Finite Group Databasebeta

← #13573

Subgroups — #13573

Order 256.

235 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 45 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×#456C211yesmaximal, characteristic
2C2×#356C211yesmaximal, characteristic, complemented
3#1558C211yesmaximal, characteristic, complemented
4#1405C214yesmaximal, complemented
5C2×2−1+4C411yescharacteristic
6C2×#137—21no—
7C2×C4⋊Q8C2211yescharacteristic
8#456—21no—
9#456—21no—
10#456C412yes—
11#456C2212yes—
12#432—21no—
13C2×#104—21no—
14C2×#101—21no—
15C2×#101C411yescharacteristic
16#356—21no—
17#356C2212yescomplemented
18#333—21no—
19#333C2212yescomplemented
202−1+4—21no—
212−1+4—41no—
222−1+4—42no—
232−1+4C2×C412yes—
24C2×#41—21no—
25C2×#41—41no—
26C2×#41—41no—
27C22×Q8—41no—
28C22×Q8C2×C411yescharacteristic
29C22×Q8D411yescharacteristic
30#137—22no—
31#137—22no—
32#137—42no—
33C2×Q16—41no—
34C2×SD16—41no—
35C2×M16—21no—
36C2×M16—41no—
37C4⋊Q8—41no—
38C4⋊Q8C2×C411yescharacteristic
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8C2311yescharacteristic
42C2×C4⋊C4—41no—
43C2×C42D411yescharacteristic
44C4⋊C8—22no—
45#104—22no—
46#104—22no—
47#101—21no—
48#101—22no—
49#101—41no—
50#101C2×C412yes—
51C22⋊C8—41no—
52#97—41no—
53#41—22no—
54#41—22no—
55#41—41no—
56#41—42no—
57#41—42no—
58#41—42no—
59#41—42no—
60#41—44no—
61#41—42no—
62#41—42no—
63#41—41no—
64C2×Q8—21no—
65C2×Q8—21no—
66C2×Q8—21no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8—42no—
72C2×Q8—42no—
73C2×Q8—44no—
74C2×Q8—41no—
75C2×Q8C22⋊C411yescharacteristic
76C2×Q8C22⋊C411yescharacteristic
77C2×Q8C22×C411yescharacteristic
78C2×Q8C2×D411yescharacteristic
79C2×D4—21no—
80C2×D4—41no—
81C2×D4—41no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—41no—
86C22×C4—42no—
87C22×C4C22⋊C411yescharacteristic
88Q16—42no—
89Q16—42no—
90SD16—42no—
91SD16—42no—
92M16—22no—
93M16—41no—
94M16—42no—
95M16—81no—
96C2×C8—41no—
97C2×C8—42no—
98C2×C8—81no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C42—21no—
104C42C22⋊C411yescharacteristic
105C42C2×D411yescharacteristic
106C23—41no—
107C23#9911yescharacteristic
108Q8—22no—
109Q8—41no—
110Q8—41no—
111Q8—41no—
112Q8—41no—
113Q8—42no—
114Q8—41no—
115Q8—42no—
116Q8—42no—
117Q8—81no—
118D4—22no—
119D4—41no—
120D4—42no—
121D4—42no—
122D4—42no—
123D4—42no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—44no—
136C2×C4—44no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4#9911yescharacteristic
143C2×C4C2×C22⋊C411yescharacteristic
144C8—42no—
145C8—82no—
146C8—82no—
147C22—41no—
148C22—42no—
149C22—82no—
150C22AΣL(1,𝔽16)11yescharacteristic
151C22AΣL(1,𝔽16)11yescharacteristic
152C22C2×#9911yescharacteristic
153C4—22no—
154C4—21no—
155C4—21no—
156C4—41no—
157C4—41no—
158C4—41no—
159C4—41no—
160C4—41no—
161C4—42no—
162C2—21no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2C2×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
center, socleC2C2×AΣL(1,𝔽16)
commutatorC2×Q8C22×C4
Frattini, cosocleC4⋊Q8C23
prime radicalC2×2−1+4C4
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G
LECSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#356C2semidirect
3. #1558C2semidirect
4. #1405C2semidirect
17. #356C22semidirect
19. #333C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4562
C2×#3562
#15582
#14052