Finite Group Databasebeta

← C3×#1871 #72586

Subgroups — C3×#1871 #72586

Order 384.

376 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C22×C3×M16C211yesmaximal, characteristic, complemented
2C3×#525C211yesmaximal, characteristic
3C2×C3×#104C212yesmaximal, complemented
4C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#339C212yesmaximal, complemented
6#1871C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41C2211yescharacteristic
8C23×C3×C4C2211yescharacteristic
9C2×C3×M16—21no—
10C2×C3×M16—24no—
11C2×C3×M16C2212yes—
12C22×C3×C8—21no—
13C3×#123—21no—
14C3×C22⋊Q8C411yescharacteristic, complemented
15C3×C22⋊Q8C411yescharacteristic, complemented
16C3×#121C411yescharacteristic, complemented
17C3×#121C411yescharacteristic, complemented
18C3×#120—21no—
19C3×C4×D4—21no—
20C3×C4×D4—21no—
21C3×#117C2211yescharacteristic
22C2×C3×C22⋊C4—21no—
23C2×C3×C42C2212yes—
24C3×#104—24no—
25C3×#95—21no—
26C22×M16C2×C311yescharacteristic, complemented
27#525C2×C311yescharacteristic
28C2×#104C2×C312yescomplemented
29C4×C22⋊C4C2×C311yescharacteristic, complemented
30#339C2×C312yescomplemented
31C24×C3D411yescharacteristic
32C3×#41—22no—
33C3×#41—22no—
34C2×C3×Q8C2×C411yescharacteristic
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4C2×C411yescharacteristic
39C22×C3×C4—22no—
40C22×C3×C4—22no—
41C22×C3×C4—22no—
42C22×C3×C4—21no—
43C22×C3×C4C2×C411yescharacteristic
44C22×C3×C4C2×C411yescharacteristic
45C22×C3×C4D411yescharacteristic
46C22×C3×C4D411yescharacteristic
47C22×C3×C4D411yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C3×M16—24no—
50C3×M16—22no—
51C3×M16D414yescomplemented
52C2×C3×C8—24no—
53C2×C3×C8—22no—
54C3×C4⋊C4—21no—
55C3×C4⋊C4—21no—
56C3×C4⋊C4C2×C411yescharacteristic
57C3×C4⋊C4C2×C411yescharacteristic
58C3×C22⋊C4—24no—
59C3×C22⋊C4—21no—
60C3×C22⋊C4—21no—
61C3×C22⋊C4—21no—
62C3×C22⋊C4—21no—
63C3×C22⋊C4—21no—
64C3×C42—21no—
65C3×C42—22no—
66C3×C42—22no—
67C2×#41C22×C311yescharacteristic
68C23×C4C22×C311yescharacteristic
69C2×M16—21no—
70C2×M16—24no—
71C2×M16C22×C312yes—
72C22×C8—21no—
73#123—21no—
74C22⋊Q8C3×C411yescharacteristic, complemented
75C22⋊Q8C3×C411yescharacteristic, complemented
76#121C3×C411yescharacteristic, complemented
77#121C3×C411yescharacteristic, complemented
78#120—21no—
79C4×D4—21no—
80C4×D4—21no—
81#117C22×C311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C42C22×C312yes—
84#104—24no—
85#95—21no—
86C23×C3—22no—
87C23×C3—22no—
88C23×C3—21no—
89C23×C3—41no—
90C23×C3C22⋊C411yescharacteristic
91C23×C3C22⋊C411yescharacteristic
92C23×C3C2×D411yescharacteristic
93C3×Q8—22no—
94C3×D4—22no—
95C3×D4—42no—
96C3×D4—42no—
97C3×D4—42no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—24no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—42no—
110C2×C3×C4—42no—
111C2×C3×C4—41no—
112C2×C3×C4—41no—
113C2×C3×C4C22⋊C411yescharacteristic
114C2×C3×C4C22⋊C411yescharacteristic
115C2×C3×C4C22×C411yescharacteristic
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4#4112yes—
119C3×C8—24no—
120C24C3×D411yescharacteristic
121#41—22no—
122#41—22no—
123C2×Q8C2×C3×C411yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C2×C3×C411yescharacteristic
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4C2×C3×C411yescharacteristic
133C22×C4C2×C3×C411yescharacteristic
134C22×C4C3×D411yescharacteristic
135C22×C4C3×D411yescharacteristic
136C22×C4C3×D411yescharacteristic
137C22×C4C23×C311yescharacteristic
138M16—24no—
139M16—22no—
140M16C3×D414yescomplemented
141C2×C8—24no—
142C2×C8—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4C2×C3×C411yescharacteristic
146C4⋊C4C2×C3×C411yescharacteristic
147C22⋊C4—24no—
148C22⋊C4—21no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—21no—
153C42—21no—
154C42—22no—
155C42—22no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—44no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3#10412yescomplemented
168C22×C3C2×C22⋊C411yescharacteristic
169C22×C3#12112yes—
170C3×C4—21no—
171C3×C4—21no—
172C3×C4—44no—
173C3×C4—41no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4C4×D412yes—
177C3×C4#12011yescharacteristic
178C3×C4#12311yescharacteristic
179C23—22no—
180C23—22no—
181C23—21no—
182C23—41no—
183C23C3×C22⋊C411yescharacteristic
184C23C3×C22⋊C411yescharacteristic
185C23C2×C3×D411yescharacteristic
186Q8—22no—
187D4—22no—
188D4—42no—
189D4—42no—
190D4—42no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4C3×C22⋊C411yescharacteristic
207C2×C4C3×C22⋊C411yescharacteristic
208C2×C4C22×C3×C411yescharacteristic
209C2×C4C2×C3×D412yes—
210C2×C4C2×C3×D411yescharacteristic
211C2×C4C3×#4112yes—
212C8—24no—
213C2×C3—22no—
214C2×C3—22no—
215C2×C3—41no—
216C2×C3—81no—
217C2×C3#38611yescharacteristic
218C2×C3C2×#10411yescharacteristic
219C2×C3#42111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22C3×#10412yescomplemented
232C22C2×C3×C22⋊C411yescharacteristic
233C22C3×#12112yes—
234C4—21no—
235C4—21no—
236C4—44no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4C3×C4×D412yes—
241C4C3×#12011yescharacteristic
242C4C3×#12311yescharacteristic
243C3#187111yescharacteristic, direct product factor, Sylow
244C2—22no—
245C2—22no—
246C2—41no—
247C2—81no—
248C2C3×#38611yescharacteristic
249C2C2×C3×#10411yescharacteristic
250C2C3×#42111yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC3×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×M16C2semidirect
3. C2×C3×#104C2semidirect
4. C3×C4×C22⋊C4C2semidirect
5. C3×#339C2semidirect
6. #1871C3direct (N × H)
14. C3×C22⋊Q8C4semidirect
15. C3×C22⋊Q8C4semidirect
16. C3×#121C4semidirect
17. C3×#121C4semidirect
26. C22×M16C2×C3semidirect
28. C2×#104C2×C3semidirect
29. C4×C22⋊C4C2×C3semidirect
30. #339C2×C3semidirect
51. C3×M16D4semidirect
74. C22⋊Q8C3×C4semidirect
75. C22⋊Q8C3×C4semidirect
76. #121C3×C4semidirect
77. #121C3×C4semidirect
140. M16C3×D4semidirect
167. C22×C3#104semidirect
231. C22C3×#104semidirect
243. C3#1871direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×M162
C3×#5252
C2×C3×#1042
C3×C4×C22⋊C42
C3×#3392
#18713