Finite Group Databasebeta

← #16855829

Subgroups — #16855829

Order 3968.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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×Dic62C211yesmaximal, characteristic, complemented
2#14521360C212yesmaximal, complemented
3#14521341C212yesmaximal, complemented
4#14521246C212yesmaximal, complemented
5#14521187C212yesmaximal, complemented
6#14521149C214yesmaximal, complemented
7#14521106C211yesmaximal, characteristic, complemented
8C31×#532C211yesmaximal, characteristic
9C2×#41×C31—21no—
10C22×Dic62C2211yescharacteristic
11C31×C4⋊Q8C2211yescharacteristic
12C31×#127C2211yescharacteristic
13C31×#124—21no—
14C31×#121—21no—
15C31×#121C2212yes—
16C4×D4×C31—21no—
17C4×D4×C31C2212yes—
18C2×C42×C31C2211yescharacteristic
19C2×#92480C2214yes—
20C2×C4⋊Dic31C2211yescharacteristic
21C2×Dic31⋊C4—21no—
22C2×C4×Dic31—21no—
23C4×Dic62—21no—
24C4×Dic62—21no—
25C4×Dic62C2212yes—
26C4×Dic62C2212yes—
27#11758420C2214yes—
28C22⋊(C31⋊C8)C2212yes—
29#11758398C2214yes—
30#11758395C2214yes—
31C4⋊(C31⋊C8)C2212yes—
32#11758391C2212yes—
33#41×C31—44no—
34C2×Q8×C31—21no—
35C2×D4×C31—22no—
36C2×D4×C31—21no—
37C2×D4×C31—21no—
38C2×D4×C31C2312yes—
39C22×C4×C31—22no—
40C22×C4×C31D411yescharacteristic
41C22×C4×C31D411yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C22×Dic31—21no—
44C22×Dic31—21no—
45C2×Dic62—21no—
46C2×Dic62—24no—
47C2×Dic62C2312yes—
48C4⋊C4×C31—21no—
49C4⋊C4×C31C2312yes—
50C22⋊C4×C31—22no—
51C22⋊C4×C31—22no—
52C42×C31D411yescharacteristic
53C42×C31D411yescharacteristic
54C42×C31C2311yescharacteristic
55C42×C31C2311yescharacteristic
56#92480—28no—
57C4⋊Dic31—21no—
58C4⋊Dic31C2312yes—
59Dic31⋊C4—22no—
60Dic31⋊C4—22no—
61C4×Dic31—22no—
62C4×Dic31—22no—
63C2×C31⋊C8C2314yes—
64C23×C31—22no—
65C23×C31C2×D411yescharacteristic
66Q8×C31—42no—
67D4×C31—24no—
68D4×C31—42no—
69D4×C31—44no—
70D4×C31—42no—
71C2×C4×C31—22no—
72C2×C4×C31—22no—
73C2×C4×C31—22no—
74C2×C4×C31—22no—
75C2×C4×C31—44no—
76C2×C4×C31C2×D411yescharacteristic
77C2×C4×C31C2×D411yescharacteristic
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31C2×D411yescharacteristic
80C2×C4×C31C2×D411yescharacteristic
81C2×C4×C31C2411yescharacteristic
82C2×Dic31—24no—
83C2×Dic31—22no—
84C2×Dic31—22no—
85C2×Dic31—42no—
86Dic62—24no—
87Dic62—22no—
88Dic62#4114yescomplemented
89C31⋊C8—24no—
90#3112—311nomaximal, Sylow
91C22×C31—21no—
92C22×C31—21no—
93C22×C31—42no—
94C22×C31—42no—
95C22×C31—42no—
96C22×C31AGL(1,ℤ/8)12yes—
97C22×C31C22×D411yescharacteristic
98C4×C31—21no—
99C4×C31—21no—
100C4×C31—21no—
101C4×C31—21no—
102C4×C31—42no—
103C4×C31#13712yes—
104C4×C31C2×#4112yes—
105Dic31—24no—
106Dic31—42no—
107#532D3111yescharacteristic
108C2×C4×Q8—311no—
109#480—312no—
110#477—312no—
111#460—312no—
112#448—312no—
113#440—314no—
114#432—311no—
115C2×C31—22no—
116C2×C31—82no—
117C2×C31#52511yescharacteristic
118C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
119C2×C31C2×#13711yescharacteristic
120C2×#41—21no—
121C22×Q8—311no—
122C2×SD16—314no—
123C4⋊Q8C2×D3111yescharacteristic
124#127C2×D3111yescharacteristic
125#124—21no—
126#121—21no—
127#121C2×D3112yes—
128C4×Q8—312no—
129C4×Q8—312no—
130C4×Q8—621no—
131C4×Q8—621no—
132C4×D4—21no—
133C4×D4C2×D3112yes—
134C2×C4⋊C4—311no—
135C2×C4⋊C4—621no—
136C2×C42—621no—
137C2×C42C2×D3111yescharacteristic
138#107—314no—
139C4⋊C8—312no—
140#103—314no—
141#102—314no—
142C22⋊C8—312no—
143#97—312no—
144C31#311211yescharacteristic, complemented, Sylow
145#41—44no—
146C2×Q8—21no—
147C2×Q8—312no—
148C2×Q8—624no—
149C2×Q8—621no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D3112yes—
154C22×C4—22no—
155C22×C4—621no—
156C22×C4—621no—
157C22×C4C22⋊D3111yescharacteristic
158C22×C4C22⋊D3111yescharacteristic
159C22×C4C22×D3111yescharacteristic
160SD16—628no—
161C2×C8—314no—
162C4⋊C4—21no—
163C4⋊C4—312no—
164C4⋊C4—622no—
165C4⋊C4—622no—
166C4⋊C4—621no—
167C4⋊C4C22×D3112yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—622no—
171C42—622no—
172C42C22⋊D3111yescharacteristic
173C42C22⋊D3111yescharacteristic
174C42C22×D3111yescharacteristic
175C42C22×D3111yescharacteristic
176C23—22no—
177C23C2×C22⋊D3111yescharacteristic
178Q8—42no—
179Q8—314no—
180Q8—624no—
181Q8—622no—
182D4—24no—
183D4—42no—
184D4—44no—
185D4—42no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—624no—
192C2×C4—622no—
193C2×C4—622no—
194C2×C4—1242no—
195C2×C4C2×C22⋊D3111yescharacteristic
196C2×C4C2×C22⋊D3111yescharacteristic
197C2×C4C2×C22⋊D3111yescharacteristic
198C2×C4C2×C22⋊D3111yescharacteristic
199C2×C4C2×C22⋊D3111yescharacteristic
200C2×C4C23×D3111yescharacteristic
201C8—624no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22#1175850912yes—
208C22C22×C22⋊D3111yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—624no—
215C4—1242no—
216C4#1175852812yes—
217C4#41×D3112yes—
218C2—22no—
219C2—82no—
220C2#1452155611yescharacteristic
221C2C2×#1175850911yescharacteristic
222C2C2×#1175852811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC31×#532C2
socleC22×C31C22×D4
hypercenter#532D31
hypocenterC31#3112
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic62C2semidirect
2. #14521360C2semidirect
3. #14521341C2semidirect
4. #14521246C2semidirect
5. #14521187C2semidirect
6. #14521149C2semidirect
7. #14521106C2semidirect
88. Dic62#41semidirect
144. C31#3112semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic622
#145213602
#145213412
#145212462
#145211872
#145211492
#145211062
C31×#5322
#311231