Finite Group Databasebeta

← #6436927

Subgroups — #6436927

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#29942C214yesmaximal, complemented
4#29925C214yesmaximal, complemented
5#29912C214yesmaximal, complemented
6#29892C212yesmaximal, complemented
7#29890C212yesmaximal, complemented
8#23764C218yesmaximal, complemented
9#21395C212yesmaximal, complemented
10#21365C212yesmaximal, complemented
11#20715C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#525—21no—
14C23×C42C2211yescharacteristic
15C2×#432C2212yes—
16#2857C2212yes—
17#2856C2212yes—
18#2854—21no—
19#2854C2212yes—
20#2850—22no—
21#2847—21no—
22#2847C2212yes—
23C4×#121—22no—
24C4×#121C2214yes—
25#2297—21no—
26#2285C2211yescharacteristic
27#2283—22no—
28#2283—22no—
29#2283C2214yes—
30#2283C2214yes—
31C2×#383—21no—
32C2×#382C2212yes—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4—21no—
38C2×#376—21no—
39#2101C2218yescomplemented
40#2092C2218yescomplemented
41#2085C2218yescomplemented
42#2077C2218yes—
43#2075C2218yes—
44#2073C2218yescomplemented
45#2034C2218yes—
46#2033C2218yes—
47#1987C2214yescomplemented
48#1982C2214yescomplemented
49#1948C2214yescomplemented
50#1947C2214yescomplemented
51#1874C2214yes—
52#1874C2214yes—
53#1853C2214yes—
54#1852C2214yes—
55#1817C2212yes—
56#1797C2214yes—
57#1779C2212yes—
58#1776C2218yescomplemented
59#1762C2218yescomplemented
60#1749C2214yes—
61C22×#41—21no—
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4C2311yescharacteristic
65#532—28no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#124—21no—
69#525—48no—
70C2×#123—22no—
71C2×C22⋊Q8—22no—
72C2×#121—21no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121C2312yes—
76C2×#121C2314yes—
77C2×#120—22no—
78C2×C4×D4—24no—
79C2×C4×D4—21no—
80C2×C4×D4C2312yes—
81C2×#117—21no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—28no—
87C22×C42—22no—
88C22×C42D412yes—
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D412yes—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#432—28no—
97C2×#107C2318yes—
98C2×#106C2318yes—
99C2×C4⋊C8C2314yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2318yes—
104C2×#97C2314yes—
105#391—22no—
106#390—22no—
107#390—22no—
108#390C2314yes—
109#389—22no—
110#389C2314yes—
111#388—24no—
112#388—24no—
113#386—24no—
114#386—22no—
115#386—24no—
116#386—22no—
117#384C2312yes—
118#383—24no—
119#383—24no—
120#382—21no—
121#382—22no—
122#382—22no—
123#382C2314yes—
124#382C2314yes—
125#381—24no—
126#381—24no—
127#380—21no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380C2314yes—
132#380C2314yes—
133#380C2314yes—
134#380C2312yes—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C4×C4⋊C4C2312yes—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—24no—
144#376—24no—
145#376—24no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2318yes—
156C25C2×D411yescharacteristic
157C2×#41—44no—
158C2×#41—44no—
159C2×#41—44no—
160C22×Q8—21no—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C2412yes—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C22×C8C2418yes—
189C4⋊Q8—24no—
190#127—24no—
191#124—44no—
192#123—48no—
193C22⋊Q8—48no—
194#121—28no—
195#121—48no—
196#121—48no—
197#121—48no—
198#121—44no—
199#120—48no—
200C4×D4—28no—
201C4×D4—416no—
202C4×D4—44no—
203#117—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2414yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—28no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—48no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4118yes—
246C2×C42C2412yes—
247C2×C42C2411yescharacteristic
248C2×C42C2411yescharacteristic
249#107—48no—
250#106—48no—
251C4⋊C8—28no—
252#102—48no—
253#102—48no—
254C22⋊C8—28no—
255C22⋊C8—48no—
256#97—28no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—42no—
273C24—42no—
274C24AGL(1,ℤ/8)14yes—
275C24C22×D412yes—
276C24C22×D411yescharacteristic
277#41—816no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×D4—24no—
282C2×D4—28no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—44no—
288C2×D4—48no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—44no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—416no—
328C22×C4—416no—
329C22×C4—416no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4#12314yes—
340C22×C4#12314yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4114yes—
350C22×C4C2511yescharacteristic
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—28no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—24no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—44no—
372C22⋊C4—48no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—48no—
382C42—44no—
383C42—48no—
384C42#12314yes—
385C42#12314yes—
386C42C2×#4114yes—
387C42C2×#4114yes—
388C23—22no—
389C23—22no—
390C23—28no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—48no—
398C23—44no—
399C23—44no—
400C23—48no—
401C23—42no—
402C23—44no—
403C23—42no—
404C23—44no—
405C23—42no—
406C23C2×#12312yes—
407C23C2×AGL(1,ℤ/8)14yes—
408C23C2×AGL(1,ℤ/8)12yes—
409C23#57512yes—
410C23C23×D411yescharacteristic
411Q8—84no—
412D4—48no—
413D4—84no—
414D4—816no—
415D4—88no—
416D4—84no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—88no—
457C2×C4—88no—
458C2×C4—88no—
459C2×C4C2×#12312yes—
460C2×C4C2×#12312yes—
461C2×C4C2×#12312yes—
462C2×C4C2×#12312yes—
463C2×C4C2×#12312yes—
464C2×C4#57514yes—
465C2×C4C22×#4112yes—
466C2×C4#58514yes—
467C8—48no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—22no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—416no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—84no—
481C22—84no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22#309414yes—
486C22C22×#12311yescharacteristic
487C22#344912yes—
488C22C22×AGL(1,ℤ/8)11yescharacteristic
489C22C2×#57512yes—
490C22C2×#57511yescharacteristic
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—84no—
499C4—84no—
500C4—84no—
501C4—82no—
502C4#309314yes—
503C4#345514yes—
504C2—24no—
505C2—44no—
506C2—162no—
507C2#3463511yescharacteristic
508C2C2×#309311yescharacteristic
509C2C2×#309411yescharacteristic
510C2#6095812yes—
511C2#6097012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
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
2. C2×#2283C2semidirect
3. #29942C2semidirect
4. #29925C2semidirect
5. #29912C2semidirect
6. #29892C2semidirect
7. #29890C2semidirect
8. #23764C2semidirect
9. #21395C2semidirect
10. #21365C2semidirect
11. #20715C2semidirect
39. #2101C22semidirect
40. #2092C22semidirect
41. #2085C22semidirect
44. #2073C22semidirect
47. #1987C22semidirect
48. #1982C22semidirect
49. #1948C22semidirect
50. #1947C22semidirect
58. #1776C22semidirect
59. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#22832
#299422
#299252
#299122
#298922
#298902
#237642
#213952
#213652
#207152