Finite Group Databasebeta

← #6571383

Subgroups — #6571383

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36315C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30064C212yesmaximal, complemented
4#29922C211yesmaximal, characteristic, complemented
5#29395C212yesmaximal, complemented
6#29350C212yesmaximal, complemented
7#27393C214yesmaximal
8#27327C214yesmaximal, complemented
9#26441C212yesmaximal, complemented
10#25902C211yesmaximal, characteristic, complemented
11#25273C212yesmaximal, complemented
12#24299C211yesmaximal, characteristic
13#23776C212yesmaximal, complemented
14#23754C211yesmaximal, characteristic
15#23217C212yesmaximal, complemented
16#23061C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#487C2212yes—
20C2×#476C2212yes—
21#2861C2212yes—
22#2857C2211yescharacteristic
23#2552C2211yescharacteristic
24#2547C2211yescharacteristic
25#2531—22no—
26#2529—22no—
27#2497—21no—
28#2489C2212yes—
29#2485—21no—
30#2426C2212yes—
31#2369C2212yes—
32#2369C2212yes—
33#2365—21no—
34#2365—21no—
35#2365C2211yescharacteristic
36#2354—21no—
37#2354C2211yescharacteristic
38#2346—21no—
39#2345C2214yes—
40Q8×C22⋊C4—22no—
41#2331—22no—
42#2316—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46#2295C2212yes—
47#2285C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2099C2211yescharacteristic
51#2096C2212yes—
52#2093C2211yescharacteristic
53#2088C2214yes—
54#2083C2211yescharacteristic
55#2075C2212yes—
56#2072C2212yes—
57#2071C2211yescharacteristic
58#2066C2214yes—
59#2059C2214yes—
60#2058C2212yes—
61#2042C2214yes—
62#2035C2214yes—
63#2033C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2212yes—
67#1990C2212yes—
68#1989C2214yes—
69#1987C2212yes—
70#1983C2214yes—
71#1981C2212yes—
72#1948C2212yes—
73#1939C2212yes—
74#1934C2214yes—
75#1933C2214yes—
76#1928C2214yes—
77#1927C2214yes—
78#1926C2214yes—
79#1923C2214yes—
80#1920C2214yes—
81#1918C2214yes—
82#1874C2212yes—
83#1872C2214yes—
84#1863C2214yes—
85#1853C2212yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88C4×#107C2212yes—
89C4×#106C2214yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#125C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×#121C2311yescharacteristic
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—21no—
110C2×C4×D4C2311yescharacteristic
111C2×#117—21no—
112C2×#117—21no—
113C2×#117D411yescharacteristic
114C2×#117D411yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C42C2311yescharacteristic
118#487—28no—
119#476—28no—
120C2×#107C2312yes—
121C2×#107C2312yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×#102C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#400—22no—
138#398—22no—
139#397—22no—
140#397—22no—
141#396—22no—
142#394—22no—
143#391—22no—
144#391—22no—
145#391—24no—
146#391—24no—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#390—22no—
151#390C2311yescharacteristic
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389C2314yes—
162#389C2314yes—
163#389C2311yescharacteristic
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—21no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—24no—
176#387—24no—
177#387—22no—
178#386—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382—21no—
184#382—22no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—24no—
192#380C2311yescharacteristic
193#380C2312yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—24no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214#376—22no—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231C22×Q8—28no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—82no—
235C22×Q8—82no—
236C22×Q8C2412yes—
237C22×D4—21no—
238C22×D4C2411yescharacteristic
239C23×C4—22no—
240C23×C4—21no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8#4118yes—
253#125—44no—
254#124—44no—
255C22⋊Q8—44no—
256#121—22no—
257#121—41no—
258C4×Q8—24no—
259C4×Q8—42no—
260C4×Q8—84no—
261C4×Q8—84no—
262C4×D4—24no—
263C4×D4—44no—
264#117—24no—
265#117—24no—
266#117—44no—
267#117—44no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—24no—
331#107—42no—
332#106—28no—
333#106—44no—
334C4⋊C8—24no—
335C4⋊C8—44no—
336#103—28no—
337#103—28no—
338#103—44no—
339#103—44no—
340#102—28no—
341#102—42no—
342#102—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C4×C8—24no—
346C4×C8—44no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—21no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24—21no—
369C24C22×D411yescharacteristic
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—82no—
379C2×Q8—82no—
380C2×Q8—816no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—81no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4#13514yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2511yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—82no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4#13714yes—
463C4⋊C4C2×#4118yes—
464C4⋊C42−1+412yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—42no—
488C42—44no—
489C42—44no—
490C42—42no—
491C42—44no—
492C42—84no—
493C42—84no—
494C42C2×#4114yes—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C2×#13512yes—
509C23C23×D411yescharacteristic
510Q8—416no—
511Q8—44no—
512Q8—44no—
513Q8—88no—
514D4—42no—
515D4—42no—
516D4—84no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—88no—
566C2×C4—84no—
567C2×C4#52512yes—
568C2×C4#52512yes—
569C2×C4C2×#13512yes—
570C2×C4C2×#13512yes—
571C2×C4C2×#13714yes—
572C2×C4C2×#13712yes—
573C2×C4#57712yes—
574C2×C4C22×#4112yes—
575C2×C4C2×2−1+411yescharacteristic
576C2×C4#58512yes—
577C2×C4#58511yescharacteristic
578C8—42no—
579C8—42no—
580C8—44no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C2×#52511yescharacteristic
596C22#348911yescharacteristic
597C22#350011yescharacteristic
598C22C22×#13511yescharacteristic
599C22C22×#13711yescharacteristic
600C22C2×#57711yescharacteristic
601C22#358511yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—48no—
611C4—41no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4#310814yes—
619C4#351812yes—
620C4#352412yes—
621C2—42no—
622C2—161no—
623C2#3607711yescharacteristic
624C2C2×#310811yescharacteristic
625C2#6057911yescharacteristic
626C2#6209811yescharacteristic
627C2#6213111yescharacteristic
628C2#6253211yescharacteristic
629C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #30064C2semidirect
4. #29922C2semidirect
5. #29395C2semidirect
6. #29350C2semidirect
8. #27327C2semidirect
9. #26441C2semidirect
10. #25902C2semidirect
11. #25273C2semidirect
13. #23776C2semidirect
15. #23217C2semidirect
16. #23061C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#363152
#342162
#300642
#299222
#293952
#293502
#273932
#273272
#264412
#259022
#252732
#242992
#237762
#237542
#232172
#230612
#226392