Finite Group Databasebeta

← C31×#2816 #16842749

Subgroups — C31×#2816 #16842749

Order 3968.

528 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 184 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×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C31×#121C211yesmaximal, characteristic
3C31×#399C212yesmaximal, complemented
4C31×#398C212yesmaximal, complemented
5C31×#394C212yesmaximal, complemented
6C31×#394C214yesmaximal, complemented
7C31×#393C212yesmaximal, complemented
8C31×#381C211yesmaximal, characteristic, complemented
9C22×Q8×C31C2211yescharacteristic
10C22×D4×C31C2212yes—
11C22×D4×C31C2211yescharacteristic
12C23×C4×C31C2211yescharacteristic
13C31×C22⋊Q8—24no—
14C31×#121—24no—
15C2×C4⋊C4×C31C2211yescharacteristic
16C2×C4⋊C4×C31C2212yes—
17C2×C4⋊C4×C31C2211yescharacteristic
18C2×C4⋊C4×C31C2214yes—
19C2×C22⋊C4×C31C2212yes—
20C2×C22⋊C4×C31C2212yes—
21C2×C22⋊C4×C31C2214yes—
22C2×C22⋊C4×C31C2212yes—
23C2×C22⋊C4×C31C2214yes—
24C31×#95C2212yes—
25C31×#95C2214yes—
26C31×#95C2212yes—
27C24×C31C2312yes—
28C24×C31C2311yescharacteristic
29C2×Q8×C31—22no—
30C2×Q8×C31—42no—
31C2×D4×C31—24no—
32C2×D4×C31—22no—
33C2×D4×C31—42no—
34C2×D4×C31—44no—
35C22×C4×C31—42no—
36C22×C4×C31D414yes—
37C22×C4×C31C2314yes—
38C22×C4×C31C2312yes—
39C22×C4×C31C2312yes—
40C22×C4×C31C2312yes—
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C4⋊C4×C31—22no—
44C4⋊C4×C31—22no—
45C4⋊C4×C31—24no—
46C4⋊C4×C31—44no—
47C22⋊C4×C31—24no—
48C22⋊C4×C31—24no—
49C22⋊C4×C31—42no—
50C22⋊C4×C31—44no—
51C22⋊C4×C31—44no—
52C23×C31—22no—
53C23×C31—22no—
54C23×C31—22no—
55C23×C31—22no—
56C23×C31—24no—
57C23×C31—24no—
58C23×C31—24no—
59C23×C31C2×D412yes—
60C23×C31C2411yescharacteristic
61Q8×C31—44no—
62D4×C31—44no—
63D4×C31—48no—
64C2×C4×C31—22no—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—22no—
68C2×C4×C31—22no—
69C2×C4×C31—22no—
70C2×C4×C31—24no—
71C2×C4×C31—24no—
72C2×C4×C31—24no—
73C2×C4×C31—22no—
74C2×C4×C31—24no—
75C2×C4×C31—24no—
76C2×C4×C31—44no—
77C2×C4×C31C2×D412yes—
78C2×C4×C31C2×D412yes—
79#2816C3111yesmaximal, characteristic, direct product factor, Sylow
80C22×C31—22no—
81C22×C31—22no—
82C22×C31—22no—
83C22×C31—44no—
84C22×C31—42no—
85C22×C31—42no—
86C22×C31—42no—
87C22×C31—48no—
88C22×C31C22×D411yescharacteristic
89C22×C31H(5,𝔽2)12yes—
90C22×C31H(5,𝔽2)12yes—
91C22×C312−1+412yes—
92C4×C31—22no—
93C4×C31—22no—
94C4×C31—42no—
95C4×C31—42no—
96C4×C31—42no—
97C4×C31—44no—
98C2×C22⋊Q8C2×C3111yescharacteristic, complemented
99C2×#121C2×C3111yescharacteristic
100#399C2×C3112yescomplemented
101#398C2×C3112yescomplemented
102#394C2×C3112yescomplemented
103#394C2×C3114yescomplemented
104#393C2×C3112yescomplemented
105#381C2×C3111yescharacteristic, complemented
106C2×C31—42no—
107C2×C31—82no—
108C2×C31#53411yescharacteristic
109C2×C31#53511yescharacteristic
110C2×C31#53611yescharacteristic
111C2×C31#56214yes—
112C22×Q8C22×C3111yescharacteristic
113C22×D4C22×C3112yes—
114C22×D4C22×C3111yescharacteristic
115C23×C4C22×C3111yescharacteristic
116C22⋊Q8—24no—
117#121—24no—
118C2×C4⋊C4C22×C3111yescharacteristic
119C2×C4⋊C4C22×C3112yes—
120C2×C4⋊C4C22×C3111yescharacteristic
121C2×C4⋊C4C22×C3114yes—
122C2×C22⋊C4C22×C3112yes—
123C2×C22⋊C4C22×C3112yes—
124C2×C22⋊C4C22×C3112yes—
125C2×C22⋊C4C22×C3114yes—
126C2×C22⋊C4C22×C3114yes—
127#95C22×C3112yes—
128#95C22×C3112yes—
129#95C22×C3114yes—
130C31#281611yescharacteristic, direct product factor, Sylow
131C24C23×C3112yes—
132C24C23×C3111yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—44no—
139C22×C4—42no—
140C22×C4D4×C3114yes—
141C22×C4C23×C3114yes—
142C22×C4C23×C3112yes—
143C22×C4C23×C3112yes—
144C22×C4C23×C3112yes—
145C22×C4C23×C3111yescharacteristic
146C22×C4C23×C3111yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—44no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—24no—
161C23—24no—
162C23—24no—
163C23C2×D4×C3112yes—
164C23C24×C3111yescharacteristic
165Q8—44no—
166D4—44no—
167D4—48no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—44no—
181C2×C4C2×D4×C3112yes—
182C2×C4C2×D4×C3112yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—48no—
191C22C22×D4×C3111yescharacteristic
192C22C31×H(5,𝔽2)12yes—
193C22C31×H(5,𝔽2)12yes—
194C22C31×2−1+412yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—42no—
200C4—44no—
201C2—42no—
202C2—82no—
203C2C31×#53411yescharacteristic
204C2C31×#53511yescharacteristic
205C2C31×#53611yescharacteristic
206C2C31×#56214yes—
subgroupG/H
center, socleC23×C31C24
commutator, Frattini, cosocleC23C24×C31
prime radicalC2×C31×#121C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×C22⋊Q8C2semidirect
3. C31×#399C2semidirect
4. C31×#398C2semidirect
5. C31×#394C2semidirect
6. C31×#394C2semidirect
7. C31×#393C2semidirect
8. C31×#381C2semidirect
79. #2816C31direct (N × H)
98. C2×C22⋊Q8C2×C31semidirect
100. #399C2×C31semidirect
101. #398C2×C31semidirect
102. #394C2×C31semidirect
103. #394C2×C31semidirect
104. #393C2×C31semidirect
105. #381C2×C31semidirect
130. C31#2816direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×C22⋊Q82
C2×C31×#1212
C31×#3992
C31×#3982
C31×#3942
C31×#3932
C31×#3812
#281631