Finite Group Databasebeta

← #6470815

Subgroups — #6470815

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60751C212yesmaximal
2C2×#3195C211yesmaximal, characteristic, complemented
3#60141C211yesmaximal, characteristic
4#33126C212yesmaximal, complemented
5#32946C212yesmaximal, complemented
6#29792C212yesmaximal, complemented
7#29752C212yesmaximal, complemented
8#25268C218yesmaximal, complemented
9#25142C218yesmaximal, complemented
10#21949C211yesmaximal, characteristic, complemented
11C2×#1901C212yesmaximal, complemented
12C2×#568C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C2×#525C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16#3196—24no—
17#3195—22no—
18#3195—22no—
19#3195—24no—
20#3195—22no—
21#3193—22no—
22#3190—24no—
23#3189—22no—
24#3187—22no—
25#3095C2214yes—
26#3092C2214yes—
27C2×#483C2214yescomplemented
28C2×#473C2214yescomplemented
29#3056C2218yes—
30C2×#470C2212yes—
31C2×#470C2212yes—
32#3007C2214yes—
33#3004C2214yes—
34C2×#451C2214yescomplemented
35D4×M16—24no—
36#2933C2218yes—
37#2930—22no—
38#2929—22no—
39#2905C2211yescharacteristic
40C22×#101C2214yescomplemented
41#2879C2211yescharacteristic
42C2×#407C2212yes—
43C2×#407C2211yescharacteristic
44#2873C2211yescharacteristic
45#2090C2214yescomplemented
46#2079C2214yescomplemented
47#2064C2218yescomplemented
48#2052C2218yescomplemented
49#2047C2214yescomplemented
50#2020C2214yescomplemented
51#1979C2214yescomplemented
52#1951C2214yescomplemented
53#1930C2218yescomplemented
54#1908C2218yescomplemented
55#1901C2214yes—
56#1901C2214yes—
57#1901C2218yescomplemented
58#1855C2218yescomplemented
59#1815C2214yescomplemented
60#1811C2212yes—
61#1811C2212yes—
62C2×#343C2214yescomplemented
63C22×#41C2311yescharacteristic
64C23×Q8C2312yes—
65C24×C4C2311yescharacteristic
66C2×#137C2318yes—
67C22×Q16C2312yes—
68C22×Q16C2312yes—
69#568—44no—
70C2×#131—22no—
71C2×#131C2314yes—
72C22×M16—21no—
73C22×M16—22no—
74C22×M16—22no—
75C22×M16—44no—
76C22×M16C2311yescharacteristic
77C22×M16C2314yes—
78C22×M16C2314yes—
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C22×M16C2311yescharacteristic
82C23×C8—21no—
83#525—21no—
84#525—21no—
85#525—22no—
86#525D412yes—
87#525D412yes—
88#525C2314yes—
89C2×#123—21no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8D412yes—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×#121D412yes—
104C2×#120—21no—
105C2×C4×D4—22no—
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—22no—
110#485—28no—
111#484—24no—
112#484—24no—
113#483—24no—
114#483—24no—
115#483—28no—
116#483—28no—
117#481—24no—
118#480—24no—
119#473—48no—
120#470—24no—
121#470—24no—
122#470—24no—
123#470—24no—
124#470—28no—
125#468—28no—
126#467—28no—
127#465—28no—
128#452—28no—
129#451—24no—
130#451—24no—
131#451—24no—
132#451—28no—
133#451—28no—
134#450—24no—
135#448—24no—
136#448—24no—
137#436—28no—
138#435—28no—
139#435—28no—
140C8×D4—28no—
141C2×#108C2314yes—
142C2×#108C2314yes—
143#428C2318yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#424—22no—
147#424C2314yes—
148#423—22no—
149#419C2318yes—
150#417C2318yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#101—48no—
154C2×#101C2318yes—
155C2×#101C2318yes—
156#408—22no—
157#408C2314yes—
158#407—22no—
159#407—22no—
160#407—22no—
161#407D412yes—
162#407D412yes—
163#407D414yes—
164#407C2314yes—
165#407C2314yes—
166C2×C22⋊C8—22no—
167C2×C22⋊C8—22no—
168C2×C22⋊C8C2314yes—
169#405—22no—
170#405C2314yes—
171C4×M16—22no—
172#343C2318yes—
173#343C2318yes—
174#341C2314yes—
175#341C2314yes—
176C25#4111yescharacteristic
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—42no—
180C2×#41—42no—
181C2×#41C2×D412yes—
182C2×#41C2412yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8C2×D411yescharacteristic
191C22×Q8C2414yes—
192C22×D4—21no—
193C22×D4—22no—
194C22×D4C2×D411yescharacteristic
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4#4112yes—
207C23×C4#4111yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210#137—48no—
211#137—48no—
212C2×Q16—28no—
213C2×Q16—24no—
214C2×Q16—24no—
215C2×Q16—44no—
216C2×Q16—44no—
217C2×SD16—28no—
218#131—28no—
219#131—44no—
220#131—48no—
221C2×M16—21no—
222C2×M16—24no—
223C2×M16—21no—
224C2×M16—24no—
225C2×M16—24no—
226C2×M16—24no—
227C2×M16—28no—
228C2×M16—42no—
229C2×M16—42no—
230C2×M16—44no—
231C2×M16—44no—
232C2×M16—42no—
233C2×M16—48no—
234C2×M16—48no—
235C2×M16—44no—
236C2×M16—44no—
237C2×M16C2×D412yes—
238C2×M16C2×D412yes—
239C2×M16C2×D412yes—
240C2×M16C2×D412yes—
241C2×M16C2412yes—
242C2×M16C2418yes—
243C2×M16C2412yes—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—24no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—44no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252#123—22no—
253#123—22no—
254#123—24no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—42no—
266C22⋊Q8—42no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—42no—
273C22⋊Q8C2×D414yes—
274C22⋊Q8C2×D414yes—
275#121—22no—
276#121—22no—
277#121C2×D414yes—
278#121C2×D414yes—
279#120—22no—
280#120—22no—
281#120—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286#117—21no—
287#117—21no—
288#117C2×D412yes—
289#117C2412yes—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2414yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C42—21no—
314#108—44no—
315#108—44no—
316#107—28no—
317#106—24no—
318#106—24no—
319#106—28no—
320C4⋊C8—28no—
321#103—28no—
322#103—28no—
323#103—28no—
324#103—28no—
325#103—28no—
326#102—28no—
327#101—216no—
328#101—48no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333C22⋊C8—28no—
334C22⋊C8C2×D418yes—
335C22⋊C8C2×D418yes—
336#97—24no—
337C4×C8—24no—
338C24—21no—
339C24—42no—
340C24—44no—
341C24#12111yescharacteristic
342C24#12112yes—
343C24#12111yescharacteristic
344C24C2×#4112yes—
345C24C2×#4111yescharacteristic
346#41—24no—
347#41—24no—
348#41—44no—
349#41—42no—
350#41—44no—
351#41—42no—
352C2×Q8—21no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—42no—
358C2×Q8—48no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—82no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8C22×D412yes—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—84no—
376C2×D4C22×D412yes—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—84no—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4#12112yes—
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4C2511yescharacteristic
424Q16—84no—
425Q16—84no—
426Q16—88no—
427SD16—88no—
428M16—24no—
429M16—22no—
430M16—28no—
431M16—24no—
432M16—28no—
433M16—42no—
434M16—42no—
435M16—42no—
436M16—42no—
437M16—48no—
438M16—48no—
439M16—42no—
440M16—84no—
441M16H(5,𝔽2)14yes—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—28no—
447C2×C8—22no—
448C2×C8—48no—
449C2×C8—48no—
450C2×C8—42no—
451C2×C8—84no—
452C2×C8C22×D414yes—
453C2×C8C22×D414yes—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4C22×D414yes—
466C22⋊C4—22no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C42—22no—
477C42—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—82no—
493C23—88no—
494C23C2×#12111yescharacteristic
495C23C2×#12112yes—
496C23C2×#12111yescharacteristic
497C23C2×#12112yes—
498C23C22×#4111yescharacteristic
499Q8—24no—
500Q8—42no—
501Q8—44no—
502Q8—44no—
503Q8—48no—
504Q8—84no—
505Q8—84no—
506D4—24no—
507D4—42no—
508D4—44no—
509D4—44no—
510D4—48no—
511D4—84no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12112yes—
550C2×C4C2×#12112yes—
551C2×C4C2×#12111yescharacteristic
552C2×C4C23×D411yescharacteristic
553C2×C4C23×D411yescharacteristic
554C2×C4C2×H(5,𝔽2)12yes—
555C2×C4#58512yes—
556C8—24no—
557C8—24no—
558C8—24no—
559C8—42no—
560C8—48no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—21no—
569C22—22no—
570C22—21no—
571C22—48no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—82no—
576C22—84no—
577C22—82no—
578C22—162no—
579C22#306914yes—
580C22C22×#12111yescharacteristic
581C22#345912yes—
582C4—21no—
583C4—21no—
584C4—22no—
585C4—22no—
586C4—44no—
587C4—42no—
588C4—84no—
589C4—84no—
590C4#348411yescharacteristic
591C4#348511yescharacteristic
592C4#349612yes—
593C2—22no—
594C2—24no—
595C2—42no—
596C2—41no—
597C2—41no—
598C2—82no—
599C2#3498411yescharacteristic
600C2C2×#306912yes—
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3195C2semidirect
4. #33126C2semidirect
5. #32946C2semidirect
6. #29792C2semidirect
7. #29752C2semidirect
8. #25268C2semidirect
9. #25142C2semidirect
10. #21949C2semidirect
11. C2×#1901C2semidirect
27. C2×#483C22semidirect
28. C2×#473C22semidirect
34. C2×#451C22semidirect
40. C22×#101C22semidirect
45. #2090C22semidirect
46. #2079C22semidirect
47. #2064C22semidirect
48. #2052C22semidirect
49. #2047C22semidirect
50. #2020C22semidirect
51. #1979C22semidirect
52. #1951C22semidirect
53. #1930C22semidirect
54. #1908C22semidirect
57. #1901C22semidirect
58. #1855C22semidirect
59. #1815C22semidirect
62. C2×#343C22semidirect

Maximal subgroups

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

subgroupindex
#607512
C2×#31952
#601412
#331262
#329462
#297922
#297522
#252682
#251422
#219492
C2×#19012