Finite Group Databasebeta

← #6621060

Subgroups — #6621060

Order 512.

1773 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3374C212yesmaximal
2#36560C211yesmaximal, characteristic
3C4×#557C211yesmaximal, characteristic
4#33200C212yesmaximal
5#33196C211yesmaximal, characteristic
6#28725C212yesmaximal
7#28717C211yesmaximal, characteristic
8#28255C212yesmaximal
9#28255C211yesmaximal, characteristic
10#25333C214yesmaximal
11#25323C212yesmaximal
12#24691C214yesmaximal
13#24683C212yesmaximal
14#22722C211yesmaximal, characteristic
15#22524C212yesmaximal
16#21506C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#557C2212yes—
20#3374—216no—
21C2×#491C2214yes—
22C2×#487C2212yes—
23C2×#479C2214yes—
24C2×#475C2212yes—
25C2×#462C2212yes—
26C2×#462C2211yescharacteristic
27C2×#462C2212yes—
28C2×#462C2214yes—
29C2×#446C2212yes—
30C2×#441C2214yes—
31C2×C4×Q16C2212yes—
32#2868C2211yescharacteristic
33#2859C2212yes—
34#2568C2212yes—
35#2565C2214yes—
36#2546—21no—
37#2546—21no—
38#2506—22no—
39#2506—22no—
40#2370—21no—
41#2370—22no—
42#2365—21no—
43#2365—21no—
44#2365C2211yescharacteristic
45#2365C2212yes—
46#2355C2211yescharacteristic
47#2355C2212yes—
48#2351—21no—
49Q8×C4⋊C4C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#125—21no—
55C4×#125—22no—
56C42×Q8—21no—
57C42×Q8C2212yes—
58C42×Q8C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2074C2214yes—
61#2074C2214yes—
62#2072C2214yes—
63#2056C2214yes—
64#2056C2214yes—
65#2056C2214yes—
66#1992C2214yes—
67#1990C2212yes—
68#1989C2214yes—
69#1988C2212yes—
70#1981C2211yescharacteristic
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2211yescharacteristic
75#1940C2214yes—
76#1939C2212yes—
77#1928C2212yes—
78#1927C2214yes—
79#1878C2214yes—
80#1878C2212yes—
81#1842C2211yescharacteristic
82#1842C2212yes—
83#1805C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2214yes—
91#1803C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#103C2212yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98#557—24no—
99#557—28no—
100#557—28no—
101#557D418yes—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125—21no—
110C2×#125—22no—
111C2×#125—24no—
112C2×#125C2314yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127#491—216no—
128#487—28no—
129#479—216no—
130#475—28no—
131#462—216no—
132#462—28no—
133#462—28no—
134#462—24no—
135#446—28no—
136#441—216no—
137C4×Q16—28no—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#400—24no—
159#398—24no—
160#398—24no—
161#391—22no—
162#391—22no—
163#391C2312yes—
164#391C2314yes—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#389—21no—
169#389—22no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—24no—
191#387—24no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—21no—
200#382—22no—
201#382C2314yes—
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⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D412yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2311yescharacteristic
231C43—22no—
232C43—21no—
233C43C2312yes—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244C22×Q8—22no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C2×Q16—24no—
250C2×Q16—28no—
251C2×Q16—24no—
252C2×Q16—28no—
253C2×Q16—42no—
254C2×Q16—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264C4⋊Q8—48no—
265C4⋊Q8—44no—
266C4⋊Q8C2×D414yes—
267C4⋊Q8C2×D418yes—
268C4⋊Q8#4114yes—
269#125—28no—
270#125—216no—
271#125—28no—
272#125—28no—
273#125—416no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—22no—
286C4×Q8—24no—
287C4×Q8—22no—
288C4×Q8—42no—
289C4×Q8—41no—
290C4×Q8—42no—
291C4×Q8—44no—
292C4×Q8—48no—
293C4×Q8C2×D414yes—
294C4×Q8C2×D418yes—
295C4×Q8C2×D414yes—
296C4×Q8#4114yes—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—24no—
363#106—28no—
364C4⋊C8—24no—
365C4⋊C8—28no—
366C4⋊C8—22no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—42no—
370C4⋊C8—44no—
371#103—24no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—28no—
376#103—28no—
377#103—24no—
378#103—24no—
379#103—24no—
380#97—28no—
381C4×C8—24no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8#12114yes—
423C2×Q8C22×D412yes—
424C2×Q8C22×D414yes—
425C2×Q8C2×#4114yes—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446Q16—48no—
447Q16—416no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4#12114yes—
499C4⋊C4#12118yes—
500C4⋊C4C22×D412yes—
501C4⋊C42−1+412yes—
502C42—28no—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—24no—
507C42—22no—
508C42—21no—
509C42—22no—
510C42—21no—
511C42—22no—
512C42—41no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42—44no—
518C42—48no—
519C42—44no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—81no—
525C42—82no—
526C42#13714yes—
527C42C22×D412yes—
528C42C22×D414yes—
529C42C2×#4114yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C422−1+412yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—24no—
536Q8—22no—
537Q8—24no—
538Q8—24no—
539Q8—22no—
540Q8—24no—
541Q8—42no—
542Q8—41no—
543Q8—42no—
544Q8—42no—
545Q8—41no—
546Q8—42no—
547Q8—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4C2×#12114yes—
595C2×C4C2×#12114yes—
596C2×C4C2×#12112yes—
597C2×C4C2×#12112yes—
598C2×C4#54812yes—
599C2×C4#54812yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4#57512yes—
604C2×C4#57812yes—
605C2×C4C23×D411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C2×C4#58511yescharacteristic
611C8—42no—
612C8—44no—
613C8—42no—
614C22C22×#12111yescharacteristic
615C22C2×#54811yescharacteristic
616C22#348911yescharacteristic
617C22C22×#13711yescharacteristic
618C22C2×#57511yescharacteristic
619C22C2×#57811yescharacteristic
620C22#358611yescharacteristic
621C4—24no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—24no—
626C4—42no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—81no—
637C4—84no—
638C4#303714yes—
639C4#337414yescomplemented
640C4D4×#4111yescharacteristic
641C4#348211yescharacteristic
642C4#349911yescharacteristic
643C4#351811yescharacteristic
644C2#3644411yescharacteristic
645C2C2×#303711yescharacteristic
646C2#6043211yescharacteristic
647C2C2×#337411yescharacteristic
648C2#6132011yescharacteristic
649C2#6211311yescharacteristic
650C2#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
639. C4#3374semidirect

Maximal subgroups

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

subgroupindex
C2×#33742
#365602
C4×#5572
#332002
#331962
#287252
#287172
#282552
#253332
#253232
#246912
#246832
#227222
#225242
#215062
C4×#4622