Finite Group Databasebeta

← #192702

Subgroups — #192702

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#21893C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic, complemented
4#20800C211yesmaximal, characteristic, complemented
5C2×#1484C211yesmaximal, characteristic
6#8889C212yesmaximal, complemented
7#8886C212yesmaximal, complemented
8#8885C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic, complemented
11#7903C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14C2×#431C2211yescharacteristic
15C2×#428—22no—
16C2×#423C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2361—22no—
19#2361—22no—
20#2354—21no—
21#2354C2212yes—
22D4×C4⋊C4—21no—
23D4×C4⋊C4C2212yes—
24#2326—22no—
25#2311—21no—
26C4×C4⋊Q8—21no—
27C4×#127—21no—
28#2285C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1987—21no—
32#1982—21no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1895—22no—
38#1854—21no—
39#1854—21no—
40#1847—22no—
41#1845C2212yes—
42#1844C2212yes—
43#1802C2212yes—
44#1802C2212yes—
45C4×#107—21no—
46C4×#106—21no—
47#1484C418yescomplemented
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41—21no—
55C22×M16—21no—
56C22×M16—21no—
57#532—22no—
58#532—22no—
59#532C2×C414yes—
60#532Q814yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×#127D411yescharacteristic
63C2×#124—21no—
64C2×#121—21no—
65C2×#121D412yes—
66#520—44no—
67#520—44no—
68C2×C4×Q8—22no—
69C2×C4×D4—22no—
70C2×C4×D4—24no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4C2312yes—
75C2×#117—22no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4D412yes—
83C22×C42C2311yescharacteristic
84#431—22no—
85#431C2×C414yescomplemented
86#428—48no—
87C2×#107—24no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—24no—
91C2×#106—21no—
92C2×#106—21no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#102—22no—
100C2×#102—22no—
101C2×C22⋊C8—21no—
102C4×M16—22no—
103C4×M16C2×C414yescomplemented
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2312yes—
107#391—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#389—22no—
112#389—22no—
113#388—24no—
114#388—24no—
115#386—24no—
116#385—24no—
117#385—24no—
118#384—21no—
119#384—21no—
120#384D412yes—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2312yes—
124#382—21no—
125#380—22no—
126#380—22no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139#376—21no—
140C43—21no—
141#340—22no—
142#340—22no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147D4⋊C8C2×C418yescomplemented
148D4⋊C8C2×C418yescomplemented
149C2×#41—24no—
150C2×#41—24no—
151C2×#41—42no—
152C2×#41—42no—
153C2×#41—42no—
154C2×#41—42no—
155C22×Q8—21no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C2×D412yes—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—42no—
170C2×M16—81no—
171C2×M16—81no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C4⋊Q8—22no—
177C4⋊Q8C4⋊C414yes—
178#127—22no—
179#127C4⋊C414yes—
180#124—24no—
181#124—24no—
182#121—24no—
183#121—24no—
184#121—24no—
185#121C4⋊C418yes—
186C4×Q8—48no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—48no—
192C4×D4—416no—
193C4×D4—82no—
194C4×D4—82no—
195C4×D4—84no—
196C4×D4C22×C414yes—
197C4×D4C2×Q814yes—
198#117—22no—
199#117—22no—
200#117—41no—
201#117—41no—
202#117—44no—
203#117—44no—
204#117—44no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—41no—
239C2×C42—21no—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—41no—
247C2×C42C22×C412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×Q812yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42C2411yescharacteristic
255#107—42no—
256#107—48no—
257#107—42no—
258#106—42no—
259#106—48no—
260#106—42no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—42no—
266C4⋊C8C22×C412yes—
267C4⋊C8C22×C412yes—
268#102—82no—
269#102—82no—
270C22⋊C8—24no—
271#97—24no—
272C4×C8—24no—
273C4×C8—22no—
274C4×C8C22×C414yes—
275#95—24no—
276#95—24no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—41no—
282#95—41no—
283C24—22no—
284C24#12011yescharacteristic
285#41—48no—
286#41—48no—
287#41—48no—
288#41—48no—
289C2×Q8—24no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×D4—28no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4C2×C4⋊C414yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—48no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4C2×C4⋊C412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4#12312yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349M16—216no—
350M16—84no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—24no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—48no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—82no—
390C4⋊C4C2×C4⋊C414yes—
391C4⋊C4AGL(1,ℤ/8)12yes—
392C4⋊C4AGL(1,ℤ/8)12yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—28no—
397C22⋊C4—48no—
398C22⋊C4—42no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C22⋊C4—82no—
402C42—24no—
403C42—42no—
404C42—48no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—42no—
410C42—42no—
411C42—82no—
412C42C2×C4⋊C412yes—
413C42C4×D412yes—
414C42C4×D412yes—
415C42C22⋊Q812yes—
416C42C22⋊Q812yes—
417C42#12312yes—
418C42C23×C411yescharacteristic
419C42C22×Q811yescharacteristic
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—24no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23#38512yes—
436C23C2×#12011yescharacteristic
437Q8—48no—
438D4—24no—
439D4—24no—
440D4—48no—
441D4—416no—
442D4—42no—
443D4—42no—
444D4—48no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—168no—
490C2×C4—164no—
491C2×C4#38512yes—
492C2×C4#38512yes—
493C2×C4#38512yes—
494C2×C4#45312yes—
495C2×C4#45612yes—
496C2×C4C22×C4⋊C411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C22⋊Q811yescharacteristic
500C2×C4C2×C22⋊Q811yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
504C2×C4#57511yescharacteristic
505C2×C4#57511yescharacteristic
506C2×C4#57611yescharacteristic
507C2×C4#57711yescharacteristic
508C2×C4#57711yescharacteristic
509C2×C4#57811yescharacteristic
510C8—28no—
511C8—42no—
512C8—42no—
513C8—82no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22C2×#38511yescharacteristic
529C22#300311yescharacteristic
530C22#300511yescharacteristic
531C22#301211yescharacteristic
532C22#301311yescharacteristic
533C22C2×#45311yescharacteristic
534C22C2×#45611yescharacteristic
535C4—22no—
536C4—21no—
537C4—21no—
538C4—22no—
539C4—42no—
540C4—44no—
541C4—42no—
542C4—84no—
543C4—81no—
544C4—81no—
545C4—161no—
546C4—161no—
547C4#185914yes—
548C4#291211yescharacteristic
549C4#291311yescharacteristic
550C4C4×AGL(1,ℤ/8)11yescharacteristic
551C4C4×AGL(1,ℤ/8)11yescharacteristic
552C4#307711yescharacteristic
553C4#307711yescharacteristic
554C4#309611yescharacteristic
555C4#309711yescharacteristic
556C2—42no—
557C2—84no—
558C2#1856611yescharacteristic
559C2#1856911yescharacteristic
560C2#1859011yescharacteristic
561C2#1859311yescharacteristic
562C2#2084911yescharacteristic
563C2C2×#185911yescharacteristic
564C2#2087311yescharacteristic
subgroupG/H
center, socleC23C2×#120
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
3. #21362C2semidirect
4. #20800C2semidirect
6. #8889C2semidirect
7. #8886C2semidirect
8. #8885C2semidirect
9. #8882C2semidirect
10. #8043C2semidirect
11. #7903C2semidirect
47. #1484C4semidirect
85. #431C2×C4semidirect
103. C4×M16C2×C4semidirect
147. D4⋊C8C2×C4semidirect
148. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342072
#218932
#213622
#208002
C2×#14842
#88892
#88862
#88852
#88822
#80432
#79032