Finite Group Databasebeta

← C2×C31×#456 #16842950

Subgroups — C2×C31×#456 #16842950

Order 3968.

708 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 216 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×C31×2−1+4C211yesmaximal, characteristic
2C2×C31×#137C212yesmaximal, complemented
3C2×C31×C4⋊Q8C211yesmaximal, characteristic, complemented
4C31×#456C218yesmaximal, direct product factor
5C2×C31×#104C212yesmaximal, complemented
6C2×C31×#101C211yesmaximal, characteristic, complemented
7C31×2−1+4—24no—
8C31×2−1+4—22no—
9C31×2−1+4C2214yes—
10C2×#41×C31—22no—
11C2×#41×C31—22no—
12C2×#41×C31C2212yes—
13C22×Q8×C31—22no—
14C22×Q8×C31C2211yescharacteristic
15C22×Q8×C31C2211yescharacteristic
16C31×#137—24no—
17C31×#137C2218yescomplemented
18C2×Q16×C31—22no—
19C2×SD16×C31—22no—
20C2×M16×C31C2212yes—
21C31×C4⋊Q8—22no—
22C31×C4⋊Q8C2214yescomplemented
23C2×C4⋊C4×C31—22no—
24C2×C42×C31C2211yescharacteristic
25C31×#104C2218yescomplemented
26C31×#101C2214yescomplemented
27#41×C31—28no—
28#41×C31—28no—
29#41×C31—28no—
30#41×C31—28no—
31#41×C31—22no—
32#41×C31—22no—
33#41×C31D414yes—
34#41×C31C2314yes—
35C2×Q8×C31—22no—
36C2×Q8×C31—28no—
37C2×Q8×C31—28no—
38C2×Q8×C31—22no—
39C2×Q8×C31—22no—
40C2×Q8×C31—21no—
41C2×Q8×C31—22no—
42C2×Q8×C31—22no—
43C2×Q8×C31D412yes—
44C2×Q8×C31D412yes—
45C2×Q8×C31C2312yes—
46C2×Q8×C31C2312yes—
47C2×D4×C31—22no—
48C2×D4×C31—22no—
49C2×D4×C31D412yes—
50C22×C4×C31—24no—
51C22×C4×C31—21no—
52C22×C4×C31—22no—
53C22×C4×C31—21no—
54C22×C4×C31D412yes—
55C22×C4×C31C2311yescharacteristic
56Q16×C31—28no—
57SD16×C31—28no—
58M16×C31—22no—
59M16×C31C2314yes—
60C2×C8×C31—22no—
61C4⋊C4×C31—28no—
62C42×C31—21no—
63C42×C31C2312yes—
64C23×C31—22no—
65C23×C31C2×D411yescharacteristic
66Q8×C31—28no—
67Q8×C31—22no—
68Q8×C31—24no—
69Q8×C31—24no—
70Q8×C31—24no—
71Q8×C31—42no—
72Q8×C31C2×D414yes—
73D4×C31—28no—
74D4×C31—24no—
75D4×C31—24no—
76D4×C31—22no—
77D4×C31C2×D414yes—
78C2×C4×C31—24no—
79C2×C4×C31—216no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—24no—
85C2×C4×C31—24no—
86C2×C4×C31—22no—
87C2×C4×C31—41no—
88C2×C4×C31—41no—
89C2×C4×C31C2×D414yes—
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D412yes—
93C2×C4×C31C2411yescharacteristic
94C8×C31—24no—
95C2×#456C3111yesmaximal, characteristic, direct product factor, Sylow
96C22×C31—22no—
97C22×C31—24no—
98C22×C31—44no—
99C22×C31#12011yescharacteristic
100C22×C31#12011yescharacteristic
101C22×C31C22×D411yescharacteristic
102C4×C31—24no—
103C4×C31—24no—
104C4×C31—24no—
105C4×C31—42no—
106C4×C31#12012yes—
107C4×C31C22×D412yes—
108C2×2−1+4C2×C3111yescharacteristic
109C2×#137C2×C3112yescomplemented
110C2×C4⋊Q8C2×C3111yescharacteristic, complemented
111#456C2×C3118yesdirect product factor
112C2×#104C2×C3112yescomplemented
113C2×#101C2×C3111yescharacteristic, complemented
114C2×C31—21no—
115C2×C31—21no—
116C2×C31—44no—
117C2×C31#45612yesdirect product factor
118C2×C31C2×#12011yescharacteristic
1192−1+4—24no—
1202−1+4—22no—
1212−1+4C22×C3114yes—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41C22×C3112yes—
125C22×Q8—22no—
126C22×Q8C22×C3111yescharacteristic
127C22×Q8C22×C3111yescharacteristic
128#137—24no—
129#137C22×C3118yescomplemented
130C2×Q16—22no—
131C2×SD16—22no—
132C2×M16C22×C3112yes—
133C4⋊Q8—22no—
134C4⋊Q8C22×C3114yescomplemented
135C2×C4⋊C4—22no—
136C2×C42C22×C3111yescharacteristic
137#104C22×C3118yescomplemented
138#101C22×C3114yescomplemented
139C31C2×#45611yescharacteristic, direct product factor, Sylow
140#41—28no—
141#41—28no—
142#41—28no—
143#41—28no—
144#41—22no—
145#41—22no—
146#41D4×C3114yes—
147#41C23×C3114yes—
148C2×Q8—22no—
149C2×Q8—28no—
150C2×Q8—28no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8D4×C3112yes—
157C2×Q8D4×C3112yes—
158C2×Q8C23×C3112yes—
159C2×Q8C23×C3112yes—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4D4×C3112yes—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4D4×C3112yes—
168C22×C4C23×C3111yescharacteristic
169Q16—28no—
170SD16—28no—
171M16—22no—
172M16C23×C3114yes—
173C2×C8—22no—
174C4⋊C4—28no—
175C42—21no—
176C42C23×C3112yes—
177C23—22no—
178C23C2×D4×C3111yescharacteristic
179Q8—28no—
180Q8—22no—
181Q8—24no—
182Q8—24no—
183Q8—24no—
184Q8—42no—
185Q8C2×D4×C3114yes—
186D4—28no—
187D4—24no—
188D4—24no—
189D4—22no—
190D4C2×D4×C3114yes—
191C2×C4—24no—
192C2×C4—216no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4C2×D4×C3114yes—
203C2×C4C2×D4×C3111yescharacteristic
204C2×C4C2×D4×C3112yes—
205C2×C4C2×D4×C3112yes—
206C2×C4C24×C3111yescharacteristic
207C8—24no—
208C22—22no—
209C22—24no—
210C22—44no—
211C22C31×#12011yescharacteristic
212C22C31×#12011yescharacteristic
213C22C22×D4×C3111yescharacteristic
214C4—24no—
215C4—24no—
216C4—24no—
217C4—42no—
218C4C31×#12012yes—
219C4C22×D4×C3112yes—
220C2—21no—
221C2—21no—
222C2—44no—
223C2C31×#45612yesdirect product factor
224C2C2×C31×#12011yescharacteristic
subgroupG/H
center, socleC22×C31#120
commutator, Frattini, cosocleC2×C4C24×C31
prime radicalC2×C31×2−1+4C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#137C2semidirect
3. C2×C31×C4⋊Q8C2semidirect
4. C31×#456C2direct (N × H)
5. C2×C31×#104C2semidirect
6. C2×C31×#101C2semidirect
17. C31×#137C22semidirect
22. C31×C4⋊Q8C22semidirect
25. C31×#104C22semidirect
26. C31×#101C22semidirect
95. C2×#456C31direct (N × H)
109. C2×#137C2×C31semidirect
110. C2×C4⋊Q8C2×C31semidirect
111. #456C2×C31direct (N × H)
112. C2×#104C2×C31semidirect
113. C2×#101C2×C31semidirect
117. C2×C31#456direct (N × H)
129. #137C22×C31semidirect
134. C4⋊Q8C22×C31semidirect
137. #104C22×C31semidirect
138. #101C22×C31semidirect
139. C31C2×#456direct (N × H)
223. C2C31×#456direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×2−1+42
C2×C31×#1372
C2×C31×C4⋊Q82
C31×#4562
C2×C31×#1042
C2×C31×#1012
C2×#45631