Finite Group Databasebeta

← #17686

Subgroups — #17686

Order 256.

451 conjugacy classes of subgroups.

The subgroup lattice has 259 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3510C211yesmaximal, characteristic
2#2970C212yesmaximal, complemented
3#1535C212yesmaximal, complemented
4#1533C211yesmaximal, characteristic, complemented
5#1532C212yesmaximal, complemented
6#1531C212yesmaximal, complemented
7#1529C212yesmaximal, complemented
8#1524C212yesmaximal, complemented
9C2×#329C211yesmaximal, characteristic, complemented
10#559—22no—
11#556—22no—
12#551—22no—
13#551—22no—
14#542C412yes—
15#542C412yes—
16#536C411yescharacteristic
17#535C411yescharacteristic
18#533C2211yescharacteristic
19#529C2212yes—
20#529C2212yes—
21C2×#124C2211yescharacteristic
22#519C411yescharacteristic
23#518C411yescharacteristic
24C4×#41C2211yescharacteristic
25#446—22no—
26#435—22no—
27#435—22no—
28C8×D4—22no—
29#432C2212yes—
30#432C2212yescomplemented
31#432C2212yescomplemented
32C4×M16C2212yescomplemented
33C2×#97C2212yes—
34C2×#97C2212yescomplemented
35#330C2212yescomplemented
36#330C2212yescomplemented
37#330C2212yescomplemented
38#330C2212yescomplemented
39#329C2211yescharacteristic
40#329C2212yes—
41#329C2211yescharacteristic
42#329C2212yes—
43#329C2211yescharacteristic
44#329C2211yescharacteristic
45C2×#41C2×C411yescharacteristic
46C22×Q8C2×C411yescharacteristic
47C22×D4C2×C411yescharacteristic
48C2×M16—22no—
49C22×C8—22no—
50C22×C8—22no—
51C4⋊Q8—21no—
52C4⋊Q8C2×C412yes—
53#127—21no—
54#126—22no—
55#126—22no—
56#126—22no—
57#126—22no—
58#125C2312yes—
59#124—21no—
60#124—21no—
61#124—22no—
62#124—22no—
63#124—22no—
64#124C2×C411yescharacteristic
65#124C2×C411yescharacteristic
66#124C2311yescharacteristic
67#124C2311yescharacteristic
68#123—22no—
69#123—22no—
70#123—22no—
71#123—22no—
72#123—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8C2×C412yes—
77C22⋊Q8C2×C412yes—
78#121C2×C412yes—
79#121C2×C412yes—
80#120—22no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8C2×C412yes—
84C4×Q8C2×C412yes—
85C4×Q8D412yes—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4C2×C412yes—
90C4×D4C2×C412yes—
91C4×D4D412yes—
92C4×D4D412yes—
93C4×D4D412yes—
94#117—21no—
95#117—21no—
96#117—22no—
97#117—22no—
98#117C2×C411yescharacteristic
99#117C2×C411yescharacteristic
100#117C2×C411yescharacteristic
101#117C2×C412yes—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2312yes—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113#97—22no—
114#97—22no—
115#97C2312yes—
116#97C2312yes—
117#97C2312yes—
118#97C2312yes—
119C4×C8—22no—
120C24—21no—
121#41—42no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—42no—
125C2×Q8C22⋊C411yescharacteristic
126C2×Q8C22×C411yescharacteristic
127C2×Q8C22×C411yescharacteristic
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—42no—
132C2×D4C22⋊C412yes—
133C2×D4C22⋊C411yescharacteristic
134C2×D4C22×C411yescharacteristic
135C2×D4C22×C411yescharacteristic
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4C22×C411yescharacteristic
140C22×C4C22×C412yes—
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2×D412yes—
144M16—44no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—44no—
150C2×C8—44no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4C22⋊C412yes—
159C4⋊C4C22⋊C412yes—
160C4⋊C4C22⋊C412yes—
161C4⋊C4C22×C412yes—
162C4⋊C4C22×C412yes—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4C22⋊C411yescharacteristic
176C22⋊C4C22⋊C411yescharacteristic
177C22⋊C4C22⋊C412yes—
178C22⋊C4C22⋊C412yes—
179C42—22no—
180C42—22no—
181C42—21no—
182C42—21no—
183C42C22×C412yes—
184C42C22×C411yescharacteristic
185C42C2×D412yes—
186C42C2×D412yes—
187C42C2×D412yes—
188C42C2411yescharacteristic
189C23—21no—
190C23—21no—
191C23—42no—
192C23—82no—
193C23C2×C22⋊C412yes—
194C23C2×C22⋊C411yescharacteristic
195Q8—41no—
196Q8—41no—
197Q8—41no—
198Q8—42no—
199Q8—82no—
200D4—24no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—42no—
205D4—82no—
206D4—82no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4C2×C22⋊C411yescharacteristic
220C2×C4C2×C22⋊C412yes—
221C2×C4C2×C22⋊C412yes—
222C2×C4C2×C22⋊C412yes—
223C2×C4C2×C22⋊C412yes—
224C2×C4C23×C411yescharacteristic
225C2×C4C22×D412yes—
226C8—24no—
227C8—42no—
228C8—42no—
229C8—42no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—162no—
239C22#41312yes—
240C22C22×C22⋊C411yescharacteristic
241C4—21no—
242C4—21no—
243C4—22no—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—41no—
249C4—41no—
250C4—42no—
251C4—42no—
252C2—22no—
253C2—42no—
254C2—81no—
255C2—81no—
256C2C2×#41311yescharacteristic
257C2#290112yes—
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#535C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2970C2semidirect
3. #1535C2semidirect
4. #1533C2semidirect
5. #1532C2semidirect
6. #1531C2semidirect
7. #1529C2semidirect
8. #1524C2semidirect
9. C2×#329C2semidirect
30. #432C22semidirect
31. #432C22semidirect
32. C4×M16C22semidirect
34. C2×#97C22semidirect
35. #330C22semidirect
36. #330C22semidirect
37. #330C22semidirect
38. #330C22semidirect

Maximal subgroups

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

subgroupindex
#35102
#29702
#15352
#15332
#15322
#15312
#15292
#15242
C2×#3292