Finite Group Databasebeta

← #10321

Subgroups — #10321

Order 256.

363 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 61 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#3186C211yesmaximal, characteristic
2#2854C212yesmaximal, complemented
3#1817C211yesmaximal, characteristic, complemented
4#1519C212yesmaximal, complemented
5#1289C211yesmaximal, characteristic, complemented
6C2×#123—22no—
7C2×#121C2212yes—
8C22×C42C2211yescharacteristic
9#480—22no—
10#466—22no—
11#447—22no—
12C2×C22⋊C8C2211yescharacteristic
13C2×C22⋊C8C2211yescharacteristic
14#390—22no—
15#389—22no—
16#388—24no—
17#386—22no—
18#380C2212yes—
19#336—22no—
20#327—24no—
21C22×D4—22no—
22C22×D4D412yes—
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C2×D8—22no—
27C22×C8—21no—
28C22×C8D412yes—
29#123—44no—
30#123—44no—
31#121—22no—
32#121—22no—
33#121—22no—
34#121—22no—
35C2×C4⋊C4—24no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4D412yes—
38C2×C22⋊C4—24no—
39C2×C22⋊C4—24no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C42—21no—
43C2×C42—22no—
44C2×C42—22no—
45C2×C42—21no—
46#107—22no—
47#102—24no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#95—24no—
51#95D414yes—
52C24—22no—
53C24C2×D411yescharacteristic
54C2×D4—22no—
55C2×D4—22no—
56C2×D4—44no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—42no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—42no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—24no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4D812yescomplemented
74C22×C4C2×D412yes—
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D412yes—
77C22×C4#4111yescharacteristic
78D8—82no—
79C2×C8—22no—
80C2×C8—42no—
81C2×C8—42no—
82C2×C8—41no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—44no—
86C4⋊C4—48no—
87C22⋊C4—48no—
88C22⋊C4—48no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C42—41no—
94C42—42no—
95C42—41no—
96C23—41no—
97C23—41no—
98C23—44no—
99C23—44no—
100C23—42no—
101C23—42no—
102C23—42no—
103C23#12012yes—
104C23#12111yescharacteristic
105C23C2×D811yescharacteristic
106C23#13511yescharacteristic
107C23AGL(1,ℤ/8)12yes—
108D4—44no—
109D4—82no—
110D4—82no—
111D4—82no—
112D4—82no—
113D4—82no—
114C2×C4—24no—
115C2×C4—22no—
116C2×C4—24no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—44no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4#12011yescharacteristic
135C2×C4#12112yes—
136C2×C4#12711yescharacteristic
137C8—42no—
138C8—81no—
139C22—22no—
140C22—21no—
141C22—22no—
142C22—21no—
143C22—41no—
144C22—41no—
145C22—42no—
146C22—84no—
147C22—84no—
148C22—82no—
149C22—82no—
150C22—82no—
151C22#39211yescharacteristic
152C22#44712yes—
153C22#44912yes—
154C22#46611yescharacteristic
155C22#46911yescharacteristic
156C4—21no—
157C4—22no—
158C4—42no—
159C4—42no—
160C4—84no—
161C4—82no—
162C4#45812yes—
163C2—22no—
164C2—22no—
165C2—21no—
166C2—21no—
167C2—162no—
168C2#199911yescharacteristic
169C2#201011yescharacteristic
170C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radical#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2854C2semidirect
3. #1817C2semidirect
4. #1519C2semidirect
5. #1289C2semidirect
73. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
#31862
#28542
#18172
#15192
#12892