Finite Group Databasebeta

← #2636

Subgroups — #2636

Order 128.

281 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 96 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×#126C211yesmaximal, characteristic, complemented
2C2×C4×D4C211yesmaximal, characteristic, complemented
3#400C211yesmaximal, characteristic, complemented
4#399C211yesmaximal, characteristic, complemented
5#396C212yesmaximal, complemented
6#394C211yesmaximal, characteristic, complemented
7#394C212yesmaximal, complemented
8#392C211yesmaximal, characteristic
9#386C212yesmaximal, complemented
10#385C212yesmaximal, complemented
11#382C211yesmaximal, characteristic, complemented
12C22×D4C2211yescharacteristic
13C22×D4C2212yes—
14C23×C4C2212yes—
15#126—24no—
16C4×D4—24no—
17C2×C4⋊C4C2211yescharacteristic
18C2×C4⋊C4C2211yescharacteristic
19C2×C4⋊C4C2211yescharacteristic
20C2×C4⋊C4C2211yescharacteristic
21C2×C4⋊C4C2212yes—
22C2×C4⋊C4C2212yes—
23C2×C22⋊C4C2212yes—
24C2×C22⋊C4C2211yescharacteristic
25C2×C22⋊C4C2211yescharacteristic
26C2×C22⋊C4C2211yescharacteristic
27C2×C22⋊C4C2212yes—
28C2×C22⋊C4C2212yes—
29C2×C22⋊C4C2212yes—
30C2×C22⋊C4C2212yes—
31C2×C42C2211yescharacteristic
32#95C2212yes—
33#95C2211yescharacteristic
34#95C2211yescharacteristic
35#95C2212yes—
36#95C2212yes—
37C24C2311yescharacteristic
38C24C2312yes—
39C2×D4—24no—
40C2×D4—24no—
41C2×D4—22no—
42C2×D4—42no—
43C2×D4—42no—
44C22×C4—24no—
45C22×C4—42no—
46C22×C4—42no—
47C22×C4C2312yes—
48C22×C4C2311yescharacteristic
49C22×C4C2311yescharacteristic
50C22×C4C2312yes—
51C22×C4C2312yes—
52C22×C4C2311yescharacteristic
53C22×C4C2311yescharacteristic
54C22×C4C2311yescharacteristic
55C22×C4C2311yescharacteristic
56C4⋊C4—22no—
57C4⋊C4—22no—
58C4⋊C4—22no—
59C4⋊C4—22no—
60C4⋊C4—42no—
61C4⋊C4—42no—
62C22⋊C4—24no—
63C22⋊C4—24no—
64C22⋊C4—22no—
65C22⋊C4—22no—
66C22⋊C4—22no—
67C22⋊C4—42no—
68C22⋊C4—42no—
69C22⋊C4—42no—
70C42D414yescomplemented
71C23—22no—
72C23—22no—
73C23—22no—
74C23—22no—
75C23—22no—
76C23—22no—
77C23—21no—
78C23—22no—
79C23—22no—
80C23—21no—
81C23—21no—
82C23#4114yes—
83C23C2411yescharacteristic
84D4—28no—
85D4—48no—
86C2×C4—22no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—21no—
106C2×C4—48no—
107C2×C4C2×D412yes—
108C2×C4C2×D412yes—
109C2×C4C2×D412yes—
110C22—24no—
111C22—24no—
112C22—24no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—42no—
117C22—41no—
118C22—41no—
119C22—41no—
120C22—42no—
121C22—42no—
122C22C22×D411yescharacteristic
123C22C2×#4112yes—
124C22H(5,𝔽2)12yes—
125C22H(5,𝔽2)11yescharacteristic
126C222−1+411yescharacteristic
127C4—22no—
128C4—22no—
129C4—22no—
130C4—41no—
131C4—41no—
132C4—42no—
133C4—41no—
134C4—42no—
135C4—42no—
136C2—44no—
137C2—81no—
138C2#52511yescharacteristic
139C2#53511yescharacteristic
140C2#53711yescharacteristic
141C2#53812yes—
142C2#53912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C24
prime radical#392C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#126C2semidirect
2. C2×C4×D4C2semidirect
3. #400C2semidirect
4. #399C2semidirect
5. #396C2semidirect
6. #394C2semidirect
7. #394C2semidirect
9. #386C2semidirect
10. #385C2semidirect
11. #382C2semidirect
70. C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×#1262
C2×C4×D42
#4002
#3992
#3962
#3942
#3922
#3862
#3852
#3822