Finite Group Databasebeta

← #6568689

Subgroups — #6568689

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36416C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic, complemented
4#30202C212yesmaximal, complemented
5#29338C212yesmaximal, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29123C211yesmaximal, characteristic, complemented
8#27966C212yesmaximal, complemented
9#27824C212yesmaximal, complemented
10#27738C212yesmaximal, complemented
11#27721C214yesmaximal, complemented
12#27707C214yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23745C212yesmaximal, complemented
15#22502C212yesmaximal
16#22502C212yesmaximal
17#20568C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#436C2212yes—
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2589—22no—
25#2587—22no—
26#2580—21no—
27#2571C2211yescharacteristic
28#2567C2212yes—
29#2566—21no—
30#2552—21no—
31#2552—21no—
32#2552C2211yescharacteristic
33#2502C2214yes—
34#2441C2212yes—
35#2440—22no—
36#2440—22no—
37#2439C2212yes—
38#2370—21no—
39#2355—21no—
40#2354C2211yescharacteristic
41#2354C2211yescharacteristic
42D4×C4⋊C4—21no—
43D4×C4⋊C4—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46#2296C2212yes—
47C2×#384C2212yes—
48C42×D4C2211yescharacteristic
49#2100C2211yescharacteristic
50#2100C2211yescharacteristic
51#2100C2214yes—
52#2100C2212yes—
53#2100C2212yes—
54#2100C2212yes—
55#2100C2212yes—
56#2100C2212yes—
57#2099C2214yes—
58#2099C2214yes—
59#2074C2214yes—
60#2072C2212yes—
61#2043C2211yescharacteristic
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2026C2214yes—
65#2024C2212yes—
66#1992C2214yes—
67#1990C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1987C2212yes—
71#1987C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1894C2214yes—
76#1894C2214yes—
77#1873C2214yes—
78#1873C2214yes—
79#1836C2214yes—
80#1834C2212yes—
81#1823C2214yes—
82#1823C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1770C2212yes—
86C4×C4⋊C8C2211yescharacteristic
87C4×#103C2212yes—
88C4×#103C2212yes—
89#1753C2214yes—
90#1749C2212yes—
91#1744C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×M16C2314yes—
96#554—28no—
97#554—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—22no—
104C2×#125—21no—
105C2×#125C2312yes—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8C2314yes—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D411yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117C2312yes—
115C2×#117C2312yes—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2314yes—
119C22×C42C2312yes—
120#436—28no—
121C2×C4⋊C8C2312yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2314yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#399—22no—
134#398—22no—
135#398—24no—
136#398—24no—
137#396—22no—
138#396—22no—
139#395—24no—
140#393—22no—
141#391—21no—
142#391—22no—
143#391C2312yes—
144#391C2314yes—
145#390—21no—
146#390—21no—
147#389—22no—
148#389—22no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2311yescharacteristic
160#389C2311yescharacteristic
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—24no—
176#385—24no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384—22no—
181#384—22no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2314yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—21no—
191#382C2312yes—
192#380—22no—
193#380—22no—
194#380C2312yes—
195#380C2312yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211C43C2311yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8C2412yes—
234C22×D4C2×D411yescharacteristic
235C23×C4—24no—
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2412yes—
239C2×M16—28no—
240C2×M16—44no—
241C2×M16—44no—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C22×C8C2412yes—
245C4⋊Q8—22no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—41no—
249C4⋊Q8—42no—
250C4⋊Q8—82no—
251C4⋊Q8—82no—
252#125—44no—
253#125—48no—
254#125—44no—
255C22⋊Q8—28no—
256C22⋊Q8—28no—
257C22⋊Q8—416no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—44no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—48no—
264C4×D4#4118yes—
265#117—24no—
266#117—42no—
267#117—44no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—84no—
300C2×C4⋊C4—84no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—44no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C4⋊C8—24no—
337#103—24no—
338#103—44no—
339#103—42no—
340#103—42no—
341#103—44no—
342#103—42no—
343C22⋊C8—28no—
344#97—44no—
345C4×C8—24no—
346C4×C8—42no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—24no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24C22×D412yes—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—81no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4#12414yes—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2×#4118yes—
413C22×C4C2511yescharacteristic
414M16—88no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4#12414yes—
468C4⋊C42−1+412yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4#12418yes—
477C42—21no—
478C42—21no—
479C42—24no—
480C42—22no—
481C42—42no—
482C42—42no—
483C42—41no—
484C42—41no—
485C42—44no—
486C42—44no—
487C42—42no—
488C42—42no—
489C42—82no—
490C42#13714yes—
491C42#13714yes—
492C42C2×#4114yes—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23C2×#12414yes—
500C23C23×D411yescharacteristic
501Q8—42no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—84no—
506Q8—164no—
507D4—28no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—168no—
558C2×C4C2×#12412yes—
559C2×C4C2×#12414yes—
560C2×C4C2×#12412yes—
561C2×C4C2×#13712yes—
562C2×C4C2×#13712yes—
563C2×C4C2×#13712yes—
564C2×C4C2×#13712yes—
565C2×C4C2×#13712yes—
566C2×C4C2×#13712yes—
567C2×C4C22×#4112yes—
568C2×C4C2×2−1+411yescharacteristic
569C2×C4#58512yes—
570C2×C4#58511yescharacteristic
571C8—42no—
572C8—44no—
573C8—42no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22C22×#12411yescharacteristic
582C22#348511yescharacteristic
583C22#348911yescharacteristic
584C22C22×#13711yescharacteristic
585C22C22×#13711yescharacteristic
586C22#358511yescharacteristic
587C22#358511yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—24no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—82no—
599C4—81no—
600C4—81no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4#313914yes—
606C4#351512yes—
607C4#352112yes—
608C2—44no—
609C2#3604211yescharacteristic
610C2C2×#313911yescharacteristic
611C2#6062611yescharacteristic
612C2#6171911yescharacteristic
613C2#6171911yescharacteristic
614C2#6210811yescharacteristic
615C2#6211211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30230C2semidirect
4. #30202C2semidirect
5. #29338C2semidirect
6. #29218C2semidirect
7. #29123C2semidirect
8. #27966C2semidirect
9. #27824C2semidirect
10. #27738C2semidirect
11. #27721C2semidirect
12. #27707C2semidirect
13. #23816C2semidirect
14. #23745C2semidirect

Maximal subgroups

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

subgroupindex
#364162
#361802
#302302
#302022
#293382
#292182
#291232
#279662
#278242
#277382
#277212
#277072
#238162
#237452
#225022
#205682