Finite Group Databasebeta

← #3038×C211 #41010658

Subgroups — #3038×C211 #41010658

Order 27008.

584 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 208 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×AGL(1,ℤ/8)×C211C212yesmaximal, complemented
2C2×#127×C211C211yesmaximal, characteristic, complemented
3C4×#41×C211C211yesmaximal, characteristic, complemented
4#460×C211C214yesmaximal, complemented
5#459×C211C214yesmaximal, complemented
6#423×C211C211yesmaximal, characteristic, complemented
7#418×C211C212yesmaximal, complemented
8C2×#41×C211C2211yescharacteristic
9C22×D4×C211—22no—
10C22×D4×C211C2212yes—
11AGL(1,ℤ/8)×C211—28no—
12C2×SD16×C211C2214yescomplemented
13C2×D8×C211C2214yescomplemented
14C2×M16×C211C2212yes—
15#127×C211—21no—
16#127×C211—21no—
17#127×C211C2214yes—
18C4×Q8×C211C2212yescomplemented
19C4×D4×C211—22no—
20C4×D4×C211C2212yescomplemented
21#117×C211—21no—
22#117×C211C2211yescharacteristic
23C2×C42×C211—21no—
24C2×C42×C211C2211yescharacteristic
25C4⋊C8×C211C2214yescomplemented
26#102×C211C2218yescomplemented
27C24×C211—22no—
28#41×C211—22no—
29#41×C211D414yescomplemented
30C2×Q8×C211C2311yescharacteristic
31C2×D4×C211—21no—
32C2×D4×C211—24no—
33C2×D4×C211—22no—
34C2×D4×C211—24no—
35C2×D4×C211—24no—
36C2×D4×C211—24no—
37C2×D4×C211—48no—
38C2×D4×C211C2314yes—
39C2×D4×C211C2311yescharacteristic
40C22×C4×C211—21no—
41C22×C4×C211—22no—
42C22×C4×C211D412yes—
43C22×C4×C211C2311yescharacteristic
44SD16×C211—28no—
45D8×C211—28no—
46M16×C211—24no—
47C2×C8×C211C2314yes—
48C4⋊C4×C211—22no—
49C4⋊C4×C211C2312yes—
50C22⋊C4×C211—21no—
51C22⋊C4×C211—22no—
52C42×C211—21no—
53C42×C211—22no—
54C42×C211D411yescharacteristic
55C42×C211D411yescharacteristic
56C42×C211C2312yes—
57C23×C211—21no—
58C23×C211—24no—
59C23×C211—48no—
60C23×C211—44no—
61C23×C211C2×D411yescharacteristic
62Q8×C211—21no—
63Q8×C211C2×D412yescomplemented
64D4×C211—24no—
65D4×C211—24no—
66D4×C211—21no—
67D4×C211—24no—
68D4×C211—48no—
69D4×C211—44no—
70D4×C211—48no—
71D4×C211C2×D412yescomplemented
72C2×C4×C211—21no—
73C2×C4×C211—24no—
74C2×C4×C211—24no—
75C2×C4×C211—22no—
76C2×C4×C211—44no—
77C2×C4×C211C2×D412yes—
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×D412yes—
80C2×C4×C211C2×D412yes—
81C2×C4×C211#4112yes—
82C2×C4×C211C2411yescharacteristic
83C8×C211—24no—
84C22×C211—22no—
85C22×C211—22no—
86C22×C211—42no—
87C22×C211—44no—
88C22×C211—48no—
89C22×C211—88no—
90C22×C211#12112yes—
91C22×C211C22×D411yescharacteristic
92C4×C211—22no—
93C4×C211—22no—
94C4×C211—42no—
95C4×C211#12112yes—
96C4×C211AGL(1,ℤ/8)14yescomplemented
97C4×C211C22×D411yescharacteristic
98C4×C211C2×#4111yescharacteristic
99C2×C211—22no—
100C2×C211—42no—
101C2×C211—84no—
102C2×C211C2×#12111yescharacteristic
103C2×C211C2×AGL(1,ℤ/8)12yes—
104C211#303811yescharacteristic, direct product factor, Sylow
105#3038C21111yesmaximal, characteristic, direct product factor, Sylow
106C2×AGL(1,ℤ/8)C2×C21112yescomplemented
107C2×#127C2×C21111yescharacteristic, complemented
108C4×#41C2×C21111yescharacteristic, complemented
109#460C2×C21114yescomplemented
110#459C2×C21114yescomplemented
111#423C2×C21111yescharacteristic, complemented
112#418C2×C21112yescomplemented
113C2×#41C22×C21111yescharacteristic
114C22×D4—22no—
115C22×D4C22×C21112yes—
116AGL(1,ℤ/8)—28no—
117C2×SD16C22×C21114yescomplemented
118C2×D8C22×C21114yescomplemented
119C2×M16C22×C21112yes—
120#127—21no—
121#127—21no—
122#127C22×C21114yes—
123C4×Q8C22×C21112yescomplemented
124C4×D4—22no—
125C4×D4C22×C21112yescomplemented
126#117—21no—
127#117C22×C21111yescharacteristic
128C2×C42—21no—
129C2×C42C22×C21111yescharacteristic
130C4⋊C8C22×C21114yescomplemented
131#102C22×C21118yescomplemented
132C24—22no—
133#41—22no—
134#41D4×C21114yescomplemented
135C2×Q8C23×C21111yescharacteristic
136C2×D4—21no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—48no—
143C2×D4C23×C21114yes—
144C2×D4C23×C21111yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4D4×C21112yes—
148C22×C4C23×C21111yescharacteristic
149SD16—28no—
150D8—28no—
151M16—24no—
152C2×C8C23×C21114yes—
153C4⋊C4—22no—
154C4⋊C4C23×C21112yes—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C42—21no—
158C42—22no—
159C42D4×C21111yescharacteristic
160C42D4×C21111yescharacteristic
161C42C23×C21112yes—
162C23—21no—
163C23—24no—
164C23—48no—
165C23—44no—
166C23C2×D4×C21111yescharacteristic
167Q8—21no—
168Q8C2×D4×C21112yescomplemented
169D4—24no—
170D4—24no—
171D4—21no—
172D4—24no—
173D4—48no—
174D4—44no—
175D4—48no—
176D4C2×D4×C21112yescomplemented
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4C2×D4×C21112yes—
183C2×C4C2×D4×C21111yescharacteristic
184C2×C4C2×D4×C21112yes—
185C2×C4C2×D4×C21112yes—
186C2×C4#41×C21112yes—
187C2×C4C24×C21111yescharacteristic
188C8—24no—
189C22—22no—
190C22—22no—
191C22—42no—
192C22—44no—
193C22—48no—
194C22—88no—
195C22#121×C21112yes—
196C22C22×D4×C21111yescharacteristic
197C4—22no—
198C4—22no—
199C4—42no—
200C4#121×C21112yes—
201C4AGL(1,ℤ/8)×C21114yescomplemented
202C4C22×D4×C21111yescharacteristic
203C4C2×#41×C21111yescharacteristic
204C2—22no—
205C2—42no—
206C2—84no—
207C2C2×#121×C21111yescharacteristic
208C2C2×AGL(1,ℤ/8)×C21112yes—
subgroupG/H
center, socleC22×C211C22×D4
commutator, Frattini, cosocleC2×C4C24×C211
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×AGL(1,ℤ/8)×C211C2semidirect
2. C2×#127×C211C2semidirect
3. C4×#41×C211C2semidirect
4. #460×C211C2semidirect
5. #459×C211C2semidirect
6. #423×C211C2semidirect
7. #418×C211C2semidirect
12. C2×SD16×C211C22semidirect
13. C2×D8×C211C22semidirect
18. C4×Q8×C211C22semidirect
20. C4×D4×C211C22semidirect
25. C4⋊C8×C211C22semidirect
26. #102×C211C22semidirect
29. #41×C211D4semidirect
63. Q8×C211C2×D4semidirect
71. D4×C211C2×D4semidirect
96. C4×C211AGL(1,ℤ/8)semidirect
104. C211#3038direct (N × H)
105. #3038C211direct (N × H)
106. C2×AGL(1,ℤ/8)C2×C211semidirect
107. C2×#127C2×C211semidirect
108. C4×#41C2×C211semidirect
109. #460C2×C211semidirect
110. #459C2×C211semidirect
111. #423C2×C211semidirect
112. #418C2×C211semidirect
117. C2×SD16C22×C211semidirect
118. C2×D8C22×C211semidirect
123. C4×Q8C22×C211semidirect
125. C4×D4C22×C211semidirect
130. C4⋊C8C22×C211semidirect
131. #102C22×C211semidirect
134. #41D4×C211semidirect
168. Q8C2×D4×C211semidirect
176. D4C2×D4×C211semidirect
201. C4AGL(1,ℤ/8)×C211semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,ℤ/8)×C2112
C2×#127×C2112
C4×#41×C2112
#460×C2112
#459×C2112
#423×C2112
#418×C2112
#3038211