Finite Group Databasebeta

← C3×#1483 #72198

Subgroups — C3×#1483 #72198

Order 384.

234 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3×#529C211yesmaximal, characteristic
2C2×C3×C4⋊C8C211yesmaximal, characteristic, complemented
3C3×C4×M16C211yesmaximal, characteristic, complemented
4C3×Q8⋊C8C212yesmaximal, complemented
5C3×D4⋊C8C212yesmaximal, complemented
6#1483C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×M16—21no—
8C22×C3×C8—21no—
9C3×#126—21no—
10C3×#125C411yescharacteristic
11C3×#124C411yescharacteristic
12C3×#123—21no—
13C3×C22⋊Q8C411yescharacteristic
14C3×#121C411yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16C3×C4×D4—21no—
17C3×C4×D4C2211yescharacteristic
18C3×#117—21no—
19C2×C3×C42C2211yescharacteristic
20C3×C4⋊C8—21no—
21C3×C4⋊C8C2212yes—
22C3×#97—21no—
23C3×C4×C8C2212yes—
24#529C2×C311yescharacteristic
25C2×C4⋊C8C2×C311yescharacteristic, complemented
26C4×M16C2×C311yescharacteristic, complemented
27Q8⋊C8C2×C312yescomplemented
28D4⋊C8C2×C312yescomplemented
29C2×C3×Q8C2×C411yescharacteristic
30C2×C3×D4—21no—
31C2×C3×D4C2×C411yescharacteristic
32C22×C3×C4—21no—
33C22×C3×C4C2×C411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C3×M16—24no—
37C2×C3×C8—22no—
38C2×C3×C8—22no—
39C2×C3×C8—42no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4C2×C411yescharacteristic
45C3×C4⋊C4C2×C411yescharacteristic
46C3×C22⋊C4—21no—
47C3×C22⋊C4—21no—
48C3×C22⋊C4—21no—
49C3×C22⋊C4—21no—
50C3×C22⋊C4—21no—
51C3×C42—21no—
52C3×C42C2×C411yescharacteristic
53C3×C42D411yescharacteristic
54C3×C42D411yescharacteristic
55C3×C42C2311yescharacteristic
56C2×M16—21no—
57C22×C8—21no—
58#126—21no—
59#125C3×C411yescharacteristic
60#124C3×C411yescharacteristic
61#123—21no—
62C22⋊Q8C3×C411yescharacteristic
63#121C3×C411yescharacteristic
64C4×Q8C22×C311yescharacteristic
65C4×D4—21no—
66C4×D4C22×C311yescharacteristic
67#117—21no—
68C2×C42C22×C311yescharacteristic
69C4⋊C8—21no—
70C4⋊C8C22×C312yes—
71#97—21no—
72C4×C8C22×C312yes—
73C23×C3—21no—
74C23×C3C22⋊C411yescharacteristic
75C3×Q8—22no—
76C3×D4—22no—
77C3×D4—42no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×C3×C4C22⋊C411yescharacteristic
88C2×C3×C4C22⋊C411yescharacteristic
89C2×C3×C4C22⋊C411yescharacteristic
90C2×C3×C4C22×C411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C3×C8—24no—
94C3×C8—42no—
95C2×Q8C2×C3×C411yescharacteristic
96C2×D4—21no—
97C2×D4C2×C3×C411yescharacteristic
98C22×C4—21no—
99C22×C4C2×C3×C411yescharacteristic
100C22×C4C3×D411yescharacteristic
101C22×C4C3×D411yescharacteristic
102M16—24no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—42no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4C2×C3×C411yescharacteristic
111C4⋊C4C2×C3×C411yescharacteristic
112C22⋊C4—21no—
113C22⋊C4—21no—
114C22⋊C4—21no—
115C22⋊C4—21no—
116C22⋊C4—21no—
117C42—21no—
118C42C2×C3×C411yescharacteristic
119C42C3×D411yescharacteristic
120C42C3×D411yescharacteristic
121C42C23×C311yescharacteristic
122C22×C3—21no—
123C22×C3—21no—
124C22×C3—41no—
125C22×C3—41no—
126C22×C3—41no—
127C22×C3#10412yes—
128C22×C3C2×C22⋊C411yescharacteristic
129C3×C4—21no—
130C3×C4—21no—
131C3×C4—21no—
132C3×C4—21no—
133C3×C4—41no—
134C3×C4—41no—
135C3×C4—41no—
136C3×C4#13112yes—
137C3×C4AGL(1,ℤ/8)11yescharacteristic
138C3×C4#13711yescharacteristic
139C23—21no—
140C23C3×C22⋊C411yescharacteristic
141Q8—22no—
142D4—22no—
143D4—42no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4C3×C22⋊C411yescharacteristic
154C2×C4C3×C22⋊C411yescharacteristic
155C2×C4C3×C22⋊C411yescharacteristic
156C2×C4C22×C3×C411yescharacteristic
157C2×C4C2×C3×D411yescharacteristic
158C2×C4C2×C3×D411yescharacteristic
159C8—24no—
160C8—42no—
161C2×C3—22no—
162C2×C3—81no—
163C2×C3#40811yescharacteristic
164C2×C3#41711yescharacteristic
165C2×C3C2×#10411yescharacteristic
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C3×#10412yes—
172C22C2×C3×C22⋊C411yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4C3×#13112yes—
181C4C3×AGL(1,ℤ/8)11yescharacteristic
182C4C3×#13711yescharacteristic
183C3#148311yescharacteristic, direct product factor, Sylow
184C2—22no—
185C2—81no—
186C2C3×#40811yescharacteristic
187C2C3×#41711yescharacteristic
188C2C2×C3×#10411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC42C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC3×#121C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊C8C2semidirect
3. C3×C4×M16C2semidirect
4. C3×Q8⋊C8C2semidirect
5. C3×D4⋊C8C2semidirect
6. #1483C3direct (N × H)
25. C2×C4⋊C8C2×C3semidirect
26. C4×M16C2×C3semidirect
27. Q8⋊C8C2×C3semidirect
28. D4⋊C8C2×C3semidirect
183. C3#1483direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5292
C2×C3×C4⋊C82
C3×C4×M162
C3×Q8⋊C82
C3×D4⋊C82
#14833