Finite Group Databasebeta

← #216783

Subgroups — #216783

Order 512.

1069 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#36638C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3C4×#384C211yesmaximal, characteristic, complemented
4#9887C212yesmaximal, complemented
5#9682C212yesmaximal
6#9148C212yesmaximal
7#9144C212yesmaximal, complemented
8#8578C212yesmaximal, complemented
9#8577C212yesmaximal
10#2866—21no—
11#2865—21no—
12#2864C411yescharacteristic, complemented
13#2863C411yescharacteristic, complemented
14#2861C411yescharacteristic, complemented
15#2860C411yescharacteristic, complemented
16#2855C2211yescharacteristic
17#2854C2211yescharacteristic
18#2853—21no—
19#2852—21no—
20#2851C2211yescharacteristic
21#2850C2211yescharacteristic
22C4×#125C411yescharacteristic, complemented
23C4×#126—21no—
24C4×#124C411yescharacteristic, complemented
25C4×C22⋊Q8C2211yescharacteristic
26C4×#123—21no—
27C4×#121C2211yescharacteristic
28#2298—21no—
29#2295C411yescharacteristic, complemented
30#2293C411yescharacteristic, complemented
31#2290—21no—
32C2×#384—21no—
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2212yes—
36C2×C43C2211yescharacteristic
37#1839—21no—
38#1831—21no—
39#1817C2212yes—
40#1797C2212yes—
41#1767—21no—
42#1749C2212yes—
43#1745—21no—
44C2×#328C2214yes—
45C2×#327C2214yes—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49#1294C2214yes—
50#1294C2214yes—
51C2×#126—21no—
52C2×#125C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#123—21no—
55C2×C22⋊Q8C2311yescharacteristic
56C2×#121C2311yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×#117—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C42—22no—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C22⋊C8C2314yes—
74C2×#97—22no—
75#391C2×C411yescharacteristic
76#391C2×C411yescharacteristic
77#391C2×C411yescharacteristic
78#391C2×C411yescharacteristic
79#390C2×C411yescharacteristic
80#390C2×C411yescharacteristic
81#390C2×C411yescharacteristic
82#390C2×C411yescharacteristic
83#389C2×C411yescharacteristic
84#389C2×C411yescharacteristic
85#389C2×C411yescharacteristic
86#389C2×C411yescharacteristic
87#389C2×C411yescharacteristic
88#389C2×C411yescharacteristic
89#389C2×C411yescharacteristic
90#389C2×C411yescharacteristic
91#388—21no—
92#388—21no—
93#388—21no—
94#388—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#387—21no—
108#387—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#385—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#384—28no—
132#383—21no—
133#382C2×C411yescharacteristic
134#382C2×C411yescharacteristic
135#381—21no—
136#380C2311yescharacteristic
137#380C2311yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—44no—
143C4×C4⋊C4—44no—
144C4×C4⋊C4—44no—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152#376—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95C2314yes—
156C43C2×C411yescharacteristic
157C43C2×C411yescharacteristic
158C43C2×C411yescharacteristic
159C43C2×C411yescharacteristic
160C43D411yescharacteristic
161C43D411yescharacteristic
162C43D411yescharacteristic
163C43D411yescharacteristic
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#328—44no—
169#327—44no—
170C22×Q8C22×C411yescharacteristic
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C23×C4—24no—
174C23×C4—21no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—24no—
183#126—44no—
184#125—42no—
185#124—42no—
186#123—42no—
187#123—42no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190#121—22no—
191#121—22no—
192C4×Q8—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196#117—44no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C4⋊C8—44no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286#97—44no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—24no—
312#95—24no—
313#95—44no—
314#95—44no—
315C24—21no—
316C24C22×D411yescharacteristic
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×D4—22no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—28no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4#13712yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+412yes—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—416no—
400C4⋊C4—416no—
401C4⋊C4—416no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—88no—
463C42#10414yes—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23C22×C22⋊C411yescharacteristic
476C23#53511yescharacteristic
477C23#53611yescharacteristic
478C23C2×AGL(1,ℤ/8)11yescharacteristic
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×#13711yescharacteristic
481C23C2×#13711yescharacteristic
482Q8—82no—
483D4—82no—
484D4—81no—
485D4—81no—
486D4—81no—
487D4—81no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
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—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4#41012yes—
564C2×C4#41012yes—
565C2×C4#41712yes—
566C2×C4#41712yes—
567C2×C4#41712yes—
568C2×C4#41712yes—
569C2×C4C2×#10412yes—
570C2×C4C2×#10412yes—
571C2×C4C2×#10412yes—
572C2×C4#42112yes—
573C2×C4#52012yes—
574C2×C4#53712yes—
575C8—84no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#231111yescharacteristic
596C22C2×#41011yescharacteristic
597C22C2×#41011yescharacteristic
598C22C2×#41711yescharacteristic
599C22C2×#41711yescharacteristic
600C22C22×#10411yescharacteristic
601C22C2×#42111yescharacteristic
602C22#321711yescharacteristic
603C22#321811yescharacteristic
604C22#323811yescharacteristic
605C22#323911yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—88no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—161no—
633C2#1754611yescharacteristic
634C2#1754611yescharacteristic
635C2#1756311yescharacteristic
636C2#1758111yescharacteristic
637C2#2186711yescharacteristic
638C2#2187511yescharacteristic
639C2#2189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. C4×#384C2semidirect
4. #9887C2semidirect
7. #9144C2semidirect
8. #8578C2semidirect
12. #2864C4semidirect
13. #2863C4semidirect
14. #2861C4semidirect
15. #2860C4semidirect
22. C4×#125C4semidirect
24. C4×#124C4semidirect
29. #2295C4semidirect
30. #2293C4semidirect

Maximal subgroups

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

subgroupindex
#366382
#207782
C4×#3842
#98872
#96822
#91482
#91442
#85782
#85772