Finite Group Databasebeta

← #6448211

Subgroups — #6448211

Order 512.

1617 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35628C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#26333C214yesmaximal, complemented
4#26331C214yesmaximal, complemented
5#26326C214yesmaximal, complemented
6#26317C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic
8C2×#2097C211yesmaximal, characteristic, complemented
9#23633C211yesmaximal, characteristic, complemented
10#23631C211yesmaximal, 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#20455C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#107C2211yescharacteristic
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#2368—24no—
42#2355C2214yes—
43#2355C2214yes—
44#2331—22no—
45#2317C2211yescharacteristic
46#2317C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2211yescharacteristic
49C4×#125—22no—
50#2297C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#382C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2214yes—
58#2097C2214yes—
59#2097C2212yes—
60#2095C2212yes—
61#2093C2212yes—
62#2092C2212yes—
63#2074C2214yes—
64#2074C2214yes—
65#2073C2214yes—
66#2072C2214yes—
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×#107C2214yes—
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×Q8C2314yes—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#428—42no—
125#427—24no—
126C2×#107—24no—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106—42no—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135#424—24no—
136C2×C4⋊C8C2314yes—
137#419—24no—
138#419—42no—
139#417—42no—
140#417—42no—
141#416—24no—
142#416—24no—
143C2×#103—24no—
144C2×#103—42no—
145C2×#103C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2311yescharacteristic
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#102C2311yescharacteristic
157C2×#102C2311yescharacteristic
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161#405—24no—
162C2×#97C2312yes—
163C2×C4×C8C2312yes—
164#400—24no—
165#400—24no—
166#397—24no—
167#393—24no—
168#391—24no—
169#391—24no—
170#391—24no—
171#391—22no—
172#391—21no—
173#391—21no—
174#391C2314yes—
175#391C2312yes—
176#391C2312yes—
177#391C2314yes—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#388—24no—
183#388—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384C2312yes—
193#384C2312yes—
194#382—24no—
195#382—21no—
196#382—21no—
197#382C2312yes—
198#382C2312yes—
199#382C2314yes—
200#381—22no—
201#380—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—24no—
205C4×C4⋊C4—24no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218#376—21no—
219C2×#95—22no—
220C2×#95—22no—
221C43—21no—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
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—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117#4114yes—
294#117#4114yes—
295#117#4114yes—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—24no—
345#107—24no—
346#107—24no—
347#107—42no—
348#106—24no—
349#106—44no—
350#106—42no—
351C4⋊C8—28no—
352#103—24no—
353#103—24no—
354#103—22no—
355#103—22no—
356#103—24no—
357#103—28no—
358#103—44no—
359#103—42no—
360#103—42no—
361#103—42no—
362#103—42no—
363#102—22no—
364#102—22no—
365#102—24no—
366#102—42no—
367#102—42no—
368#97—24no—
369C4×C8—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386C24—21no—
387C24C22×D411yescharacteristic
388#41—44no—
389#41—44no—
390#41—44no—
391#41—44no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
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—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C22×C4C2511yescharacteristic
459M16—44no—
460M16—44no—
461C2×C8—28no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—44no—
476C2×C8C2×#4112yes—
477C2×C8C2×#4112yes—
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4#12318yes—
502C4⋊C4C2×#4114yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—48no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C22⋊C4#12318yes—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—42no—
520C42—42no—
521C42—84no—
522C42—84no—
523C42—82no—
524C42—82no—
525C42—84no—
526C42C2×#4114yes—
527C42C2×#4114yes—
528C23—22no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×#12312yes—
536C23#54812yes—
537C23#55012yes—
538C23C23×D411yescharacteristic
539Q8—42no—
540Q8—41no—
541Q8—42no—
542Q8—41no—
543Q8—42no—
544Q8—41no—
545Q8—42no—
546Q8—41no—
547Q8—88no—
548Q8—84no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
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—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—162no—
591C2×C4—162no—
592C2×C4—162no—
593C2×C4C2×#12318yes—
594C2×C4C2×#12312yes—
595C2×C4#52914yes—
596C2×C4#54214yes—
597C2×C4#54812yes—
598C2×C4#55012yes—
599C2×C4#55614yes—
600C2×C4#55914yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C8—22no—
608C8—22no—
609C8—42no—
610C8—42no—
611C8—42no—
612C22—24no—
613C22—24no—
614C22—22no—
615C22—22no—
616C22—82no—
617C22—81no—
618C22—82no—
619C22—81no—
620C22—81no—
621C22#244814yes—
622C22C22×#12311yescharacteristic
623C22C2×#54811yescharacteristic
624C22C2×#55011yescharacteristic
625C22#358412yes—
626C22#358612yes—
627C4—28no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—44no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4#244814yes—
641C4C2×#52911yescharacteristic
642C4C2×#54211yescharacteristic
643C4C2×#55611yescharacteristic
644C4C2×#55911yescharacteristic
645C2—24no—
646C2—161no—
647C2C2×#244811yescharacteristic
648C2#6052412yes—
649C2#6135012yes—
650C2#6141311yescharacteristic
651C2#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. #34215C2semidirect
3. #26333C2semidirect
4. #26331C2semidirect
5. #26326C2semidirect
6. #26317C2semidirect
8. C2×#2097C2semidirect
9. #23633C2semidirect
11. #21304C2semidirect
14. #21174C2semidirect
16. #21066C2semidirect
17. #21061C2semidirect
19. #20455C2semidirect

Maximal subgroups

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

subgroupindex
#356282
#342152
#263332
#263312
#263262
#263172
#237372
C2×#20972
#236332
#236312
#213042
#212622
#212052
#211742
#210682
#210662
#210612
#210582
#204552