Finite Group Databasebeta

← #6627758

Subgroups — #6627758

Order 512.

1539 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×Q82C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic
4#33027C212yesmaximal
5#29347C212yesmaximal
6#29290C211yesmaximal, characteristic
7#29259C212yesmaximal
8#28748C211yesmaximal, characteristic
9#26694C214yesmaximal
10#26322C212yesmaximal
11#26171C214yesmaximal
12#25936C212yesmaximal
13#23217C212yesmaximal
14#22998C212yesmaximal
15#22821C211yesmaximal, characteristic
16C4×#477C211yesmaximal, characteristic
17C4×#462C212yesmaximal
18C2×Q82C2211yescharacteristic
19C2×#487C2212yes—
20C2×#477C2211yescharacteristic
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×C4×Q16C2214yes—
25#2868C2211yescharacteristic
26#2862C2212yes—
27#2582—21no—
28#2571—21no—
29#2570—22no—
30#2568C2212yes—
31#2511—22no—
32#2446C2214yes—
33#2369—21no—
34#2369—22no—
35#2369C2212yes—
36#2369C2211yescharacteristic
37#2365C2212yes—
38#2355—21no—
39#2355—21no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2211yescharacteristic
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2211yescharacteristic
44C4×C4⋊Q8—22no—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C42×Q8—21no—
51C42×Q8C2212yes—
52C42×Q8C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54#2095C2214yes—
55#2087C2214yes—
56#2072C2214yes—
57#2043C2214yes—
58#2024C2214yes—
59#2024C2214yes—
60#1987C2212yes—
61#1987C2212yes—
62#1987C2211yescharacteristic
63#1984C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1956C2212yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1933C2214yes—
74#1933C2214yes—
75#1928C2214yes—
76#1928C2214yes—
77#1928C2214yes—
78#1921C2212yes—
79#1916C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1863C2212yes—
84#1863C2214yes—
85#1863C2214yes—
86#1842C2211yescharacteristic
87#1831C2212yes—
88#1803C2211yescharacteristic
89C4×#107C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×C4⋊C8C2212yes—
93C4×#103C2214yes—
94C4×#103C2212yes—
95C22×Q16C2314yes—
96Q82—24no—
97Q82—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2314yes—
119C2×C4×Q8C2311yescharacteristic
120#487—28no—
121#477—24no—
122#462—28no—
123#462—28no—
124#462—44no—
125C4×Q16—48no—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—24no—
146#400—24no—
147#398—24no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2314yes—
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—22no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#387—24no—
178#387—24no—
179#387—24no—
180#384—21no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2311yescharacteristic
189#382—22no—
190#382—22no—
191#382—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
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—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C43—22no—
222C43—21no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2414yes—
241C22×Q8C2411yescharacteristic
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—28no—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C4⋊Q8—216no—
249C4⋊Q8—28no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8#4114yes—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—42no—
272C4×Q8—42no—
273C4×Q8—41no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—82no—
279C4×Q8—84no—
280C4×Q8Q1618yescomplemented
281C4×Q8#4114yes—
282C4×Q8#4114yes—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—22no—
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—24no—
348#107—42no—
349#107—44no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—42no—
355#103—24no—
356#103—24no—
357#103—28no—
358#103—28no—
359#103—28no—
360#103—28no—
361#103—44no—
362#103—44no—
363#103—44no—
364C4×C8—24no—
365C4×C8—44no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C2×Q8—28no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8—82no—
404C2×Q8—81no—
405C2×Q8—82no—
406C2×Q8C2×Q1614yes—
407C2×Q8C2×#4112yes—
408C2×Q8C2×#4112yes—
409C2×Q8H(5,𝔽2)12yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2511yescharacteristic
430Q16—416no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—84no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4C2×Q1618yes—
481C4⋊C4C2×Q1614yes—
482C4⋊C4C2×#4112yes—
483C4⋊C4C2×#4114yes—
484C4⋊C4C2×#4112yes—
485C42—28no—
486C42—24no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—22no—
496C42—21no—
497C42—41no—
498C42—42no—
499C42—42no—
500C42—44no—
501C42—42no—
502C42—44no—
503C42—42no—
504C42—44no—
505C42—44no—
506C42—42no—
507C42—82no—
508C42—81no—
509C42—84no—
510C42C2×Q1618yes—
511C42C2×Q1614yes—
512C42C2×#4112yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C422−1+412yes—
516C23C23×D411yescharacteristic
517Q8—28no—
518Q8—22no—
519Q8—22no—
520Q8—22no—
521Q8—44no—
522Q8—44no—
523Q8—44no—
524Q8—44no—
525Q8—84no—
526Q8—84no—
527Q8—88no—
528Q8#55914yes—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4#52512yes—
576C2×C4#52512yes—
577C2×C4#54812yes—
578C2×C4#54812yes—
579C2×C4C22×Q1614yes—
580C2×C4C22×Q1612yes—
581C2×C4C22×Q1614yes—
582C2×C4C22×Q1612yes—
583C2×C4C22×Q1612yes—
584C2×C4#57512yes—
585C2×C4#57712yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C2×C4#58511yescharacteristic
592C8—44no—
593C8—42no—
594C8—42no—
595C22C2×#52511yescharacteristic
596C22C2×#54811yescharacteristic
597C22#348811yescharacteristic
598C22C23×Q1611yescharacteristic
599C22C2×#57511yescharacteristic
600C22C2×#57711yescharacteristic
601C22#358611yescharacteristic
602C4—24no—
603C4—22no—
604C4—24no—
605C4—22no—
606C4—22no—
607C4—24no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—44no—
613C4—42no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4#310414yes—
620C4#336314yes—
621C4C2×#55911yescharacteristic
622C4#348011yescharacteristic
623C4#350711yescharacteristic
624C4#352211yescharacteristic
625C2#3658011yescharacteristic
626C2C2×#310411yescharacteristic
627C2#6059111yescharacteristic
628C2C2×#336311yescharacteristic
629C2#6132311yescharacteristic
630C2#6195211yescharacteristic
631C2#6198711yescharacteristic
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
280. C4×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×Q822
#331632
#330272
#293472
#292902
#292592
#287482
#266942
#263222
#261712
#259362
#232172
#229982
#228212
C4×#4772
C4×#4622