Finite Group Databasebeta

← C3×#1792 #72507

Subgroups — C3×#1792 #72507

Order 384.

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

normal subgroups only

This group has 620 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×H(5,𝔽2)C211yesmaximal, characteristic
2C2×C3×#104C212yesmaximal, complemented
3C3×#417C212yesmaximal, complemented
4C2×C3×#100C211yesmaximal, characteristic, complemented
5C3×#390C211yesmaximal, characteristic, complemented
6#1792C311yesmaximal, characteristic, direct product factor, Sylow
7C3×H(5,𝔽2)—24no—
8C3×H(5,𝔽2)—22no—
9C3×H(5,𝔽2)C414yescomplemented
10C2×C3×#41—22no—
11C2×C3×#41C2212yes—
12C22×C3×D4—22no—
13C22×C3×D4—22no—
14C22×C3×D4C2211yescharacteristic
15C2×C3×M16C2212yescomplemented
16C2×C3×C4⋊C4C2211yescharacteristic, complemented
17C2×C3×C22⋊C4—22no—
18C2×C3×C42C2211yescharacteristic, complemented
19C3×#104—24no—
20C3×#103—22no—
21C3×#102—22no—
22C3×#100—22no—
23C2×H(5,𝔽2)C2×C311yescharacteristic
24C2×#104C2×C312yescomplemented
25#417C2×C312yescomplemented
26C2×#100C2×C311yescharacteristic, complemented
27#390C2×C311yescharacteristic, complemented
28C24×C3—22no—
29C24×C3D412yes—
30C3×#41—28no—
31C3×#41—28no—
32C3×#41—22no—
33C3×#41—22no—
34C3×#41C2×C414yes—
35C3×#41D414yes—
36C2×C3×Q8D412yes—
37C2×C3×D4—24no—
38C2×C3×D4—28no—
39C2×C3×D4—28no—
40C2×C3×D4—28no—
41C2×C3×D4—22no—
42C2×C3×D4—28no—
43C2×C3×D4—22no—
44C2×C3×D4—22no—
45C2×C3×D4C2×C412yes—
46C2×C3×D4D412yes—
47C2×C3×D4D412yes—
48C22×C3×C4—22no—
49C22×C3×C4—22no—
50C22×C3×C4—21no—
51C22×C3×C4—21no—
52C22×C3×C4C2311yescharacteristic
53C3×M16—22no—
54C3×M16—22no—
55C2×C3×C8—22no—
56C3×C4⋊C4—22no—
57C3×C22⋊C4—44no—
58C3×C42—22no—
59H(5,𝔽2)—24no—
60H(5,𝔽2)—22no—
61H(5,𝔽2)C3×C414yescomplemented
62C2×#41—22no—
63C2×#41C22×C312yes—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4C22×C311yescharacteristic
67C2×M16C22×C312yescomplemented
68C2×C4⋊C4C22×C311yescharacteristic, complemented
69C2×C22⋊C4—22no—
70C2×C42C22×C311yescharacteristic, complemented
71#104—24no—
72#103—22no—
73#102—22no—
74#100—22no—
75C23×C3—28no—
76C23×C3—22no—
77C23×C3—22no—
78C23×C3—22no—
79C23×C3—42no—
80C23×C3—42no—
81C23×C3—82no—
82C23×C3—82no—
83C23×C3C22⋊C414yes—
84C23×C3C2×D411yescharacteristic
85C3×Q8—22no—
86C3×Q8C22⋊C414yescomplemented
87C3×D4—216no—
88C3×D4—24no—
89C3×D4—24no—
90C3×D4—24no—
91C3×D4—24no—
92C3×D4—22no—
93C3×D4C22⋊C414yescomplemented
94C2×C3×C4—28no—
95C2×C3×C4—22no—
96C2×C3×C4—24no—
97C2×C3×C4—24no—
98C2×C3×C4—22no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—42no—
102C2×C3×C4—41no—
103C2×C3×C4C22×C411yescharacteristic
104C2×C3×C4C2×D412yes—
105C2×C3×C4C2×D412yes—
106C2×C3×C4C2×D411yescharacteristic
107C3×C8—42no—
108C24—22no—
109C24C3×D412yes—
110#41—28no—
111#41—28no—
112#41—22no—
113#41—22no—
114#41C2×C3×C414yes—
115#41C3×D414yes—
116C2×Q8C3×D412yes—
117C2×D4—24no—
118C2×D4—28no—
119C2×D4—28no—
120C2×D4—28no—
121C2×D4—22no—
122C2×D4—28no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C2×C3×C412yes—
126C2×D4C3×D412yes—
127C2×D4C3×D412yes—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4C23×C311yescharacteristic
133M16—22no—
134M16—22no—
135C2×C8—22no—
136C4⋊C4—22no—
137C22⋊C4—44no—
138C42—22no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—24no—
142C22×C3—24no—
143C22×C3—44no—
144C22×C3—42no—
145C22×C3—42no—
146C22×C3—42no—
147C22×C3—42no—
148C22×C3—82no—
149C22×C3—88no—
150C22×C3C2×C22⋊C411yescharacteristic
151C22×C3#12011yescharacteristic
152C22×C3#12011yescharacteristic
153C3×C4—24no—
154C3×C4—42no—
155C3×C4—81no—
156C3×C4C2×C22⋊C412yes—
157C3×C4#12012yes—
158C23—28no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23—82no—
165C23—82no—
166C23C3×C22⋊C414yes—
167C23C2×C3×D411yescharacteristic
168Q8—22no—
169Q8C3×C22⋊C414yescomplemented
170D4—216no—
171D4—24no—
172D4—24no—
173D4—24no—
174D4—24no—
175D4—22no—
176D4C3×C22⋊C414yescomplemented
177C2×C4—28no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4C22×C3×C411yescharacteristic
187C2×C4C2×C3×D412yes—
188C2×C4C2×C3×D412yes—
189C2×C4C2×C3×D411yescharacteristic
190C8—42no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—44no—
194C2×C3—44no—
195C2×C3#37911yescharacteristic
196C2×C3#45311yescharacteristic
197C2×C3#45511yescharacteristic
198C22—21no—
199C22—21no—
200C22—24no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—82no—
208C22—88no—
209C22C2×C3×C22⋊C411yescharacteristic
210C22C3×#12011yescharacteristic
211C22C3×#12011yescharacteristic
212C4—24no—
213C4—42no—
214C4—81no—
215C4C2×C3×C22⋊C412yes—
216C4C3×#12012yes—
217C3#179211yescharacteristic, direct product factor, Sylow
218C2—21no—
219C2—21no—
220C2—44no—
221C2—44no—
222C2C3×#37911yescharacteristic
223C2C3×#45311yescharacteristic
224C2C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×H(5,𝔽2)C2
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×#100C2semidirect
5. C3×#390C2semidirect
6. #1792C3direct (N × H)
9. C3×H(5,𝔽2)C4semidirect
15. C2×C3×M16C22semidirect
16. C2×C3×C4⋊C4C22semidirect
18. C2×C3×C42C22semidirect
24. C2×#104C2×C3semidirect
25. #417C2×C3semidirect
26. C2×#100C2×C3semidirect
27. #390C2×C3semidirect
61. H(5,𝔽2)C3×C4semidirect
67. C2×M16C22×C3semidirect
68. C2×C4⋊C4C22×C3semidirect
70. C2×C42C22×C3semidirect
86. C3×Q8C22⋊C4semidirect
93. C3×D4C22⋊C4semidirect
169. Q8C3×C22⋊C4semidirect
176. D4C3×C22⋊C4semidirect
217. C3#1792direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×H(5,𝔽2)2
C2×C3×#1042
C3×#4172
C2×C3×#1002
C3×#3902
#17923