Finite Group Databasebeta

← #8320

Subgroups — #8320

Order 256.

397 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 81 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#3460C211yesmaximal, characteristic
2#1954C211yesmaximal, characteristic, complemented
3#1477C212yesmaximal, complemented
4#1476C212yesmaximal, complemented
5#1460C211yesmaximal, characteristic, complemented
6#544C411yescharacteristic
7#542—22no—
8#540—22no—
9#538—22no—
10#537C411yescharacteristic
11#535C412yes—
12#532—22no—
13#529—22no—
14#525—22no—
15C4×#41C2212yes—
16C2×C4×D4C2211yescharacteristic
17#431C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C22⋊C8—22no—
20C2×C4×C8C2211yescharacteristic
21Q8⋊C8—22no—
22Q8⋊C8—22no—
23D4⋊C8—22no—
24D4⋊C8—22no—
25#323—22no—
26C2×#41—22no—
27C2×#41C2×C412yes—
28C22×D4C2×C411yescharacteristic
29C23×C4D412yes—
30C22×C8—21no—
31C22×C8—21no—
32C4⋊Q8—22no—
33#127—22no—
34#126—24no—
35#125—22no—
36#124—24no—
37#124—22no—
38#123—24no—
39#123—24no—
40C22⋊Q8—22no—
41C22⋊Q8—22no—
42C22⋊Q8—22no—
43#121—22no—
44#121—24no—
45#121—22no—
46#121—22no—
47#121—22no—
48#121—22no—
49#120—24no—
50C4×Q8D412yes—
51C4×Q8D412yes—
52C4×D4—24no—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4D412yes—
58C4×D4D412yes—
59#117—22no—
60#117—22no—
61#117C2×C412yes—
62C2×C4⋊C4C2×C411yescharacteristic
63C2×C22⋊C4D412yes—
64C2×C42—22no—
65C2×C42C2311yescharacteristic
66C4⋊C8—22no—
67C4⋊C8—22no—
68C22⋊C8—22no—
69C22⋊C8—44no—
70C4×C8—22no—
71C4×C8—22no—
72C24C22⋊C412yes—
73#41—24no—
74#41—24no—
75#41—82no—
76#41—82no—
77C2×Q8—22no—
78C2×Q8C22⋊C412yes—
79C2×D4—22no—
80C2×D4—24no—
81C2×D4—24no—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4C22⋊C412yes—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4C22⋊C412yes—
95C22×C4C22×C411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C2×C8—22no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—42no—
102C2×C8—42no—
103C2×C8—41no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—24no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4C22⋊C412yes—
111C4⋊C4C22⋊C412yes—
112C22⋊C4—24no—
113C22⋊C4—24no—
114C22⋊C4—24no—
115C22⋊C4—22no—
116C22⋊C4—24no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C42—22no—
120C42—22no—
121C42—22no—
122C42C2×D412yes—
123C42C2×D412yes—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—22no—
128C23—42no—
129C23—42no—
130C23C2×C22⋊C411yescharacteristic
131Q8—24no—
132Q8—82no—
133D4—24no—
134D4—42no—
135D4—42no—
136D4—44no—
137D4—84no—
138D4—82no—
139D4—84no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4C2×C22⋊C412yes—
156C2×C4#12012yes—
157C2×C4#12011yescharacteristic
158C2×C4#12011yescharacteristic
159C2×C4#13112yes—
160C2×C4#13112yes—
161C8—42no—
162C8—82no—
163C8—81no—
164C22—21no—
165C22—21no—
166C22—44no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—82no—
174C22#37911yescharacteristic
175C22#40811yescharacteristic
176C22#40811yescharacteristic
177C4—21no—
178C4—22no—
179C4—42no—
180C4—42no—
181C4—42no—
182C4—42no—
183C4#40814yes—
184C4#45311yescharacteristic
185C4#45511yescharacteristic
186C2—21no—
187C2—21no—
188C2—82no—
189C2—82no—
190C2#178011yescharacteristic
191C2#179211yescharacteristic
192C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1954C2semidirect
3. #1477C2semidirect
4. #1476C2semidirect
5. #1460C2semidirect

Maximal subgroups

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

subgroupindex
#34602
#19542
#14772
#14762
#14602