Finite Group Databasebeta

← #7510023

Subgroups — #7510023

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35630C211yesmaximal, characteristic, complemented
2#35558C211yesmaximal, characteristic
3#32543C214yesmaximal, complemented
4#32540C214yesmaximal, complemented
5#32538C214yesmaximal, complemented
6#32534C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23735C211yesmaximal, characteristic
9#23619C211yesmaximal, characteristic, complemented
10#23618C211yesmaximal, characteristic
11#23541C211yesmaximal, characteristic
12#23541C211yesmaximal, characteristic
13C2×#1973C211yesmaximal, characteristic
14#21173C211yesmaximal, characteristic, complemented
15#21173C211yesmaximal, characteristic, complemented
16#21154C211yesmaximal, characteristic, complemented
17#21144C211yesmaximal, characteristic, complemented
18#21066C211yesmaximal, characteristic, complemented
19#21065C211yesmaximal, characteristic
20C23×SD16C2211yescharacteristic
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2514—22no—
37#2514C2214yescomplemented
38#2513—22no—
39#2513C2214yes—
40#2512—22no—
41#2512C2214yes—
42#2511—22no—
43#2511C2214yescomplemented
44#2493—24no—
45#2480—24no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2310C2211yescharacteristic
49#2310C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2296C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2285C2211yescharacteristic
57#2274C2211yescharacteristic
58#2274C2211yescharacteristic
59#2097C2212yescomplemented
60#2097C2212yescomplemented
61#2096C2212yescomplemented
62#2095C2212yescomplemented
63#2094C2212yes—
64#2093C2212yes—
65#2092C2212yes—
66#2092C2212yes—
67#2068C2214yescomplemented
68#2067C2214yes—
69#2066C2214yescomplemented
70#2065C2214yes—
71#2055C2214yes—
72#2055C2214yes—
73#2055C2214yes—
74#2055C2214yes—
75#1973C2214yes—
76#1973C2214yes—
77#1949C2214yescomplemented
78#1949C2214yescomplemented
79#1947C2214yescomplemented
80#1947C2214yescomplemented
81#1944C2212yescomplemented
82#1944C2212yescomplemented
83#1943C2212yescomplemented
84#1943C2212yescomplemented
85#1940C2214yescomplemented
86#1940C2214yescomplemented
87#1928C2214yescomplemented
88#1927C2214yescomplemented
89#1926C2214yes—
90#1925C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×SD16—24no—
100C22×SD16—42no—
101C22×SD16—42no—
102C22×SD16C2314yes—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C2×#126—24no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—22no—
117C22×C42C2311yescharacteristic
118#428—42no—
119#428—42no—
120#427—24no—
121#427—42no—
122C2×#107—42no—
123C2×#107C2311yescharacteristic
124C2×#107C2311yescharacteristic
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#106C2311yescharacteristic
130C2×#106C2311yescharacteristic
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134#424—24no—
135#424—24no—
136#423—42no—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140#419—42no—
141#419—42no—
142#418—42no—
143#418—42no—
144C2×#103—24no—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102—24no—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#97—42no—
157C2×#97C2312yes—
158C2×#97C2312yes—
159#401—24no—
160#400—24no—
161#400—24no—
162#400—24no—
163#397—24no—
164#397—24no—
165#397—24no—
166#396—24no—
167#391—22no—
168#391—22no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2312yes—
173#391C2314yes—
174#391C2314yes—
175#390—22no—
176#390—22no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#390C2314yes—
182#390C2314yes—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—24no—
190#388—24no—
191#388—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384D412yes—
200#384C2312yes—
201#384C2312yes—
202#383—24no—
203#382D412yes—
204#382C2312yes—
205#382C2312yes—
206#381—22no—
207#381—22no—
208#380D414yes—
209#379—22no—
210#379—22no—
211C4×C4⋊C4—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×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
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—22no—
249C22×D4C2412yes—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×SD16—416no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8#4114yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2411yescharacteristic
275C22×C8C2411yescharacteristic
276#126—88no—
277#117—22no—
278#117—22no—
279#117—22no—
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—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—84no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—22no—
340#107—22no—
341#107—44no—
342#107—42no—
343#107—42no—
344#107—42no—
345#106—22no—
346#106—22no—
347#106—42no—
348#106—42no—
349#106—42no—
350C4⋊C8—28no—
351C4⋊C8—28no—
352C4⋊C8—44no—
353#103—24no—
354#103—24no—
355#103—42no—
356#103—44no—
357#103—42no—
358#102—24no—
359#102—24no—
360#102—42no—
361#102—44no—
362#102—42no—
363#97—42no—
364#97—42no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—82no—
386C24—82no—
387C24C22×D411yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C22×C42−1+412yes—
444C22×C4C2511yescharacteristic
445SD16—416no—
446M16—44no—
447M16—44no—
448C2×C8—28no—
449C2×C8—28no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8C2×#4112yes—
463C2×C8C2×#4112yes—
464C2×C8C2×#4114yes—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—48no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—84no—
518C42—84no—
519C23—22no—
520C23—22no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—88no—
527C23—84no—
528C23—164no—
529C23—162no—
530C23—162no—
531C23#53712yes—
532C23#54712yes—
533C23#55012yes—
534C23C23×D411yescharacteristic
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—84no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—84no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—162no—
581C2×C4—162no—
582C2×C4#53712yes—
583C2×C4#53814yes—
584C2×C4#53914yes—
585C2×C4#54712yes—
586C2×C4#55012yes—
587C2×C4#55614yes—
588C2×C4#56314yes—
589C2×C4C22×#4111yescharacteristic
590C2×C4C22×#4111yescharacteristic
591C2×C4C2×H(5,𝔽2)11yescharacteristic
592C2×C4C2×H(5,𝔽2)11yescharacteristic
593C2×C4C2×2−1+411yescharacteristic
594C2×C4C2×2−1+411yescharacteristic
595C8—24no—
596C8—22no—
597C8—22no—
598C8—42no—
599C8—42no—
600C22—24no—
601C22—24no—
602C22—22no—
603C22—22no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—168no—
610C22#272714yes—
611C22C2×#53711yescharacteristic
612C22C2×#54711yescharacteristic
613C22C2×#55011yescharacteristic
614C22#358512yes—
615C22#358612yes—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—81no—
624C4—81no—
625C4—82no—
626C4—82no—
627C4—81no—
628C4—81no—
629C4#272714yes—
630C4C2×#53811yescharacteristic
631C4C2×#53911yescharacteristic
632C4C2×#55611yescharacteristic
633C4C2×#56311yescharacteristic
634C2—24no—
635C2—162no—
636C2C2×#272711yescharacteristic
637C2#6093812yes—
638C2#6129112yes—
639C2#6141711yescharacteristic
640C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35630C2semidirect
3. #32543C2semidirect
4. #32540C2semidirect
5. #32538C2semidirect
6. #32534C2semidirect
7. #23736C2semidirect
9. #23619C2semidirect
14. #21173C2semidirect
15. #21173C2semidirect
16. #21154C2semidirect
17. #21144C2semidirect
18. #21066C2semidirect
37. #2514C22semidirect
43. #2511C22semidirect
59. #2097C22semidirect
60. #2097C22semidirect
61. #2096C22semidirect
62. #2095C22semidirect
67. #2068C22semidirect
69. #2066C22semidirect
77. #1949C22semidirect
78. #1949C22semidirect
79. #1947C22semidirect
80. #1947C22semidirect
81. #1944C22semidirect
82. #1944C22semidirect
83. #1943C22semidirect
84. #1943C22semidirect
85. #1940C22semidirect
86. #1940C22semidirect
87. #1928C22semidirect
88. #1927C22semidirect

Maximal subgroups

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

subgroupindex
#356302
#355582
#325432
#325402
#325382
#325342
#237362
#237352
#236192
#236182
#235412
C2×#19732
#211732
#211542
#211442
#210662
#210652