Finite Group Databasebeta

← C3×#1513 #72228

Subgroups — C3×#1513 #72228

Order 384.

286 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 92 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×#525C211yesmaximal, characteristic
2C3×#407C211yesmaximal, characteristic, complemented
3C3×#380C211yesmaximal, characteristic, complemented
4C3×#328C212yesmaximal, complemented
5C3×#327C212yesmaximal, complemented
6#1513C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41C411yescharacteristic, complemented
8C23×C3×C4C2211yescharacteristic
9C2×C3×M16—21no—
10C3×#123—21no—
11C3×C22⋊Q8C411yescharacteristic, complemented
12C3×C22⋊Q8C2211yescharacteristic
13C3×#121C411yescharacteristic, complemented
14C3×#121C2211yescharacteristic
15C3×#120—21no—
16C3×C4×D4—21no—
17C3×C4×D4—21no—
18C3×#117C411yescharacteristic, complemented
19C2×C3×C4⋊C4—21no—
20C2×C3×C22⋊C4—21no—
21C3×C22⋊C8—21no—
22C3×C22⋊C8C2212yes—
23C3×#95C2212yes—
24#525C2×C311yescharacteristic
25#407C2×C311yescharacteristic, complemented
26#380C2×C311yescharacteristic, complemented
27#328C2×C312yescomplemented
28#327C2×C312yescomplemented
29C24×C3D411yescharacteristic
30C3×#41—42no—
31C2×C3×Q8C2×C411yescharacteristic
32C2×C3×D4—21no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4C2×C411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4—22no—
38C22×C3×C4—42no—
39C22×C3×C4C2×C411yescharacteristic
40C22×C3×C4C2×C411yescharacteristic
41C22×C3×C4D411yescharacteristic
42C22×C3×C4D411yescharacteristic
43C22×C3×C4D411yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C3×M16—42no—
46C2×C3×C8—22no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—42no—
50C3×C4⋊C4C2×C411yescharacteristic
51C3×C4⋊C4C2×C411yescharacteristic
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4—42no—
58C3×C42—21no—
59C2×#41C3×C411yescharacteristic, complemented
60C23×C4C22×C311yescharacteristic
61C2×M16—21no—
62#123—21no—
63C22⋊Q8C3×C411yescharacteristic, complemented
64C22⋊Q8C22×C311yescharacteristic
65#121C3×C411yescharacteristic, complemented
66#121C22×C311yescharacteristic
67#120—21no—
68C4×D4—21no—
69C4×D4—21no—
70#117C3×C411yescharacteristic, complemented
71C2×C4⋊C4—21no—
72C2×C22⋊C4—21no—
73C22⋊C8—21no—
74C22⋊C8C22×C312yes—
75#95C22×C312yes—
76C23×C3—21no—
77C23×C3—21no—
78C23×C3—21no—
79C23×C3—41no—
80C23×C3—41no—
81C23×C3C22⋊C411yescharacteristic
82C23×C3C22⋊C411yescharacteristic
83C23×C3C2×D411yescharacteristic
84C3×Q8—41no—
85C3×D4—41no—
86C3×D4—42no—
87C3×D4—42no—
88C3×D4—41no—
89C3×D4—41no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—42no—
100C2×C3×C4—42no—
101C2×C3×C4—42no—
102C2×C3×C4—42no—
103C2×C3×C4C22⋊C411yescharacteristic
104C2×C3×C4C22⋊C411yescharacteristic
105C2×C3×C4C22×C411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C3×C8—42no—
108C24C3×D411yescharacteristic
109#41—42no—
110C2×Q8C2×C3×C411yescharacteristic
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4C2×C3×C411yescharacteristic
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4C2×C3×C411yescharacteristic
119C22×C4C2×C3×C411yescharacteristic
120C22×C4C3×D411yescharacteristic
121C22×C4C3×D411yescharacteristic
122C22×C4C3×D411yescharacteristic
123C22×C4C23×C311yescharacteristic
124M16—42no—
125C2×C8—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—42no—
129C4⋊C4C2×C3×C411yescharacteristic
130C4⋊C4C2×C3×C411yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—42no—
137C42—21no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—21no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3—42no—
147C22×C3—42no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3C2×C22⋊C411yescharacteristic
152C22×C3AGL(1,ℤ/8)11yescharacteristic
153C22×C3#13711yescharacteristic
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—82no—
161C3×C4#9912yes—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23C3×C22⋊C411yescharacteristic
168C23C3×C22⋊C411yescharacteristic
169C23C2×C3×D411yescharacteristic
170Q8—41no—
171D4—41no—
172D4—42no—
173D4—42no—
174D4—41no—
175D4—41no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4C3×C22⋊C411yescharacteristic
190C2×C4C3×C22⋊C411yescharacteristic
191C2×C4C22×C3×C411yescharacteristic
192C2×C4C2×C3×D411yescharacteristic
193C8—42no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3—41no—
197C2×C3—41no—
198C2×C3—81no—
199C2×C3C2×#9911yescharacteristic
200C2×C3#41711yescharacteristic
201C2×C3#42111yescharacteristic
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22C2×C3×C22⋊C411yescharacteristic
216C22C3×AGL(1,ℤ/8)11yescharacteristic
217C22C3×#13711yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—82no—
225C4C3×#9912yes—
226C3#151311yescharacteristic, direct product factor, Sylow
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—81no—
232C2C2×C3×#9911yescharacteristic
233C2C3×#41711yescharacteristic
234C2C3×#42111yescharacteristic
subgroupG/H
center, socleC22×C3C2×C22⋊C4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC3×#525C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#407C2semidirect
3. C3×#380C2semidirect
4. C3×#328C2semidirect
5. C3×#327C2semidirect
6. #1513C3direct (N × H)
7. C2×C3×#41C4semidirect
11. C3×C22⋊Q8C4semidirect
13. C3×#121C4semidirect
18. C3×#117C4semidirect
25. #407C2×C3semidirect
26. #380C2×C3semidirect
27. #328C2×C3semidirect
28. #327C2×C3semidirect
59. C2×#41C3×C4semidirect
63. C22⋊Q8C3×C4semidirect
65. #121C3×C4semidirect
70. #117C3×C4semidirect
226. C3#1513direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5252
C3×#4072
C3×#3802
C3×#3282
C3×#3272
#15133