Finite Group Databasebeta

← #12728

Subgroups — #12728

Order 256.

357 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 69 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#3510C211yesmaximal, characteristic
2#2127C212yesmaximal
3C2×#354C212yesmaximal, complemented
4#1828C212yesmaximal, complemented
5#559—22no—
6#556—22no—
7#551—24no—
8#542—22no—
9#536C411yescharacteristic, complemented
10#535C2211yescharacteristic
11#533C411yescharacteristic, complemented
12#529—22no—
13C2×#124C2211yescharacteristic
14#519C411yescharacteristic, complemented
15#518C2211yescharacteristic
16C4×#41C411yescharacteristic, complemented
17C2×#99C2214yes—
18#354—22no—
19#354—22no—
20#354—22no—
21#353—22no—
22#342—24no—
23C2×#41C2×C411yescharacteristic
24C22×Q8C2×C411yescharacteristic
25C22×D4C2311yescharacteristic
26C4⋊Q8—21no—
27C4⋊Q8—21no—
28#127D412yes—
29#126—24no—
30#126—24no—
31#125—21no—
32#124—24no—
33#124—21no—
34#124—22no—
35#124D412yes—
36#124D412yes—
37#124D412yes—
38#123—24no—
39#123—24no—
40#123—22no—
41C22⋊Q8—24no—
42C22⋊Q8—22no—
43C22⋊Q8—22no—
44#121—22no—
45#120—22no—
46C4×Q8—21no—
47C4×Q8—22no—
48C4×Q8—22no—
49C4×Q8—22no—
50C4×D4—24no—
51C4×D4—21no—
52C4×D4—22no—
53C4×D4—22no—
54C4×D4—22no—
55#117—24no—
56#117—22no—
57#117—21no—
58#117C2×C411yescharacteristic
59#117C2×C411yescharacteristic
60#117C2×C411yescharacteristic
61C2×C22⋊C4—22no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4D414yes—
64C2×C42—21no—
65C2×C42C2×C411yescharacteristic
66#99—24no—
67#99—24no—
68C24C2×D412yes—
69#41—42no—
70C2×Q8—22no—
71C2×Q8—21no—
72C2×Q8—42no—
73C2×Q8C22⋊C412yes—
74C2×Q8C22⋊C411yescharacteristic
75C2×D4—21no—
76C2×D4—24no—
77C2×D4—42no—
78C2×D4C22⋊C411yescharacteristic
79C2×D4C2×D412yes—
80C2×D4C2×D412yes—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—24no—
85C22×C4—21no—
86C22×C4—44no—
87C22×C4C22×C411yescharacteristic
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C22⋊C4—22no—
98C22⋊C4—24no—
99C22⋊C4—21no—
100C22⋊C4—24no—
101C22⋊C4—24no—
102C22⋊C4—24no—
103C22⋊C4—24no—
104C22⋊C4—44no—
105C22⋊C4—44no—
106C22⋊C4C22⋊C412yes—
107C22⋊C4C22⋊C412yes—
108C42—22no—
109C42—24no—
110C42—21no—
111C42—21no—
112C42—21no—
113C42C22⋊C411yescharacteristic
114C42C22⋊C411yescharacteristic
115C42C22⋊C412yes—
116C23—24no—
117C23—21no—
118C23—82no—
119C23C2×C22⋊C411yescharacteristic
120C23#12014yes—
121Q8—42no—
122Q8—41no—
123Q8—42no—
124Q8—82no—
125D4—42no—
126D4—42no—
127D4—41no—
128D4—42no—
129D4—84no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—82no—
141C2×C4—84no—
142C2×C4—81no—
143C2×C4—84no—
144C2×C4—164no—
145C2×C4C2×C22⋊C411yescharacteristic
146C2×C4C2×C22⋊C411yescharacteristic
147C22—24no—
148C22—41no—
149C22—44no—
150C22—42no—
151C22—41no—
152C22—84no—
153C22#37911yescharacteristic
154C22#41012yes—
155C4—21no—
156C4—21no—
157C4—42no—
158C4—41no—
159C4—42no—
160C4—41no—
161C4—42no—
162C4—41no—
163C4—42no—
164C4—164no—
165C2—21no—
166C2—22no—
167C2—44no—
168C2—81no—
169C2#178211yescharacteristic
subgroupG/H
center, socleC2#1782
commutatorC22×C4C22×C4
Frattini, cosocleC22×D4C23
prime radical#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#354C2semidirect
4. #1828C2semidirect
9. #536C4semidirect
11. #533C4semidirect
14. #519C4semidirect
16. C4×#41C4semidirect

Maximal subgroups

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

subgroupindex
#35102
#21272
C2×#3542
#18282