Finite Group Databasebeta

← #6477606

Subgroups — #6477606

Order 512.

2234 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
1C22×#483C211yesmaximal, characteristic, complemented
2C22×#480C211yesmaximal, characteristic
3#35019C211yesmaximal, characteristic
4C2×#2344C211yesmaximal, characteristic, complemented
5#33111C214yesmaximal, complemented
6#29760C214yesmaximal, complemented
7#24581C214yesmaximal, complemented
8#24577C214yesmaximal, complemented
9#24481C214yesmaximal, complemented
10#24477C214yesmaximal, complemented
11#21930C211yesmaximal, characteristic, complemented
12#21759C211yesmaximal, characteristic, complemented
13#21758C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117—21no—
17C23×C4⋊C4C2211yescharacteristic
18C2×#483—28no—
19C2×#483C2214yes—
20C2×#483C2212yescomplemented
21C2×#483C2212yescomplemented
22C2×#480—28no—
23C2×#480C2214yes—
24C2×#480C2212yes—
25C2×#480C2212yes—
26C22×#107C2212yes—
27C22×#106C2212yescomplemented
28C22×#103C2212yescomplemented
29C22×#102C2212yes—
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2850C2214yes—
33#2639—22no—
34#2591—22no—
35#2561—22no—
36#2504C2212yes—
37#2503C2212yes—
38#2436C2212yes—
39#2435C2212yes—
40#2344—22no—
41#2344—22no—
42#2344C2214yes—
43#2344C2214yescomplemented
44#2308—21no—
45#2308—21no—
46C2×#389C2212yescomplemented
47C2×#387—22no—
48C2×#381—21no—
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2212yescomplemented
51C2×C4×C22⋊C4C2211yescharacteristic
52#2085C2218yescomplemented
53#2068C2218yescomplemented
54#2065C2218yescomplemented
55#2042C2218yescomplemented
56#1983C2218yescomplemented
57#1949C2218yescomplemented
58#1928C2218yescomplemented
59#1926C2218yescomplemented
60#1893C2214yes—
61#1892C2214yes—
62#1874C2214yes—
63#1872C2214yescomplemented
64#1850C2214yes—
65#1850C2214yescomplemented
66#1824C2212yescomplemented
67#1823C2212yes—
68#1797C2214yescomplemented
69#1787C2212yescomplemented
70#1786C2212yes—
71C23×Q8C2311yescharacteristic
72C23×D4—21no—
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C24×C4C2311yescharacteristic
76C23×C8C2312yes—
77C23×C8C2312yes—
78C2×#124C2314yes—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8C2312yes—
84C2×C22⋊Q8C2312yes—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121C2312yes—
90C2×#121C2312yes—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—48no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4D418yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D411yescharacteristic
112C22×C22⋊C4D411yescharacteristic
113C22×C42—22no—
114C22×C42C2312yes—
115#483—216no—
116#483—28no—
117#483—28no—
118#480—216no—
119#480—28no—
120#480—28no—
121C2×#107—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106—28no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2318yes—
128C2×#103—28no—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2318yes—
132C2×#102—28no—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2318yes—
136C2×C22⋊C8—24no—
137C2×C22⋊C8—24no—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142#400—24no—
143#397—24no—
144#394—24no—
145#393—24no—
146#391C2314yes—
147#390C2314yes—
148#389—24no—
149#389C2318yes—
150#388—24no—
151#388—24no—
152#387—28no—
153#387—28no—
154#386—24no—
155#386—24no—
156#386—22no—
157#386—22no—
158#386—22no—
159#386—22no—
160#382C2314yes—
161#381—22no—
162#381—22no—
163#381—24no—
164#381—24no—
165#380—21no—
166#380—21no—
167#380C2312yes—
168#380C2312yes—
169#379—22no—
170#379—22no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4C2318yes—
173C4×C22⋊C4D412yes—
174C4×C22⋊C4D412yes—
175C4×C22⋊C4C2312yes—
176C4×C22⋊C4C2312yes—
177C2×#95—22no—
178C2×#95—21no—
179C2×#95—22no—
180C2×#95D412yes—
181#340C2318yes—
182#340C2318yes—
183#336C2318yes—
184C25—21no—
185C25C2×D411yescharacteristic
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—44no—
190C22×Q8C2412yes—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4C2412yes—
201C23×C4—28no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C22×C8—24no—
224C22×C8—24no—
225C22×C8—44no—
226C22×C8—44no—
227C22×C8C2414yes—
228C22×C8C2414yes—
229#124—48no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—44no—
235C22⋊Q8—44no—
236#121—24no—
237#121—24no—
238#121—24no—
239#121—24no—
240#121—44no—
241#121—44no—
242#117—416no—
243#117—416no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—48no—
299C2×C42C2×D414yes—
300C2×C42C2414yes—
301#107—28no—
302#107—28no—
303#106—28no—
304#106—28no—
305C4⋊C8—48no—
306#103—28no—
307#103—28no—
308#103—48no—
309#102—28no—
310#102—28no—
311#102—48no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316#95—28no—
317#95—22no—
318#95—22no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95C2×D414yes—
325#95C2×D414yes—
326C24—22no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—82no—
333C24—82no—
334C24#12114yes—
335C24#13514yes—
336C24AGL(1,ℤ/8)14yes—
337C24C22×D411yescharacteristic
338C24C22×D411yescharacteristic
339C24C22×D411yescharacteristic
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—28no—
343C2×Q8—48no—
344C2×Q8—42no—
345C2×Q8—82no—
346C2×Q8—84no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—28no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—84no—
359C2×D4—82no—
360C2×D4—816no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—432no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4#12114yes—
405C22×C4#12114yes—
406C22×C4#12114yes—
407C22×C4C22×D414yes—
408C22×C4C22×D414yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2511yescharacteristic
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—28no—
422C2×C8—28no—
423C2×C8—48no—
424C2×C8—48no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—28no—
436C4⋊C4—44no—
437C4⋊C4—416no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4C22×D414yes—
443C4⋊C4C22×D414yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C42—28no—
459C42—44no—
460C42—416no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—44no—
468C23—44no—
469C23—48no—
470C23—48no—
471C23—44no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23—164no—
480C23—164no—
481C23#463116yes—
482C23C2×#12112yes—
483C23C2×#12112yes—
484C23C2×#12112yes—
485C23C2×#13512yes—
486C23C2×#13512yes—
487C23C2×#13512yes—
488C23C2×AGL(1,ℤ/8)12yes—
489C23C2×AGL(1,ℤ/8)12yes—
490C23C2×AGL(1,ℤ/8)12yes—
491C23C23×D411yescharacteristic
492Q8—44no—
493Q8—44no—
494Q8—84no—
495D4—44no—
496D4—44no—
497D4—84no—
498D4—88no—
499D4—88no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—48no—
521C2×C4—416no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—416no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4—88no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4C2×#12112yes—
545C2×C4C2×#12112yes—
546C2×C4C2×#12112yes—
547C2×C4C23×D411yescharacteristic
548C2×C4C22×#4111yescharacteristic
549C2×C4#58514yes—
550C8—44no—
551C8—44no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—416no—
565C22—84no—
566C22—82no—
567C22—84no—
568C22—82no—
569C22—82no—
570C22—168no—
571C22C2×#46314yes—
572C22C2×#46314yes—
573C22C2×#46314yes—
574C22C22×#12111yescharacteristic
575C22#348512yes—
576C22C22×#13511yescharacteristic
577C22C22×AGL(1,ℤ/8)11yescharacteristic
578C22#358612yes—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—44no—
586C4—82no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4#344912yes—
591C4#349112yes—
592C2—24no—
593C2—24no—
594C2—24no—
595C2—162no—
596C2#3508911yescharacteristic
597C2C22×#46311yescharacteristic
598C2#6042411yescharacteristic
599C2#6169612yes—
600C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#483C2semidirect
4. C2×#2344C2semidirect
5. #33111C2semidirect
6. #29760C2semidirect
7. #24581C2semidirect
8. #24577C2semidirect
9. #24481C2semidirect
10. #24477C2semidirect
11. #21930C2semidirect
12. #21759C2semidirect
20. C2×#483C22semidirect
21. C2×#483C22semidirect
27. C22×#106C22semidirect
28. C22×#103C22semidirect
43. #2344C22semidirect
46. C2×#389C22semidirect
50. C2×C4×C4⋊C4C22semidirect
52. #2085C22semidirect
53. #2068C22semidirect
54. #2065C22semidirect
55. #2042C22semidirect
56. #1983C22semidirect
57. #1949C22semidirect
58. #1928C22semidirect
59. #1926C22semidirect
63. #1872C22semidirect
65. #1850C22semidirect
66. #1824C22semidirect
68. #1797C22semidirect
69. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C22×#4802
#350192
C2×#23442
#331112
#297602
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