Finite Group Databasebeta

← #6345255

Subgroups — #6345255

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic
2#60268C212yesmaximal, complemented
3#17979C212yesmaximal, complemented
4#17977C211yesmaximal, characteristic, complemented
5#17975C211yesmaximal, characteristic, complemented
6#17960C212yesmaximal, complemented
7#17958C211yesmaximal, characteristic, complemented
8#17956C211yesmaximal, characteristic, complemented
9#17954C218yesmaximal, complemented
10#17926C218yesmaximal, complemented
11#17814C211yesmaximal, characteristic, complemented
12#17504C212yesmaximal, complemented
13C2×#1453C211yesmaximal, 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#2994—22no—
26#2994—22no—
27#2993C2214yes—
28#2993C2214yes—
29#2991—22no—
30#2990—22no—
31#2989C2214yes—
32#2988C2214yes—
33#2960C2212yes—
34#2960C2212yes—
35#2933—22no—
36#2933—22no—
37#2931C2212yes—
38#2931C2212yes—
39#2925—22no—
40C2×#431C2211yescharacteristic
41C2×#431C2211yescharacteristic
42#2906—22no—
43C2×#423C2212yes—
44C2×C4×M16C2211yescharacteristic
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#1493C2212yes—
60#1493C2212yes—
61#1492C2212yes—
62#1492C2212yes—
63#1489C2218yes—
64#1483C2218yes—
65#1453C2214yes—
66#1453C2214yes—
67C2×#322C2211yescharacteristic
68C2×#322C2211yescharacteristic
69C2×#322C2211yescharacteristic
70C2×C82C2211yescharacteristic
71C2×#131—22no—
72C2×#131—22no—
73C22×M16—21no—
74C22×M16—22no—
75#533C2×C412yes—
76#533C2×C412yes—
77#533C2312yes—
78#532C2×C412yes—
79#532C2×C412yes—
80#532C2312yes—
81#529—22no—
82#529—22no—
83#529—22no—
84#529—22no—
85#529C2×C414yes—
86#529C2314yes—
87#525—22no—
88#525—22no—
89#525C2×C414yes—
90C4×#41—22no—
91C4×#41—22no—
92C4×#41C2314yes—
93C22×C42C2311yescharacteristic
94#500C2×C411yescharacteristic
95#500C2×C411yescharacteristic
96#498C2×C412yes—
97#494C2×C411yescharacteristic
98#493C2×C411yescharacteristic
99#492C2×C412yes—
100#487C2×C412yes—
101#487C2×C412yes—
102#486C2×C412yes—
103#486C2×C412yes—
104#485C2×C418yes—
105#482C2×C418yes—
106#479C2×C418yes—
107#478C2×C418yes—
108#467C2×C414yes—
109#466C2×C414yes—
110#465C2×C418yes—
111#464C2×C418yes—
112#463C2×C418yes—
113#452C2×C418yes—
114#449C2×C418yes—
115#446—22no—
116#446—22no—
117#446C2314yes—
118#446C2314yes—
119C8×Q8—22no—
120C8×Q8—22no—
121C8×Q8C2314yes—
122C8×Q8C2314yes—
123C4×Q16C2314yes—
124C4×SD16C2318yes—
125C4×D8C2314yes—
126#436—24no—
127#436—24no—
128#436—22no—
129#436—22no—
130#436C2314yes—
131#436C2314yes—
132#435—24no—
133#435—24no—
134#435—24no—
135#435—24no—
136C8×D4—24no—
137C8×D4—24no—
138C8×D4—22no—
139C8×D4—22no—
140C8×D4C2314yes—
141C8×D4C2314yes—
142#433—24no—
143#433—24no—
144#432—24no—
145#431—24no—
146#431C2×C412yes—
147#431C2×C412yes—
148#431D412yes—
149#431D412yes—
150#431C2314yes—
151#431C2312yes—
152#431C2312yes—
153#424—22no—
154#424—22no—
155#424—24no—
156#424—24no—
157#423—22no—
158#423—22no—
159#423—24no—
160#423C2312yes—
161#423C2312yes—
162#423C2314yes—
163C2×C4⋊C8—22no—
164C2×C4⋊C8—24no—
165C2×C4⋊C8C2×C412yes—
166C2×C4⋊C8D411yescharacteristic
167C2×C4⋊C8D411yescharacteristic
168C2×C4⋊C8C2314yes—
169#408—22no—
170#408—22no—
171C2×C22⋊C8C2×C412yes—
172C2×C22⋊C8D411yescharacteristic
173C2×C22⋊C8D411yescharacteristic
174#405—22no—
175#405—22no—
176C4×M16—22no—
177C4×M16C2314yes—
178C2×#97—21no—
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#322C2312yes—
193#322C2314yes—
194#322C2314yes—
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—24no—
215C2×M16—44no—
216C2×M16—84no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8C22×C412yes—
223C22×C8C22×C412yes—
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C4⋊Q8C22×C412yes—
229C4⋊Q8C22×C412yes—
230C4⋊Q8C22×C412yes—
231#127C22×C412yes—
232#126—24no—
233#126—24no—
234#125—22no—
235#125—22no—
236#125C22×C414yes—
237#124—22no—
238#124—22no—
239#124C22×C414yes—
240#123—24no—
241#123—24no—
242#123—24no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8C22×C414yes—
248C22⋊Q8C22×C414yes—
249#121—22no—
250#121—22no—
251#121—22no—
252#121—22no—
253#121C22×C414yes—
254#121C22×C414yes—
255#120—24no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8C22×C414yes—
261C4×Q8C2414yes—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4C22×C414yes—
273C4×D4C2414yes—
274#117—22no—
275#117—22no—
276#117—24no—
277#117—24no—
278#117—24no—
279#117C22×C414yes—
280C2×C42—24no—
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42#4111yescharacteristic
286C2×C42#4112yes—
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289#107C22×C414yes—
290#107C22×C414yes—
291#106C22×C418yes—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8C22×C414yes—
301C4⋊C8C22×C414yes—
302C4⋊C8C2×D412yes—
303C4⋊C8C2×D412yes—
304C4⋊C8C2×D412yes—
305C4⋊C8C2×D412yes—
306C4⋊C8C2414yes—
307C4⋊C8C2414yes—
308#103C22×C418yes—
309#103C22×C418yes—
310#102C22×C418yes—
311#102C22×C418yes—
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—
324#97—24no—
325C4×C8—24no—
326C4×C8—24no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8C22×C411yescharacteristic
330C4×C8C22×C412yes—
331C4×C8C22×C411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D412yes—
334C4×C8C2×D411yescharacteristic
335C4×C8C2412yes—
336C4×C8C2412yes—
337C4×C8C2412yes—
338C4×C8C2412yes—
339C4×C8C2411yescharacteristic
340C4×C8C2411yescharacteristic
341C4×C8C2412yes—
342C24C4×D411yescharacteristic
343#41—48no—
344#41—44no—
345#41—44no—
346#41—84no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8C23×C414yes—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4C23×C414yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D412yes—
369C22×C4C23×C412yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375Q16—44no—
376SD16—48no—
377D8—44no—
378M16—48no—
379M16—44no—
380M16—44no—
381M16—84no—
382M16—84no—
383M16—84no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—84no—
397C2×C8—84no—
398C2×C8—84no—
399C2×C8C23×C412yes—
400C2×C8C23×C414yes—
401C2×C8C23×C412yes—
402C2×C8C22×D411yescharacteristic
403C2×C8C22×D412yes—
404C2×C8C22×D411yescharacteristic
405C2×C8C22×D411yescharacteristic
406C2×C8C22×D412yes—
407C2×C8C22×D411yescharacteristic
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4C23×C414yes—
419C4⋊C4C23×C414yes—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42C4×D412yes—
433C42C4×D412yes—
434C42C4×D412yes—
435C42C23×C411yescharacteristic
436C42C23×C412yes—
437C42C22×D411yescharacteristic
438C42C22×D412yes—
439C42C2×#4111yescharacteristic
440C42C2×#4112yes—
441C42C2511yescharacteristic
442C23—22no—
443C23—24no—
444C23—41no—
445C23—41no—
446C23C2×C4×D412yes—
447C23C2×C4×D411yescharacteristic
448Q8—28no—
449Q8—44no—
450Q8—44no—
451D4—28no—
452D4—48no—
453D4—48no—
454D4—44no—
455D4—44no—
456D4—84no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—84no—
476C2×C4C2×C4×D412yes—
477C2×C4C2×C4×D412yes—
478C2×C4C2×C4×D412yes—
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4#57612yes—
483C2×C4#57812yes—
484C2×C4C24×C411yescharacteristic
485C2×C4C23×D411yescharacteristic
486C2×C4C22×#4111yescharacteristic
487C8—24no—
488C8—22no—
489C8—24no—
490C8—22no—
491C8—21no—
492C8—22no—
493C8—21no—
494C8—21no—
495C8—22no—
496C8—21no—
497C8—42no—
498C8—44no—
499C8—44no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—82no—
509C22#295112yes—
510C22C22×C4×D411yescharacteristic
511C4—22no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—21no—
516C4—44no—
517C4—44no—
518C4—44no—
519C4#295112yes—
520C4#295612yes—
521C4#357312yes—
522C4C2×#57611yescharacteristic
523C4C2×#57811yescharacteristic
524C2—22no—
525C2—42no—
526C2—84no—
527C2#6014011yescharacteristic
528C2C2×#295111yescharacteristic
529C2C2×#295611yescharacteristic
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. #17979C2semidirect
4. #17977C2semidirect
5. #17975C2semidirect
6. #17960C2semidirect
7. #17958C2semidirect
8. #17956C2semidirect
9. #17954C2semidirect
10. #17926C2semidirect
11. #17814C2semidirect
12. #17504C2semidirect
13. C2×#1453C2semidirect
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
#179792
#179772
#179752
#179602
#179582
#179562
#179542
#179262
#178142
#175042
C2×#14532