Finite Group Databasebeta

← #2674

Subgroups — #2674

Order 128.

281 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 104 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×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#124C211yesmaximal, characteristic
3C2×#123C211yesmaximal, characteristic
4C2×C22⋊Q8C211yesmaximal, characteristic, complemented
5#399C211yesmaximal, characteristic, complemented
6#397C211yesmaximal, characteristic, complemented
7#396C211yesmaximal, characteristic, complemented
8#395C211yesmaximal, characteristic, complemented
9#394C211yesmaximal, characteristic
10#393C211yesmaximal, characteristic, complemented
11#389C211yesmaximal, characteristic, complemented
12#388C211yesmaximal, characteristic, complemented
13#388C211yesmaximal, characteristic, complemented
14#387C211yesmaximal, characteristic, complemented
15#385C211yesmaximal, characteristic, complemented
16C22×Q8C2211yescharacteristic
17C22×Q8C2211yescharacteristic
18C22×D4C2211yescharacteristic
19C23×C4C2211yescharacteristic
20C4⋊Q8—24no—
21#124—24no—
22#123—24no—
23C22⋊Q8—24no—
24C2×C4⋊C4C2211yescharacteristic
25C2×C4⋊C4C2211yescharacteristic
26C2×C4⋊C4C2211yescharacteristic
27C2×C4⋊C4C2211yescharacteristic
28C2×C4⋊C4C2211yescharacteristic
29C2×C4⋊C4C2211yescharacteristic
30C2×C4⋊C4C2211yescharacteristic
31C2×C4⋊C4C2211yescharacteristic
32C2×C4⋊C4C2211yescharacteristic
33C2×C4⋊C4C2211yescharacteristic
34C2×C22⋊C4C2211yescharacteristic
35C2×C22⋊C4C2211yescharacteristic
36C2×C22⋊C4C2211yescharacteristic
37C2×C22⋊C4C2211yescharacteristic
38C2×C22⋊C4C2211yescharacteristic
39C2×C22⋊C4C2211yescharacteristic
40C2×C22⋊C4C2211yescharacteristic
41C2×C22⋊C4C2211yescharacteristic
42C2×C22⋊C4C2211yescharacteristic
43C2×C22⋊C4C2211yescharacteristic
44C2×C22⋊C4C2211yescharacteristic
45C2×C42C2211yescharacteristic
46C2×C42C2211yescharacteristic
47#95C2211yescharacteristic
48#95C2211yescharacteristic
49#95C2211yescharacteristic
50#95C2211yescharacteristic
51#95C2211yescharacteristic
52#95C2211yescharacteristic
53#95C2211yescharacteristic
54#95C2211yescharacteristic
55C24C2311yescharacteristic
56C24C2311yescharacteristic
57C2×Q8—22no—
58C2×Q8—22no—
59C2×Q8—22no—
60C2×Q8—22no—
61C2×Q8—41no—
62C2×Q8—41no—
63C2×D4—22no—
64C2×D4—22no—
65C2×D4—41no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—41no—
69C22×C4C2311yescharacteristic
70C22×C4C2311yescharacteristic
71C22×C4C2311yescharacteristic
72C22×C4C2311yescharacteristic
73C22×C4C2311yescharacteristic
74C22×C4C2311yescharacteristic
75C22×C4C2311yescharacteristic
76C22×C4C2311yescharacteristic
77C22×C4C2311yescharacteristic
78C22×C4C2311yescharacteristic
79C22×C4C2311yescharacteristic
80C22×C4C2311yescharacteristic
81C22×C4C2311yescharacteristic
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—41no—
91C4⋊C4D414yes—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C22⋊C4—22no—
98C22⋊C4—22no—
99C22⋊C4—22no—
100C22⋊C4—22no—
101C22⋊C4—41no—
102C22⋊C4D414yes—
103C42—22no—
104C42—22no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23C2×D412yes—
119C23C2411yescharacteristic
120Q8—44no—
121Q8—44no—
122D4—44no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—44no—
157C2×C4C2×D412yes—
158C2×C4C2×D412yes—
159C2×C4C2×D412yes—
160C2×C4C2×D412yes—
161C2×C4C2×D412yes—
162C2×C4#4112yes—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22C22×D411yescharacteristic
178C22C22×D411yescharacteristic
179C22C2×#4111yescharacteristic
180C22H(5,𝔽2)11yescharacteristic
181C222−1+411yescharacteristic
182C222−1+411yescharacteristic
183C222−1+411yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—22no—
188C4—22no—
189C4—22no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—41no—
194C4—41no—
195C4—41no—
196C4—41no—
197C2—42no—
198C2—81no—
199C2#53611yescharacteristic
200C2#53611yescharacteristic
201C2#54211yescharacteristic
202C2D4211yescharacteristic
203C2#54711yescharacteristic
204C2#54811yescharacteristic
205C2#56311yescharacteristic
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
1. C2×C4⋊Q8C2semidirect
4. C2×C22⋊Q8C2semidirect
5. #399C2semidirect
6. #397C2semidirect
7. #396C2semidirect
8. #395C2semidirect
10. #393C2semidirect
11. #389C2semidirect
12. #388C2semidirect
13. #388C2semidirect
14. #387C2semidirect
15. #385C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q82
C2×#1242
C2×#1232
C2×C22⋊Q82
#3992
#3972
#3962
#3952
#3942
#3932
#3892
#3882
#3872
#3852