Finite Group Databasebeta

← #6623886

Subgroups — #6623886

Order 512.

1521 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#33027C212yesmaximal
5#28511C211yesmaximal, characteristic
6#28373C212yesmaximal
7#28255C211yesmaximal, characteristic
8#28206C212yesmaximal
9#27492C214yesmaximal
10#27068C214yesmaximal
11#25062C212yesmaximal
12#24191C212yesmaximal
13#23096C212yesmaximal
14#22821C211yesmaximal, characteristic
15#22808C212yesmaximal
16C4×#477C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#477C2212yes—
20C2×#477C2212yes—
21C2×#477C2212yes—
22C2×#462C2212yes—
23C2×#462C2211yescharacteristic
24C2×#462C2212yes—
25C2×C4×Q16C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2582C2212yes—
29#2571—21no—
30#2570—22no—
31#2568—21no—
32#2511C2214yes—
33#2446—22no—
34#2370—21no—
35#2370—22no—
36#2365—21no—
37#2365C2211yescharacteristic
38#2365C2212yes—
39#2355—21no—
40#2355C2211yescharacteristic
41#2355C2212yes—
42#2355C2212yes—
43#2351—21no—
44Q8×C4⋊C4C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#125—22no—
50C4×#125—21no—
51C42×Q8—21no—
52C42×Q8C2212yes—
53C42×Q8C2211yescharacteristic
54C42×Q8C2211yescharacteristic
55#2095C2214yes—
56#2095C2214yes—
57#2083C2214yes—
58#2056C2214yes—
59#2043C2214yes—
60#2016C2214yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2211yescharacteristic
64#1969C2214yes—
65#1969C2214yes—
66#1956C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1947C2211yescharacteristic
70#1928C2214yes—
71#1928C2214yes—
72#1928C2214yes—
73#1928C2214yes—
74#1928C2212yes—
75#1921C2214yes—
76#1916C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1878C2212yes—
80#1863C2214yes—
81#1845C2212yes—
82#1842C2211yescharacteristic
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89C4×#107C2214yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×C4⋊C8C2212yes—
93C4×#103C2214yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97#557—24no—
98#557—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—21no—
106C2×#125—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120#477—28no—
121#477—44no—
122#477—44no—
123#462—28no—
124#462—24no—
125#462—44no—
126C4×Q16—28no—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#400—24no—
149#398—24no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#391C2314yes—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389C2314yes—
173#387—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—22no—
191#382—22no—
192#382—22no—
193#382—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C43—22no—
225C43—21no—
226C43C2312yes—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C22×Q8C2411yescharacteristic
242C2×Q16—24no—
243C2×Q16—28no—
244C2×Q16—24no—
245C2×Q16—42no—
246C2×Q16—44no—
247C2×Q16—44no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#125—216no—
262#125—28no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—42no—
278C4×Q8—42no—
279C4×Q8—41no—
280C4×Q8—44no—
281C4×Q8—84no—
282C4×Q8Q1618yescomplemented
283C4×Q8#4114yes—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#107—24no—
349#107—44no—
350#107—44no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8—42no—
355C4⋊C8#4114yes—
356#103—24no—
357#103—28no—
358#103—24no—
359#103—28no—
360#103—24no—
361#103—24no—
362#103—44no—
363#103—44no—
364#103—44no—
365C4×C8—24no—
366C4×C8—44no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8C2×Q1614yes—
411C2×Q8C2×#4114yes—
412C2×Q8C2×#4112yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433Q16—48no—
434Q16—88no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8C2×#4114yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C2×Q1618yes—
484C4⋊C4C2×Q1614yes—
485C4⋊C4C2×#4112yes—
486C4⋊C42−1+412yes—
487C42—28no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—22no—
497C42—21no—
498C42—42no—
499C42—42no—
500C42—41no—
501C42—44no—
502C42—42no—
503C42—44no—
504C42—44no—
505C42—42no—
506C42—44no—
507C42—42no—
508C42—42no—
509C42—82no—
510C42—81no—
511C42—84no—
512C42C2×Q1618yes—
513C42C2×Q1614yes—
514C42C2×#4112yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C42H(5,𝔽2)12yes—
518C23C23×D411yescharacteristic
519Q8—28no—
520Q8—22no—
521Q8—24no—
522Q8—22no—
523Q8—24no—
524Q8—42no—
525Q8—41no—
526Q8—42no—
527Q8—41no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—88no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4#55012yes—
582C2×C4#55012yes—
583C2×C4C22×Q1614yes—
584C2×C4C22×Q1612yes—
585C2×C4C22×Q1614yes—
586C2×C4C22×Q1612yes—
587C2×C4C22×Q1612yes—
588C2×C4#57512yes—
589C2×C4#57812yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×H(5,𝔽2)11yescharacteristic
594C2×C4C2×2−1+411yescharacteristic
595C2×C4#58511yescharacteristic
596C8—24no—
597C8—42no—
598C8—44no—
599C22C2×#52511yescharacteristic
600C22C2×#55011yescharacteristic
601C22#348911yescharacteristic
602C22C23×Q1611yescharacteristic
603C22C2×#57511yescharacteristic
604C22C2×#57811yescharacteristic
605C22#358511yescharacteristic
606C4—24no—
607C4—24no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—44no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—82no—
619C4—82no—
620C4—81no—
621C4—82no—
622C4—84no—
623C4#310414yes—
624C4#339514yes—
625C4#347511yescharacteristic
626C4#348211yescharacteristic
627C4#351111yescharacteristic
628C4#351411yescharacteristic
629C2#3650411yescharacteristic
630C2C2×#310411yescharacteristic
631C2#6060011yescharacteristic
632C2C2×#339511yescharacteristic
633C2#6140511yescharacteristic
634C2#6210011yescharacteristic
635C2#6213311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
282. C4×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×#5572
#331982
#330272
#285112
#283732
#282552
#282062
#274922
#270682
#250622
#241912
#230962
#228212
#228082
C4×#4772
C4×#4622