Finite Group Databasebeta

← #6355465

Subgroups — #6355465

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60958C212yesmaximal, complemented
2#60940C211yesmaximal, characteristic, complemented
3C2×#2994C211yesmaximal, characteristic, complemented
4#19591C218yesmaximal, complemented
5#19582C218yesmaximal, complemented
6#19289C212yesmaximal, complemented
7#19283C212yesmaximal, complemented
8#19264C212yesmaximal, complemented
9#19258C212yesmaximal, complemented
10#17818C211yesmaximal, characteristic, complemented
11C2×#1551C212yesmaximal, complemented
12C2×#532C2212yes—
13C2×#529C2211yescharacteristic
14#3247C2218yes—
15#3245—21no—
16#3245—21no—
17#3238C2214yescomplemented
18#3234—22no—
19#3226C2214yescomplemented
20#3217C2214yescomplemented
21#3213—22no—
22#3205C2214yescomplemented
23#3187—22no—
24#3186—21no—
25#3186—21no—
26#3186—22no—
27#3180—24no—
28#3169C2212yes—
29#3169C2212yes—
30#3114—24no—
31#3112C2214yescomplemented
32#3109C2214yescomplemented
33#3100—24no—
34#2994—24no—
35#2994C2218yescomplemented
36C2×#446C2211yescharacteristic, complemented
37C2×C8×Q8C2211yescharacteristic, complemented
38C2×#436C2211yescharacteristic, complemented
39C2×#435—21no—
40C2×C8×D4—21no—
41C2×C8×D4C2211yescharacteristic, complemented
42C2×#433—21no—
43C2×#432C2212yes—
44C2×#431C2211yescharacteristic
45C2×#424—21no—
46C22×C4⋊C8C2211yescharacteristic
47#1698C2218yescomplemented
48#1697C2218yescomplemented
49#1696C2214yescomplemented
50#1695C2214yescomplemented
51#1688C2214yescomplemented
52#1687C2214yescomplemented
53#1678C2218yescomplemented
54#1677C2218yescomplemented
55#1676C2214yescomplemented
56#1675C2214yescomplemented
57#1668C2214yescomplemented
58#1667C2214yescomplemented
59#1573C2218yescomplemented
60#1568C2212yescomplemented
61#1568C2212yescomplemented
62#1551C2214yescomplemented
63#1551C2214yescomplemented
64#1551C2218yescomplemented
65C2×#333C2214yescomplemented
66C2×#331C2214yescomplemented
67C22×#41—22no—
68C22×M16—21no—
69C23×C8—21no—
70C23×C8—21no—
71#532—22no—
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532C2314yes—
78#532C2314yes—
79C2×C4⋊Q8C2312yes—
80C2×#127C2312yes—
81#529—22no—
82#529—22no—
83#529D414yes—
84#529C2314yes—
85C2×#126—21no—
86C2×#125D411yescharacteristic
87C2×#124—22no—
88C2×#124D411yescharacteristic
89C2×#123—21no—
90C2×C22⋊Q8D411yescharacteristic
91C2×#121—22no—
92C2×#121—22no—
93C2×#121D411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C22×C42C2311yescharacteristic
101#501—24no—
102#500C2312yes—
103#500C2312yes—
104#496—24no—
105#493C2312yes—
106#493C2312yes—
107#490—24no—
108#489—24no—
109#486C2314yes—
110#486C2314yes—
111#483—24no—
112#481—24no—
113#480—24no—
114#480—24no—
115#480—24no—
116#480—24no—
117#478—28no—
118#477C2318yes—
119#476C2318yes—
120#474—24no—
121#474—24no—
122#466—22no—
123#466—22no—
124#466—22no—
125#466—22no—
126#466—28no—
127#465—28no—
128#463—28no—
129#460C2318yes—
130#459—24no—
131#459—24no—
132#459C2318yes—
133#450—24no—
134#448—24no—
135#447—24no—
136#447—24no—
137#447—24no—
138#447—24no—
139#446—22no—
140#446C2314yes—
141C8×Q8—22no—
142C8×Q8C2314yes—
143#442—28no—
144#440—28no—
145C4×D8—22no—
146C4×D8—22no—
147C4×D8—22no—
148C4×D8—22no—
149#436—22no—
150#436C2314yes—
151#435—28no—
152C8×D4—28no—
153C8×D4—22no—
154C8×D4C2314yes—
155#433—28no—
156#432—22no—
157#432—22no—
158#432C2314yes—
159#432C2314yes—
160#431D414yes—
161#431C2314yes—
162#424—28no—
163C2×C4⋊C8—21no—
164C2×C4⋊C8—21no—
165C2×C4⋊C8—21no—
166C2×C4⋊C8—22no—
167C2×C4⋊C8D411yescharacteristic
168C2×C4⋊C8D411yescharacteristic
169C2×C4⋊C8C2314yes—
170C2×C4⋊C8C2314yes—
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×C22⋊C8—22no—
174C2×C22⋊C8—21no—
175C2×C22⋊C8—21no—
176C2×C22⋊C8D411yescharacteristic
177C2×C22⋊C8D411yescharacteristic
178C2×#97—21no—
179C2×#97—22no—
180C2×C4×C8—21no—
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183#335C2314yes—
184#335C2314yes—
185#334C2314yes—
186#334C2314yes—
187#333C2318yes—
188#333C2318yes—
189#331C2318yes—
190#331C2318yes—
191C2×#41—24no—
192C2×#41—24no—
193C2×#41—42no—
194C2×#41—42no—
195C2×#41—88no—
196C22×Q8—22no—
197C22×Q8C2×D411yescharacteristic
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4C2×D411yescharacteristic
204C23×C4—21no—
205C23×C4—42no—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4#4111yescharacteristic
209C2×SD16—28no—
210C2×D8—28no—
211C2×D8—22no—
212C2×D8—22no—
213C2×D8—22no—
214C2×D8—22no—
215C2×M16—24no—
216C2×M16—82no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8—82no—
223C22×C8—82no—
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8C2414yes—
232#127—22no—
233#127—22no—
234#127—22no—
235#127C2414yes—
236#126—24no—
237#126—24no—
238#125—22no—
239#125C2×D414yes—
240#124—22no—
241#124—24no—
242#124—24no—
243#124—42no—
244#124—42no—
245#124C2×D414yes—
246#123—24no—
247#123—24no—
248C22⋊Q8—22no—
249C22⋊Q8C2×D414yes—
250#121—22no—
251#121—24no—
252#121—24no—
253#121—24no—
254#121—24no—
255#121—42no—
256#121—42no—
257#121—42no—
258#121—42no—
259#121C2×D414yes—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8C2×D412yes—
263C4×Q8C2412yes—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—21no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—21no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4C2×D412yes—
277C4×D4C2412yes—
278#117—24no—
279#117—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C42—21no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D411yescharacteristic
300C2×C42#4112yes—
301C2×C42#4111yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2412yes—
304#107—22no—
305#107—22no—
306#107—22no—
307#107—22no—
308#107—28no—
309#106—28no—
310C4⋊C8—24no—
311C4⋊C8—24no—
312C4⋊C8—24no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—24no—
316C4⋊C8C2×D414yes—
317C4⋊C8C2×D414yes—
318C4⋊C8C2412yes—
319C4⋊C8C2412yes—
320C4⋊C8C2418yes—
321#103—28no—
322#102—28no—
323#102—24no—
324#102—24no—
325#102—24no—
326#102—24no—
327#102—28no—
328#102—28no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333C22⋊C8C2×D414yes—
334C22⋊C8C2×D414yes—
335#97—24no—
336#97—24no—
337#97—24no—
338C4×C8—24no—
339C4×C8C2×D412yes—
340C4×C8C2×D412yes—
341C4×C8C2412yes—
342C4×C8C2412yes—
343C24—21no—
344C24—42no—
345C24#12111yescharacteristic
346#41—88no—
347#41—88no—
348#41—88no—
349#41—88no—
350C2×Q8—21no—
351C2×Q8—24no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—84no—
355C2×Q8C22×D412yes—
356C2×D4—24no—
357C2×D4—21no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—84no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—84no—
372C2×D4—88no—
373C2×D4C22×D412yes—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—88no—
389C22×C4#12111yescharacteristic
390C22×C4#12112yes—
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4112yes—
401SD16—88no—
402D8—82no—
403D8—82no—
404D8—82no—
405D8—82no—
406D8—88no—
407M16—84no—
408C2×C8—22no—
409C2×C8—28no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—84no—
418C2×C8—84no—
419C2×C8—84no—
420C2×C8C22×D412yes—
421C2×C8C22×D412yes—
422C2×C8C22×D412yes—
423C2×C8C22×D412yes—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4C22×D412yes—
436C4⋊C4C22×D412yes—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C42—22no—
448C42—22no—
449C42#12112yes—
450C42#12112yes—
451C42#12112yes—
452C42C22×D411yescharacteristic
453C42C22×D411yescharacteristic
454C42C22×D412yes—
455C42C22×D412yes—
456C42C2×#4111yescharacteristic
457C42C2×#4112yes—
458C42C2511yescharacteristic
459C23—22no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23#47214yes—
470C23C2×#12112yes—
471C23C2×#12111yescharacteristic
472Q8—24no—
473Q8—42no—
474Q8—84no—
475Q8—84no—
476D4—24no—
477D4—42no—
478D4—48no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—84no—
483D4—84no—
484D4—84no—
485D4—84no—
486D4—88no—
487D4—88no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—88no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×#12111yescharacteristic
517C2×C4C2×#12112yes—
518C2×C4C2×#12112yes—
519C2×C4C2×#12112yes—
520C2×C4C2×#12111yescharacteristic
521C2×C4C23×D411yescharacteristic
522C2×C4C23×D411yescharacteristic
523C2×C4C22×#4111yescharacteristic
524C8—22no—
525C8—22no—
526C8—22no—
527C8—22no—
528C8—48no—
529C8—42no—
530C8—42no—
531C22—21no—
532C22—22no—
533C22—21no—
534C22—22no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—168no—
544C22C2×#47212yes—
545C22C2×#47214yes—
546C22C22×#12111yescharacteristic
547C4—21no—
548C4—22no—
549C4—21no—
550C4—22no—
551C4—21no—
552C4—22no—
553C4—21no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—84no—
558C4#358412yes—
559C4#358511yescharacteristic
560C4#358611yescharacteristic
561C2—22no—
562C2—24no—
563C2—82no—
564C2—164no—
565C2#6042511yescharacteristic
566C2#6045311yescharacteristic
567C2C22×#47211yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60958C2semidirect
2. #60940C2semidirect
3. C2×#2994C2semidirect
4. #19591C2semidirect
5. #19582C2semidirect
6. #19289C2semidirect
7. #19283C2semidirect
8. #19264C2semidirect
9. #19258C2semidirect
10. #17818C2semidirect
11. C2×#1551C2semidirect
17. #3238C22semidirect
19. #3226C22semidirect
20. #3217C22semidirect
22. #3205C22semidirect
31. #3112C22semidirect
32. #3109C22semidirect
35. #2994C22semidirect
36. C2×#446C22semidirect
37. C2×C8×Q8C22semidirect
38. C2×#436C22semidirect
41. C2×C8×D4C22semidirect
47. #1698C22semidirect
48. #1697C22semidirect
49. #1696C22semidirect
50. #1695C22semidirect
51. #1688C22semidirect
52. #1687C22semidirect
53. #1678C22semidirect
54. #1677C22semidirect
55. #1676C22semidirect
56. #1675C22semidirect
57. #1668C22semidirect
58. #1667C22semidirect
59. #1573C22semidirect
60. #1568C22semidirect
61. #1568C22semidirect
62. #1551C22semidirect
63. #1551C22semidirect
64. #1551C22semidirect
65. C2×#333C22semidirect
66. C2×#331C22semidirect

Maximal subgroups

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

subgroupindex
#609582
#609402
C2×#29942
#195912
#195822
#192892
#192832
#192642
#192582
#178182
C2×#15512