Finite Group Databasebeta

← #7537801

Subgroups — #7537801

Order 512.

1863 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3398C212yesmaximal
2#59828C211yesmaximal, characteristic, complemented
3#36188C211yesmaximal, characteristic
4#30754C214yesmaximal
5#28471C212yesmaximal
6#28440C211yesmaximal, characteristic, complemented
7#28440C211yesmaximal, characteristic, complemented
8#26607C214yesmaximal, complemented
9#25634C212yesmaximal, complemented
10#25370C212yesmaximal, complemented
11#25302C212yesmaximal, complemented
12#24729C212yesmaximal
13#24189C212yesmaximal
14#23153C211yesmaximal, characteristic, complemented
15#23004C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic
17C4×#461C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#556C2212yes—
20#3398—216no—
21C2×#497C2214yes—
22C2×#494C2212yes—
23C2×#494C2212yes—
24C2×#464C2214yes—
25C2×#462C2212yes—
26C2×#462C2212yes—
27C2×#462C2214yes—
28C2×#462C2212yes—
29C2×#462C2212yes—
30C2×#461C2211yescharacteristic
31C2×#461C2211yescharacteristic
32C2×#446C2212yes—
33C2×#441C2214yes—
34C2×C4×SD16C2212yes—
35C2×C4×SD16C2212yes—
36#2868C2211yescharacteristic
37#2857—21no—
38#2726—22no—
39#2582C2212yes—
40#2568C2212yes—
41#2552—21no—
42#2547C2212yes—
43#2547C2212yes—
44#2546C2212yes—
45#2506—22no—
46#2499—22no—
47#2496—22no—
48#2496—22no—
49#2453—22no—
50#2446C2214yes—
51#2365—21no—
52#2365C2212yes—
53#2365C2212yes—
54Q8×C4⋊C4C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58#2285—21no—
59C42×D4C2211yescharacteristic
60#2066C2214yes—
61#2056C2214yes—
62#2042C2214yes—
63#2016C2214yes—
64#2016C2214yes—
65#2016C2214yes—
66#2016C2214yes—
67#2015C2214yes—
68#1984C2212yes—
69#1981C2211yescharacteristic
70#1981C2211yescharacteristic
71#1971C2214yes—
72#1956C2212yes—
73#1956C2212yes—
74#1948C2211yescharacteristic
75#1948C2211yescharacteristic
76#1939C2212yes—
77#1935C2214yes—
78#1927C2212yes—
79#1927C2212yes—
80#1917C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1877C2212yes—
85#1877C2212yes—
86#1865C2214yes—
87#1802C2211yescharacteristic
88#1802C2211yescharacteristic
89#1770C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2212yes—
94C22×Q16C2314yes—
95C22×Q16C2314yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C22×SD16C2312yes—
100#557—28no—
101#557—44no—
102#556—28no—
103#556—44no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×#126—24no—
112C2×#125—22no—
113C2×#125—24no—
114C2×#124C2314yes—
115C2×C22⋊Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C22×C42—21no—
127#497—216no—
128#494—28no—
129#494—44no—
130#464—216no—
131#462—28no—
132#462—28no—
133#462—216no—
134#462—44no—
135#462—44no—
136#461—24no—
137#461D418yes—
138#446—28no—
139#441—216no—
140C4×SD16—28no—
141C4×SD16—44no—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#97C2314yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2312yes—
164#400—24no—
165#400—24no—
166#398—24no—
167#398—24no—
168#398—24no—
169#397—24no—
170#397—24no—
171#391—22no—
172#391—22no—
173#391D412yes—
174#391C2312yes—
175#391C2312yes—
176#391C2314yes—
177#391C2314yes—
178#391C2314yes—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—24no—
189#389D412yes—
190#389D412yes—
191#389C2314yes—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384D411yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8C2×D412yes—
234C22×Q8C2414yes—
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C2×Q16—28no—
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C2×SD16—42no—
257C2×SD16—42no—
258C2×SD16C2×D418yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274C4⋊Q8—44no—
275C4⋊Q8C2×D414yes—
276#126—416no—
277#125—416no—
278#125—48no—
279#124—28no—
280#124—44no—
281C22⋊Q8—48no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—48no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—41no—
297C4×D4—41no—
298C4×D4C2×D414yes—
299#117—48no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2414yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#106—24no—
354#106—42no—
355C4⋊C8—28no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8C2×D414yes—
359C4⋊C8#4114yes—
360#103—28no—
361#103—24no—
362#103—28no—
363#103—24no—
364#103—42no—
365#103—42no—
366#103—42no—
367#103—44no—
368#103—44no—
369#103C2×D418yes—
370#102—24no—
371#102—42no—
372#97—28no—
373C4×C8—24no—
374C4×C8—24no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—21no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8C22×D414yes—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C22×D412yes—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445Q16—416no—
446Q16—88no—
447Q16—88no—
448Q16—88no—
449SD16—48no—
450SD16—48no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8C22×D414yes—
462C2×C8C2×#4114yes—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—84no—
499C4⋊C4C22×D412yes—
500C22⋊C4—24no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—48no—
513C42—48no—
514C42—44no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42#13714yes—
520C42C22×D412yes—
521C42C2×#4112yes—
522C42H(5,𝔽2)12yes—
523C42H(5,𝔽2)12yes—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—24no—
534Q8—42no—
535Q8—48no—
536Q8—42no—
537Q8—42no—
538Q8—44no—
539Q8—42no—
540Q8—44no—
541Q8—42no—
542Q8—44no—
543Q8—42no—
544Q8—44no—
545D4—22no—
546D4—22no—
547D4—41no—
548D4—41no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4D4212yes—
592C2×C4D4212yes—
593C2×C4#55012yes—
594C2×C4#55012yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4#57812yes—
599C2×C4#57812yes—
600C2×C4C23×D411yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4#58511yescharacteristic
605C2×C4#58511yescharacteristic
606C8—24no—
607C8—24no—
608C8—42no—
609C8—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×D4211yescharacteristic
618C22C2×#55011yescharacteristic
619C22#348711yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57811yescharacteristic
622C22C2×#57811yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—41no—
632C4—82no—
633C4—84no—
634C4—82no—
635C4—81no—
636C4—81no—
637C4—82no—
638C4#327614yes—
639C4#339814yescomplemented
640C4D4×#4111yescharacteristic
641C4#349611yescharacteristic
642C4#351711yescharacteristic
643C4#354611yescharacteristic
644C2—82no—
645C2#5905111yescharacteristic
646C2C2×#327611yescharacteristic
647C2#6109711yescharacteristic
648C2C2×#339811yescharacteristic
649C2#6142211yescharacteristic
650C2#6188911yescharacteristic
651C2#6193611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
2. #59828C2semidirect
6. #28440C2semidirect
7. #28440C2semidirect
8. #26607C2semidirect
9. #25634C2semidirect
10. #25370C2semidirect
11. #25302C2semidirect
14. #23153C2semidirect
15. #23004C2semidirect
639. C4#3398semidirect

Maximal subgroups

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

subgroupindex
C2×#33982
#598282
#361882
#307542
#284712
#284402
#266072
#256342
#253702
#253022
#247292
#241892
#231532
#230042
C4×#4612