Finite Group Databasebeta

← #7511792

Subgroups — #7511792

Order 512.

2216 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34935C212yesmaximal, complemented
2#32648C212yesmaximal, complemented
3#32544C212yesmaximal, complemented
4#32458C211yesmaximal, characteristic, complemented
5#31994C214yesmaximal, complemented
6#31666C212yesmaximal, complemented
7#31056C211yesmaximal, characteristic, complemented
8#30396C214yesmaximal, complemented
9#23754C214yesmaximal, complemented
10#23696C212yesmaximal, complemented
11C4×#466C212yesmaximal, complemented
12C4×#447C214yesmaximal, complemented
13#21373C211yesmaximal, characteristic, complemented
14C2×#525—22no—
15C2×C4×#41C2211yescharacteristic
16C22×C4×D4C2212yescomplemented
17C2×#466C2212yes—
18C2×#447C2214yes—
19C2×C4×D8C2212yes—
20C2×C4×D8C2212yes—
21C2×#431C2211yescharacteristic, complemented
22C22×C4×C8C2211yescharacteristic, complemented
23#2538C2214yescomplemented
24#2537C2214yes—
25#2536—22no—
26#2535—22no—
27#2534—22no—
28#2534—22no—
29#2486C2212yes—
30#2486C2212yescomplemented
31#2408C2212yes—
32#2408C2212yescomplemented
33#2392—22no—
34#2392—22no—
35C4×#121C2214yes—
36C4×#121C2214yescomplemented
37#2285C2212yescomplemented
38#2280—22no—
39#2099C2214yescomplemented
40#2093C2214yescomplemented
41#2086C2212yescomplemented
42#2083C2214yescomplemented
43#2075C2212yescomplemented
44#2075C2214yescomplemented
45#2075C2214yescomplemented
46#2071C2214yescomplemented
47#2067C2214yescomplemented
48#2054C2212yes—
49#2033C2214yes—
50#2033C2214yes—
51#2033C2214yescomplemented
52#1999C2212yes—
53#1984C2211yescharacteristic, complemented
54#1984C2211yescharacteristic, complemented
55#1970C2212yes—
56#1954C2211yescharacteristic
57#1954C2211yescharacteristic
58#1942C2212yescomplemented
59#1934C2214yescomplemented
60#1893C2214yescomplemented
61#1879C2214yes—
62#1879C2214yes—
63#1874C2214yes—
64#1869C2214yescomplemented
65#1862C2214yescomplemented
66#1862C2214yescomplemented
67#1852C2214yescomplemented
68#1808C2212yescomplemented
69#1786C2214yescomplemented
70C4×#107C2212yescomplemented
71C4×#107C2212yescomplemented
72C4×#102C2214yescomplemented
73C4×#102C2214yescomplemented
74C4×C22⋊C8C2214yescomplemented
75C22×#41D411yescharacteristic
76C23×D4C2312yes—
77C24×C4—22no—
78C22×D8C2312yes—
79C22×D8C2312yes—
80C23×C8C2312yes—
81#525—416no—
82C2×#123—22no—
83C2×#123—22no—
84C2×C22⋊Q8—24no—
85C2×#121—24no—
86C2×#121C2314yes—
87C2×#121C2314yes—
88C2×#120—22no—
89C2×#120—22no—
90C4×#41—42no—
91C4×#41—42no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8C2311yescharacteristic
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—44no—
99C2×C4×D4—44no—
100C2×C4×D4D411yescharacteristic
101C2×C4×D4D412yes—
102C2×C4×D4C2314yes—
103C2×C4×D4C2314yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2311yescharacteristic
106C2×#117—22no—
107C2×#117D411yescharacteristic
108C2×#117D412yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—22no—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113#466—28no—
114#447—216no—
115C4×D8—44no—
116C4×D8—44no—
117#431—42no—
118C2×#107C2312yes—
119C2×#107C2312yes—
120C2×#107C2312yes—
121C2×#107C2314yes—
122C2×#107C2312yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2314yes—
134C2×C4×C8—42no—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#398—22no—
140#398—22no—
141#397—22no—
142#397—22no—
143#396—24no—
144#396—24no—
145#394—22no—
146#394—22no—
147#394—24no—
148#394—24no—
149#392—22no—
150#392—22no—
151#390C2314yes—
152#390C2314yes—
153#389C2314yes—
154#389C2314yes—
155#388—24no—
156#388—24no—
157#386—24no—
158#386—24no—
159#384C2314yes—
160#382—22no—
161#381—22no—
162#381—22no—
163#380—22no—
164#380—24no—
165#380—24no—
166#380C2314yes—
167#380C2314yes—
168#380C2312yes—
169#380C2312yes—
170#379—22no—
171#379—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4—24no—
175C4×C22⋊C4—24no—
176#340C2312yes—
177#340C2314yes—
178#340C2314yes—
179#340C2312yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2314yes—
185C25—22no—
186C2×#41—42no—
187C2×#41—42no—
188C2×#41D818yescomplemented
189C22×Q8C2×D411yescharacteristic
190C22×D4—28no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4C2×D411yescharacteristic
198C22×D4C2×D412yes—
199C22×D4C2414yes—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—48no—
204C23×C4—48no—
205C23×C4—48no—
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C2×D8—28no—
213C2×D8—42no—
214C2×D8—42no—
215C2×D8—42no—
216C2×D8#4118yescomplemented
217C22×C8—24no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8C2412yes—
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224#123—48no—
225#123—48no—
226C22⋊Q8—416no—
227#121—28no—
228#121—28no—
229#121—44no—
230#121—44no—
231#121—416no—
232#120—48no—
233#120—48no—
234C4×Q8—42no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—42no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—48no—
248C4×D4—48no—
249C4×D4—416no—
250C4×D4—416no—
251#117—44no—
252#117—42no—
253#117—48no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C22⋊C4C2×D412yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2412yes—
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—24no—
297#107—44no—
298#107—42no—
299#107—42no—
300#107—42no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303#102—28no—
304#102—28no—
305#102—28no—
306#102—44no—
307#102—44no—
308#102—44no—
309C22⋊C8—28no—
310C22⋊C8—44no—
311C22⋊C8—42no—
312C4×C8—24no—
313C4×C8—24no—
314C4×C8—41no—
315C4×C8—41no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—24no—
323C24—24no—
324C24—48no—
325C24—44no—
326C24—44no—
327C24—44no—
328C24—84no—
329C24C22×D412yes—
330C24C22×D411yescharacteristic
331#41—416no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8C2×D814yes—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—416no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4C2×D814yes—
356C2×D4C2×D818yes—
357C2×D4C2×#4118yes—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—832no—
387C22×C4C2×D818yes—
388C22×C4C2×D814yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C2511yescharacteristic
398D8—48no—
399D8—48no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—48no—
410C2×C8C2×#4114yes—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—84no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—22no—
442C42—24no—
443C42—24no—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—22no—
448C42—22no—
449C42—48no—
450C42H(5,𝔽2)12yes—
451C42H(5,𝔽2)12yes—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—21no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—21no—
460C23—28no—
461C23—416no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—816no—
469C23—88no—
470C23—816no—
471C23—88no—
472C23C22×D814yes—
473C23C22×D812yes—
474C23C23×D411yescharacteristic
475Q8—44no—
476D4—28no—
477D4—28no—
478D4—48no—
479D4—44no—
480D4—416no—
481D4—44no—
482D4—44no—
483D4—816no—
484D4—816no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—216no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—84no—
517C2×C4—816no—
518C2×C4—88no—
519C2×C4—816no—
520C2×C4#53412yes—
521C2×C4#53412yes—
522C2×C4C22×D812yes—
523C2×C4C22×D814yes—
524C2×C4C22×D812yes—
525C2×C4#57512yes—
526C2×C4C22×#4112yes—
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C2×C4C2×H(5,𝔽2)11yescharacteristic
529C2×C4#58512yes—
530C8—24no—
531C8—42no—
532C8—42no—
533C8—42no—
534C22—24no—
535C22—22no—
536C22—24no—
537C22—28no—
538C22—42no—
539C22—44no—
540C22—41no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—48no—
545C22—48no—
546C22—48no—
547C22—832no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22C2×#53411yescharacteristic
553C22D4×#4112yes—
554C22C23×D811yescharacteristic
555C22C2×#57511yescharacteristic
556C22#358412yes—
557C4—24no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—42no—
562C4—41no—
563C4—42no—
564C4—84no—
565C4—84no—
566C4—84no—
567C4#318614yes—
568C4#345912yes—
569C4#348111yescharacteristic
570C4#352911yescharacteristic
571C2—42no—
572C2—44no—
573C2—88no—
574C2#5216311yescharacteristic
575C2C2×#318611yescharacteristic
576C2#6074911yescharacteristic
577C2D8×#4112yes—
578C2#6144212yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34935C2semidirect
2. #32648C2semidirect
3. #32544C2semidirect
4. #32458C2semidirect
5. #31994C2semidirect
6. #31666C2semidirect
7. #31056C2semidirect
8. #30396C2semidirect
9. #23754C2semidirect
10. #23696C2semidirect
11. C4×#466C2semidirect
12. C4×#447C2semidirect
13. #21373C2semidirect
16. C22×C4×D4C22semidirect
21. C2×#431C22semidirect
22. C22×C4×C8C22semidirect
23. #2538C22semidirect
30. #2486C22semidirect
32. #2408C22semidirect
36. C4×#121C22semidirect
37. #2285C22semidirect
39. #2099C22semidirect
40. #2093C22semidirect
41. #2086C22semidirect
42. #2083C22semidirect
43. #2075C22semidirect
44. #2075C22semidirect
45. #2075C22semidirect
46. #2071C22semidirect
47. #2067C22semidirect
51. #2033C22semidirect
53. #1984C22semidirect
54. #1984C22semidirect
58. #1942C22semidirect
59. #1934C22semidirect
60. #1893C22semidirect
64. #1869C22semidirect
65. #1862C22semidirect
66. #1862C22semidirect
67. #1852C22semidirect
68. #1808C22semidirect
69. #1786C22semidirect
70. C4×#107C22semidirect
71. C4×#107C22semidirect
72. C4×#102C22semidirect
73. C4×#102C22semidirect
74. C4×C22⋊C8C22semidirect
188. C2×#41D8semidirect
216. C2×D8#41semidirect

Maximal subgroups

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

subgroupindex
#349352
#326482
#325442
#324582
#319942
#316662
#310562
#303962
#237542
#236962
C4×#4662
C4×#4472
#213732