Finite Group Databasebeta

← C3×#1793 #72508

Subgroups — C3×#1793 #72508

Order 384.

552 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 136 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×C3×2−1+4C211yesmaximal, characteristic
2C2×C3×#104C212yesmaximal, complemented
3C3×#417C212yesmaximal, complemented
4C2×C3×#101C211yesmaximal, characteristic, complemented
5C3×#391C211yesmaximal, characteristic, complemented
6#1793C311yesmaximal, characteristic, direct product factor, Sylow
7C3×2−1+4—24no—
8C3×2−1+4—22no—
9C3×2−1+4C414yescomplemented
10C2×C3×#41—22no—
11C2×C3×#41—22no—
12C2×C3×#41C2212yes—
13C22×C3×Q8—22no—
14C22×C3×Q8C2211yescharacteristic
15C2×C3×M16C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2211yescharacteristic
18C3×#104—24no—
19C3×#103—22no—
20C3×#102—22no—
21C3×#101—22no—
22C3×#95—22no—
23C2×2−1+4C2×C311yescharacteristic
24C2×#104C2×C312yescomplemented
25#417C2×C312yescomplemented
26C2×#101C2×C311yescharacteristic, complemented
27#391C2×C311yescharacteristic, complemented
28C3×#41—28no—
29C3×#41—28no—
30C3×#41—28no—
31C3×#41—28no—
32C3×#41—22no—
33C3×#41—22no—
34C3×#41C2×C414yes—
35C3×#41D414yes—
36C2×C3×Q8—22no—
37C2×C3×Q8—28no—
38C2×C3×Q8—28no—
39C2×C3×Q8—22no—
40C2×C3×Q8—22no—
41C2×C3×Q8C2×C412yes—
42C2×C3×Q8D412yes—
43C2×C3×Q8D412yes—
44C2×C3×D4—22no—
45C2×C3×D4—22no—
46C2×C3×D4D412yes—
47C22×C3×C4—24no—
48C22×C3×C4—22no—
49C22×C3×C4—21no—
50C22×C3×C4—21no—
51C22×C3×C4D412yes—
52C22×C3×C4C2311yescharacteristic
53C3×M16—22no—
54C3×M16—22no—
55C2×C3×C8—22no—
56C3×C4⋊C4—22no—
57C3×C42—22no—
582−1+4—24no—
592−1+4—22no—
602−1+4C3×C414yescomplemented
61C2×#41—22no—
62C2×#41—22no—
63C2×#41C22×C312yes—
64C22×Q8—22no—
65C22×Q8C22×C311yescharacteristic
66C2×M16C22×C312yes—
67C2×C4⋊C4C22×C311yescharacteristic
68C2×C42C22×C311yescharacteristic
69#104—24no—
70#103—22no—
71#102—22no—
72#101—22no—
73#95—22no—
74C23×C3—22no—
75C23×C3C2×D411yescharacteristic
76C3×Q8—28no—
77C3×Q8—24no—
78C3×Q8—22no—
79C3×Q8—24no—
80C3×Q8C22⋊C414yes—
81C3×D4—28no—
82C3×D4—24no—
83C3×D4—24no—
84C3×D4—22no—
85C3×D4C22⋊C414yes—
86C2×C3×C4—216no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—24no—
91C2×C3×C4—24no—
92C2×C3×C4—22no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—42no—
96C2×C3×C4—41no—
97C2×C3×C4C22⋊C414yes—
98C2×C3×C4C22×C411yescharacteristic
99C2×C3×C4C2×D412yes—
100C2×C3×C4C2×D412yes—
101C2×C3×C4C2×D411yescharacteristic
102C3×C8—42no—
103#41—28no—
104#41—28no—
105#41—28no—
106#41—28no—
107#41—22no—
108#41—22no—
109#41C2×C3×C414yes—
110#41C3×D414yes—
111C2×Q8—22no—
112C2×Q8—28no—
113C2×Q8—28no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8C2×C3×C412yes—
117C2×Q8C3×D412yes—
118C2×Q8C3×D412yes—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4C3×D412yes—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4C3×D412yes—
127C22×C4C23×C311yescharacteristic
128M16—22no—
129M16—22no—
130C2×C8—22no—
131C4⋊C4—22no—
132C42—22no—
133C22×C3—21no—
134C22×C3—21no—
135C22×C3—24no—
136C22×C3—42no—
137C22×C3—42no—
138C22×C3C2×C22⋊C411yescharacteristic
139C22×C3#12011yescharacteristic
140C22×C3#12011yescharacteristic
141C3×C4—24no—
142C3×C4—24no—
143C3×C4—42no—
144C3×C4—81no—
145C3×C4C2×C22⋊C412yes—
146C3×C4#12012yes—
147C23—22no—
148C23C2×C3×D411yescharacteristic
149Q8—28no—
150Q8—24no—
151Q8—22no—
152Q8—24no—
153Q8C3×C22⋊C414yes—
154D4—28no—
155D4—24no—
156D4—24no—
157D4—22no—
158D4C3×C22⋊C414yes—
159C2×C4—216no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4C3×C22⋊C414yes—
171C2×C4C22×C3×C411yescharacteristic
172C2×C4C2×C3×D412yes—
173C2×C4C2×C3×D412yes—
174C2×C4C2×C3×D411yescharacteristic
175C8—42no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—44no—
179C2×C3#37911yescharacteristic
180C2×C3#45411yescharacteristic
181C2×C3#45611yescharacteristic
182C22—21no—
183C22—21no—
184C22—24no—
185C22—42no—
186C22—42no—
187C22C2×C3×C22⋊C411yescharacteristic
188C22C3×#12011yescharacteristic
189C22C3×#12011yescharacteristic
190C4—24no—
191C4—24no—
192C4—42no—
193C4—81no—
194C4C2×C3×C22⋊C412yes—
195C4C3×#12012yes—
196C3#179311yescharacteristic, direct product factor, Sylow
197C2—21no—
198C2—21no—
199C2—44no—
200C2C3×#37911yescharacteristic
201C2C3×#45411yescharacteristic
202C2C3×#45611yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×2−1+4C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#104C2semidirect
3. C3×#417C2semidirect
4. C2×C3×#101C2semidirect
5. C3×#391C2semidirect
6. #1793C3direct (N × H)
9. C3×2−1+4C4semidirect
24. C2×#104C2×C3semidirect
25. #417C2×C3semidirect
26. C2×#101C2×C3semidirect
27. #391C2×C3semidirect
60. 2−1+4C3×C4semidirect
196. C3#1793direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+42
C2×C3×#1042
C3×#4172
C2×C3×#1012
C3×#3912
#17933