Finite Group Databasebeta

← #2413

Subgroups — #2413

Order 128.

305 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 112 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×#123C211yesmaximal, characteristic
2C2×#123C211yesmaximal, characteristic
3C2×#117C211yesmaximal, characteristic, complemented
4C22×C4⋊C4C211yesmaximal, characteristic, complemented
5#398C212yesmaximal, complemented
6#396C212yesmaximal, complemented
7#389C212yesmaximal, complemented
8#388C212yesmaximal, complemented
9#386C211yesmaximal, characteristic
10#385C211yesmaximal, characteristic, complemented
11#381C211yesmaximal, characteristic, complemented
12C22×D4C2211yescharacteristic
13C23×C4C2211yescharacteristic
14C23×C4C2211yescharacteristic
15C23×C4C2211yescharacteristic
16#123—24no—
17#123—24no—
18#117—24no—
19C2×C4⋊C4—24no—
20C2×C4⋊C4C2212yes—
21C2×C4⋊C4C2212yes—
22C2×C4⋊C4C2212yes—
23C2×C4⋊C4C2211yescharacteristic
24C2×C4⋊C4C2212yes—
25C2×C4⋊C4C2211yescharacteristic
26C2×C4⋊C4C2212yes—
27C2×C22⋊C4C2211yescharacteristic
28C2×C22⋊C4C2211yescharacteristic
29C2×C22⋊C4C2211yescharacteristic
30C2×C22⋊C4C2212yes—
31C2×C22⋊C4C2211yescharacteristic
32C2×C22⋊C4C2212yes—
33C2×C22⋊C4C2211yescharacteristic
34C2×C42C2212yes—
35#95C2212yes—
36#95C2212yes—
37#95C2212yes—
38#95C2212yes—
39C24C2311yescharacteristic
40C24C2311yescharacteristic
41C2×D4—22no—
42C2×D4—22no—
43C2×D4—22no—
44C22×C4—22no—
45C22×C4—22no—
46C22×C4—22no—
47C22×C4—24no—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4—22no—
51C22×C4D414yes—
52C22×C4C2311yescharacteristic
53C22×C4C2312yes—
54C22×C4C2311yescharacteristic
55C22×C4C2312yes—
56C22×C4C2311yescharacteristic
57C22×C4C2312yes—
58C22×C4C2311yescharacteristic
59C22×C4C2311yescharacteristic
60C22×C4C2312yes—
61C4⋊C4—24no—
62C4⋊C4—24no—
63C4⋊C4—24no—
64C4⋊C4—22no—
65C4⋊C4—22no—
66C4⋊C4—24no—
67C4⋊C4—24no—
68C22⋊C4—22no—
69C22⋊C4—22no—
70C22⋊C4—22no—
71C22⋊C4—22no—
72C22⋊C4—24no—
73C22⋊C4—24no—
74C22⋊C4D414yes—
75C42—24no—
76C23—22no—
77C23—22no—
78C23—21no—
79C23—21no—
80C23—22no—
81C23—21no—
82C23—22no—
83C23C2×D412yes—
84C23C2×D412yes—
85C23#4112yes—
86C23C2411yescharacteristic
87D4—44no—
88C2×C4—28no—
89C2×C4—28no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—44no—
112C2×C4C2×D414yes—
113C2×C4C2×D412yes—
114C2×C4C2×D412yes—
115C2×C4#4114yes—
116C22—22no—
117C22—22no—
118C22—24no—
119C22—24no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—42no—
124C22—42no—
125C22#12114yes—
126C22C22×D411yescharacteristic
127C22C22×D411yescharacteristic
128C22C2×#4111yescharacteristic
129C22C2×#4112yes—
130C222−1+412yes—
131C4—22no—
132C4—22no—
133C4—24no—
134C4—24no—
135C4—42no—
136C4—41no—
137C4—41no—
138C4—41no—
139C4—42no—
140C2—24no—
141C2—81no—
142C2C2×#12111yescharacteristic
143C2#52511yescharacteristic
144C2#53611yescharacteristic
145C2#54712yes—
146C2#55212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C24
prime radicalC22×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#117C2semidirect
4. C22×C4⋊C4C2semidirect
5. #398C2semidirect
6. #396C2semidirect
7. #389C2semidirect
8. #388C2semidirect
10. #385C2semidirect
11. #381C2semidirect

Maximal subgroups

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

subgroupindex
C2×#1232
C2×#1172
C22×C4⋊C42
#3982
#3962
#3892
#3882
#3862
#3852
#3812