Finite Group Databasebeta

← #205463

Subgroups — #205463

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic, complemented
3#21258C211yesmaximal, characteristic
4#20535C211yesmaximal, characteristic, complemented
5C2×#1574C211yesmaximal, characteristic, complemented
6C2×#1559C211yesmaximal, characteristic, complemented
7C2×#1553C211yesmaximal, characteristic
8#9747C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9741C211yesmaximal, characteristic, complemented
11#9740C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9735C211yesmaximal, characteristic, complemented
14#9731C211yesmaximal, characteristic, complemented
15#9729C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3228—24no—
19#3228C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#482—21no—
25C2×#482—21no—
26C2×#478—21no—
27C2×#478—21no—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C2×#417—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1987C2211yescharacteristic
35#1987C2211yescharacteristic
36#1983—22no—
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1981C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1981C2211yescharacteristic
43#1872—22no—
44#1771—21no—
45#1771—21no—
46#1770C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C4×#103—22no—
51#1749—21no—
52C4×C22⋊C8—21no—
53#1574—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1559—24no—
59#1553—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×M16—21no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533C2×C414yes—
74#532—22no—
75#532—22no—
76#532—22no—
77#532C2×C414yes—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×#127C2311yescharacteristic
82C2×#125—21no—
83C2×#124—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×#121—21no—
87C2×#121—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×#117—21no—
93C22×C4⋊C4D412yes—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#500—22no—
97#500C2×C414yescomplemented
98#498—22no—
99#498C2×C414yescomplemented
100#486—22no—
101#486—22no—
102#486C2×C414yescomplemented
103#486C2×C414yescomplemented
104#482—28no—
105#482—28no—
106#478—28no—
107#478—28no—
108#432—22no—
109#432D414yes—
110#431—22no—
111#431—22no—
112#431C2×C414yes—
113#431D414yes—
114C2×#107—21no—
115C2×#107—21no—
116C2×#106—21no—
117C2×#106—21no—
118#423—22no—
119#423—22no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129#417—48no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—22no—
133C2×#103—22no—
134C2×#102—21no—
135C2×#102—21no—
136C2×#102—21no—
137C2×#102—21no—
138C2×#102—22no—
139C2×#102—22no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×#97—21no—
145C2×#97—21no—
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4D412yes—
155C4×C4⋊C4C2311yescharacteristic
156C4×C4⋊C4C2311yescharacteristic
157C2×#95—21no—
158#340—22no—
159#340—22no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#335—22no—
165#335—22no—
166#334—22no—
167#334—22no—
168#333—22no—
169#333—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#331—22no—
175#331—22no—
176C2×#41—24no—
177C2×#41—44no—
178C2×#41—44no—
179C2×#41—42no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—41no—
187C23×C4—22no—
188C23×C4—41no—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4111yescharacteristic
192C2×M16—44no—
193C2×M16—42no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C22×C412yes—
214C4⋊Q8C22×C412yes—
215C4⋊Q8C22×C412yes—
216#127—21no—
217#127—21no—
218#127—21no—
219#127C22×C412yes—
220#125—24no—
221#125—42no—
222#124—24no—
223#124—42no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—42no—
227C22⋊Q8—42no—
228#121—24no—
229#121—24no—
230#121—42no—
231#121—42no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—42no—
235C4×Q8—42no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—42no—
239C4×D4—42no—
240#117—24no—
241#117—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—41no—
262C2×C42—81no—
263C2×C42—81no—
264C2×C42C22×C412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42#4111yescharacteristic
270C2×C42#4112yes—
271C2×C42C2411yescharacteristic
272#107—24no—
273#107—24no—
274#106—24no—
275#106—24no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—42no—
287C4⋊C8C22⋊C414yes—
288C4⋊C8C22⋊C414yes—
289C4⋊C8C2×D412yes—
290C4⋊C8C2×D412yes—
291#103—44no—
292#103—44no—
293#103—44no—
294#103—44no—
295#102—24no—
296#102—24no—
297#102—24no—
298#102—24no—
299#102—44no—
300#102—44no—
301C22⋊C8—24no—
302C22⋊C8—42no—
303C22⋊C8C22⋊C414yes—
304C22⋊C8C22⋊C414yes—
305#97—24no—
306#97—81no—
307#97—81no—
308C4×C8—42no—
309C4×C8—81no—
310C4×C8—81no—
311C4×C8C22×C412yes—
312C4×C8C22×C412yes—
313C4×C8C2×D412yes—
314C4×C8C2×D412yes—
315#95—22no—
316#95—21no—
317#95—21no—
318C24—41no—
319C24#12111yescharacteristic
320#41—416no—
321#41—84no—
322#41—84no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—82no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—44no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—82no—
348C2×D4—84no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—44no—
364C22×C4—84no—
365C22×C4—82no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4#12112yes—
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12412yes—
374C22×C4#12712yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378M16—84no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8C2×C22⋊C412yes—
402C2×C8C2×C22⋊C412yes—
403C2×C8C2×C22⋊C412yes—
404C2×C8C2×C22⋊C412yes—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4AGL(1,ℤ/8)14yes—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—44no—
432C42—82no—
433C42C2×C22⋊C412yes—
434C42C4×D412yes—
435C42C4×D412yes—
436C42#12112yes—
437C42#12412yes—
438C42#12712yes—
439C42C23×C411yescharacteristic
440C42C22×D411yescharacteristic
441C42C22×D411yescharacteristic
442C42C2×#4111yescharacteristic
443C23—21no—
444C23—21no—
445C23—22no—
446C23—22no—
447C23—42no—
448C23—81no—
449C23—81no—
450C23—82no—
451C23—82no—
452C23#39012yes—
453C23C2×#12111yescharacteristic
454Q8—44no—
455Q8—44no—
456Q8—82no—
457Q8—82no—
458Q8—81no—
459Q8—81no—
460Q8—81no—
461Q8—81no—
462D4—42no—
463D4—42no—
464D4—44no—
465D4—82no—
466D4—82no—
467D4—88no—
468D4—81no—
469D4—81no—
470D4—81no—
471D4—81no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—88no—
496C2×C4—88no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#39012yes—
501C2×C4#39012yes—
502C2×C4#39012yes—
503C2×C4C22×C22⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×#12411yescharacteristic
508C2×C4C2×#12711yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)12yes—
510C2×C4#57512yes—
511C2×C4#57711yescharacteristic
512C2×C4#57711yescharacteristic
513C2×C4#57811yescharacteristic
514C2×C4#57811yescharacteristic
515C8—22no—
516C8—22no—
517C8—22no—
518C8—22no—
519C8—42no—
520C8—42no—
521C8—81no—
522C8—81no—
523C8—81no—
524C8—81no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—162no—
534C22—162no—
535C22C2×#39011yescharacteristic
536C22#304211yescharacteristic
537C22#304411yescharacteristic
538C22#305512yes—
539C22#306712yes—
540C4—21no—
541C4—21no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—44no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4#289811yescharacteristic
551C4#289811yescharacteristic
552C4C4×AGL(1,ℤ/8)12yes—
553C4#304611yescharacteristic
554C4#304611yescharacteristic
555C4#315111yescharacteristic
556C4#315311yescharacteristic
557C2—42no—
558C2—162no—
559C2#1946712yes—
560C2#1947912yes—
561C2#2123611yescharacteristic
562C2#2124811yescharacteristic
563C2#2126211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21385C2semidirect
4. #20535C2semidirect
5. C2×#1574C2semidirect
6. C2×#1559C2semidirect
8. #9747C2semidirect
9. #9746C2semidirect
10. #9741C2semidirect
11. #9740C2semidirect
12. #9738C2semidirect
13. #9735C2semidirect
14. #9731C2semidirect
15. #9729C2semidirect
19. #3228C4semidirect
97. #500C2×C4semidirect
99. #498C2×C4semidirect
102. #486C2×C4semidirect
103. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#213852
#212582
#205352
C2×#15742
C2×#15592
C2×#15532
#97472
#97462
#97412
#97402
#97382
#97352
#97312
#97292