Finite Group Databasebeta

← #17628

Subgroups — #17628

Order 256.

471 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 183 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
1#3507C211yesmaximal, characteristic
2#2983C211yesmaximal, characteristic, complemented
3#2351C211yesmaximal, characteristic, complemented
4#1518C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic
6#1512C214yesmaximal, complemented
7#1509C214yesmaximal, complemented
8#1503C211yesmaximal, characteristic, complemented
9#1502C211yesmaximal, characteristic
10#559C412yescomplemented
11#557C412yescomplemented
12#556—22no—
13#556C412yescomplemented
14#554C412yescomplemented
15#552—22no—
16#539—22no—
17#536—21no—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20#529C2214yes—
21#520—21no—
22C2×C4×Q8—21no—
23C2×C4×Q8C2211yescharacteristic
24#446—22no—
25#446—22no—
26#432C2214yes—
27#431C2211yescharacteristic
28#431C2211yescharacteristic
29#389—21no—
30#389—22no—
31#382C2212yes—
32#382C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34C4×C4⋊C4C2212yes—
35#328C2212yes—
36#328C2212yes—
37#328C2214yes—
38#327C2212yes—
39#327C2212yes—
40#327C2214yes—
41C2×#41—21no—
42C22×Q8D411yescharacteristic
43C4⋊Q8—22no—
44C4⋊Q8C2×C411yescharacteristic
45C4⋊Q8C2×C411yescharacteristic
46C4⋊Q8C2×C411yescharacteristic
47#127C2×C411yescharacteristic
48#126—24no—
49#126—24no—
50#125—24no—
51#125—21no—
52#125C2×C414yes—
53#124—22no—
54#124—21no—
55#124C2×C414yes—
56#123—24no—
57#123—24no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—21no—
61C22⋊Q8—24no—
62C22⋊Q8C2312yes—
63#121—21no—
64#121C2312yes—
65C4×Q8—22no—
66C4×Q8—21no—
67C4×Q8—44no—
68C4×Q8C2×C414yes—
69C4×Q8C2×C412yes—
70C4×Q8C2×C412yes—
71C4×Q8C2×C412yes—
72C4×Q8D412yes—
73C4×Q8D412yes—
74C4×D4—24no—
75C4×D4—22no—
76C4×D4—21no—
77C4×D4C2×C414yes—
78C4×D4C2×C412yes—
79#117—21no—
80#117—22no—
81#117—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D412yes—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42C2312yes—
92C2×C42C2311yescharacteristic
93C4⋊C8—21no—
94C4⋊C8—21no—
95C4⋊C8—24no—
96C22⋊C8C2312yes—
97C22⋊C8C2312yes—
98#97—24no—
99C4×C8—21no—
100C4×C8—21no—
101#95—22no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2311yescharacteristic
105#95C2312yes—
106#41—42no—
107#41—42no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8C22⋊C412yes—
113C2×Q8C22⋊C412yes—
114C2×Q8C22×C412yes—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C22×C412yes—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2411yescharacteristic
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—21no—
132C4⋊C4—24no—
133C4⋊C4—21no—
134C4⋊C4—24no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—42no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4C22⋊C412yes—
145C4⋊C4C22⋊C412yes—
146C4⋊C4C22⋊C414yes—
147C4⋊C4C22⋊C414yes—
148C4⋊C4C22×C412yes—
149C4⋊C4C22×C412yes—
150C22⋊C4—22no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—22no—
154C22⋊C4—24no—
155C22⋊C4—22no—
156C42—24no—
157C42—22no—
158C42—44no—
159C42—42no—
160C42C22×C411yescharacteristic
161C42C22×C414yes—
162C42C22×C411yescharacteristic
163C42C2×D411yescharacteristic
164C42C2×D411yescharacteristic
165C42C2×D414yes—
166C23—22no—
167C23C22×D411yescharacteristic
168Q8—24no—
169Q8—24no—
170Q8—42no—
171Q8—42no—
172Q8—81no—
173D4—42no—
174D4—42no—
175D4—42no—
176D4—42no—
177D4—81no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—84no—
197C2×C4C2×C22⋊C414yes—
198C2×C4C2×C22⋊C414yes—
199C2×C4C2×C22⋊C411yescharacteristic
200C2×C4C2×C22⋊C411yescharacteristic
201C2×C4C2×C22⋊C411yescharacteristic
202C2×C4C2×C22⋊C411yescharacteristic
203C2×C4AGL(1,ℤ/8)12yes—
204C2×C4#13712yes—
205C2×C4C23×C411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C8—42no—
208C8—42no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C22×C22⋊C411yescharacteristic
215C22C2×AGL(1,ℤ/8)11yescharacteristic
216C22C2×#13711yescharacteristic
217C4—21no—
218C4—21no—
219C4—24no—
220C4—24no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—82no—
228C4—81no—
229C4#41712yes—
230C4#41712yes—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#288411yescharacteristic
235C2C2×#41711yescharacteristic
236C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2983C2semidirect
3. #2351C2semidirect
4. #1518C2semidirect
6. #1512C2semidirect
7. #1509C2semidirect
8. #1503C2semidirect
10. #559C4semidirect
11. #557C4semidirect
13. #556C4semidirect
14. #554C4semidirect

Maximal subgroups

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

subgroupindex
#35072
#29832
#23512
#15182
#15172
#15122
#15092
#15032
#15022