Finite Group Databasebeta

← #6640541

Subgroups — #6640541

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3#32982C212yesmaximal, complemented
4#32259C214yesmaximal, complemented
5#32027C214yesmaximal, complemented
6#31998C218yesmaximal, complemented
7#29893C212yesmaximal, complemented
8#27613C212yesmaximal, complemented
9#24551C214yesmaximal, complemented
10#24063C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#123—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#483C2214yes—
17C22×C22⋊C8C2212yes—
18#2848C2212yes—
19#2848C2212yes—
20#2626—21no—
21#2626—21no—
22#2580—22no—
23#2566—22no—
24#2563C2212yes—
25#2542C2212yes—
26#2538C2214yes—
27#2486—22no—
28#2486—22no—
29#2486C2214yes—
30#2486C2214yes—
31C2×#398—21no—
32C2×#397—21no—
33#2317—21no—
34#2309—21no—
35C2×#389C2212yes—
36#2285C2211yescharacteristic
37C2×#381—21no—
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40#2100C2214yescomplemented
41#2084C2214yescomplemented
42#2077C2218yescomplemented
43#2077C2218yescomplemented
44#2058C2214yescomplemented
45#2043C2214yescomplemented
46#2035C2218yescomplemented
47#2035C2218yescomplemented
48#1919C2218yescomplemented
49#1892C2218yescomplemented
50#1872C2214yes—
51#1872C2218yescomplemented
52#1869C2214yes—
53#1869C2214yes—
54#1865C2218yescomplemented
55#1853C2214yes—
56#1853C2218yescomplemented
57#1823C2214yes—
58#1817C2212yes—
59#1807C2218yescomplemented
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×#123—24no—
71C2×#123—24no—
72C2×#123—48no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8C2312yes—
75C2×C22⋊Q8C2314yes—
76C2×C22⋊Q8C2314yes—
77C2×C4×Q8—21no—
78C2×C4×Q8C2312yes—
79C2×C4×D4—44no—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×C4×D4C2312yes—
83C2×C4×D4C2312yes—
84C2×#117—21no—
85C2×#117—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42C2311yescharacteristic
99#483—48no—
100C2×#106C2318yes—
101C2×#106C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104C2×C22⋊C8—44no—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C2×C22⋊C8C2318yes—
108#399—22no—
109#399—22no—
110#398—24no—
111#398—24no—
112#397—24no—
113#397—24no—
114#396—24no—
115#396—24no—
116#393—22no—
117#393—22no—
118#390—22no—
119#390—22no—
120#390—24no—
121#389—24no—
122#389—22no—
123#389—22no—
124#389C2314yes—
125#389C2314yes—
126#385—24no—
127#385—24no—
128#385—22no—
129#385—22no—
130#384—22no—
131#384C2312yes—
132#382—22no—
133#382—21no—
134#381—24no—
135#381—24no—
136#380—21no—
137#380—24no—
138#380—22no—
139#380C2314yes—
140#380C2314yes—
141#380C2318yes—
142#380C2314yes—
143#380C2312yes—
144#379—22no—
145#379—22no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4C2312yes—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2314yes—
156#336C2314yes—
157C25C2×D411yescharacteristic
158C25C2×D411yescharacteristic
159C22×Q8—21no—
160C22×Q8C2412yes—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4SD1618yes—
164C22×D4C2×D412yes—
165C22×D4C2×D412yes—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C22×C8—48no—
186C22×C8—44no—
187C22×C8C2418yes—
188C4⋊Q8—42no—
189C4⋊Q8—42no—
190#125—44no—
191#123—416no—
192#123—416no—
193C22⋊Q8—28no—
194C22⋊Q8—44no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C4×Q8—44no—
198C4×Q8—44no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205#117—44no—
206#117—44no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2412yes—
234C2×C4⋊C4C2414yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4C2×D412yes—
248C2×C22⋊C4C2×D412yes—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×D412yes—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2411yescharacteristic
257C2×C42C2411yescharacteristic
258#106—48no—
259#106—48no—
260#103—48no—
261#103—48no—
262C22⋊C8—28no—
263C22⋊C8—44no—
264C22⋊C8—48no—
265#95—28no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—24no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24C2×SD1614yes—
282C24C2×SD1614yes—
283C24C22×D412yes—
284C24C22×D412yes—
285C24C22×D411yescharacteristic
286C2×Q8—24no—
287C2×Q8—28no—
288C2×Q8—42no—
289C2×Q8—82no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—416no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4C2×SD1618yes—
296C2×D4C2×SD1618yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—416no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—48no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—48no—
330C22×C4—44no—
331C22×C4—88no—
332C22×C4—816no—
333C22×C4C2×SD1614yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4H(5,𝔽2)14yes—
342C22×C4C2511yescharacteristic
343C2×C8—28no—
344C2×C8—28no—
345C2×C8—28no—
346C2×C8—88no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—28no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—48no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—82no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—44no—
373C22⋊C4—48no—
374C22⋊C4—416no—
375C22⋊C4—44no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—44no—
385C42—44no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—28no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—28no—
398C23—48no—
399C23—48no—
400C23—44no—
401C23—48no—
402C23—44no—
403C23—82no—
404C23—84no—
405C23—82no—
406C23#53412yes—
407C23C22×SD1614yes—
408C23C22×SD1614yes—
409C23C22×SD1612yes—
410C23#57512yes—
411C23C23×D411yescharacteristic
412Q8—48no—
413Q8—84no—
414D4—48no—
415D4—48no—
416C2×C4—28no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—28no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
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—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—88no—
454C2×C4—816no—
455C2×C4—88no—
456C2×C4#53412yes—
457C2×C4C22×SD1612yes—
458C2×C4C22×SD1612yes—
459C2×C4C2×H(5,𝔽2)12yes—
460C2×C4#58514yes—
461C8—48no—
462C22—24no—
463C22—22no—
464C22—22no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—28no—
469C22—28no—
470C22—416no—
471C22—44no—
472C22—44no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—88no—
478C22#318714yes—
479C22C2×#53411yescharacteristic
480C22#348512yes—
481C22C23×SD1611yescharacteristic
482C22C2×#57511yescharacteristic
483C22#358612yes—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—41no—
488C4—41no—
489C4—42no—
490C4—42no—
491C4—84no—
492C4—84no—
493C4—82no—
494C4—82no—
495C4—84no—
496C4#353314yes—
497C2—24no—
498C2—44no—
499C2—44no—
500C2#3695511yescharacteristic
501C2C2×#318711yescharacteristic
502C2#6075411yescharacteristic
503C2#6169512yes—
504C2#6171112yes—
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. #32982C2semidirect
4. #32259C2semidirect
5. #32027C2semidirect
6. #31998C2semidirect
7. #29893C2semidirect
8. #27613C2semidirect
9. #24551C2semidirect
10. #24063C2semidirect
40. #2100C22semidirect
41. #2084C22semidirect
42. #2077C22semidirect
43. #2077C22semidirect
44. #2058C22semidirect
45. #2043C22semidirect
46. #2035C22semidirect
47. #2035C22semidirect
48. #1919C22semidirect
49. #1892C22semidirect
51. #1872C22semidirect
54. #1865C22semidirect
56. #1853C22semidirect
59. #1807C22semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#24862
#329822
#322592
#320272
#319982
#298932
#276132
#245512
#240632
#217382