Finite Group Databasebeta

← #6544591

Subgroups — #6544591

Order 512.

1462 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36459C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30129C212yesmaximal, complemented
4#29691C211yesmaximal, characteristic, complemented
5#29521C212yesmaximal, complemented
6#29278C212yesmaximal, complemented
7#27961C211yesmaximal, characteristic, complemented
8#27678C211yesmaximal, characteristic
9#27050C214yesmaximal, complemented
10#26742C214yesmaximal
11#26027C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#23825C212yesmaximal, complemented
14#23568C211yesmaximal, characteristic
15#22971C212yesmaximal, complemented
16#22759C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#477C2212yes—
20C2×C4×SD16C2212yes—
21#2860C2212yes—
22#2857C2211yescharacteristic
23#2563C2211yescharacteristic
24#2560—22no—
25#2552C2211yescharacteristic
26#2532—22no—
27#2496—21no—
28#2495C2212yes—
29#2490C2212yes—
30#2485—21no—
31#2369C2211yescharacteristic
32#2360—22no—
33#2360C2214yes—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2211yescharacteristic
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40#2347—21no—
41#2330—22no—
42#2317—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#125C2212yes—
46C4×#121C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2094C2212yes—
54#2074C2214yes—
55#2066C2214yes—
56#2058C2211yescharacteristic
57#2057C2212yes—
58#2055C2214yes—
59#2043C2211yescharacteristic
60#2043C2212yes—
61#2036C2214yes—
62#2035C2212yes—
63#2026C2214yes—
64#2023C2211yescharacteristic
65#2013C2211yescharacteristic
66#2000C2212yes—
67#1990C2212yes—
68#1989C2214yes—
69#1949C2214yes—
70#1948C2212yes—
71#1943C2214yes—
72#1939C2212yes—
73#1936C2214yes—
74#1925C2214yes—
75#1924C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1918C2214yes—
79#1880C2214yes—
80#1874C2212yes—
81#1865C2214yes—
82#1853C2212yes—
83#1834C2212yes—
84#1831C2212yes—
85#1824C2214yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1770C2212yes—
90#1763C2214yes—
91#1763C2214yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C22×SD16C2312yes—
96#533—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#126—22no—
101C2×#125—21no—
102C2×#125C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×D4—21no—
112C2×C4×D4C2311yescharacteristic
113C2×#117—21no—
114C2×#117—21no—
115C2×#117D411yescharacteristic
116C2×#117D411yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121#477—28no—
122C4×SD16—28no—
123C2×#107C2314yes—
124C2×#106C2312yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#398—22no—
141#397—22no—
142#397—22no—
143#394—22no—
144#393—22no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391C2314yes—
149#390—22no—
150#390C2312yes—
151#390C2311yescharacteristic
152#390C2312yes—
153#389—21no—
154#389—21no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—21no—
159#389—24no—
160#389—24no—
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
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#386—21no—
178#385—24no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#383—22no—
187#382—22no—
188#382C2312yes—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380C2312yes—
194#380C2311yescharacteristic
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—24no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2314yes—
229#336C2314yes—
230#336C2311yescharacteristic
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8C2412yes—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×SD16—24no—
242C2×SD16—42no—
243C2×SD16—42no—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250#126—84no—
251#125—28no—
252#125—24no—
253#125—42no—
254C22⋊Q8—28no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—82no—
258#121—22no—
259#121—41no—
260C4×Q8—28no—
261C4×Q8—24no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—84no—
265C4×Q8#4118yes—
266C4×D4—24no—
267C4×D4—44no—
268#117—28no—
269#117—24no—
270#117—42no—
271#117—44no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—28no—
338#106—24no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—24no—
343#103—28no—
344#103—24no—
345#103—44no—
346#103—42no—
347#102—24no—
348#102—44no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351#97—44no—
352C4×C8—24no—
353#95—21no—
354#95—24no—
355#95—24no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—21no—
373C24C22×D411yescharacteristic
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×Q8C2×#4114yes—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—81no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4#13714yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427SD16—48no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—28no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—81no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C2×#4114yes—
472C4⋊C42−1+412yes—
473C22⋊C4—28no—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C42—22no—
487C42—22no—
488C42—28no—
489C42—24no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—42no—
499C42—82no—
500C42—84no—
501C42—84no—
502C42C2×#4114yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×#13712yes—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—48no—
521Q8—42no—
522Q8—42no—
523Q8—162no—
524D4—42no—
525D4—42no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—84no—
577C2×C4—164no—
578C2×C4#52512yes—
579C2×C4#52512yes—
580C2×C4C2×#13712yes—
581C2×C4C2×#13712yes—
582C2×C4#57512yes—
583C2×C4#57712yes—
584C2×C4C22×#4112yes—
585C2×C4C2×2−1+411yescharacteristic
586C2×C4#58511yescharacteristic
587C2×C4#58512yes—
588C8—44no—
589C8—42no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22C2×#52511yescharacteristic
606C22#348911yescharacteristic
607C22#350011yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57511yescharacteristic
610C22C2×#57711yescharacteristic
611C22#358511yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—41no—
621C4—44no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—84no—
626C4—82no—
627C4—82no—
628C4#311714yes—
629C4#349312yes—
630C4#350712yes—
631C2—42no—
632C2—161no—
633C2#3575911yescharacteristic
634C2C2×#311711yescharacteristic
635C2#6058811yescharacteristic
636C2#6210811yescharacteristic
637C2#6213511yescharacteristic
638C2#6254011yescharacteristic
639C2#6259011yescharacteristic
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. #34214C2semidirect
3. #30129C2semidirect
4. #29691C2semidirect
5. #29521C2semidirect
6. #29278C2semidirect
7. #27961C2semidirect
9. #27050C2semidirect
11. #26027C2semidirect
12. #24679C2semidirect
13. #23825C2semidirect
15. #22971C2semidirect
16. #22759C2semidirect
17. #22636C2semidirect

Maximal subgroups

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

subgroupindex
#364592
#342142
#301292
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