Finite Group Databasebeta

← #2813

Subgroups — #2813

Order 128.

289 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 92 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, complemented
2C2×#121C211yesmaximal, characteristic, complemented
3#400C211yesmaximal, characteristic, complemented
4#399C211yesmaximal, characteristic, complemented
5#396C212yesmaximal, complemented
6#394C211yesmaximal, characteristic, complemented
7#394C212yesmaximal, complemented
8#394C212yesmaximal, complemented
9#392C211yesmaximal, characteristic, complemented
10#392C212yesmaximal, complemented
11#380C211yesmaximal, characteristic, complemented
12C22×D4C2212yes—
13C22×D4C2211yescharacteristic
14C22×D4C2211yescharacteristic
15C22×D4C2212yes—
16C23×C4C2211yescharacteristic
17#123—24no—
18#121—24no—
19C2×C4⋊C4C2211yescharacteristic
20C2×C4⋊C4C2211yescharacteristic
21C2×C4⋊C4C2211yescharacteristic
22C2×C4⋊C4C2212yes—
23C2×C4⋊C4C2212yes—
24C2×C22⋊C4C2212yes—
25C2×C22⋊C4C2211yescharacteristic
26C2×C22⋊C4C2211yescharacteristic
27C2×C22⋊C4C2211yescharacteristic
28C2×C22⋊C4C2212yes—
29C2×C22⋊C4C2212yes—
30C2×C22⋊C4C2212yes—
31C2×C22⋊C4C2212yes—
32C2×C22⋊C4C2212yes—
33#95C2211yescharacteristic
34#95C2211yescharacteristic
35#95C2212yes—
36#95C2212yes—
37C24C2311yescharacteristic
38C24C2312yes—
39C24C2311yescharacteristic
40C2×D4—24no—
41C2×D4—22no—
42C2×D4—22no—
43C2×D4—42no—
44C2×D4—42no—
45C2×D4—42no—
46C2×D4—42no—
47C2×D4—42no—
48C2×D4—42no—
49C2×D4—42no—
50C22×C4—42no—
51C22×C4D414yescomplemented
52C22×C4C2312yes—
53C22×C4C2311yescharacteristic
54C22×C4C2312yes—
55C22×C4C2311yescharacteristic
56C22×C4C2312yes—
57C22×C4C2311yescharacteristic
58C22×C4C2311yescharacteristic
59C22×C4C2311yescharacteristic
60C4⋊C4—22no—
61C4⋊C4—22no—
62C4⋊C4—22no—
63C4⋊C4—42no—
64C4⋊C4—42no—
65C22⋊C4—22no—
66C22⋊C4—24no—
67C22⋊C4—22no—
68C22⋊C4—22no—
69C22⋊C4—42no—
70C22⋊C4—42no—
71C22⋊C4—42no—
72C22⋊C4—42no—
73C22⋊C4—42no—
74C23—21no—
75C23—22no—
76C23—22no—
77C23—21no—
78C23—21no—
79C23—22no—
80C23—22no—
81C23—22no—
82C23—22no—
83C23—22no—
84C23—22no—
85C23—22no—
86C23—22no—
87C23—21no—
88C23—21no—
89C23—22no—
90C23C2×D412yes—
91C23C2411yescharacteristic
92D4—44no—
93D4—44no—
94D4—48no—
95D4—48no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—44no—
115C2×C4C2×D412yes—
116C2×C4C2×D412yes—
117C22—22no—
118C22—22no—
119C22—22no—
120C22—42no—
121C22—42no—
122C22—42no—
123C22—41no—
124C22—42no—
125C22—41no—
126C22—41no—
127C22—42no—
128C22—42no—
129C22—42no—
130C22—42no—
131C22—42no—
132C22—42no—
133C22—42no—
134C22C22×D411yescharacteristic
135C22H(5,𝔽2)12yes—
136C22H(5,𝔽2)12yes—
137C22H(5,𝔽2)11yescharacteristic
138C222−1+411yescharacteristic
139C4—22no—
140C4—22no—
141C4—41no—
142C4—41no—
143C4—42no—
144C4—41no—
145C4—42no—
146C4—42no—
147C2—42no—
148C2—82no—
149C2—81no—
150C2#53411yescharacteristic
151C2#53511yescharacteristic
152C2#53711yescharacteristic
153C2#56012yes—
154C2#56212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C24
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×#123C2semidirect
2. C2×#121C2semidirect
3. #400C2semidirect
4. #399C2semidirect
5. #396C2semidirect
6. #394C2semidirect
7. #394C2semidirect
8. #394C2semidirect
9. #392C2semidirect
10. #392C2semidirect
11. #380C2semidirect
51. C22×C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#1232
C2×#1212
#4002
#3992
#3962
#3942
#3922
#3802