Finite Group Databasebeta

← C3×#1867 #72582

Subgroups — C3×#1867 #72582

Order 384.

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

normal subgroups only

This group has 374 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
1C3×#518C211yesmaximal, characteristic
2C2×C3×#99C211yesmaximal, characteristic
3C3×#386C211yesmaximal, characteristic
4C3×#385C212yesmaximal, complemented
5C3×#342C212yesmaximal, complemented
6#1867C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4C2211yescharacteristic
8C23×C3×C4C2211yescharacteristic
9C3×C4×D4—22no—
10C3×C4×D4—22no—
11C3×#117—21no—
12C2×C3×C4⋊C4—22no—
13C2×C3×C22⋊C4—21no—
14C2×C3×C22⋊C4—21no—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2212yes—
18C3×#99C414yescomplemented
19C3×#95—22no—
20C3×#95—21no—
21#518C2×C311yescharacteristic
22C2×#99C2×C311yescharacteristic
23#386C2×C311yescharacteristic
24#385C2×C312yescomplemented
25#342C2×C312yescomplemented
26C24×C3D411yescharacteristic
27C24×C3C2311yescharacteristic
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4C2×C412yes—
31C2×C3×D4Q812yes—
32C22×C3×C4—21no—
33C22×C3×C4—22no—
34C22×C3×C4—22no—
35C22×C3×C4—21no—
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4—42no—
39C22×C3×C4D411yescharacteristic
40C22×C3×C4D412yes—
41C22×C3×C4D412yes—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4—44no—
45C3×C22⋊C4—22no—
46C3×C22⋊C4—22no—
47C3×C22⋊C4—22no—
48C3×C22⋊C4—22no—
49C3×C22⋊C4—22no—
50C3×C22⋊C4—42no—
51C3×C22⋊C4C2×C414yescomplemented
52C3×C42—22no—
53C22×D4C22×C311yescharacteristic
54C23×C4C22×C311yescharacteristic
55C4×D4—22no—
56C4×D4—22no—
57#117—21no—
58C2×C4⋊C4—22no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4C22×C311yescharacteristic
62C2×C22⋊C4C22×C312yes—
63C2×C22⋊C4C22×C312yes—
64#99C3×C414yescomplemented
65#95—22no—
66#95—21no—
67C23×C3—21no—
68C23×C3—21no—
69C23×C3—41no—
70C23×C3—41no—
71C23×C3—41no—
72C23×C3—41no—
73C23×C3C4⋊C412yescomplemented
74C23×C3C22×C411yescharacteristic
75C23×C3C2×D411yescharacteristic
76C23×C3C2×D412yes—
77C23×C3C2×Q811yescharacteristic
78C23×C3#4112yes—
79C3×D4—24no—
80C3×D4—24no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—22no—
84C2×C3×C4—24no—
85C2×C3×C4—24no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—42no—
89C2×C3×C4—42no—
90C2×C3×C4—42no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93C2×C3×C4—42no—
94C2×C3×C4—42no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—81no—
99C2×C3×C4—81no—
100C2×C3×C4C4⋊C412yescomplemented
101C24C3×D411yescharacteristic
102C24C23×C311yescharacteristic
103C2×D4—22no—
104C2×D4—22no—
105C2×D4C2×C3×C412yes—
106C2×D4C3×Q812yes—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4C3×D411yescharacteristic
115C22×C4C3×D412yes—
116C22×C4C3×D412yes—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—44no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—42no—
126C22⋊C4C2×C3×C414yescomplemented
127C42—22no—
128C22×C3—22no—
129C22×C3—21no—
130C22×C3—22no—
131C22×C3—21no—
132C22×C3—22no—
133C22×C3—41no—
134C22×C3—41no—
135C22×C3—42no—
136C22×C3—42no—
137C22×C3—41no—
138C22×C3—41no—
139C22×C3—81no—
140C22×C3C2×C4⋊C411yescharacteristic
141C22×C3C4×D412yes—
142C22×C3#12011yescharacteristic
143C22×C3C22⋊Q812yes—
144C22×C3#12311yescharacteristic
145C3×C4—22no—
146C3×C4—44no—
147C3×C4—42no—
148C3×C4—42no—
149C3×C4—42no—
150C3×C4—81no—
151C23—21no—
152C23—21no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23C3×C4⋊C412yescomplemented
158C23C22×C3×C411yescharacteristic
159C23C2×C3×D411yescharacteristic
160C23C2×C3×D412yes—
161C23C2×C3×Q811yescharacteristic
162C23C3×#4112yes—
163D4—24no—
164D4—24no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4C3×C4⋊C412yescomplemented
185C2×C3—22no—
186C2×C3—21no—
187C2×C3—21no—
188C2×C3—22no—
189C2×C3—42no—
190C2×C3#38511yescharacteristic
191C2×C3B(4,𝔽2)11yescharacteristic
192C2×C3#45811yescharacteristic
193C22—22no—
194C22—21no—
195C22—22no—
196C22—21no—
197C22—22no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—81no—
205C22C2×C3×C4⋊C411yescharacteristic
206C22C3×C4×D412yes—
207C22C3×#12011yescharacteristic
208C22C3×C22⋊Q812yes—
209C22C3×#12311yescharacteristic
210C4—22no—
211C4—44no—
212C4—42no—
213C4—42no—
214C4—42no—
215C4—81no—
216C3#186711yescharacteristic, direct product factor, Sylow
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2C3×#38511yescharacteristic
223C2C3×B(4,𝔽2)11yescharacteristic
224C2C3×#45811yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutatorC23C22×C3×C4
Frattini, cosocleC24C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#385C2semidirect
5. C3×#342C2semidirect
6. #1867C3direct (N × H)
18. C3×#99C4semidirect
24. #385C2×C3semidirect
25. #342C2×C3semidirect
51. C3×C22⋊C4C2×C4semidirect
64. #99C3×C4semidirect
73. C23×C3C4⋊C4semidirect
100. C2×C3×C4C4⋊C4semidirect
126. C22⋊C4C2×C3×C4semidirect
157. C23C3×C4⋊C4semidirect
184. C2×C4C3×C4⋊C4semidirect
216. C3#1867direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5182
C2×C3×#992
C3×#3862
C3×#3852
C3×#3422
#18673