Finite Group Databasebeta

← #6355371

Subgroups — #6355371

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic, complemented
2C2×#3110C212yesmaximal
3C2×#2974C211yesmaximal, characteristic
4#19464C218yesmaximal, complemented
5#19459C218yesmaximal, complemented
6#19284C214yesmaximal, complemented
7#19259C214yesmaximal, complemented
8#17818C211yesmaximal, characteristic, complemented
9#17755C212yesmaximal, complemented
10C2×#533C2212yes—
11C22×C4×D4C2211yescharacteristic
12#3249—22no—
13#3230C2214yescomplemented
14#3209C2214yescomplemented
15#3189—21no—
16#3189—21no—
17#3170C2214yescomplemented
18#3110—22no—
19#3110—22no—
20#3110—24no—
21#3110C2218yes—
22#3110C2214yes—
23#3110C2214yes—
24#3104—24no—
25C2×#484—22no—
26C2×#474C2214yes—
27C2×#461C2214yes—
28C2×#450—22no—
29#2974—24no—
30#2974C2218yes—
31C2×#440—24no—
32C2×#436C2212yes—
33C2×#435—22no—
34C2×C8×D4C2212yes—
35C2×#432C2211yescharacteristic
36C2×#432C2212yes—
37C2×#431C2211yescharacteristic
38C2×#407—21no—
39C22×C22⋊C8—21no—
40#1699C22116yescomplemented
41#1693C2218yescomplemented
42#1689C2218yescomplemented
43#1679C22116yescomplemented
44#1673C2218yescomplemented
45#1669C2218yescomplemented
46#1573C2218yescomplemented
47#1568C2214yescomplemented
48#1556C2214yescomplemented
49#1556C2214yescomplemented
50#1552C2218yescomplemented
51#1552C2218yescomplemented
52C23×D4D411yescharacteristic
53C24×C4—21no—
54C22×SD16—24no—
55C22×M16—22no—
56C23×C8—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533—22no—
63#533C2314yes—
64#533C2314yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—21no—
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D414yes—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2314yes—
82C2×#117—22no—
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C42C2311yescharacteristic
86#500C2314yescomplemented
87#494C2314yescomplemented
88#487C2318yescomplemented
89#484—28no—
90#484—24no—
91#484—24no—
92#484—24no—
93#484—24no—
94#477—28no—
95#474—24no—
96#474—24no—
97#474C2318yes—
98#474C2318yes—
99#467—24no—
100#467—24no—
101#462—28no—
102#461—24no—
103#461—24no—
104#461C2318yes—
105#461C2318yes—
106#451—24no—
107#451—24no—
108#450—24no—
109#450—28no—
110#450—24no—
111#440—28no—
112#440—28no—
113#440—28no—
114#440—28no—
115C4×Q16—24no—
116C4×Q16—24no—
117#436—24no—
118#436C2318yes—
119#435—216no—
120C8×D4—24no—
121C8×D4C2318yes—
122#432—22no—
123#432—22no—
124#432—22no—
125#432C2314yes—
126#432C2314yes—
127#432C2314yes—
128#431D414yes—
129#431C2314yes—
130C2×#107—24no—
131C2×C4⋊C8D412yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103—24no—
135C2×#102—24no—
136#407—28no—
137C2×C22⋊C8—22no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—22no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—28no—
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×#97—21no—
145C2×#97—22no—
146C2×C4×C8C2312yes—
147#335C2318yescomplemented
148#334C2318yescomplemented
149#332C23116yescomplemented
150#332C23116yescomplemented
151C25—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—44no—
157C22×D4C2×D411yescharacteristic
158C22×D4C2×D414yes—
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—84no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C2×Q16—24no—
172C2×Q16—24no—
173C2×SD16—28no—
174C2×SD16—28no—
175C2×SD16—416no—
176C2×M16—28no—
177C2×M16—84no—
178C22×C8—22no—
179C22×C8—24no—
180C22×C8—28no—
181C22×C8—84no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8C2414yes—
191C4⋊Q8C2414yes—
192#125—24no—
193#125—24no—
194#125—42no—
195#125—42no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—21no—
217C4×D4—24no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—24no—
222C4×D4C2×D412yes—
223C4×D4C2×D412yes—
224C4×D4C2×D414yes—
225C4×D4C2×D414yes—
226C4×D4C2412yes—
227C4×D4C2412yes—
228#117—24no—
229#117—24no—
230#117—42no—
231#117—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C42—42no—
251C2×C42C2×D412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4111yescharacteristic
257C2×C42C2412yes—
258C2×C42C2411yescharacteristic
259#107—24no—
260#107—24no—
261#107—28no—
262#107—28no—
263C4⋊C8—22no—
264C4⋊C8—24no—
265C4⋊C8C2×D418yes—
266C4⋊C8C2414yes—
267C4⋊C8C2418yes—
268#103—28no—
269#103—28no—
270#103—28no—
271#103—28no—
272#102—28no—
273#102—28no—
274C22⋊C8—24no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C22⋊C8—28no—
278C22⋊C8—28no—
279C22⋊C8C2×D414yes—
280C22⋊C8C2×D414yes—
281#97—24no—
282#97—24no—
283#97—24no—
284C4×C8C2×D414yes—
285C4×C8C2414yes—
286C24—21no—
287C24—21no—
288C24—24no—
289C24—44no—
290C24—44no—
291C24—82no—
292C24#12111yescharacteristic
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—48no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—84no—
299C2×D4—24no—
300C2×D4—28no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—24no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4C22×D412yes—
309C2×D4C22×D412yes—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4#12112yes—
332C22×C4#12111yescharacteristic
333C22×C4#12112yes—
334C22×C4#12111yescharacteristic
335C22×C4#12112yes—
336C22×C4#12111yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4111yescharacteristic
343Q16—84no—
344Q16—84no—
345SD16—416no—
346SD16—416no—
347M16—88no—
348C2×C8—24no—
349C2×C8—28no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—44no—
353C2×C8—48no—
354C2×C8—88no—
355C2×C8C22×D414yes—
356C2×C8C22×D414yes—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4C22×D412yes—
373C4⋊C4C22×D412yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C42—44no—
383C42—44no—
384C42#12112yes—
385C42#12112yes—
386C42#12112yes—
387C42C22×D411yescharacteristic
388C42C22×D411yescharacteristic
389C42C22×D412yes—
390C42C22×D412yes—
391C42C2×#4111yescharacteristic
392C42C2×#4112yes—
393C42C2511yescharacteristic
394C23—24no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—48no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—48no—
405C23—42no—
406C23—84no—
407C23—816no—
408C23#47314yes—
409C23C2×#12111yescharacteristic
410C23C2×#12112yes—
411Q8—48no—
412Q8—48no—
413Q8—84no—
414Q8—84no—
415D4—24no—
416D4—22no—
417D4—24no—
418D4—48no—
419D4—42no—
420D4#57514yes—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4C2×#12111yescharacteristic
449C2×C4C2×#12112yes—
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×#12112yes—
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12112yes—
454C2×C4C23×D411yescharacteristic
455C2×C4C23×D411yescharacteristic
456C2×C4C22×#4111yescharacteristic
457C8—24no—
458C8—24no—
459C8—44no—
460C8—48no—
461C22—21no—
462C22—22no—
463C22—22no—
464C22—21no—
465C22—24no—
466C22—44no—
467C22—48no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—816no—
474C22—84no—
475C22C2×#47312yes—
476C22C2×#47314yes—
477C22C22×#12111yescharacteristic
478C4—21no—
479C4—22no—
480C4—21no—
481C4—22no—
482C4—21no—
483C4—22no—
484C4—21no—
485C4—42no—
486C4—42no—
487C4—84no—
488C4—84no—
489C4C2×#57512yes—
490C4#358512yes—
491C2—22no—
492C2—24no—
493C2—44no—
494C2—82no—
495C2#6042311yescharacteristic
496C2#6044411yescharacteristic
497C2C22×#47311yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60943C2semidirect
4. #19464C2semidirect
5. #19459C2semidirect
6. #19284C2semidirect
7. #19259C2semidirect
8. #17818C2semidirect
9. #17755C2semidirect
13. #3230C22semidirect
14. #3209C22semidirect
17. #3170C22semidirect
40. #1699C22semidirect
41. #1693C22semidirect
42. #1689C22semidirect
43. #1679C22semidirect
44. #1673C22semidirect
45. #1669C22semidirect
46. #1573C22semidirect
47. #1568C22semidirect
48. #1556C22semidirect
49. #1556C22semidirect
50. #1552C22semidirect
51. #1552C22semidirect
86. #500C23semidirect
87. #494C23semidirect
88. #487C23semidirect
147. #335C23semidirect
148. #334C23semidirect
149. #332C23semidirect
150. #332C23semidirect

Maximal subgroups

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

subgroupindex
#609432
C2×#31102
C2×#29742
#194642
#194592
#192842
#192592
#178182
#177552