Finite Group Databasebeta

← #6481858

Subgroups — #6481858

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3255C211yesmaximal, characteristic, complemented
2C2×#2920C211yesmaximal, characteristic
3#36625C212yesmaximal
4#33144C214yesmaximal, complemented
5#29743C214yesmaximal
6#24710C214yesmaximal, complemented
7#24703C214yesmaximal, complemented
8#24660C214yesmaximal, complemented
9#24652C214yesmaximal, complemented
10#21964C211yesmaximal, characteristic, complemented
11#21894C212yesmaximal, complemented
12C2×#533C2212yes—
13C23×C42C2211yescharacteristic
14#3255—28no—
15C2×#475C2214yescomplemented
16C2×#470C2214yes—
17C2×#462C2214yescomplemented
18#2920—28no—
19C2×#432C2214yes—
20C2×#431C2214yes—
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2855—22no—
30#2855C2214yes—
31#2851—22no—
32#2851—22no—
33#2848—22no—
34#2848C2214yes—
35C4×C22⋊Q8—22no—
36C4×C22⋊Q8—22no—
37C4×C22⋊Q8C2214yes—
38C4×C22⋊Q8C2214yes—
39#2296—22no—
40#2285C2212yes—
41#2280—22no—
42#2077C2218yes—
43#2036C2218yes—
44#2035C2218yes—
45#2001C2218yes—
46#1988C2214yescomplemented
47#1981C2214yescomplemented
48#1971C2214yescomplemented
49#1969C2214yescomplemented
50#1948C2214yescomplemented
51#1947C2214yescomplemented
52#1943C2214yescomplemented
53#1941C2214yescomplemented
54#1935C2218yescomplemented
55#1933C2218yescomplemented
56#1894C2218yes—
57#1809C2214yes—
58#1803C2214yescomplemented
59#1803C2214yescomplemented
60C24×C4D412yes—
61C24×C4C2311yescharacteristic
62C22×Q16C2314yes—
63C22×M16C2312yes—
64C22×M16C2312yes—
65#533—28no—
66#533—28no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×#125—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2314yes—
75C2×C22⋊Q8C2314yes—
76C2×C4×Q8—22no—
77C2×C4×Q8C2314yes—
78C2×#117—22no—
79C2×#117—22no—
80C22×C42—24no—
81C22×C42—24no—
82C22×C42D412yes—
83C22×C42D412yes—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2312yes—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#475—216no—
90#470—216no—
91#462—216no—
92#432—216no—
93#431—216no—
94C2×#107C2314yes—
95C2×#106C2314yes—
96C2×#106C2314yes—
97#423—24no—
98#423D418yes—
99C2×C4⋊C8C2314yes—
100C2×C4⋊C8C2314yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104#407—28no—
105C2×C22⋊C8C2318yes—
106C4×M16—28no—
107C2×#97C2314yes—
108C2×C4×C8C2314yes—
109#391—22no—
110#391—22no—
111#391C2314yes—
112#391C2314yes—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389C2314yes—
120#389C2314yes—
121#387—24no—
122#387—24no—
123#387—24no—
124#387—24no—
125#385—24no—
126#385—24no—
127#385—24no—
128#385—24no—
129#385—24no—
130#385—24no—
131#384C2312yes—
132#384C2312yes—
133#382—22no—
134#381—24no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—22no—
141#380—22no—
142#380C2314yes—
143#380C2314yes—
144#380C2314yes—
145#379—24no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4C2314yes—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155C25C2×D411yescharacteristic
156C22×Q8—22no—
157C22×Q8C2414yes—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—44no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C2×Q16—28no—
178C2×Q16—44no—
179C2×Q16—44no—
180C2×M16—24no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16C2×D418yes—
186C22×C8C2414yes—
187C22×C8C2414yes—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192#125—48no—
193C22⋊Q8—28no—
194C22⋊Q8—28no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—48no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C4×Q8—28no—
203C4×Q8—28no—
204C4×Q8—48no—
205#117—48no—
206#117—48no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2414yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C42—28no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D414yes—
248C2×C42#4114yes—
249C2×C42C2411yescharacteristic
250C2×C42C2411yescharacteristic
251C2×C42C2412yes—
252#107—44no—
253#106—28no—
254#106—44no—
255C4⋊C8—28no—
256C4⋊C8C2×D4116yes—
257#103—216no—
258#103—48no—
259#103—48no—
260C22⋊C8—216no—
261#97—28no—
262C4×C8—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273C24—22no—
274C24—24no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24C22×D411yescharacteristic
279C24C22×D412yes—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—44no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—432no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4#12118yes—
331C22×C4#13718yes—
332C22×C4C22×D412yes—
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C2×#4112yes—
340C22×C4C2511yescharacteristic
341Q16—88no—
342M16—48no—
343M16—48no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8C22×D418yes—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—48no—
368C4⋊C4—44no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—48no—
385C42—48no—
386C42#12114yes—
387C42#12114yes—
388C42C22×D414yes—
389C42C2×#4114yes—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—21no—
394C23—21no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—48no—
401C23—44no—
402C23—44no—
403C23—48no—
404C23—44no—
405C23C2×#12112yes—
406C23C2×#13714yes—
407C23C23×D411yescharacteristic
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—88no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—216no—
424C2×C4—216no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—48no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4C2×#12112yes—
454C2×C4C2×#12114yes—
455C2×C4C2×#12114yes—
456C2×C4C2×#13714yes—
457C2×C4C2×#13714yes—
458C2×C4#57514yes—
459C2×C4C23×D411yescharacteristic
460C2×C4C22×#4111yescharacteristic
461C2×C4#58511yescharacteristic
462C2×C4#58511yescharacteristic
463C2×C4#58511yescharacteristic
464C2×C4#58511yescharacteristic
465C8—28no—
466C8—44no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—22no—
472C22—416no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—41no—
478C22—41no—
479C22—42no—
480C22C22×#12111yescharacteristic
481C22#344911yescharacteristic
482C22#344911yescharacteristic
483C22C22×#13712yes—
484C22C2×#57512yes—
485C4—24no—
486C4—24no—
487C4—22no—
488C4—42no—
489C4—42no—
490C4—84no—
491C4—84no—
492C4—84no—
493C4—84no—
494C4#305618yes—
495C4#348511yescharacteristic
496C4#348711yescharacteristic
497C4#349311yescharacteristic
498C4#349311yescharacteristic
499C2—44no—
500C2—42no—
501C2#3515011yescharacteristic
502C2C2×#305612yes—
503C2#6095212yes—
504C2#6096711yescharacteristic
505C2#6097111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3255C2semidirect
4. #33144C2semidirect
6. #24710C2semidirect
7. #24703C2semidirect
8. #24660C2semidirect
9. #24652C2semidirect
10. #21964C2semidirect
11. #21894C2semidirect
15. C2×#475C22semidirect
17. C2×#462C22semidirect
46. #1988C22semidirect
47. #1981C22semidirect
48. #1971C22semidirect
49. #1969C22semidirect
50. #1948C22semidirect
51. #1947C22semidirect
52. #1943C22semidirect
53. #1941C22semidirect
54. #1935C22semidirect
55. #1933C22semidirect
58. #1803C22semidirect
59. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32552
C2×#29202
#366252
#331442
#297432
#247102
#247032
#246602
#246522
#219642
#218942