Finite Group Databasebeta

← C3×#2044 #72759

Subgroups — C3×#2044 #72759

Order 384.

382 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 84 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×#534C211yesmaximal, characteristic
2C2×C3×#99C212yesmaximal, complemented
3C3×#386C212yesmaximal, complemented
4C3×#385C211yesmaximal, characteristic, complemented
5C3×#342C211yesmaximal, characteristic, complemented
6#2044C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4C2212yes—
8C23×C3×C4C2211yescharacteristic
9C3×#123—22no—
10C3×#121—21no—
11C3×#121—21no—
12C3×#120—22no—
13C2×C3×C4⋊C4—21no—
14C2×C3×C22⋊C4—22no—
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C3×#99—24no—
19C3×#95—21no—
20C3×#95—22no—
21#534C2×C311yescharacteristic
22C2×#99C2×C312yescomplemented
23#386C2×C312yescomplemented
24#385C2×C311yescharacteristic, complemented
25#342C2×C311yescharacteristic, complemented
26C24×C3D412yes—
27C24×C3C2311yescharacteristic
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4—22no—
33C2×C3×D4—22no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C22×C3×C4—22no—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4—22no—
40C22×C3×C4—42no—
41C22×C3×C4D412yes—
42C22×C3×C4D411yescharacteristic
43C22×C3×C4D411yescharacteristic
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—42no—
47C3×C22⋊C4—22no—
48C3×C22⋊C4—22no—
49C3×C22⋊C4—22no—
50C3×C22⋊C4—21no—
51C3×C22⋊C4—22no—
52C3×C22⋊C4—22no—
53C3×C22⋊C4—44no—
54C3×C22⋊C4D412yescomplemented
55C22×D4C22×C312yes—
56C23×C4C22×C311yescharacteristic
57#123—22no—
58#121—21no—
59#121—21no—
60#120—22no—
61C2×C4⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4C22×C312yes—
64C2×C22⋊C4C22×C311yescharacteristic
65C2×C22⋊C4C22×C311yescharacteristic
66#99—24no—
67#95—21no—
68#95—22no—
69C23×C3—22no—
70C23×C3—22no—
71C23×C3—22no—
72C23×C3—41no—
73C23×C3—41no—
74C23×C3—42no—
75C23×C3—42no—
76C23×C3C2×D412yes—
77C23×C3C2×D411yescharacteristic
78C23×C3C2×D411yescharacteristic
79C23×C3#4111yescharacteristic
80C23×C3#4112yes—
81C3×D4—42no—
82C3×D4—42no—
83C3×D4—42no—
84C3×D4—42no—
85C3×D4—81no—
86C3×D4—81no—
87C2×C3×C4—24no—
88C2×C3×C4—22no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—42no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4—42no—
103C2×C3×C4—42no—
104C2×C3×C4—42no—
105C24C3×D412yes—
106C24C23×C311yescharacteristic
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4C3×D412yes—
121C22×C4C3×D411yescharacteristic
122C22×C4C3×D411yescharacteristic
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—42no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C22⋊C4—21no—
130C22⋊C4—22no—
131C22⋊C4—22no—
132C22⋊C4—44no—
133C22⋊C4C3×D412yescomplemented
134C22×C3—21no—
135C22×C3—22no—
136C22×C3—21no—
137C22×C3—22no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—42no—
141C22×C3—44no—
142C22×C3—42no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—82no—
146C22×C3#12011yescharacteristic
147C22×C3#12111yescharacteristic
148C22×C3#12112yes—
149C22×C3#12312yes—
150C22×C3#12411yescharacteristic
151C3×C4—42no—
152C3×C4—42no—
153C3×C4—42no—
154C3×C4—41no—
155C3×C4—41no—
156C3×C4—82no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—41no—
161C23—41no—
162C23—42no—
163C23—42no—
164C23C2×C3×D412yes—
165C23C2×C3×D411yescharacteristic
166C23C2×C3×D411yescharacteristic
167C23C3×#4111yescharacteristic
168C23C3×#4112yes—
169D4—42no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—81no—
174D4—81no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C3—21no—
194C2×C3—22no—
195C2×C3—22no—
196C2×C3—21no—
197C2×C3—82no—
198C2×C3#39411yescharacteristic
199C2×C3B(4,𝔽2)11yescharacteristic
200C2×C3#45811yescharacteristic
201C22—21no—
202C22—22no—
203C22—21no—
204C22—22no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22—44no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—82no—
213C22C3×#12011yescharacteristic
214C22C3×#12111yescharacteristic
215C22C3×#12112yes—
216C22C3×#12312yes—
217C22C3×#12411yescharacteristic
218C4—42no—
219C4—42no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—82no—
224C3#204411yescharacteristic, direct product factor, Sylow
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2C3×#39411yescharacteristic
231C2C3×B(4,𝔽2)11yescharacteristic
232C2C3×#45811yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC24C23×C3
prime radicalC3×#534C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#99C2semidirect
3. C3×#386C2semidirect
4. C3×#385C2semidirect
5. C3×#342C2semidirect
6. #2044C3direct (N × H)
22. C2×#99C2×C3semidirect
23. #386C2×C3semidirect
24. #385C2×C3semidirect
25. #342C2×C3semidirect
54. C3×C22⋊C4D4semidirect
133. C22⋊C4C3×D4semidirect
224. C3#2044direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5342
C2×C3×#992
C3×#3862
C3×#3852
C3×#3422
#20443