Finite Group Databasebeta

← #6357474

Subgroups — #6357474

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60971C211yesmaximal, characteristic, complemented
2#60846C212yesmaximal
3C2×#3230C212yesmaximal, complemented
4#60800C211yesmaximal, characteristic
5#19732C214yesmaximal, complemented
6#19731C214yesmaximal, complemented
7#19706C214yesmaximal, complemented
8#19704C214yesmaximal, complemented
9#19664C214yesmaximal, complemented
10#19657C214yesmaximal, complemented
11#17325C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13C2×#533C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15#3255—22no—
16#3233—24no—
17#3232—24no—
18#3231C2214yes—
19#3230—24no—
20#3230—24no—
21#3230C2218yes—
22#3230C2214yes—
23#3230C2214yes—
24#3230C2212yes—
25#3229C2214yes—
26#3229C2212yes—
27#3228C2214yes—
28#3227C2212yes—
29#3226C2212yes—
30#3218—24no—
31#3209C2214yes—
32#3209C2214yes—
33#3206C2214yes—
34#3195—22no—
35#3185C2214yes—
36#3160C2214yes—
37C2×#500C2214yes—
38#3152C2212yes—
39#3139C2212yes—
40#3131C2214yes—
41C2×#487C2214yes—
42#3113—24no—
43#3091—24no—
44C2×#484—24no—
45#3076—24no—
46C2×#477—24no—
47#3039—22no—
48#3004—22no—
49#2920C2211yescharacteristic
50C2×#431C2212yes—
51#1723C2218yescomplemented
52#1723C2218yescomplemented
53#1722C2214yescomplemented
54#1719C2218yescomplemented
55#1718C2218yescomplemented
56#1718C2218yescomplemented
57#1716C2214yescomplemented
58#1713C2214yescomplemented
59#1710C2218yescomplemented
60#1708C2214yescomplemented
61#1453C2214yes—
62#1452C2214yes—
63#1450C2214yes—
64C23×Q8—21no—
65C2×#137—24no—
66#533—22no—
67#533—22no—
68#533—22no—
69#533—22no—
70#533—22no—
71#533—22no—
72#533C2314yes—
73#533C2311yescharacteristic
74#533C2312yes—
75#533C2314yes—
76#532C2312yes—
77#532C2311yescharacteristic
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2314yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87#529—22no—
88#529—22no—
89#529—22no—
90C2×#125—22no—
91#525—22no—
92#525—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C4×#41—22no—
96C4×#41—21no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×#117—22no—
100C22×C4⋊C4—22no—
101C22×C42C2311yescharacteristic
102#500—28no—
103#500C2318yes—
104#500C2318yes—
105#500C2314yes—
106#500C2314yes—
107#499—28no—
108#498C2314yes—
109#498C2314yes—
110#498C2314yes—
111#497—24no—
112#494C2314yes—
113#494C2314yes—
114#492C2314yes—
115#491—24no—
116#488—28no—
117#487—28no—
118#487C2318yes—
119#487C2318yes—
120#487C2314yes—
121#487C2314yes—
122#487C2318yes—
123#487C2318yes—
124#486C2314yes—
125#486C2314yes—
126#485—28no—
127#484—28no—
128#484—28no—
129#484—216no—
130#484—24no—
131#483—24no—
132#483—28no—
133#482—28no—
134#481—24no—
135#480—24no—
136#479—28no—
137#478—28no—
138#477—28no—
139#477—28no—
140#477—216no—
141#477—24no—
142#476—24no—
143#475—24no—
144#475—28no—
145#474—24no—
146#470—28no—
147#462—28no—
148#462—28no—
149#461—28no—
150#451—28no—
151#451—28no—
152#448—28no—
153#441—28no—
154#432D414yes—
155#431D414yes—
156#431D414yes—
157#431C2312yes—
158#431C2312yes—
159#431C2314yes—
160#431C2314yes—
161#427—28no—
162C2×#107—28no—
163#423D412yes—
164C2×C4⋊C8D414yes—
165#419—24no—
166#416—28no—
167C2×#103—28no—
168#407D412yes—
169C2×C22⋊C8D414yes—
170C4×M16C2312yes—
171C2×C4×C8C2314yes—
172#322C2318yes—
173#322C2314yes—
174C82C2314yes—
175C2×#41—21no—
176C2×#41—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—42no—
182C22×Q8—41no—
183C22×Q8—84no—
184C23×C4—41no—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187#137—88no—
188C2×Q16—28no—
189C2×Q16—28no—
190C2×SD16—28no—
191C2×M16C2×D414yes—
192C22×C8C2×D418yes—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—41no—
205C4⋊Q8—42no—
206C4⋊Q8—41no—
207C4⋊Q8—42no—
208C4⋊Q8C2414yes—
209C4⋊Q8C2414yes—
210C4⋊Q8C2414yes—
211C4⋊Q8C2412yes—
212C4⋊Q8C2411yescharacteristic
213#127C2411yescharacteristic
214#126—42no—
215#125—24no—
216#125—24no—
217#125—21no—
218#125—22no—
219#125—42no—
220#125—42no—
221#124—22no—
222#124—21no—
223#123—42no—
224#123—41no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—42no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—42no—
236#121—22no—
237#121—22no—
238#121—22no—
239#120—41no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×Q8—42no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257#117—24no—
258#117—24no—
259#117—21no—
260#117—22no—
261#117—42no—
262#117—42no—
263#117—42no—
264#117—41no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C42—42no—
279C2×C42—41no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2412yes—
286C2×C42C2411yescharacteristic
287#107—216no—
288#107—216no—
289#107—28no—
290#107—28no—
291#106—28no—
292#106—28no—
293#106—28no—
294C4⋊C8C2×D414yes—
295C4⋊C8C2×D414yes—
296C4⋊C8C2×D418yes—
297C4⋊C8C2×D418yes—
298#103—216no—
299#103—216no—
300#103—28no—
301#103—28no—
302#103—216no—
303#103—216no—
304#102—28no—
305#102—28no—
306C22⋊C8C2×D414yes—
307C22⋊C8C2×D414yes—
308C22⋊C8C2×D418yes—
309C22⋊C8C2×D418yes—
310#97C2×D414yes—
311C4×C8C2×D418yes—
312C4×C8C2414yes—
313C4×C8C2414yes—
314C4×C8C2414yes—
315C24#12711yescharacteristic
316#41—44no—
317#41—48no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—21no—
323C2×Q8—48no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—44no—
328C2×Q8—48no—
329C2×Q8—84no—
330C2×Q8—88no—
331C2×Q8—84no—
332C2×Q8—88no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—42no—
337C22×C4—28no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—84no—
346C22×C4#12712yes—
347C22×C4#12712yes—
348C22×C4#12712yes—
349C22×C4#12712yes—
350C22×C4#12711yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356Q16—88no—
357Q16—88no—
358SD16—88no—
359M16—44no—
360C2×C8—216no—
361C2×C8C22×D418yes—
362C2×C8C22×D418yes—
363C2×C8C22×D418yes—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—28no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—41no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C42—41no—
394C42—42no—
395C42—44no—
396C42—44no—
397C42#12711yescharacteristic
398C42#12712yes—
399C42#12711yescharacteristic
400C42#12712yes—
401C42C22×D412yes—
402C42C22×D411yescharacteristic
403C42C22×D412yes—
404C42C22×D412yes—
405C42C22×D412yes—
406C42C2511yescharacteristic
407C23—24no—
408C23—41no—
409C23—41no—
410C23C2×#12711yescharacteristic
411C23C2×#12712yes—
412Q8—48no—
413Q8—42no—
414Q8—48no—
415Q8—48no—
416Q8—44no—
417Q8—84no—
418Q8—84no—
419Q8—88no—
420Q8—84no—
421D4—42no—
422D4—44no—
423D4—82no—
424D4—84no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—88no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4C2×#12712yes—
446C2×C4C2×#12711yescharacteristic
447C2×C4C2×#12712yes—
448C2×C4C2×#12712yes—
449C2×C4C2×#12712yes—
450C2×C4#57818yes—
451C2×C4C23×D411yescharacteristic
452C2×C4C23×D412yes—
453C8—28no—
454C8—28no—
455C8—28no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—44no—
461C22—82no—
462C22—81no—
463C22#315614yes—
464C22C22×#12711yescharacteristic
465C4—22no—
466C4—22no—
467C4—21no—
468C4—22no—
469C4—82no—
470C4—84no—
471C4—84no—
472C4—81no—
473C4—84no—
474C4#315614yes—
475C4C2×#57814yes—
476C4#358512yes—
477C2—24no—
478C2—41no—
479C2—161no—
480C2C2×#315612yes—
481C2#6064411yescharacteristic
subgroupG/H
center, socleC22C22×#127
commutator, Frattini, cosocleC42C25
prime radical#3444C22
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. #60971C2semidirect
3. C2×#3230C2semidirect
5. #19732C2semidirect
6. #19731C2semidirect
7. #19706C2semidirect
8. #19704C2semidirect
9. #19664C2semidirect
10. #19657C2semidirect
11. #17325C2semidirect
51. #1723C22semidirect
52. #1723C22semidirect
53. #1722C22semidirect
54. #1719C22semidirect
55. #1718C22semidirect
56. #1718C22semidirect
57. #1716C22semidirect
58. #1713C22semidirect
59. #1710C22semidirect
60. #1708C22semidirect

Maximal subgroups

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

subgroupindex
#609712
#608462
C2×#32302
#608002
#197322
#197312
#197062
#197042
#196642
#196572
#173252