Finite Group Databasebeta

← #205495

Subgroups — #205495

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60673C211yesmaximal, characteristic
2C2×#1770C211yesmaximal, characteristic, complemented
3#17793C211yesmaximal, characteristic, complemented
4#17546C212yesmaximal, complemented
5#17466C212yesmaximal, complemented
6#9754C214yesmaximal, complemented
7#9753C212yesmaximal, complemented
8#9752C212yesmaximal, complemented
9#3444C2212yes—
10#3170C411yescharacteristic, complemented
11#3170C411yescharacteristic, complemented
12#3169C411yescharacteristic, complemented
13#3169C411yescharacteristic, complemented
14#3168—21no—
15#3168—21no—
16#3168—21no—
17#3168—21no—
18#3167C412yescomplemented
19#3167C412yescomplemented
20#3160C2211yescharacteristic
21#3147C2211yescharacteristic
22#3131C2212yes—
23#3050—21no—
24#3050—21no—
25C4×#135—21no—
26C4×#135—21no—
27#2920C2212yes—
28C22×C4⋊C8C2211yescharacteristic
29C22×#97—21no—
30C22×C4×C8C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1770—21no—
33#1770—21no—
34#1770C2212yes—
35#1770C2212yes—
36C2×#336—21no—
37#1568—21no—
38#1568—21no—
39#1567—21no—
40#1566—21no—
41#1564C2212yes—
42#1564C2212yes—
43#1509—24no—
44#1504—24no—
45#1503C2214yes—
46#1502C2214yes—
47#1483—24no—
48#1481C2214yes—
49#1480C2214yes—
50#1477—24no—
51C2×#135—21no—
52C2×#135—21no—
53C23×C8—21no—
54C23×C8D411yescharacteristic
55C23×C8D411yescharacteristic
56#533C2×C412yes—
57#533C2×C412yes—
58#533C2312yes—
59#532C2×C412yes—
60#532C2×C412yes—
61#532C2312yes—
62#529—22no—
63#529—22no—
64#529—22no—
65#529—22no—
66#529—22no—
67#529—22no—
68#525—22no—
69#525—22no—
70#525—22no—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C22×C4⋊C4—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#500C2×C411yescharacteristic, complemented
78#500C2×C411yescharacteristic, complemented
79#499—21no—
80#499—21no—
81#498C2×C412yescomplemented
82#495—21no—
83#495—21no—
84#494C2×C411yescharacteristic, complemented
85#493C2×C411yescharacteristic, complemented
86#492C2×C412yescomplemented
87#488—22no—
88#488—22no—
89#487C2×C412yescomplemented
90#487C2×C412yescomplemented
91#486C2×C412yescomplemented
92#486C2×C412yescomplemented
93#467—21no—
94#467—21no—
95#467—21no—
96#467—21no—
97#466—21no—
98#466—21no—
99#466—21no—
100#466—21no—
101#465—22no—
102#465—22no—
103#465—22no—
104#465—22no—
105C4×Q16—21no—
106C4×Q16—21no—
107C4×Q16—21no—
108C4×Q16—21no—
109C4×SD16—22no—
110C4×SD16—22no—
111C4×SD16—22no—
112C4×SD16—22no—
113C4×D8—21no—
114C4×D8—21no—
115C4×D8—21no—
116C4×D8—21no—
117#432—22no—
118#432—22no—
119#431—22no—
120#431C2314yes—
121#427—21no—
122#427—21no—
123#423—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8D412yes—
128C2×C4⋊C8D412yes—
129C2×C4⋊C8C2312yes—
130#416—22no—
131#416—22no—
132#407—22no—
133C4×M16—22no—
134C2×#97—22no—
135C2×#97—22no—
136C2×#97—44no—
137C2×C4×C8C2×C411yescharacteristic
138C2×C4×C8C2×C411yescharacteristic
139C2×C4×C8C2×C411yescharacteristic
140C2×C4×C8C2×C411yescharacteristic
141C2×C4×C8D411yescharacteristic
142C2×C4×C8D411yescharacteristic
143C2×C4×C8D411yescharacteristic
144C2×C4×C8D411yescharacteristic
145C2×C4×C8D411yescharacteristic
146C2×C4×C8D411yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4D412yes—
152C4×C4⋊C4C2312yes—
153C2×#95D412yes—
154#336—22no—
155#336—22no—
156#335—22no—
157#335—22no—
158#334—22no—
159#334—22no—
160#328—28no—
161#327—28no—
162Q8⋊C8—28no—
163D4⋊C8—28no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C23×C4—22no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171#135—42no—
172#135—42no—
173#135—42no—
174#135—42no—
175C2×Q16—21no—
176C2×Q16—21no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×D8—21no—
180C2×D8—21no—
181C2×M16—42no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—41no—
188C22×C8—41no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8C22⋊C411yescharacteristic
192C22×C8C22⋊C411yescharacteristic
193C22×C8C22⋊C411yescharacteristic
194C22×C8C22⋊C411yescharacteristic
195C22×C8C22⋊C411yescharacteristic
196C22×C8C22⋊C411yescharacteristic
197C22×C8C22⋊C411yescharacteristic
198C22×C8C22⋊C411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C4⋊Q8C22×C412yes—
204C4⋊Q8C22×C412yes—
205C4⋊Q8C22×C412yes—
206#127C22×C412yes—
207#126—42no—
208#126—42no—
209#125—22no—
210#125—22no—
211#125—22no—
212#124—22no—
213#124—22no—
214#124—22no—
215#123—42no—
216#123—42no—
217#123—42no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—22no—
228#121—22no—
229#121—22no—
230#120—42no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—42no—
249#117—22no—
250#117—22no—
251#117—22no—
252#117—42no—
253#117—42no—
254#117—42no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—44no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42C2411yescharacteristic
275#107—21no—
276#107—21no—
277#107—21no—
278#107—21no—
279#106—22no—
280#106—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—24no—
284C4⋊C8—22no—
285C4⋊C8—42no—
286C4⋊C8C2×D414yes—
287#103—22no—
288#103—22no—
289#103—22no—
290#103—22no—
291#102—22no—
292#102—22no—
293#102—22no—
294#102—22no—
295C22⋊C8—24no—
296C22⋊C8—42no—
297#97—44no—
298#97—44no—
299#97—42no—
300C4×C8—24no—
301C4×C8C22⋊C411yescharacteristic
302C4×C8C22⋊C411yescharacteristic
303C4×C8C22⋊C411yescharacteristic
304C4×C8C22⋊C411yescharacteristic
305C4×C8C22⋊C411yescharacteristic
306C4×C8C22⋊C411yescharacteristic
307C4×C8C22⋊C411yescharacteristic
308C4×C8C22⋊C411yescharacteristic
309C4×C8C22×C411yescharacteristic
310C4×C8C22×C411yescharacteristic
311C4×C8C22×C411yescharacteristic
312C4×C8C22×C411yescharacteristic
313C4×C8C2×D411yescharacteristic
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316C4×C8C2×D411yescharacteristic
317#95—22no—
318#95C2×D414yes—
319C24#12111yescharacteristic
320#41—42no—
321#41—42no—
322#41—42no—
323#41—42no—
324#41—48no—
325#41—84no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12411yescharacteristic
360C22×C4#12411yescharacteristic
361C22×C4#12711yescharacteristic
362C22×C4#12711yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366Q16—42no—
367Q16—42no—
368SD16—44no—
369SD16—44no—
370D8—42no—
371D8—42no—
372M16—84no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—44no—
389C2×C8—44no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—44no—
393C2×C8C2×C22⋊C411yescharacteristic
394C2×C8C2×C22⋊C411yescharacteristic
395C2×C8C2×C22⋊C411yescharacteristic
396C2×C8C2×C22⋊C411yescharacteristic
397C2×C8C2×C22⋊C411yescharacteristic
398C2×C8C2×C22⋊C411yescharacteristic
399C2×C8C2×C22⋊C411yescharacteristic
400C2×C8C2×C22⋊C411yescharacteristic
401C2×C8AGL(1,ℤ/8)12yes—
402C2×C8#13712yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C42—44no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42C2×C22⋊C411yescharacteristic
433C42C2×C22⋊C411yescharacteristic
434C42C4×D411yescharacteristic
435C42C4×D411yescharacteristic
436C42C4×D411yescharacteristic
437C42C4×D411yescharacteristic
438C42#12111yescharacteristic
439C42#12111yescharacteristic
440C42#12411yescharacteristic
441C42#12411yescharacteristic
442C42#12711yescharacteristic
443C42#12711yescharacteristic
444C42C23×C411yescharacteristic
445C42C22×D411yescharacteristic
446C42C22×D411yescharacteristic
447C42C2×#4111yescharacteristic
448C23—22no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—42no—
453C23#39011yescharacteristic
454C23#39011yescharacteristic
455C23C2×#12111yescharacteristic
456Q8—42no—
457Q8—42no—
458Q8—44no—
459Q8—82no—
460D4—42no—
461D4—42no—
462D4—44no—
463D4—82no—
464D4—84no—
465D4—84no—
466D4—82no—
467D4—82no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—88no—
497C2×C4—84no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#39011yescharacteristic
501C2×C4#39011yescharacteristic
502C2×C4#39011yescharacteristic
503C2×C4#39011yescharacteristic
504C2×C4#39011yescharacteristic
505C2×C4#39011yescharacteristic
506C2×C4#44414yes—
507C2×C4C22×C22⋊C411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×#12111yescharacteristic
511C2×C4C2×#12411yescharacteristic
512C2×C4C2×#12711yescharacteristic
513C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
514C2×C4C2×#13711yescharacteristic
515C2×C4#57511yescharacteristic
516C2×C4#57511yescharacteristic
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—21no—
522C8—21no—
523C8—44no—
524C8—84no—
525C8#41712yes—
526C8#41712yes—
527C22—24no—
528C22—24no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22#197512yes—
539C22#197612yes—
540C22C2×#39011yescharacteristic
541C22#303611yescharacteristic
542C22#303711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—84no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4#197512yes—
556C4C2×#41711yescharacteristic
557C4#289611yescharacteristic
558C4C2×#44412yes—
559C4#303611yescharacteristic
560C4#303711yescharacteristic
561C2—21no—
562C2—22no—
563C2—22no—
564C2—21no—
565C2—162no—
566C2C2×#197511yescharacteristic
567C2C2×#197611yescharacteristic
568C2#2125811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60673C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1770C2semidirect
3. #17793C2semidirect
4. #17546C2semidirect
5. #17466C2semidirect
6. #9754C2semidirect
7. #9753C2semidirect
8. #9752C2semidirect
10. #3170C4semidirect
11. #3170C4semidirect
12. #3169C4semidirect
13. #3169C4semidirect
18. #3167C4semidirect
19. #3167C4semidirect
77. #500C2×C4semidirect
78. #500C2×C4semidirect
81. #498C2×C4semidirect
84. #494C2×C4semidirect
85. #493C2×C4semidirect
86. #492C2×C4semidirect
89. #487C2×C4semidirect
90. #487C2×C4semidirect
91. #486C2×C4semidirect
92. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#606732
C2×#17702
#177932
#175462
#174662
#97542
#97532
#97522