Finite Group Databasebeta

← C3×#1999 #72714

Subgroups — C3×#1999 #72714

Order 384.

496 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C3×D8C211yesmaximal, characteristic, complemented
2C2×C3×#121C212yesmaximal, complemented
3C2×C3×#102C212yesmaximal, complemented
4C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
5C3×#340C211yesmaximal, characteristic, complemented
6#1999C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4—22no—
8C22×C3×D4C2212yes—
9C23×C3×C4C2211yescharacteristic
10C2×C3×D8—24no—
11C2×C3×D8—22no—
12C22×C3×C8C2211yescharacteristic
13C22×C3×C8C2211yescharacteristic
14C3×#121—24no—
15C3×#121—24no—
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C22⋊C4—22no—
18C3×#102—24no—
19C3×C22⋊C8—22no—
20C22×D8C2×C311yescharacteristic, complemented
21C2×#121C2×C312yescomplemented
22C2×#102C2×C312yescomplemented
23C2×C22⋊C8C2×C311yescharacteristic, complemented
24#340C2×C311yescharacteristic, complemented
25C24×C3—22no—
26C24×C3D411yescharacteristic
27C2×C3×D4—28no—
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4—22no—
33C2×C3×D4—44no—
34C2×C3×D4—44no—
35C2×C3×D4D414yescomplemented
36C22×C3×C4—22no—
37C22×C3×C4—21no—
38C22×C3×C4—22no—
39C22×C3×C4—22no—
40C22×C3×C4—21no—
41C22×C3×C4D411yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C3×D8—28no—
44C2×C3×C8—22no—
45C2×C3×C8—21no—
46C2×C3×C8—22no—
47C2×C3×C8D412yescomplemented
48C3×C4⋊C4—22no—
49C3×C4⋊C4D414yescomplemented
50C3×C22⋊C4—28no—
51C22×D4—22no—
52C22×D4C22×C312yes—
53C23×C4C22×C311yescharacteristic
54C2×D8—24no—
55C2×D8—22no—
56C22×C8C22×C311yescharacteristic
57C22×C8C22×C311yescharacteristic
58#121—24no—
59#121—24no—
60C2×C4⋊C4C22×C312yes—
61C2×C22⋊C4—22no—
62#102—24no—
63C22⋊C8—22no—
64C23×C3—22no—
65C23×C3—21no—
66C23×C3—21no—
67C23×C3—22no—
68C23×C3—24no—
69C23×C3—42no—
70C23×C3—42no—
71C23×C3—42no—
72C23×C3—42no—
73C23×C3—42no—
74C23×C3—42no—
75C23×C3D812yescomplemented
76C23×C3C2×D411yescharacteristic
77C3×D4—24no—
78C3×D4—24no—
79C3×D4—416no—
80C3×D4—42no—
81C3×D4—42no—
82C2×C3×C4—22no—
83C2×C3×C4—24no—
84C2×C3×C4—21no—
85C2×C3×C4—42no—
86C2×C3×C4—42no—
87C2×C3×C4—41no—
88C2×C3×C4—42no—
89C2×C3×C4—41no—
90C2×C3×C4C2×D412yes—
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D412yes—
93C2×C3×C4#4111yescharacteristic
94C3×C8—22no—
95C3×C8—41no—
96C24—22no—
97C24C3×D411yescharacteristic
98C2×D4—28no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4C3×D414yescomplemented
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4C3×D411yescharacteristic
113C22×C4C23×C311yescharacteristic
114D8—28no—
115C2×C8—22no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8C3×D412yescomplemented
119C4⋊C4—22no—
120C4⋊C4C3×D414yescomplemented
121C22⋊C4—28no—
122C22×C3—24no—
123C22×C3—22no—
124C22×C3—41no—
125C22×C3—42no—
126C22×C3—41no—
127C22×C3—44no—
128C22×C3—44no—
129C22×C3—44no—
130C22×C3—82no—
131C22×C3—82no—
132C22×C3—82no—
133C22×C3—82no—
134C22×C3#12012yes—
135C22×C3#12111yescharacteristic
136C22×C3C2×D811yescharacteristic
137C22×C3#13511yescharacteristic
138C22×C3AGL(1,ℤ/8)12yes—
139C3×C4—41no—
140C3×C4—44no—
141C3×C4#12011yescharacteristic
142C3×C4#12112yes—
143C3×C4#12711yescharacteristic
144C23—22no—
145C23—21no—
146C23—21no—
147C23—22no—
148C23—24no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23C3×D812yescomplemented
156C23C2×C3×D411yescharacteristic
157D4—24no—
158D4—24no—
159D4—416no—
160D4—42no—
161D4—42no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4C2×C3×D412yes—
171C2×C4C2×C3×D411yescharacteristic
172C2×C4C2×C3×D412yes—
173C2×C4C3×#4111yescharacteristic
174C8—22no—
175C8—41no—
176C2×C3—42no—
177C2×C3—84no—
178C2×C3#39211yescharacteristic
179C2×C3#44712yes—
180C2×C3#44912yes—
181C2×C3#46611yescharacteristic
182C2×C3#46911yescharacteristic
183C22—24no—
184C22—22no—
185C22—41no—
186C22—42no—
187C22—41no—
188C22—44no—
189C22—44no—
190C22—44no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22—82no—
195C22C3×#12012yes—
196C22C3×#12111yescharacteristic
197C22C2×C3×D811yescharacteristic
198C22C3×#13511yescharacteristic
199C22C3×AGL(1,ℤ/8)12yes—
200C4—41no—
201C4—44no—
202C4C3×#12011yescharacteristic
203C4C3×#12112yes—
204C4C3×#12711yescharacteristic
205C3#199911yescharacteristic, direct product factor, Sylow
206C2—42no—
207C2—84no—
208C2C3×#39211yescharacteristic
209C2C3×#44712yes—
210C2C3×#44912yes—
211C2C3×#46611yescharacteristic
212C2C3×#46911yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×D8C2semidirect
2. C2×C3×#121C2semidirect
3. C2×C3×#102C2semidirect
4. C2×C3×C22⋊C8C2semidirect
5. C3×#340C2semidirect
6. #1999C3direct (N × H)
20. C22×D8C2×C3semidirect
21. C2×#121C2×C3semidirect
22. C2×#102C2×C3semidirect
23. C2×C22⋊C8C2×C3semidirect
24. #340C2×C3semidirect
35. C2×C3×D4D4semidirect
47. C2×C3×C8D4semidirect
49. C3×C4⋊C4D4semidirect
75. C23×C3D8semidirect
106. C2×D4C3×D4semidirect
118. C2×C8C3×D4semidirect
120. C4⋊C4C3×D4semidirect
155. C23C3×D8semidirect
205. C3#1999direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D82
C2×C3×#1212
C2×C3×#1022
C2×C3×C22⋊C82
C3×#3402
#19993