Finite Group Databasebeta

← C3×#1996 #72711

Subgroups — C3×#1996 #72711

Order 384.

324 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 116 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×C3×C22⋊Q8C211yesmaximal, characteristic
2C3×#380C211yesmaximal, characteristic
3C3×C4×C22⋊C4C211yesmaximal, characteristic
4C3×#342C214yesmaximal
5#1996C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×Q8C411yescharacteristic, complemented
7C23×C3×C4C2211yescharacteristic
8C3×C22⋊Q8—21no—
9C3×C22⋊Q8—21no—
10C3×C22⋊Q8—21no—
11C3×C22⋊Q8—21no—
12C2×C3×C4⋊C4—21no—
13C2×C3×C4⋊C4C411yescharacteristic, complemented
14C2×C3×C4⋊C4C412yescomplemented
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C42—21no—
19C3×#95—21no—
20C3×#95—21no—
21C2×C22⋊Q8C2×C311yescharacteristic
22#380C2×C311yescharacteristic
23C4×C22⋊C4C2×C311yescharacteristic
24#342C2×C314yes—
25C24×C3C2311yescharacteristic
26C2×C3×Q8—21no—
27C2×C3×Q8—21no—
28C2×C3×Q8—42no—
29C22×C3×C4—22no—
30C22×C3×C4—22no—
31C22×C3×C4—42no—
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4C2×C412yes—
34C22×C3×C4C2×C412yes—
35C22×C3×C4C2×C411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4D411yescharacteristic
38C22×C3×C4D411yescharacteristic
39C22×C3×C4D411yescharacteristic
40C3×C4⋊C4—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4—22no—
43C3×C4⋊C4—22no—
44C3×C4⋊C4—42no—
45C3×C22⋊C4—22no—
46C3×C22⋊C4—22no—
47C3×C22⋊C4—22no—
48C3×C22⋊C4—24no—
49C3×C22⋊C4Q814yescomplemented
50C3×C42—42no—
51C22×Q8C3×C411yescharacteristic, complemented
52C23×C4C22×C311yescharacteristic
53C22⋊Q8—21no—
54C22⋊Q8—21no—
55C22⋊Q8—21no—
56C22⋊Q8—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C3×C411yescharacteristic, complemented
59C2×C4⋊C4C3×C412yescomplemented
60C2×C22⋊C4C22×C312yes—
61C2×C22⋊C4C22×C312yes—
62C2×C22⋊C4C22×C312yes—
63C2×C42—21no—
64#95—21no—
65#95—21no—
66C23×C3—41no—
67C23×C3—41no—
68C23×C3C22×C411yescharacteristic
69C23×C3C2×D411yescharacteristic
70C23×C3C2×D411yescharacteristic
71C23×C3C2×Q812yes—
72C23×C3#4112yes—
73C3×Q8—42no—
74C3×Q8—42no—
75C2×C3×C4—22no—
76C2×C3×C4—22no—
77C2×C3×C4—22no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4—24no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4—42no—
87C2×C3×C4—42no—
88C2×C3×C4—42no—
89C2×C3×C4—42no—
90C2×C3×C4—42no—
91C2×C3×C4C22⋊C411yescharacteristic
92C2×C3×C4C22⋊C411yescharacteristic
93C2×C3×C4C22⋊C411yescharacteristic
94C2×C3×C4C22⋊C411yescharacteristic
95C24C23×C311yescharacteristic
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—42no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—42no—
102C22×C4C2×C3×C411yescharacteristic
103C22×C4C2×C3×C412yes—
104C22×C4C2×C3×C412yes—
105C22×C4C2×C3×C411yescharacteristic
106C22×C4C3×D411yescharacteristic
107C22×C4C3×D411yescharacteristic
108C22×C4C3×D411yescharacteristic
109C22×C4C3×D411yescharacteristic
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—42no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—24no—
119C22⋊C4C3×Q814yescomplemented
120C42—42no—
121C22×C3—21no—
122C22×C3—21no—
123C22×C3—21no—
124C22×C3—21no—
125C22×C3—21no—
126C22×C3—21no—
127C22×C3—42no—
128C22×C3—42no—
129C22×C3C2×C22⋊C411yescharacteristic
130C22×C3C4×Q812yes—
131C22×C3C22⋊Q812yes—
132C22×C3#12411yescharacteristic
133C22×C3C4⋊Q811yescharacteristic
134C3×C4—21no—
135C3×C4—21no—
136C3×C4—21no—
137C3×C4—44no—
138C3×C4—42no—
139C3×C4—42no—
140C3×C4—82no—
141C3×C4#9912yes—
142C23—41no—
143C23—41no—
144C23C22×C3×C411yescharacteristic
145C23C2×C3×D411yescharacteristic
146C23C2×C3×D411yescharacteristic
147C23C2×C3×Q812yes—
148C23C3×#4112yes—
149Q8—42no—
150Q8—42no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4C3×C22⋊C411yescharacteristic
168C2×C4C3×C22⋊C411yescharacteristic
169C2×C4C3×C22⋊C411yescharacteristic
170C2×C4C3×C22⋊C411yescharacteristic
171C2×C3—22no—
172C2×C3—21no—
173C2×C3—22no—
174C2×C3—21no—
175C2×C3#39111yescharacteristic
176C2×C3C2×#9911yescharacteristic
177C2×C3#41011yescharacteristic
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—42no—
185C22—42no—
186C22C2×C3×C22⋊C411yescharacteristic
187C22C3×C4×Q812yes—
188C22C3×C22⋊Q812yes—
189C22C3×#12411yescharacteristic
190C22C3×C4⋊Q811yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—44no—
195C4—42no—
196C4—42no—
197C4—82no—
198C4C3×#9912yes—
199C3#199611yescharacteristic, direct product factor, Sylow
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2C3×#39111yescharacteristic
205C2C2×C3×#9911yescharacteristic
206C2C3×#41011yescharacteristic
subgroupG/H
center, socleC22×C3C2×C22⋊C4
commutatorC23C22×C3×C4
Frattini, cosocleC24C23×C3
prime radicalC24×C3C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
5. #1996C3direct (N × H)
6. C22×C3×Q8C4semidirect
13. C2×C3×C4⋊C4C4semidirect
14. C2×C3×C4⋊C4C4semidirect
49. C3×C22⋊C4Q8semidirect
51. C22×Q8C3×C4semidirect
58. C2×C4⋊C4C3×C4semidirect
59. C2×C4⋊C4C3×C4semidirect
119. C22⋊C4C3×Q8semidirect
199. C3#1996direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C3×#3802
C3×C4×C22⋊C42
C3×#3422
#19963