Finite Group Databasebeta

← #7505189

Subgroups — #7505189

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35627C211yesmaximal, characteristic, complemented
2#35626C211yesmaximal, characteristic
3#31585C214yesmaximal, complemented
4#31582C214yesmaximal, complemented
5#31581C214yesmaximal, complemented
6#31578C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23735C211yesmaximal, characteristic
9#23659C211yesmaximal, characteristic, complemented
10#23633C211yesmaximal, characteristic, complemented
11#23615C211yesmaximal, characteristic, complemented
12#23506C211yesmaximal, characteristic
13C2×#1958C211yesmaximal, characteristic
14#21151C211yesmaximal, characteristic, complemented
15#21066C211yesmaximal, characteristic, complemented
16#21063C211yesmaximal, characteristic
17#21020C211yesmaximal, characteristic, complemented
18#20879C211yesmaximal, characteristic
19#20650C211yesmaximal, characteristic
20C22×#124C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2571C2214yescomplemented
37#2570C2214yescomplemented
38#2509—22no—
39#2508—22no—
40#2505—24no—
41#2483—24no—
42#2447—22no—
43#2447—22no—
44#2447C2214yes—
45#2447C2214yes—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2309C2211yescharacteristic
52#2296C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2094C2212yes—
62#2093C2212yes—
63#2092C2212yes—
64#2092C2212yes—
65#2085C2214yescomplemented
66#2085C2214yescomplemented
67#2074C2214yescomplemented
68#2072C2214yescomplemented
69#2068C2214yescomplemented
70#2066C2214yescomplemented
71#2042C2214yes—
72#2042C2214yes—
73#1958C2214yes—
74#1958C2214yes—
75#1942C2212yescomplemented
76#1942C2212yescomplemented
77#1941C2212yescomplemented
78#1941C2212yescomplemented
79#1928C2214yescomplemented
80#1927C2214yescomplemented
81#1926C2214yes—
82#1925C2214yes—
83#1916C2214yescomplemented
84#1916C2214yescomplemented
85#1864C2214yes—
86#1862C2214yes—
87#1804C2214yes—
88#1802C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C2×#124—24no—
104C2×#124—21no—
105C2×#124—21no—
106C2×#124—42no—
107C2×#124C2314yes—
108#518—24no—
109#518—42no—
110C2×C4×D4—22no—
111C2×C4×D4C2314yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C42C2311yescharacteristic
123#428—42no—
124#428—42no—
125#427—24no—
126#427—42no—
127C2×#107C2311yescharacteristic
128C2×#107C2311yescharacteristic
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#106C2311yescharacteristic
133C2×#106C2311yescharacteristic
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137#424—42no—
138C2×C4⋊C8C2314yes—
139#419—42no—
140#419—42no—
141#418—24no—
142#418—42no—
143C2×#103—24no—
144C2×#103—42no—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102—24no—
152C2×#102—42no—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×C4×C8—42no—
160C2×C4×C8C2312yes—
161C2×C4×C8C2312yes—
162#400—24no—
163#400—24no—
164#400—24no—
165#398—24no—
166#397—24no—
167#397—24no—
168#397—24no—
169#397—24no—
170#391—22no—
171#391—22no—
172#391C2312yes—
173#391C2312yes—
174#391C2314yes—
175#390—22no—
176#390C2312yes—
177#390C2312yes—
178#389—24no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389D412yes—
183#389D412yes—
184#389C2312yes—
185#389C2314yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#387—24no—
190#387—24no—
191#385—22no—
192#385—22no—
193#385—24no—
194#385—24no—
195#385—22no—
196#385—22no—
197#385D414yes—
198#384C2312yes—
199#384C2312yes—
200#382—24no—
201#382—21no—
202#382—21no—
203#382C2312yes—
204#382C2314yes—
205#382C2312yes—
206#381—22no—
207#381—22no—
208#381—24no—
209#381—24no—
210#380—22no—
211#379—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C2×#95—22no—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8C2412yes—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4#4114yes—
249C22×D4C2412yes—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—24no—
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8#4114yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2411yescharacteristic
272C22×C8C2411yescharacteristic
273#124—28no—
274#124—44no—
275#124—44no—
276#124—44no—
277C4×D4—28no—
278C4×D4—44no—
279C4×D4—48no—
280#117—22no—
281#117—22no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342#107—22no—
343#107—22no—
344#107—42no—
345#107—42no—
346#107—42no—
347#106—22no—
348#106—22no—
349#106—42no—
350#106—42no—
351#106—42no—
352C4⋊C8—44no—
353#103—24no—
354#103—24no—
355#103—44no—
356#103—42no—
357#103—42no—
358#103—44no—
359#102—28no—
360#102—24no—
361#102—24no—
362#102—44no—
363#102—42no—
364#102—42no—
365C4×C8—42no—
366C4×C8—42no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95C2×D414yes—
382#95C2×D414yes—
383C24—22no—
384C24—24no—
385C24—42no—
386C24—42no—
387C24—82no—
388C24—82no—
389C24C22×D411yescharacteristic
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×D4—28no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4C2×#4114yes—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—48no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+412yes—
448C22×C42−1+412yes—
449C22×C4C2511yescharacteristic
450M16—44no—
451M16—44no—
452C2×C8—28no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—44no—
467C2×C8C2×#4112yes—
468C2×C8C2×#4112yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—84no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—48no—
530C23—88no—
531C23—164no—
532C23—162no—
533C23—162no—
534C23#53612yes—
535C23#54612yes—
536C23#55012yes—
537C23C23×D411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—84no—
543D4—24no—
544D4—24no—
545D4—42no—
546D4—48no—
547D4—42no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—84no—
583C2×C4—168no—
584C2×C4—162no—
585C2×C4—162no—
586C2×C4#53612yes—
587C2×C4#53914yes—
588C2×C4#53914yes—
589C2×C4#54612yes—
590C2×C4#55012yes—
591C2×C4#55214yes—
592C2×C4#56214yes—
593C2×C4C22×#4111yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C8—22no—
600C8—22no—
601C8—42no—
602C8—42no—
603C8—42no—
604C22—24no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—44no—
609C22—48no—
610C22—84no—
611C22—82no—
612C22—82no—
613C22—168no—
614C22#269414yes—
615C22C2×#53611yescharacteristic
616C22C2×#54611yescharacteristic
617C22C2×#55011yescharacteristic
618C22#358412yes—
619C22#358412yes—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—44no—
624C4—82no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4—81no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#269414yes—
634C4C2×#53911yescharacteristic
635C4C2×#53911yescharacteristic
636C4C2×#55211yescharacteristic
637C4C2×#56211yescharacteristic
638C2—24no—
639C2—84no—
640C2C2×#269411yescharacteristic
641C2#6087912yes—
642C2#6116912yes—
643C2#6141211yescharacteristic
644C2#6141311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35627C2semidirect
3. #31585C2semidirect
4. #31582C2semidirect
5. #31581C2semidirect
6. #31578C2semidirect
7. #23736C2semidirect
9. #23659C2semidirect
10. #23633C2semidirect
11. #23615C2semidirect
14. #21151C2semidirect
15. #21066C2semidirect
17. #21020C2semidirect
36. #2571C22semidirect
37. #2570C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
65. #2085C22semidirect
66. #2085C22semidirect
67. #2074C22semidirect
68. #2072C22semidirect
69. #2068C22semidirect
70. #2066C22semidirect
75. #1942C22semidirect
76. #1942C22semidirect
77. #1941C22semidirect
78. #1941C22semidirect
79. #1928C22semidirect
80. #1927C22semidirect
83. #1916C22semidirect
84. #1916C22semidirect

Maximal subgroups

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

subgroupindex
#356272
#356262
#315852
#315822
#315812
#315782
#237362
#237352
#236592
#236332
#236152
#235062
C2×#19582
#211512
#210662
#210632
#210202
#208792
#206502