Finite Group Databasebeta

← #6345630

Subgroups — #6345630

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic
2#60268C212yesmaximal, complemented
3#18033C218yesmaximal, complemented
4#18027C218yesmaximal, complemented
5#17979C212yesmaximal, complemented
6#17977C211yesmaximal, characteristic, complemented
7#17975C211yesmaximal, characteristic, complemented
8#17960C212yesmaximal, complemented
9#17958C211yesmaximal, characteristic, complemented
10#17956C211yesmaximal, characteristic, complemented
11#17814C211yesmaximal, characteristic, complemented
12#17481C212yesmaximal, complemented
13C2×#1454C211yesmaximal, characteristic, complemented
14#3444C2212yes—
15#3250C414yes—
16#3231C412yes—
17#3228C412yes—
18#3210C412yes—
19#3207C412yes—
20#3191C414yes—
21#3170C2211yescharacteristic
22#3169C2211yescharacteristic
23#3167C2212yes—
24#3102C2218yes—
25#2994C2214yes—
26#2994C2214yes—
27#2993—22no—
28#2993—22no—
29#2991C2214yes—
30#2990C2214yes—
31#2989—22no—
32#2988—22no—
33#2960C2212yes—
34#2960C2212yes—
35#2933—22no—
36#2933—22no—
37#2931C2212yes—
38#2931C2212yes—
39#2925—22no—
40C2×#431C2211yescharacteristic
41C2×#431C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#2906—22no—
44C2×#423C2212yes—
45#1592C2218yescomplemented
46#1591C2214yescomplemented
47#1589C2214yescomplemented
48#1588C2218yescomplemented
49#1585C2218yescomplemented
50#1584C2218yescomplemented
51#1581C2218yescomplemented
52SD16⋊C8C2218yescomplemented
53C8×Q16C2214yescomplemented
54C8×D8C2214yescomplemented
55#1571C2218yes—
56#1567C2211yescharacteristic
57#1567C2211yescharacteristic
58#1567C2212yes—
59#1495C2218yes—
60#1493C2212yes—
61#1493C2212yes—
62#1492C2212yes—
63#1492C2212yes—
64#1477C2218yes—
65#1454C2214yes—
66#1454C2214yes—
67C2×#322C2211yescharacteristic
68C2×#322C2211yescharacteristic
69C2×#322C2211yescharacteristic
70C2×C82C2211yescharacteristic
71C2×#131—22no—
72C2×#131—22no—
73C22×M16—22no—
74#533C2×C412yes—
75#533C2×C412yes—
76#533C2312yes—
77#532C2×C412yes—
78#532C2×C412yes—
79#532C2312yes—
80#529—22no—
81#529—22no—
82#529—22no—
83#529—22no—
84#529C2×C414yes—
85#529C2314yes—
86#525—22no—
87#525—22no—
88#525C2×C414yes—
89C4×#41—22no—
90C4×#41—22no—
91C4×#41C2314yes—
92C22×C42C2311yescharacteristic
93#500C2×C411yescharacteristic
94#500C2×C411yescharacteristic
95#498C2×C412yes—
96#494C2×C411yescharacteristic
97#493C2×C411yescharacteristic
98#492C2×C412yes—
99#487C2×C412yes—
100#487C2×C412yes—
101#486C2×C412yes—
102#486C2×C412yes—
103#485C2×C418yes—
104#482C2×C418yes—
105#479C2×C418yes—
106#478C2×C418yes—
107#467C2×C414yes—
108#466C2×C414yes—
109#465C2×C418yes—
110#464C2×C418yes—
111#463C2×C418yes—
112#452C2×C418yes—
113#449C2×C418yes—
114#446—22no—
115#446—22no—
116#446C2314yes—
117#446C2314yes—
118C8×Q8—22no—
119C8×Q8—22no—
120C8×Q8C2314yes—
121C8×Q8C2314yes—
122C4×Q16C2314yes—
123C4×SD16C2318yes—
124C4×D8C2314yes—
125#436—24no—
126#436—24no—
127#436—22no—
128#436—22no—
129#436C2314yes—
130#436C2314yes—
131#435—24no—
132#435—24no—
133#435—24no—
134#435—24no—
135C8×D4—24no—
136C8×D4—24no—
137C8×D4—22no—
138C8×D4—22no—
139C8×D4C2314yes—
140C8×D4C2314yes—
141#433—24no—
142#433—24no—
143#432—24no—
144#431—22no—
145#431—24no—
146#431C2×C412yes—
147#431C2×C412yes—
148#431D412yes—
149#431D412yes—
150#431C2314yes—
151#431C2314yes—
152#431C2312yes—
153#431C2312yes—
154#424—22no—
155#424—22no—
156#424—24no—
157#424—24no—
158#423—22no—
159#423—22no—
160#423—24no—
161#423C2312yes—
162#423C2312yes—
163#423C2314yes—
164C2×C4⋊C8—22no—
165C2×C4⋊C8—21no—
166C2×C4⋊C8—24no—
167C2×C4⋊C8C2×C412yes—
168C2×C4⋊C8D411yescharacteristic
169C2×C4⋊C8D411yescharacteristic
170C2×C4⋊C8C2314yes—
171#408—22no—
172#408—22no—
173C2×C22⋊C8—21no—
174C2×C22⋊C8C2×C412yes—
175C2×C22⋊C8D411yescharacteristic
176C2×C22⋊C8D411yescharacteristic
177#405—22no—
178#405—22no—
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2311yescharacteristic
185#335C2314yes—
186#335C2314yes—
187#335C2318yes—
188Q8⋊C8C2318yes—
189Q8⋊C8C2318yes—
190D4⋊C8C2318yes—
191D4⋊C8C2318yes—
192#322C2314yes—
193#322C2314yes—
194#322C2312yes—
195#322C2312yes—
196C82C2312yes—
197C82C2312yes—
198C2×#41—22no—
199C2×#41—22no—
200C2×#41C22×C414yes—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4111yescharacteristic
204C2×Q16C22×C414yes—
205C2×SD16C22×C418yes—
206C2×D8C22×C414yes—
207#131—84no—
208#131—84no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—84no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8C22×C412yes—
221C22×C8C22×C412yes—
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C4⋊Q8C22×C412yes—
227C4⋊Q8C22×C412yes—
228C4⋊Q8C22×C412yes—
229#127C22×C412yes—
230#126—24no—
231#126—24no—
232#125—22no—
233#125—22no—
234#125C22×C414yes—
235#124—22no—
236#124—22no—
237#124C22×C414yes—
238#123—24no—
239#123—24no—
240#123—24no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8C22×C414yes—
246C22⋊Q8C22×C414yes—
247#121—22no—
248#121—22no—
249#121—22no—
250#121—22no—
251#121C22×C414yes—
252#121C22×C414yes—
253#120—24no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8C22×C414yes—
259C4×Q8C2414yes—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4C22×C414yes—
271C4×D4C2414yes—
272#117—22no—
273#117—22no—
274#117—24no—
275#117—24no—
276#117—24no—
277#117C22×C414yes—
278C2×C42—24no—
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42#4111yescharacteristic
284C2×C42#4112yes—
285C2×C42C2412yes—
286C2×C42C2411yescharacteristic
287#107C22×C414yes—
288#107C22×C414yes—
289#106C22×C418yes—
290C4⋊C8—24no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—24no—
298C4⋊C8—24no—
299C4⋊C8C22×C414yes—
300C4⋊C8C22×C414yes—
301C4⋊C8C2×D412yes—
302C4⋊C8C2×D412yes—
303C4⋊C8C2×D412yes—
304C4⋊C8C2×D412yes—
305C4⋊C8C2414yes—
306C4⋊C8C2414yes—
307#103C22×C418yes—
308#103C22×C418yes—
309#102C22×C418yes—
310#102C22×C418yes—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8C22×C414yes—
317C22⋊C8C22×C414yes—
318C22⋊C8C2×D412yes—
319C22⋊C8C2×D412yes—
320C22⋊C8C2×D412yes—
321C22⋊C8C2×D412yes—
322#97—24no—
323#97—24no—
324C4×C8—24no—
325C4×C8—24no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8C22×C411yescharacteristic
329C4×C8C22×C412yes—
330C4×C8C22×C411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D412yes—
333C4×C8C2×D411yescharacteristic
334C4×C8C2412yes—
335C4×C8C2412yes—
336C4×C8C2412yes—
337C4×C8C2412yes—
338C4×C8C2411yescharacteristic
339C4×C8C2411yescharacteristic
340C4×C8C2412yes—
341C24C4×D411yescharacteristic
342#41—48no—
343#41—48no—
344#41—44no—
345#41—44no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8C23×C414yes—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4C23×C414yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D412yes—
368C22×C4C23×C412yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374Q16—44no—
375SD16—48no—
376D8—44no—
377M16—44no—
378M16—44no—
379M16—84no—
380M16—84no—
381M16—84no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—84no—
395C2×C8—84no—
396C2×C8—84no—
397C2×C8C23×C412yes—
398C2×C8C23×C414yes—
399C2×C8C23×C412yes—
400C2×C8C22×D411yescharacteristic
401C2×C8C22×D412yes—
402C2×C8C22×D411yescharacteristic
403C2×C8C22×D411yescharacteristic
404C2×C8C22×D412yes—
405C2×C8C22×D411yescharacteristic
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4C23×C414yes—
417C4⋊C4C23×C414yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42C4×D412yes—
431C42C4×D412yes—
432C42C4×D412yes—
433C42C23×C411yescharacteristic
434C42C23×C412yes—
435C42C22×D411yescharacteristic
436C42C22×D412yes—
437C42C2×#4111yescharacteristic
438C42C2×#4112yes—
439C42C2511yescharacteristic
440C23—22no—
441C23—24no—
442C23—41no—
443C23—41no—
444C23C2×C4×D412yes—
445C23C2×C4×D411yescharacteristic
446Q8—28no—
447Q8—44no—
448Q8—44no—
449D4—28no—
450D4—48no—
451D4—48no—
452D4—44no—
453D4—44no—
454D4—84no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—84no—
474C2×C4C2×C4×D412yes—
475C2×C4C2×C4×D412yes—
476C2×C4C2×C4×D412yes—
477C2×C4C2×C4×D411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4#57612yes—
481C2×C4#57812yes—
482C2×C4C24×C411yescharacteristic
483C2×C4C23×D411yescharacteristic
484C2×C4C22×#4111yescharacteristic
485C8—22no—
486C8—24no—
487C8—22no—
488C8—21no—
489C8—22no—
490C8—21no—
491C8—21no—
492C8—22no—
493C8—21no—
494C8—42no—
495C8—42no—
496C8—44no—
497C8—44no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—82no—
507C22#295112yes—
508C22C22×C4×D411yescharacteristic
509C4—22no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—21no—
514C4—44no—
515C4—44no—
516C4—44no—
517C4#295112yes—
518C4#295512yes—
519C4#357312yes—
520C4C2×#57611yescharacteristic
521C4C2×#57811yescharacteristic
522C2—22no—
523C2—42no—
524C2—84no—
525C2#6014311yescharacteristic
526C2C2×#295111yescharacteristic
527C2C2×#295511yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60944C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60268C2semidirect
3. #18033C2semidirect
4. #18027C2semidirect
5. #17979C2semidirect
6. #17977C2semidirect
7. #17975C2semidirect
8. #17960C2semidirect
9. #17958C2semidirect
10. #17956C2semidirect
11. #17814C2semidirect
12. #17481C2semidirect
13. C2×#1454C2semidirect
45. #1592C22semidirect
46. #1591C22semidirect
47. #1589C22semidirect
48. #1588C22semidirect
49. #1585C22semidirect
50. #1584C22semidirect
51. #1581C22semidirect
52. SD16⋊C8C22semidirect
53. C8×Q16C22semidirect
54. C8×D8C22semidirect

Maximal subgroups

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

subgroupindex
#609442
#602682
#180332
#180272
#179792
#179772
#179752
#179602
#179582
#179562
#178142
#174812
C2×#14542