Finite Group Databasebeta

← #60415

Subgroups — #60415

Order 256.

1421 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#575C212yesmaximal, complemented
2C2×#532C211yesmaximal, characteristic, complemented
3C2×C4×#41C211yesmaximal, characteristic, complemented
4#3040C214yesmaximal, complemented
5#3039C212yesmaximal, complemented
6#3038C212yesmaximal, complemented
7#3037C214yesmaximal, complemented
8#3036C214yesmaximal, complemented
9#3035C218yesmaximal, complemented
10C2×#423C211yesmaximal, characteristic, complemented
11#2896C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×#41C2211yescharacteristic
14#575—216no—
15C2×#137C2214yes—
16C2×#137C2214yescomplemented
17C2×AGL(1,ℤ/8)C2214yes—
18C2×AGL(1,ℤ/8)C2214yescomplemented
19C2×#135C2218yescomplemented
20C22×M16C2212yes—
21#532—21no—
22#532—21no—
23#532—21no—
24#532—21no—
25#532C2214yes—
26#532C2214yescomplemented
27C2×C4⋊Q8C2211yescharacteristic, complemented
28C2×#127C2211yescharacteristic, complemented
29C2×#124C2212yescomplemented
30C2×#121—22no—
31C4×#41—21no—
32C4×#41—21no—
33C4×#41—22no—
34C4×#41C2212yes—
35C4×#41C2212yes—
36C4×#41C2214yescomplemented
37C2×C4×Q8C2212yescomplemented
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4C2212yescomplemented
41C2×#117—21no—
42C2×#117C2211yescharacteristic
43C22×C42—21no—
44C22×C42C2211yescharacteristic
45#464C22116yescomplemented
46#463C22116yescomplemented
47#462C2218yescomplemented
48#461C2218yescomplemented
49#460C2218yescomplemented
50#459C2218yescomplemented
51#423C2214yescomplemented
52#423C2212yescomplemented
53#423C2212yescomplemented
54C2×C4⋊C8C2214yescomplemented
55#419C2214yescomplemented
56#418C2214yescomplemented
57#417C2218yescomplemented
58#416C2218yescomplemented
59C2×#41—21no—
60C2×#41—21no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—24no—
64C2×#41—28no—
65C2×#41—28no—
66C2×#41D418yescomplemented
67C2×#41C2314yes—
68C2×#41C2314yes—
69C2×#41C2312yes—
70C2×#41C2312yes—
71C22×Q8C2312yes—
72C22×Q8C2311yescharacteristic
73C22×D4—21no—
74C22×D4—22no—
75C22×D4—22no—
76C22×D4C2312yes—
77C22×D4C2311yescharacteristic
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4D412yes—
82C23×C4C2311yescharacteristic
83#137—216no—
84#137—216no—
85AGL(1,ℤ/8)—216no—
86AGL(1,ℤ/8)—216no—
87#135—232no—
88C2×Q16C2318yes—
89C2×SD16C2318yes—
90C2×SD16C2318yes—
91C2×D8C2318yes—
92C2×M16—28no—
93C2×M16C2314yes—
94C2×M16C2314yes—
95C22×C8C2314yes—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8C2314yes—
99#127—21no—
100#127—21no—
101#127C2314yes—
102#124—22no—
103#124—22no—
104#124C2318yes—
105#121—24no—
106#121—24no—
107#121—24no—
108#121—24no—
109C4×Q8—22no—
110C4×Q8—22no—
111C4×Q8C2314yes—
112C4×Q8C2314yes—
113C4×D4—24no—
114C4×D4—24no—
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—24no—
118C4×D4—24no—
119C4×D4—24no—
120C4×D4—24no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4C2314yes—
124C4×D4C2314yes—
125#117—21no—
126#117—21no—
127#117—22no—
128#117—22no—
129#117—24no—
130#117C2312yes—
131#117C2312yes—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C2312yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—24no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—24no—
143C2×C42D411yescharacteristic
144C2×C42D412yes—
145C2×C42D411yescharacteristic
146C2×C42D411yescharacteristic
147C2×C42D411yescharacteristic
148C2×C42C2312yes—
149C2×C42C2311yescharacteristic
150C2×C42C2311yescharacteristic
151C2×C42C2312yes—
152C4⋊C8C2318yes—
153C4⋊C8C2318yes—
154#103C23116yes—
155#102C23116yes—
156C24—21no—
157C24—22no—
158C24C2×D411yescharacteristic
159#41—28no—
160#41—28no—
161#41—24no—
162#41—24no—
163#41—28no—
164#41—28no—
165#41—28no—
166#41—28no—
167#41—48no—
168#41—48no—
169#41C2×D418yes—
170#41C2×D418yescomplemented
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8C2×D414yescomplemented
177C2×Q8C2414yes—
178C2×Q8C2412yes—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—21no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—24no—
191C2×D4—48no—
192C2×D4—48no—
193C2×D4C2×D414yescomplemented
194C2×D4C2414yes—
195C2×D4C2412yes—
196C22×C4—21no—
197C22×C4—28no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4C2×D412yes—
210C22×C4C2×D412yes—
211C22×C4C2×D412yes—
212C22×C4C2×D412yes—
213C22×C4C2×D412yes—
214C22×C4C2×D412yes—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D414yes—
219C22×C4#4114yes—
220C22×C4C2411yescharacteristic
221C22×C4C2411yescharacteristic
222C22×C4C2411yescharacteristic
223Q16—216no—
224SD16—216no—
225SD16—216no—
226D8—216no—
227M16—28no—
228M16—28no—
229C2×C8—28no—
230C2×C8C2418yes—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4C2414yes—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—28no—
242C22⋊C4—28no—
243C42—22no—
244C42—24no—
245C42—24no—
246C42—22no—
247C42C2×D411yescharacteristic
248C42C2×D411yescharacteristic
249C42C2×D411yescharacteristic
250C42C2×D411yescharacteristic
251C42C2×D412yes—
252C42C2×D412yes—
253C42C2×D412yes—
254C42C2×D412yes—
255C42C2412yes—
256C42C2412yes—
257C23—24no—
258C23—22no—
259C23—24no—
260C23—42no—
261C23—44no—
262C23—48no—
263C23—44no—
264C23#12114yes—
265C23C22×D411yescharacteristic
266C23C22×D411yescharacteristic
267C23C22×D411yescharacteristic
268Q8—24no—
269Q8—22no—
270Q8—24no—
271Q8—24no—
272Q8—44no—
273Q8C22×D414yes—
274D4—28no—
275D4—28no—
276D4—24no—
277D4—24no—
278D4—22no—
279D4—24no—
280D4—48no—
281D4—48no—
282D4—48no—
283D4—44no—
284D4—48no—
285D4C22×D414yes—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—28no—
289C2×C4—28no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4#12114yes—
305C2×C4#12114yes—
306C2×C4#12114yes—
307C2×C4C22×D412yes—
308C2×C4C22×D412yes—
309C2×C4C22×D412yes—
310C2×C4C22×D411yescharacteristic
311C2×C4C22×D411yescharacteristic
312C2×C4C22×D412yes—
313C2×C4C22×D412yes—
314C2×C4C22×D412yes—
315C2×C4C22×D412yes—
316C2×C4C22×D411yescharacteristic
317C2×C4C2×#4112yes—
318C2×C4C2×#4112yes—
319C2×C4C2×#4112yes—
320C2×C4C2511yescharacteristic
321C8—28no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—24no—
326C22—44no—
327C22—48no—
328C22—44no—
329C22—44no—
330C22—48no—
331C22—88no—
332C22C2×#12112yes—
333C22C2×#12112yes—
334C22C2×#12112yes—
335C22C23×D411yescharacteristic
336C4—24no—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—44no—
341C4—44no—
342C4C2×#12112yes—
343C4C2×#12112yes—
344C4C2×#12112yes—
345C4#57514yescomplemented
346C4C23×D411yescharacteristic
347C4C22×#4111yescharacteristic
348C2—22no—
349C2—22no—
350C2—22no—
351C2—44no—
352C2—84no—
353C2C22×#12111yescharacteristic
354C2C2×#57512yes—
subgroupG/H
centerC2×C4C22×D4
commutator, Frattini, cosocleC2×C4C25
socleC22C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#575C2semidirect
2. C2×#532C2semidirect
3. C2×C4×#41C2semidirect
4. #3040C2semidirect
5. #3039C2semidirect
6. #3038C2semidirect
7. #3037C2semidirect
8. #3036C2semidirect
9. #3035C2semidirect
10. C2×#423C2semidirect
11. #2896C2semidirect
16. C2×#137C22semidirect
18. C2×AGL(1,ℤ/8)C22semidirect
19. C2×#135C22semidirect
26. #532C22semidirect
27. C2×C4⋊Q8C22semidirect
28. C2×#127C22semidirect
29. C2×#124C22semidirect
36. C4×#41C22semidirect
37. C2×C4×Q8C22semidirect
40. C2×C4×D4C22semidirect
45. #464C22semidirect
46. #463C22semidirect
47. #462C22semidirect
48. #461C22semidirect
49. #460C22semidirect
50. #459C22semidirect
51. #423C22semidirect
52. #423C22semidirect
53. #423C22semidirect
54. C2×C4⋊C8C22semidirect
55. #419C22semidirect
56. #418C22semidirect
57. #417C22semidirect
58. #416C22semidirect
66. C2×#41D4semidirect
170. #41C2×D4semidirect
176. C2×Q8C2×D4semidirect
193. C2×D4C2×D4semidirect
345. C4#575semidirect

Maximal subgroups

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

subgroupindex
C2×#5752
C2×#5322
C2×C4×#412
#30402
#30392
#30382
#30372
#30362
#30352
C2×#4232
#28962