Finite Group Databasebeta

← #7541643

Subgroups — #7541643

Order 512.

1345 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32432C214yesmaximal
4#29208C211yesmaximal, characteristic
5#29141C211yesmaximal, characteristic
6#27729C212yesmaximal
7#27132C214yesmaximal
8#26322C212yesmaximal
9#26092C212yesmaximal
10#25294C212yesmaximal
11#24996C212yesmaximal
12#24682C212yesmaximal
13#23149C211yesmaximal, characteristic
14#22808C212yesmaximal
15#22740C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#477C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2212yes—
21C2×C4×Q16C2212yes—
22#2868C2211yescharacteristic
23#2859—21no—
24#2805—22no—
25#2582C2212yes—
26#2581—22no—
27#2571—21no—
28#2568C2212yes—
29#2567C2212yes—
30#2567C2212yes—
31#2552C2212yes—
32#2550—22no—
33#2511C2214yes—
34#2509—22no—
35#2507—22no—
36#2451—22no—
37#2369C2212yes—
38#2365—21no—
39#2355C2212yes—
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#2095C2214yes—
50#2072C2214yes—
51#2072C2214yes—
52#2068C2214yes—
53#2057C2214yes—
54#2056C2214yes—
55#2024C2214yes—
56#1990C2212yes—
57#1987C2211yescharacteristic
58#1984C2212yes—
59#1981C2211yescharacteristic
60#1969C2214yes—
61#1956C2212yes—
62#1947C2212yes—
63#1947C2212yes—
64#1947C2211yescharacteristic
65#1947C2214yes—
66#1947C2214yes—
67#1947C2211yescharacteristic
68#1939C2212yes—
69#1933C2214yes—
70#1928C2214yes—
71#1928C2214yes—
72#1928C2214yes—
73#1928C2212yes—
74#1921C2212yes—
75#1921C2214yes—
76#1916C2214yes—
77#1878C2212yes—
78#1863C2212yes—
79#1845C2212yes—
80#1834C2214yes—
81#1834C2212yes—
82#1803C2211yescharacteristic
83#1803C2212yes—
84#1803C2212yes—
85#1803C2211yescharacteristic
86C4×#107C2214yes—
87C4×#107C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
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×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109#477—24no—
110#462—24no—
111C8×Q8—28no—
112C4×Q16—28no—
113C2×#107C2312yes—
114C2×#107C2312yes—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2312yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×C4×C8C2314yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2312yes—
134#400—24no—
135#400—24no—
136#400—24no—
137#398—24no—
138#398—24no—
139#395—24no—
140#395—24no—
141#391—22no—
142#391—22no—
143#391—21no—
144#391—24no—
145#391D412yes—
146#391C2312yes—
147#391C2314yes—
148#391C2314yes—
149#391C2314yes—
150#389—22no—
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#389—21no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#389C2314yes—
168#387—24no—
169#387—24no—
170#387—24no—
171#387—24no—
172#384—21no—
173#384—21no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2314yes—
178#384C2314yes—
179#384C2312yes—
180#384C2312yes—
181#382—22no—
182#382—22no—
183#382—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C43—21no—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
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—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—42no—
234C4⋊Q8—42no—
235C4⋊Q8—82no—
236C4⋊Q8—82no—
237C4⋊Q8Q1618yescomplemented
238#125—84no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—24no—
246C4×Q8—41no—
247C4×Q8—41no—
248C4×Q8—82no—
249C4×Q8—84no—
250C4×Q8—84no—
251C4×Q8#4114yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
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—24no—
306#107—24no—
307#107—44no—
308#107—44no—
309#107—42no—
310C4⋊C8—28no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8#4114yes—
315#103—24no—
316#103—24no—
317#103—24no—
318#103—24no—
319#103—42no—
320#103—42no—
321#103—44no—
322#103—44no—
323C4×C8—28no—
324C4×C8—24no—
325C4×C8—24no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95C2×D414yes—
343#95C2×D414yes—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×Q8—82no—
362C2×Q8—82no—
363C2×Q8C2×Q1618yes—
364C2×Q8C2×#4112yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
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×C8C2×#4114yes—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—81no—
426C4⋊C4—84no—
427C4⋊C4—81no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—82no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4C2×Q1618yes—
436C4⋊C4C2×Q1618yes—
437C4⋊C4C2×#4112yes—
438C42—24no—
439C42—22no—
440C42—22no—
441C42—24no—
442C42—24no—
443C42—21no—
444C42—21no—
445C42—41no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—44no—
450C42—42no—
451C42—44no—
452C42—42no—
453C42—84no—
454C42—84no—
455C42C2×Q1614yes—
456C42C2×#4112yes—
457C42C2×#4112yes—
458C422−1+412yes—
459C422−1+412yes—
460C23C23×D411yescharacteristic
461Q8—216no—
462Q8—22no—
463Q8—22no—
464Q8—41no—
465Q8—41no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—84no—
471Q8—164no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4#54712yes—
514C2×C4#54712yes—
515C2×C4#54812yes—
516C2×C4#54812yes—
517C2×C4C22×Q1612yes—
518C2×C4C22×Q1618yes—
519C2×C4C22×Q1612yes—
520C2×C4C22×Q1612yes—
521C2×C4#57612yes—
522C2×C4#57712yes—
523C2×C4C22×#4111yescharacteristic
524C2×C4C22×#4111yescharacteristic
525C2×C4C2×2−1+411yescharacteristic
526C2×C4C2×2−1+411yescharacteristic
527C2×C4#58511yescharacteristic
528C2×C4#58511yescharacteristic
529C8—24no—
530C8—42no—
531C8—42no—
532C8—42no—
533C22C2×#54711yescharacteristic
534C22C2×#54811yescharacteristic
535C22#348711yescharacteristic
536C22C23×Q1611yescharacteristic
537C22C2×#57611yescharacteristic
538C22C2×#57711yescharacteristic
539C22#358611yescharacteristic
540C4—28no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—81no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—81no—
553C4—82no—
554C4—84no—
555C4#333814yes—
556C4#336314yes—
557C4#347511yescharacteristic
558C4#350711yescharacteristic
559C4#351811yescharacteristic
560C4#355511yescharacteristic
561C2#5906511yescharacteristic
562C2C2×#333811yescharacteristic
563C2#6128111yescharacteristic
564C2C2×#336311yescharacteristic
565C2#6134511yescharacteristic
566C2#6188411yescharacteristic
567C2#6193711yescharacteristic
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
237. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598702
#365632
#324322
#292082
#291412
#277292
#271322
#263222
#260922
#252942
#249962
#246822
#231492
#228082
#227402
C4×#4772
C4×#4622