Finite Group Databasebeta

← #7598804

Subgroups — #7598804

Order 512.

1452 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic
2#59832C211yesmaximal, characteristic
3#31808C214yesmaximal
4#29290C211yesmaximal, characteristic
5#29073C214yesmaximal
6#28255C211yesmaximal, characteristic
7#27815C212yesmaximal
8#26026C212yesmaximal
9#25936C212yesmaximal
10#25324C212yesmaximal
11#25294C212yesmaximal
12#24996C212yesmaximal
13#23096C212yesmaximal
14Q8×#103C212yesmaximal
15C8×C4⋊Q8C211yesmaximal, characteristic
16C4×#487C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#487C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×C8×Q8C2212yes—
24#2805—22no—
25#2734—22no—
26#2582C2212yes—
27#2581—22no—
28#2571—21no—
29#2568C2212yes—
30#2567C2212yes—
31#2551—21no—
32#2551C2212yes—
33#2546—22no—
34#2511C2214yes—
35#2509—22no—
36#2451—22no—
37#2451C2214yes—
38#2365—21no—
39#2365—21no—
40#2365C2212yes—
41#2355C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2100C2214yes—
49#2087C2214yes—
50#2072C2214yes—
51#2068C2214yes—
52#2056C2214yes—
53#2056C2214yes—
54#2024C2214yes—
55#2024C2214yes—
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1990C2214yes—
59#1987C2214yes—
60#1987C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1969C2214yes—
64#1956C2211yescharacteristic
65#1956C2211yescharacteristic
66#1947C2212yes—
67#1947C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1933C2212yes—
71#1933C2212yes—
72#1933C2212yes—
73#1933C2212yes—
74#1933C2214yes—
75#1928C2214yes—
76#1921C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1842C2211yescharacteristic
80#1834C2211yescharacteristic
81#1834C2211yescharacteristic
82#1803C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85C8×C4⋊C4C2212yes—
86C8×C4⋊C4C2212yes—
87C4×#103C2212yes—
88C4×#103C2214yes—
89C4×#103C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93C22×Q16C2314yes—
94Q82—28no—
95Q82—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2312yes—
113#487—24no—
114#487—24no—
115#462—28no—
116#462—44no—
117C8×Q8—44no—
118C2×#107C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2312yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2314yes—
140#400—24no—
141#400—24no—
142#400—24no—
143#398—24no—
144#398—24no—
145#395—24no—
146#395—24no—
147#395—24no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—21no—
152#391—24no—
153#391D412yes—
154#391C2312yes—
155#391C2312yes—
156#391C2314yes—
157#391C2314yes—
158#391C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—24no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389D412yes—
173#389D412yes—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#384D411yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2314yes—
188#382—22no—
189#382—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⋊C4C2312yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2312yes—
207#340C2314yes—
208#340C2312yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C2×Q16—28no—
225C2×Q16—44no—
226C2×Q16—44no—
227C22×C8C2414yes—
228C22×C8C2411yescharacteristic
229C22×C8C2411yescharacteristic
230C22×C8C2411yescharacteristic
231C22×C8C2411yescharacteristic
232C4⋊Q8—28no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—44no—
240C4⋊Q8—48no—
241C4⋊Q8—416no—
242C4⋊Q8—84no—
243C4⋊Q8Q1618yescomplemented
244C4⋊Q8#4114yes—
245C4⋊Q8#4114yes—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—28no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—44no—
254C4×Q8—48no—
255C4×Q8—82no—
256C4×Q8—82no—
257C4×Q8—84no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2414yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—44no—
308C4⋊C8—24no—
309C4⋊C8—24no—
310C4⋊C8—42no—
311C4⋊C8—44no—
312#103—28no—
313#103—24no—
314#103—24no—
315#103—24no—
316#103—24no—
317#103—24no—
318#103—24no—
319#103—42no—
320#103—42no—
321#103—44no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—22no—
326C4×C8—22no—
327C4×C8—41no—
328C4×C8—44no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95C2×D414yes—
346#95C2×D414yes—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—48no—
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8C2×Q1618yes—
370C2×Q8C2×#4114yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D414yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2511yescharacteristic
388Q16—88no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—24no—
396C2×C8—21no—
397C2×C8—24no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—24no—
403C2×C82−1+412yes—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—84no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—84no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4C2×Q1618yes—
443C4⋊C4C2×Q1618yes—
444C4⋊C4C2×#4114yes—
445C42—22no—
446C42—22no—
447C42—24no—
448C42—24no—
449C42—21no—
450C42—21no—
451C42—41no—
452C42—48no—
453C42—44no—
454C42—44no—
455C42—44no—
456C42—44no—
457C42—84no—
458C42C2×Q1614yes—
459C42C2×#4112yes—
460C42C2×#4112yes—
461C42H(5,𝔽2)12yes—
462C422−1+412yes—
463C23C23×D411yescharacteristic
464Q8—216no—
465Q8—24no—
466Q8—24no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—42no—
472Q8—42no—
473Q8—84no—
474Q8—84no—
475Q8—168no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4#54612yes—
515C2×C4#54612yes—
516C2×C4#54812yes—
517C2×C4#54812yes—
518C2×C4C22×Q1612yes—
519C2×C4C22×Q1618yes—
520C2×C4C22×Q1612yes—
521C2×C4C22×Q1612yes—
522C2×C4#57712yes—
523C2×C4#57712yes—
524C2×C4C22×#4111yescharacteristic
525C2×C4C22×#4111yescharacteristic
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4C2×2−1+411yescharacteristic
528C2×C4C2×2−1+411yescharacteristic
529C2×C4#58511yescharacteristic
530C8—22no—
531C8—41no—
532C8—41no—
533C8—44no—
534C8—41no—
535C22C2×#54611yescharacteristic
536C22C2×#54811yescharacteristic
537C22#349711yescharacteristic
538C22C23×Q1611yescharacteristic
539C22C2×#57711yescharacteristic
540C22C2×#57711yescharacteristic
541C22#358411yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—28no—
546C4—44no—
547C4—44no—
548C4—44no—
549C4—41no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4#329914yes—
557C4#336314yes—
558C4#348211yescharacteristic
559C4#351411yescharacteristic
560C4#351811yescharacteristic
561C4#355211yescharacteristic
562C2#5942111yescharacteristic
563C2C2×#329911yescharacteristic
564C2#6121211yescharacteristic
565C2C2×#336311yescharacteristic
566C2#6132711yescharacteristic
567C2#6236011yescharacteristic
568C2#6237211yescharacteristic
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
243. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598702
#598322
#318082
#292902
#290732
#282552
#278152
#260262
#259362
#253242
#252942
#249962
#230962
Q8×#1032
C8×C4⋊Q82
C4×#4872