Finite Group Databasebeta

← #9300

Subgroups — #9300

Order 256.

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

normal subgroups only

This group has 254 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#3226C211yesmaximal, characteristic
2#2327C211yesmaximal, characteristic, complemented
3#1963C211yesmaximal, characteristic
4#1546C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic
6#1286C211yesmaximal, characteristic, complemented
7#1274C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4×Q8—21no—
11#500C412yescomplemented
12#486C412yescomplemented
13#480—22no—
14#474—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18C2×#104—21no—
19C2×#104—21no—
20#391—21no—
21#387—22no—
22#387—22no—
23#382C2211yescharacteristic
24#376C2211yescharacteristic
25#332—22no—
26#327—22no—
27C2×#41—21no—
28C22×Q8—21no—
29C2×M16—21no—
30C4⋊Q8—21no—
31C4⋊Q8—21no—
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C412yes—
34C4⋊Q8D412yes—
35#127C2×C411yescharacteristic
36#124—21no—
37#121—21no—
38#121—21no—
39C4×Q8—42no—
40C4×Q8—42no—
41C4×D4—21no—
42C4×D4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4D412yes—
47C2×C42—22no—
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51#107—24no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56#104—42no—
57#104—42no—
58#102—24no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C4×C8—21no—
62C4×C8C2×C412yes—
63#95—24no—
64#95—21no—
65#95—21no—
66#95—21no—
67#95D412yes—
68#41—41no—
69#41—41no—
70#41—41no—
71#41—41no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—21no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—41no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91M16—41no—
92M16—41no—
93C2×C8—22no—
94C2×C8—41no—
95C2×C8—41no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—44no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4C22⋊C414yes—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42—21no—
108C42—21no—
109C42—21no—
110C42—44no—
111C42D812yes—
112C42C22×C411yescharacteristic
113C42C2×D411yescharacteristic
114C42#4111yescharacteristic
115C42#4111yescharacteristic
116C23—41no—
117C23#12011yescharacteristic
118Q8—42no—
119Q8—42no—
120Q8—42no—
121Q8—42no—
122Q8—41no—
123Q8—41no—
124D4—42no—
125D4—41no—
126D4—41no—
127D4—81no—
128D4—81no—
129D4—81no—
130C2×C4—24no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4C2×C22⋊C411yescharacteristic
148C2×C4C4×D411yescharacteristic
149C2×C4C4×D411yescharacteristic
150C2×C4#12111yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12311yescharacteristic
153C2×C4C2×D811yescharacteristic
154C2×C4#13511yescharacteristic
155C2×C4AGL(1,ℤ/8)11yescharacteristic
156C2×C4AGL(1,ℤ/8)11yescharacteristic
157C8—42no—
158C8—81no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22#38611yescharacteristic
166C22#44711yescharacteristic
167C22#44911yescharacteristic
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—42no—
173C4—44no—
174C4—82no—
175C4—81no—
176C4—81no—
177C4C4×D811yescharacteristic
178C4#44211yescharacteristic
179C4#45911yescharacteristic
180C4#46311yescharacteristic
181C2—21no—
182C2—21no—
183C2—161no—
184C2#187911yescharacteristic
185C2#190511yescharacteristic
186C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2327C2semidirect
4. #1546C2semidirect
6. #1286C2semidirect
7. #1274C2semidirect
11. #500C4semidirect
12. #486C4semidirect

Maximal subgroups

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

subgroupindex
#32262
#23272
#19632
#15462
#15172
#12862
#12742