Finite Group Databasebeta

← #7533371

Subgroups — #7533371

Order 512.

1469 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59828C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#31640C214yesmaximal
4#29141C211yesmaximal, characteristic
5#29141C211yesmaximal, characteristic
6#28187C212yesmaximal
7#27749C212yesmaximal
8#26694C214yesmaximal
9#26112C212yesmaximal
10#24996C212yesmaximal
11#24632C212yesmaximal
12#24191C212yesmaximal
13#23149C211yesmaximal, characteristic
14#22771C212yesmaximal
15#22542C212yesmaximal
16C4×#462C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#462C2211yescharacteristic
20C2×#462C2211yescharacteristic
21C2×C8×Q8C2212yes—
22C2×C4×Q16C2212yes—
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2859—21no—
26#2726—22no—
27#2582C2212yes—
28#2582C2212yes—
29#2567C2212yes—
30#2552—21no—
31#2552C2212yes—
32#2550—22no—
33#2546C2212yes—
34#2507—22no—
35#2506—22no—
36#2453—22no—
37#2451—22no—
38#2446C2214yes—
39#2369C2212yes—
40#2365—21no—
41#2355C2212yes—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2100C2214yes—
50#2087C2214yes—
51#2072C2214yes—
52#2056C2214yes—
53#2043C2214yes—
54#2024C2214yes—
55#2024C2214yes—
56#2016C2214yes—
57#1984C2212yes—
58#1981C2211yescharacteristic
59#1981C2214yes—
60#1981C2211yescharacteristic
61#1956C2214yes—
62#1956C2212yes—
63#1956C2212yes—
64#1947C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1933C2214yes—
68#1933C2214yes—
69#1928C2214yes—
70#1928C2212yes—
71#1928C2212yes—
72#1921C2214yes—
73#1916C2214yes—
74#1878C2214yes—
75#1878C2212yes—
76#1878C2212yes—
77#1863C2214yes—
78#1834C2212yes—
79#1834C2212yes—
80#1803C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2214yes—
92C4×#103C2212yes—
93C22×Q16C2312yes—
94C22×Q16C2312yes—
95#557—28no—
96#557—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×#125—24no—
106C2×#125—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116#462—24no—
117#462—24no—
118C8×Q8—44no—
119C4×Q16—28no—
120C4×Q16—44no—
121C2×#107C2312yes—
122C2×#107C2312yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—24no—
144#398—24no—
145#398—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#395—24no—
150#391—22no—
151#391—22no—
152#391—24no—
153#391D412yes—
154#391C2312yes—
155#391C2312yes—
156#391C2314yes—
157#391C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—24no—
173#389D412yes—
174#389D412yes—
175#389C2314yes—
176#389C2314yes—
177#387—24no—
178#387—24no—
179#387—24no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384D411yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2314yes—
187#384C2312yes—
188#384C2312yes—
189#382—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C43—21no—
207C43C2311yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C22×Q8—22no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C22×Q8C2411yescharacteristic
225C2×Q16—24no—
226C2×Q16—24no—
227C2×Q16—24no—
228C2×Q16—42no—
229C2×Q16—42no—
230C2×Q16—42no—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C4⋊Q8—28no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241C4⋊Q8—44no—
242C4⋊Q8—82no—
243C4⋊Q8—82no—
244C4⋊Q8Q1618yescomplemented
245C4⋊Q8#4114yes—
246#125—416no—
247#125—48no—
248#125—84no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—22no—
252C4×Q8—28no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—22no—
256C4×Q8—24no—
257C4×Q8—22no—
258C4×Q8—24no—
259C4×Q8—41no—
260C4×Q8—41no—
261C4×Q8—44no—
262C4×Q8—48no—
263C4×Q8—84no—
264C4×Q8#4114yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—24no—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2414yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—24no—
321#107—42no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—44no—
327#103—24no—
328#103—24no—
329#103—24no—
330#103—24no—
331#103—42no—
332#103—42no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#103—42no—
338#103—42no—
339C4×C8—24no—
340C4×C8—24no—
341C4×C8—44no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—44no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8C2×Q1618yes—
380C2×Q8C2×#4112yes—
381C2×Q8C2×#4114yes—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D414yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D414yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2511yescharacteristic
400Q16—48no—
401Q16—48no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—42no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4C2×Q1618yes—
455C4⋊C4C2×Q1618yes—
456C4⋊C4C2×#4112yes—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—24no—
461C42—24no—
462C42—21no—
463C42—21no—
464C42—41no—
465C42—48no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42—44no—
471C42—42no—
472C42—42no—
473C42—84no—
474C42C2×Q1614yes—
475C42C2×#4112yes—
476C42C2×#4112yes—
477C42H(5,𝔽2)12yes—
478C422−1+412yes—
479C23C23×D411yescharacteristic
480Q8—216no—
481Q8—22no—
482Q8—24no—
483Q8—22no—
484Q8—24no—
485Q8—42no—
486Q8—41no—
487Q8—42no—
488Q8—41no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—164no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4#54612yes—
536C2×C4#54612yes—
537C2×C4#54812yes—
538C2×C4#54812yes—
539C2×C4C22×Q1612yes—
540C2×C4C22×Q1618yes—
541C2×C4C22×Q1612yes—
542C2×C4C22×Q1612yes—
543C2×C4#57712yes—
544C2×C4#57712yes—
545C2×C4C22×#4111yescharacteristic
546C2×C4C22×#4111yescharacteristic
547C2×C4C2×H(5,𝔽2)11yescharacteristic
548C2×C4C2×2−1+411yescharacteristic
549C2×C4#58511yescharacteristic
550C2×C4#58511yescharacteristic
551C8—42no—
552C8—42no—
553C8—42no—
554C8—42no—
555C22C2×#54611yescharacteristic
556C22C2×#54811yescharacteristic
557C22#348711yescharacteristic
558C22C23×Q1611yescharacteristic
559C22C2×#57711yescharacteristic
560C22C2×#57711yescharacteristic
561C22#358611yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—28no—
566C4—44no—
567C4—44no—
568C4—42no—
569C4—42no—
570C4—41no—
571C4—81no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—81no—
576C4—84no—
577C4#329914yes—
578C4#336314yes—
579C4#347511yescharacteristic
580C4#350611yescharacteristic
581C4#351511yescharacteristic
582C4#354711yescharacteristic
583C2#5903411yescharacteristic
584C2C2×#329911yescharacteristic
585C2#6120211yescharacteristic
586C2C2×#336311yescharacteristic
587C2#6134611yescharacteristic
588C2#6188411yescharacteristic
589C2#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
244. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598282
#365632
#316402
#291412
#281872
#277492
#266942
#261122
#249962
#246322
#241912
#231492
#227712
#225422
C4×#4622