2211yescharacteristic
61#1984C2214yes—
62#1981C2211yescharacteristic
63#1947C2211yescharacteristic
64#1947C2214yes—
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1939C2212yes—
68#1933C2214yes—
69#1933C2214yes—
70#1928C2214yes—
71#1928C2212yes—
72#1921C2212yes—
73#1921C2214yes—
74#1878C2212yes—
75#1878C2214yes—
76#1863C2214yes—
77#1863C2212yes—
78#1834C2212yes—
79#1834C2214yes—
80#1803C2212yes—
81#1803C2214yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#107C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93C22×Q16C2312yes—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
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#487—28no—
111#477—24no—
112#462—24no—
113C4×Q16—28no—
114C2×#107C2312yes—
115C2×#107C2312yes—
116C2×#107C2312yes—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2314yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×#103C2314yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132C2×C4×C8C2314yes—
133#398—24no—
134#398—24no—
135#398—24no—
136#398—24no—
137#398—24no—
138#395—24no—
139#395—24no—
140#391—22no—
141#391—24no—
142#391—21no—
143#391—24no—
144#391D412yes—
145#391C2312yes—
146#391C2314yes—
147#391C2314yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—24no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—24no—
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#384—22no—
172#384—21no—
173#384—21no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2314yes—
178#384C2314yes—
179#384C2312yes—
180#384C2312yes—
181#382—22no—
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⋊C4—21no—
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#340C2314yes—
202#340C2312yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216#336C2312yes—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8C2×D412yes—
220C22×Q8C2412yes—
221C22×Q8C2411yescharacteristic
222C2×Q16—24no—
223C2×Q16—24no—
224C2×Q16—42no—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C4⋊Q8—22no—
230C4⋊Q8—24no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—24no—
234C4⋊Q8—82no—
235C4⋊Q8—82no—
236C4⋊Q8Q1618yescomplemented
237C4⋊Q8#4114yes—
238#125—84no—
239#125—84no—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—24no—
244C4×Q8—22no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8—41no—
248C4×Q8—41no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8—82no—
252C4×Q8—84no—
253C4×Q8#4114yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2411yescharacteristic
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2414yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309#107—24no—
310#107—24no—
311#107—42no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—44no—
317#103—24no—
318#103—24no—
319#103—24no—
320#103—28no—
321#103—28no—
322#103—24no—
323#103—42no—
324#103—42no—
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—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95C2×D414yes—
343#95C2×D414yes—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—81no—
359C2×Q8—81no—
360C2×Q8—82no—
361C2×Q8—82no—
362C2×Q8C2×Q1618yes—
363C2×Q8C2×#4114yes—
364C2×Q8C2×#4112yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4C22×D414yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D414yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2511yescharacteristic
383Q16—48no—
384C2×C8—22no—
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—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—81no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—81no—
430C4⋊C4—82no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4C2×Q1618yes—
438C4⋊C4C2×Q1618yes—
439C4⋊C4C2×#4112yes—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—24no—
444C42—24no—
445C42—21no—
446C42—21no—
447C42—41no—
448C42—42no—
449C42—42no—
450C42—44no—
451C42—42no—
452C42—42no—
453C42—44no—
454C42—42no—
455C42—84no—
456C42—84no—
457C42C2×Q1614yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C422−1+412yes—
461C422−1+412yes—
462C23C23×D411yescharacteristic
463Q8—22no—
464Q8—24no—
465Q8—22no—
466Q8—24no—
467Q8—48no—
468Q8—42no—
469Q8—41no—
470Q8—41no—
471Q8—42no—
472Q8—84no—
473Q8—164no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
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—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4#54812yes—
516C2×C4#54812yes—
517C2×C4#54812yes—
518C2×C4#54812yes—
519C2×C4C22×Q1612yes—
520C2×C4C22×Q1618yes—
521C2×C4C22×Q1612yes—
522C2×C4C22×Q1612yes—
523C2×C4#57712yes—
524C2×C4#57712yes—
525C2×C4C22×#4111yescharacteristic
526C2×C4C22×#4111yescharacteristic
527C2×C4C2×2−1+411yescharacteristic
528C2×C4C2×2−1+411yescharacteristic
529C2×C4#58511yescharacteristic
530C2×C4#58511yescharacteristic
531C8—42no—
532C8—42no—
533C8—42no—
534C8—42no—
535C22C2×#54811yescharacteristic
536C22C2×#54811yescharacteristic
537C22#348511yescharacteristic
538C22C23×Q1611yescharacteristic
539C22C2×#57711yescharacteristic
540C22C2×#57711yescharacteristic
541C22#358511yescharacteristic
542C4—28no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—41no—
547C4—44no—
548C4—42no—
549C4—42no—
550C4—44no—
551C4—81no—
552C4—82no—
553C4—81no—
554C4—84no—
555C4—82no—
556C4—82no—
557C4#336314yes—
558C4#336314yes—
559C4#347511yescharacteristic
560C4#350711yescharacteristic
561C4#351511yescharacteristic
562C4#355511yescharacteristic
563C2#5907911yescharacteristic
564C2C2×#336311yescharacteristic
565C2C2×#336311yescharacteristic
566C2#6134111yescharacteristic
567C2#6134711yescharacteristic
568C2#6169811yescharacteristic
569C2#6179211yescharacteristic
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
236. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598712
#365632
#325482
#292182
#287172
#277292
#265122
#261122
#253242
#249962
#246822
#232472
#232172
#227712
C4×#4772
C4×#4622