Finite Group Databasebeta

← #6448212

Subgroups — #6448212

Order 512.

1667 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35628C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#26335C214yesmaximal, complemented
4#26328C214yesmaximal, complemented
5#26324C214yesmaximal, complemented
6#26320C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8C2×#2096C211yesmaximal, characteristic, complemented
9#23633C211yesmaximal, characteristic, complemented
10#23630C211yesmaximal, characteristic
11#21304C211yesmaximal, characteristic, complemented
12#21262C211yesmaximal, characteristic
13#21205C211yesmaximal, characteristic
14#21174C211yesmaximal, characteristic, complemented
15#21068C211yesmaximal, characteristic
16#21066C211yesmaximal, characteristic, complemented
17#21061C211yesmaximal, characteristic, complemented
18#21058C211yesmaximal, characteristic
19#20454C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#405C2211yescharacteristic
36#2592C2214yes—
37#2568C2214yes—
38#2492—22no—
39#2489—22no—
40#2488—24no—
41#2360—24no—
42#2355C2214yes—
43Q8×C4⋊C4C2214yes—
44#2330—22no—
45#2317C2211yescharacteristic
46#2317C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2211yescharacteristic
49C4×C4⋊Q8—22no—
50#2297C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yes—
58#2096C2214yes—
59#2096C2214yes—
60#2096C2212yes—
61#2094C2212yes—
62#2092C2212yes—
63#2074C2214yes—
64#2072C2214yes—
65#2072C2214yes—
66#2071C2214yes—
67#1985C2214yes—
68#1984C2214yes—
69#1983C2214yes—
70#1981C2214yes—
71#1960C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1956C2212yes—
75#1949C2214yes—
76#1948C2214yes—
77#1928C2212yes—
78#1928C2214yes—
79#1928C2214yes—
80#1927C2212yes—
81#1927C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1926C2212yes—
85#1926C2214yes—
86#1925C2212yes—
87#1776C2214yes—
88C4×#106C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—24no—
104#536—24no—
105#533—24no—
106#533—24no—
107C2×C4⋊Q8D412yes—
108C2×C4⋊Q8C2312yes—
109C2×#125D412yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8D414yes—
114C2×C4×Q8—24no—
115C2×C4×Q8C2314yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#428—42no—
127#427—24no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106—24no—
131C2×#106—42no—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137#424—24no—
138C2×C4⋊C8C2314yes—
139#419—24no—
140#419—42no—
141#417—42no—
142#417—42no—
143#416—24no—
144#416—24no—
145C2×#103—24no—
146C2×#103—42no—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2311yescharacteristic
150C2×#103C2311yescharacteristic
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102C2311yescharacteristic
159C2×#102C2311yescharacteristic
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163#405—24no—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166#400—24no—
167#400—24no—
168#397—24no—
169#393—24no—
170#391—24no—
171#391—24no—
172#391—22no—
173#391—21no—
174#391—21no—
175#391C2314yes—
176#391C2312yes—
177#391C2312yes—
178#391C2314yes—
179#390C2312yes—
180#390C2312yes—
181#389—22no—
182#389—22no—
183#389—24no—
184#389—24no—
185#389—24no—
186#388—24no—
187#388—24no—
188#387—24no—
189#387—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2312yes—
198#384C2314yes—
199#382—24no—
200#382C2312yes—
201#382C2312yes—
202#381—22no—
203#380—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219#376—21no—
220C2×#95—22no—
221C43—21no—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2×D414yes—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8#4114yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2411yescharacteristic
274C22×C8C2411yescharacteristic
275C4⋊Q8—28no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278#125—28no—
279#124—24no—
280#124—24no—
281#123—48no—
282C22⋊Q8—216no—
283C22⋊Q8—48no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—88no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117#4114yes—
295#117#4114yes—
296#117#4114yes—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—24no—
349#107—42no—
350#106—24no—
351#106—24no—
352#106—24no—
353#106—44no—
354#106—42no—
355C4⋊C8—28no—
356#103—24no—
357#103—24no—
358#103—22no—
359#103—22no—
360#103—24no—
361#103—28no—
362#103—44no—
363#103—42no—
364#103—42no—
365#103—42no—
366#103—42no—
367#102—22no—
368#102—22no—
369#102—24no—
370#102—42no—
371#102—42no—
372#97—24no—
373C4×C8—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388C24—21no—
389C24C22×D411yescharacteristic
390#41—44no—
391#41—44no—
392#41—44no—
393#41—44no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461M16—44no—
462M16—44no—
463C2×C8—28no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8C2×#4112yes—
479C2×C8C2×#4112yes—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4#12318yes—
506C4⋊C4C2×#4114yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—84no—
515C22⋊C4#12318yes—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—44no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—84no—
528C42—84no—
529C42—84no—
530C42C2×#4114yes—
531C42C2×#4114yes—
532C23—22no—
533C23—22no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23C2×#12312yes—
540C23#54812yes—
541C23#55012yes—
542C23C23×D411yescharacteristic
543Q8—416no—
544Q8—42no—
545Q8—41no—
546Q8—42no—
547Q8—41no—
548Q8—42no—
549Q8—41no—
550Q8—42no—
551Q8—41no—
552Q8—84no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—84no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—162no—
594C2×C4—162no—
595C2×C4—162no—
596C2×C4C2×#12318yes—
597C2×C4C2×#12312yes—
598C2×C4#52914yes—
599C2×C4#54214yes—
600C2×C4#54812yes—
601C2×C4#55012yes—
602C2×C4#55614yes—
603C2×C4#55914yes—
604C2×C4#57714yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C8—22no—
612C8—22no—
613C8—42no—
614C8—42no—
615C8—42no—
616C22—24no—
617C22—24no—
618C22—22no—
619C22—22no—
620C22—82no—
621C22—81no—
622C22—82no—
623C22—81no—
624C22—81no—
625C22#244814yes—
626C22C22×#12311yescharacteristic
627C22C2×#54811yescharacteristic
628C22C2×#55011yescharacteristic
629C22C2×#57712yes—
630C22#358612yes—
631C4—24no—
632C4—28no—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4#244814yes—
645C4C2×#52911yescharacteristic
646C4C2×#54211yescharacteristic
647C4C2×#55611yescharacteristic
648C4C2×#55911yescharacteristic
649C2—24no—
650C2—161no—
651C2C2×#244811yescharacteristic
652C2#6052212yes—
653C2#6134612yes—
654C2#6141011yescharacteristic
655C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #26335C2semidirect
4. #26328C2semidirect
5. #26324C2semidirect
6. #26320C2semidirect
8. C2×#2096C2semidirect
9. #23633C2semidirect
11. #21304C2semidirect
14. #21174C2semidirect
16. #21066C2semidirect
17. #21061C2semidirect
19. #20454C2semidirect

Maximal subgroups

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

subgroupindex
#356282
#342142
#263352
#263282
#263242
#263202
#237382
C2×#20962
#236332
#236302
#213042
#212622
#212052
#211742
#210682
#210662
#210612
#210582
#204542