Finite Group Databasebeta

← #33121

Subgroups — #33121

Order 256.

436 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 131 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×#481C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3#2620C211yesmaximal, characteristic
4#2274C211yesmaximal, characteristic, complemented
5#1983C212yesmaximal, complemented
6#1948C212yesmaximal, complemented
7#1927C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1875C211yesmaximal, characteristic
10#1873C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#483—24no—
19#481—24no—
20C2×#106C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#396—22no—
30#394—22no—
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#384C2211yescharacteristic
34C4×C4⋊C4—22no—
35C4×C4⋊C4C2212yes—
36C4×C4⋊C4C2212yes—
37C4×C22⋊C4—22no—
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C23×C4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2312yes—
48#124—24no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51#121—22no—
52#121—22no—
53#117—24no—
54#117—24no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4D412yes—
60C2×C4⋊C4D414yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4D412yes—
71C2×C42—21no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#106—24no—
77#106—24no—
78C4⋊C8—24no—
79#103—24no—
80#103—24no—
81#102—24no—
82#102—24no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88C24—21no—
89C24C2×D411yescharacteristic
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—22no—
98C2×D4—42no—
99C2×D4—82no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—82no—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4#4112yes—
114C22×C4C2411yescharacteristic
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C4⋊C4C2×D414yes—
132C4⋊C4#4114yes—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—44no—
138C22⋊C4—82no—
139C42—24no—
140C42—22no—
141C42—82no—
142C42—82no—
143C42#4114yes—
144C23—21no—
145C23—21no—
146C23—22no—
147C23—22no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23#12112yes—
154C23C22×D411yescharacteristic
155Q8—42no—
156Q8—42no—
157D4—42no—
158D4—42no—
159D4—84no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4#12112yes—
184C2×C4#12314yes—
185C2×C4C22×D411yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C4C2×#4112yes—
188C2×C4C2×#4112yes—
189C8—42no—
190C8—42no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—82no—
197C22—81no—
198C22—82no—
199C22—81no—
200C22—81no—
201C22C2×#12111yescharacteristic
202C22C2×#12311yescharacteristic
203C22#52511yescharacteristic
204C22#57712yes—
205C22#57712yes—
206C4—24no—
207C4—22no—
208C4—42no—
209C4—44no—
210C4—44no—
211C4—81no—
212C4—81no—
213C4—81no—
214C4#52912yes—
215C4#53212yes—
216C2—42no—
217C2—161no—
218C2#285411yescharacteristic
219C2#304211yescharacteristic
220C2#304311yescharacteristic
221C2#309912yes—
222C2#312512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#483C2semidirect
4. #2274C2semidirect
5. #1983C2semidirect
6. #1948C2semidirect
7. #1927C2semidirect
8. #1926C2semidirect
10. #1873C2semidirect
11. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4832
#26202
#22742
#19832
#19482
#19272
#19262
#18752
#18732
#17982