Finite Group Databasebeta

← #18536

Subgroups — #18536

Order 256.

470 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic, complemented
2#3141C211yesmaximal, characteristic, complemented
3#3112C211yesmaximal, characteristic, complemented
4#3109C211yesmaximal, characteristic
5#1638C214yesmaximal, complemented
6#1628C214yesmaximal, complemented
7#1556C211yesmaximal, characteristic, complemented
8#1491C211yesmaximal, characteristic, complemented
9#1490C211yesmaximal, characteristic, complemented
10#556—22no—
11#556—22no—
12#556—22no—
13#556C2214yes—
14#532C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#126—22no—
17C2×#124—21no—
18C2×C22⋊Q8—21no—
19C2×C4×Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23#489—22no—
24#487C2212yescomplemented
25#486C2212yescomplemented
26#481—21no—
27#480—21no—
28#477C2212yescomplemented
29#476C2212yes—
30#460C2212yescomplemented
31#459C2212yes—
32#448—21no—
33#447—21no—
34#442—22no—
35#440—22no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#417—22no—
39C4×M16C2211yescharacteristic
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—21no—
45C22×Q8D411yescharacteristic
46C22×D4D411yescharacteristic
47C23×C4—21no—
48C2×SD16—22no—
49C2×D8—22no—
50C2×M16—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8D412yes—
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2312yes—
56#127C2311yescharacteristic
57#126—28no—
58#126—28no—
59#124—24no—
60#124—24no—
61#124—21no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64#121—21no—
65#121—21no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8D412yes—
71C4×Q8C2312yes—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4—21no—
76C4×D4D412yes—
77C4×D4C2312yes—
78#117—24no—
79#117—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D412yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#107—22no—
93#106—22no—
94C4⋊C8C2312yes—
95C4⋊C8C2312yes—
96#103—22no—
97#103—24no—
98#102—22no—
99#102—24no—
100#102—22no—
101#102—22no—
102C22⋊C8—21no—
103C22⋊C8—21no—
104#97—21no—
105#97—21no—
106#97—21no—
107C4×C8C2312yes—
108C24—21no—
109#41—44no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—42no—
115C2×Q8—44no—
116C2×Q8C2×D412yes—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—42no—
122C2×D4—41no—
123C2×D4C2×D412yes—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—44no—
131C22×C4—41no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135SD16—44no—
136D8—44no—
137M16—42no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C4⋊C4—28no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4C2×D414yes—
152C4⋊C4C2×D412yes—
153C4⋊C4C2×D412yes—
154C22⋊C4—28no—
155C22⋊C4—28no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C42—28no—
161C42—22no—
162C42—22no—
163C42C2×D411yescharacteristic
164C42C2×D411yescharacteristic
165C42C2×D411yescharacteristic
166C42C2411yescharacteristic
167C23—22no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—41no—
172C23#12011yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—42no—
176Q8—44no—
177Q8—42no—
178Q8—44no—
179D4—22no—
180D4—22no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—82no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4#12011yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4C22×D411yescharacteristic
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C8—42no—
208C8—42no—
209C8—42no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—84no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#45412yes—
220C22C2×#12011yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—42no—
226C4—42no—
227C4—44no—
228C4—42no—
229C4—81no—
230C4#57512yes—
231C4#57512yes—
232C2—22no—
233C2—82no—
234C2—161no—
235C2#300511yescharacteristic
236C2#300511yescharacteristic
237C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3109C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#556C2semidirect
2. #3141C2semidirect
3. #3112C2semidirect
5. #1638C2semidirect
6. #1628C2semidirect
7. #1556C2semidirect
8. #1491C2semidirect
9. #1490C2semidirect
24. #487C22semidirect
25. #486C22semidirect
28. #477C22semidirect
30. #460C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
#31412
#31122
#31092
#16382
#16282
#15562
#14912
#14902