Finite Group Databasebeta

← #6470474

Subgroups — #6470474

Order 512.

2300 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic
2#34173C211yesmaximal, characteristic
3#32967C212yesmaximal, complemented
4#32964C212yesmaximal, complemented
5#29963C212yesmaximal, complemented
6#29960C212yesmaximal, complemented
7#25965C214yesmaximal, complemented
8#25963C214yesmaximal, complemented
9#25957C214yesmaximal, complemented
10#25955C214yesmaximal, complemented
11#21941C212yesmaximal
12#21931C211yesmaximal, characteristic
13C2×#1787C212yesmaximal
14C24×Q8C2211yescharacteristic
15C2×#519C2211yescharacteristic
16C2×#519C2211yescharacteristic
17C2×#479C2214yescomplemented
18C2×#477C2212yescomplemented
19C2×#475C2212yescomplemented
20C22×#103C2214yes—
21C22×#103C2214yes—
22C22×C22⋊C8C2212yes—
23#2841—22no—
24#2659—22no—
25#2646—22no—
26#2629C2212yes—
27#2628C2212yes—
28#2442C2214yes—
29#2440C2214yes—
30#2437—22no—
31Q8×C22⋊C4—22no—
32Q8×C22⋊C4—22no—
33Q8×C22⋊C4C2214yes—
34Q8×C22⋊C4C2214yes—
35#2310—22no—
36#2310C2211yescharacteristic
37#2310C2211yescharacteristic
38#2280—22no—
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41C2×#380C2212yes—
42#2097C2214yescomplemented
43#2096C2212yescomplemented
44#2095C2212yescomplemented
45#2068C2214yescomplemented
46#2068C2214yescomplemented
47#2066C2214yescomplemented
48#2066C2214yescomplemented
49#2026C2212yescomplemented
50#2026C2212yescomplemented
51#2024C2214yescomplemented
52#1865C2214yescomplemented
53#1865C2214yescomplemented
54#1863C2214yescomplemented
55#1863C2214yescomplemented
56#1854C2214yes—
57#1854C2214yes—
58#1853C2212yes—
59#1853C2212yes—
60#1852C2212yes—
61#1852C2212yes—
62#1850C2214yes—
63#1850C2214yes—
64#1808C2214yes—
65#1807C2214yes—
66#1805C2212yescomplemented
67#1805C2212yescomplemented
68#1803C2214yescomplemented
69#1798C2212yes—
70#1798C2212yes—
71#1787C2214yes—
72#1787C2214yes—
73#1787C2214yes—
74#1787C2214yes—
75C23×Q8—21no—
76C23×Q8—22no—
77C23×Q8—24no—
78C23×Q8—44no—
79C23×Q8—44no—
80C23×Q8Q818yescomplemented
81C23×Q8C2311yescharacteristic
82C23×Q8C2314yes—
83C23×Q8C2311yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C23×C8C2314yes—
87C2×C4⋊Q8C2312yes—
88C2×#125C2312yes—
89C2×C22⋊Q8—24no—
90#519—44no—
91#519—44no—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2314yes—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107#479—48no—
108#477—44no—
109#475—44no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2314yes—
117C2×#103—48no—
118C2×#103—48no—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×C22⋊C8—28no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131#400—22no—
132#400—24no—
133#399—22no—
134#398—24no—
135#398—24no—
136#398—24no—
137#396—24no—
138#395—22no—
139#395—24no—
140#393—22no—
141#391—21no—
142#391—22no—
143#391—21no—
144#391—24no—
145#391—24no—
146#391—24no—
147#391—24no—
148#391C2312yes—
149#391C2314yes—
150#391C2312yes—
151#389C2314yes—
152#389C2314yes—
153#385—24no—
154#385—24no—
155#384C2312yes—
156#382C2312yes—
157#381—21no—
158#381—22no—
159#381—22no—
160#381D412yes—
161#380—22no—
162#380—22no—
163#380—24no—
164#380—24no—
165#380—24no—
166#380D412yes—
167#380D412yes—
168#380C2312yes—
169#380C2314yes—
170#380C2312yes—
171#380C2314yes—
172#379—21no—
173#379D412yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4—24no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4C2314yes—
178C2×#95—22no—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2312yes—
184#336C2312yes—
185#336C2314yes—
186C25C2×D411yescharacteristic
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—28no—
190C22×Q8—216no—
191C22×Q8—21no—
192C22×Q8—24no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—28no—
198C22×Q8—28no—
199C22×Q8—28no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—432no—
207C22×Q8—432no—
208C22×Q8—48no—
209C22×Q8—48no—
210C22×Q8C2×Q814yes—
211C22×Q8C2×Q8116yes—
212C22×Q8C2×Q814yes—
213C22×Q8C2412yes—
214C22×Q8C2412yes—
215C23×C4—21no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×Q814yes—
228C23×C4#4114yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C22×C8—28no—
232C22×C8—44no—
233C22×C8—44no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238#125—44no—
239C22⋊Q8—88no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—28no—
243C4×Q8—44no—
244C4×Q8—42no—
245C4×Q8—42no—
246C4×Q8—44no—
247C4×Q8—48no—
248C4×Q8—44no—
249#117—42no—
250#117—42no—
251#117—42no—
252#117—42no—
253#117—48no—
254#117—48no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2411yescharacteristic
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2411yescharacteristic
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—84no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C2412yes—
292C2×C42C2412yes—
293#107—44no—
294#107—44no—
295#106—44no—
296#106—44no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8—44no—
300#103—28no—
301#103—28no—
302#103—28no—
303#103—28no—
304#103—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308C22⋊C8—28no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—24no—
326C24—24no—
327C24—41no—
328C24—41no—
329C24C22⋊Q814yes—
330C24C22×D411yescharacteristic
331C24C22×D412yes—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—216no—
335C2×Q8—216no—
336C2×Q8—24no—
337C2×Q8—216no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—232no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—416no—
349C2×Q8—48no—
350C2×Q8—416no—
351C2×Q8—416no—
352C2×Q8—432no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—48no—
356C2×Q8—464no—
357C2×Q8—84no—
358C2×Q8—82no—
359C2×Q8C22×Q814yes—
360C2×Q8C22×Q814yes—
361C2×Q8H(5,𝔽2)14yes—
362C2×Q8H(5,𝔽2)14yes—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—216no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—432no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4C22⋊Q814yes—
397C22×C4C22⋊Q818yes—
398C22×C4C22×D412yes—
399C22×C4C22×D414yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q812yes—
404C22×C4C22×Q814yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4114yes—
407C22×C4C2511yescharacteristic
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—88no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—82no—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—84no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—44no—
449C42—48no—
450C42—44no—
451C42—42no—
452C42—42no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—216no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—44no—
463C23—48no—
464C23—44no—
465C23—44no—
466C23C2×C22⋊Q812yes—
467C23C2×C22⋊Q814yes—
468C23#57714yes—
469C23#57814yes—
470C23C23×D411yescharacteristic
471Q8—216no—
472Q8—24no—
473Q8—24no—
474Q8—24no—
475Q8—28no—
476Q8—432no—
477Q8—48no—
478Q8—44no—
479Q8—164no—
480Q8#54318yes—
481C2×C4—216no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—432no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—164no—
521C2×C4—164no—
522C2×C4C2×C22⋊Q812yes—
523C2×C4C2×C22⋊Q814yes—
524C2×C4C23×Q811yescharacteristic
525C2×C4C22×#4111yescharacteristic
526C2×C4C2×H(5,𝔽2)12yes—
527C2×C4C2×H(5,𝔽2)12yes—
528C8—44no—
529C8—44no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—416no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22#308218yes—
543C22C22×C22⋊Q811yescharacteristic
544C22#348411yescharacteristic
545C22#348411yescharacteristic
546C22C2×#57712yes—
547C22C2×#57812yes—
548C4—28no—
549C4—22no—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—84no—
557C4—81no—
558C4—84no—
559C4—81no—
560C4C2×#54312yes—
561C4#349012yes—
562C2—28no—
563C2—42no—
564C2#3498211yescharacteristic
565C2C2×#308212yes—
566C2#6163212yes—
567C2#6163311yescharacteristic
568C2#6163411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32967C2semidirect
4. #32964C2semidirect
5. #29963C2semidirect
6. #29960C2semidirect
7. #25965C2semidirect
8. #25963C2semidirect
9. #25957C2semidirect
10. #25955C2semidirect
17. C2×#479C22semidirect
18. C2×#477C22semidirect
19. C2×#475C22semidirect
42. #2097C22semidirect
43. #2096C22semidirect
44. #2095C22semidirect
45. #2068C22semidirect
46. #2068C22semidirect
47. #2066C22semidirect
48. #2066C22semidirect
49. #2026C22semidirect
50. #2026C22semidirect
51. #2024C22semidirect
52. #1865C22semidirect
53. #1865C22semidirect
54. #1863C22semidirect
55. #1863C22semidirect
66. #1805C22semidirect
67. #1805C22semidirect
68. #1803C22semidirect
80. C23×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#374772
#341732
#329672
#329642
#299632
#299602
#259652
#259632
#259572
#259552
#219412
#219312
C2×#17872