Finite Group Databasebeta

← #6640544

Subgroups — #6640544

Order 512.

1776 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3#32983C212yesmaximal, complemented
4#32261C214yesmaximal, complemented
5#32029C214yesmaximal, complemented
6#32000C218yesmaximal, complemented
7#29895C212yesmaximal, complemented
8#27614C212yesmaximal, complemented
9#24552C214yesmaximal, complemented
10#24064C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#484C2214yes—
16C22×C22⋊C8C2212yes—
17#2848C2212yes—
18#2848C2212yes—
19#2626—21no—
20#2626—21no—
21#2580—22no—
22#2566—22no—
23#2563C2212yes—
24#2542C2212yes—
25#2538C2214yes—
26#2487—22no—
27#2487—22no—
28#2487C2214yes—
29#2487C2214yes—
30C2×#400—21no—
31C2×#399—21no—
32#2317—21no—
33#2309—21no—
34C2×#389C2212yes—
35C2×#386—21no—
36C2×#385—21no—
37#2285C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40#2100C2214yescomplemented
41#2083C2214yescomplemented
42#2076C2218yescomplemented
43#2076C2218yescomplemented
44#2057C2214yescomplemented
45#2043C2214yescomplemented
46#2036C2218yescomplemented
47#2036C2218yescomplemented
48#1921C2218yescomplemented
49#1892C2218yescomplemented
50#1872C2214yes—
51#1872C2218yescomplemented
52#1869C2214yes—
53#1869C2214yes—
54#1863C2218yescomplemented
55#1852C2214yes—
56#1852C2218yescomplemented
57#1824C2214yes—
58#1817C2212yes—
59#1808C2218yescomplemented
60#1779C2212yes—
61C23×D4D411yescharacteristic
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C23×C8C2314yes—
66#533—44no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8C2312yes—
72C2×C22⋊Q8C2314yes—
73C2×C22⋊Q8C2314yes—
74C2×C4×Q8—21no—
75C2×C4×Q8C2312yes—
76C2×C4×D4—44no—
77C2×C4×D4D412yes—
78C2×C4×D4D412yes—
79C2×C4×D4C2312yes—
80C2×C4×D4C2312yes—
81C2×#117—21no—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42C2311yescharacteristic
96#484—48no—
97C2×#107C2318yes—
98C2×#107C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×C22⋊C8—44no—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2318yes—
105#400—24no—
106#400—24no—
107#399—22no—
108#399—22no—
109#399—24no—
110#399—24no—
111#396—24no—
112#396—24no—
113#393—22no—
114#393—22no—
115#390—22no—
116#390—22no—
117#390—24no—
118#389—24no—
119#389—22no—
120#389—22no—
121#389C2314yes—
122#389C2314yes—
123#386—24no—
124#386—24no—
125#385—24no—
126#385—24no—
127#385—22no—
128#385—22no—
129#385—24no—
130#385—24no—
131#384—22no—
132#384C2312yes—
133#382—22no—
134#382—21no—
135#380—21no—
136#380—24no—
137#380—22no—
138#380C2314yes—
139#380C2314yes—
140#380C2318yes—
141#380C2314yes—
142#380C2312yes—
143#379—22no—
144#379—22no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2312yes—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—22no—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C25C2×D411yescharacteristic
157C25C2×D411yescharacteristic
158C22×Q8—21no—
159C22×Q8C2412yes—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4Q1618yes—
163C22×D4C2×D412yes—
164C22×D4C2×D412yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—84no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C22×C8—48no—
184C22×C8—44no—
185C22×C8C2418yes—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188#125—44no—
189C22⋊Q8—28no—
190C22⋊Q8—44no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C4×Q8—44no—
194C4×Q8—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201#117—44no—
202#117—44no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—84no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2412yes—
230C2×C4⋊C4C2414yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—84no—
243C2×C22⋊C4C2×D412yes—
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2411yescharacteristic
253C2×C42C2411yescharacteristic
254#107—48no—
255#107—48no—
256#103—48no—
257#103—48no—
258C22⋊C8—28no—
259C22⋊C8—44no—
260C22⋊C8—48no—
261#95—28no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—24no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24C2×Q1614yes—
278C24C2×Q1614yes—
279C24C22×D412yes—
280C24C22×D412yes—
281C24C22×D411yescharacteristic
282C2×Q8—24no—
283C2×Q8—28no—
284C2×Q8—42no—
285C2×Q8—82no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—416no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4C2×Q1618yes—
292C2×D4C2×Q1618yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—416no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—816no—
328C22×C4C2×Q1614yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4H(5,𝔽2)14yes—
337C22×C4C2511yescharacteristic
338C2×C8—28no—
339C2×C8—28no—
340C2×C8—28no—
341C2×C8—88no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—28no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—82no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—48no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C22⋊C4—88no—
373C22⋊C4—84no—
374C22⋊C4—84no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—44no—
380C42—44no—
381C23—22no—
382C23—22no—
383C23—24no—
384C23—24no—
385C23—28no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—28no—
393C23—48no—
394C23—48no—
395C23—44no—
396C23—44no—
397C23—48no—
398C23—82no—
399C23—84no—
400C23—82no—
401C23#53412yes—
402C23C22×Q1614yes—
403C23C22×Q1614yes—
404C23C22×Q1612yes—
405C23#57512yes—
406C23C23×D411yescharacteristic
407Q8—48no—
408Q8—84no—
409D4—48no—
410D4—48no—
411C2×C4—28no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—48no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
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—44no—
444C2×C4—44no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—88no—
449C2×C4—88no—
450C2×C4—816no—
451C2×C4#53412yes—
452C2×C4C22×Q1612yes—
453C2×C4C22×Q1612yes—
454C2×C4C2×H(5,𝔽2)12yes—
455C2×C4#58514yes—
456C8—48no—
457C22—24no—
458C22—24no—
459C22—22no—
460C22—22no—
461C22—28no—
462C22—28no—
463C22—24no—
464C22—24no—
465C22—416no—
466C22—44no—
467C22—44no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—88no—
473C22#318914yes—
474C22C2×#53411yescharacteristic
475C22#348512yes—
476C22C23×Q1611yescharacteristic
477C22C2×#57511yescharacteristic
478C22#358512yes—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—42no—
485C4—42no—
486C4—84no—
487C4—84no—
488C4—84no—
489C4—82no—
490C4—82no—
491C4#353314yes—
492C2—24no—
493C2—44no—
494C2—44no—
495C2#3695511yescharacteristic
496C2C2×#318911yescharacteristic
497C2#6075311yescharacteristic
498C2#6169812yes—
499C2#6171312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32983C2semidirect
4. #32261C2semidirect
5. #32029C2semidirect
6. #32000C2semidirect
7. #29895C2semidirect
8. #27614C2semidirect
9. #24552C2semidirect
10. #24064C2semidirect
40. #2100C22semidirect
41. #2083C22semidirect
42. #2076C22semidirect
43. #2076C22semidirect
44. #2057C22semidirect
45. #2043C22semidirect
46. #2036C22semidirect
47. #2036C22semidirect
48. #1921C22semidirect
49. #1892C22semidirect
51. #1872C22semidirect
54. #1863C22semidirect
56. #1852C22semidirect
59. #1808C22semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#24872
#329832
#322612
#320292
#320002
#298952
#276142
#245522
#240642
#217382