Finite Group Databasebeta

← C3×#1606 #72321

Subgroups — C3×#1606 #72321

Order 384.

260 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 64 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×#538C211yesmaximal, characteristic
2C3×#451C211yesmaximal, characteristic, complemented
3C3×#448C211yesmaximal, characteristic, complemented
4C3×#393C211yesmaximal, characteristic, complemented
5C3×#328C211yesmaximal, characteristic, complemented
6C3×#327C211yesmaximal, characteristic, complemented
7C3×#323C211yesmaximal, characteristic, complemented
8#1606C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q8C2211yescharacteristic
10C2×C3×Q16—21no—
11C2×C3×SD16—21no—
12C3×#126—21no—
13C3×#124—21no—
14C3×#123—21no—
15C3×C22⋊Q8C2211yescharacteristic
16C3×#121—21no—
17C3×#121C2211yescharacteristic
18C3×#120—21no—
19C3×C4×D4—21no—
20C2×C3×C22⋊C4—21no—
21C2×C3×C22⋊C4C2211yescharacteristic
22C3×#103—21no—
23C3×#103—21no—
24C3×C22⋊C8C2211yescharacteristic
25C3×C22⋊C8C2211yescharacteristic
26C3×#95—21no—
27C3×#95C2211yescharacteristic
28#538C2×C311yescharacteristic
29#451C2×C311yescharacteristic, complemented
30#448C2×C311yescharacteristic, complemented
31#393C2×C311yescharacteristic, complemented
32#328C2×C311yescharacteristic, complemented
33#327C2×C311yescharacteristic, complemented
34#323C2×C311yescharacteristic, complemented
35C24×C3D411yescharacteristic
36C2×C3×Q8—21no—
37C2×C3×Q8—21no—
38C2×C3×Q8—42no—
39C2×C3×Q8D411yescharacteristic
40C2×C3×D4—21no—
41C2×C3×D4—21no—
42C2×C3×D4—21no—
43C2×C3×D4D411yescharacteristic
44C22×C3×C4—21no—
45C22×C3×C4—21no—
46C22×C3×C4—21no—
47C22×C3×C4D411yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C3×Q16—42no—
50C3×SD16—42no—
51C2×C3×C8—21no—
52C2×C3×C8—21no—
53C3×C4⋊C4—21no—
54C3×C4⋊C4—21no—
55C3×C4⋊C4D411yescharacteristic
56C3×C4⋊C4D411yescharacteristic
57C3×C22⋊C4—21no—
58C3×C22⋊C4—21no—
59C3×C22⋊C4—21no—
60C3×C22⋊C4—21no—
61C3×C22⋊C4—21no—
62C3×C22⋊C4—21no—
63C3×C22⋊C4—21no—
64C3×C22⋊C4—42no—
65C3×C42—21no—
66C22×Q8C22×C311yescharacteristic
67C2×Q16—21no—
68C2×SD16—21no—
69#126—21no—
70#124—21no—
71#123—21no—
72C22⋊Q8C22×C311yescharacteristic
73#121—21no—
74#121C22×C311yescharacteristic
75#120—21no—
76C4×D4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4C22×C311yescharacteristic
79#103—21no—
80#103—21no—
81C22⋊C8C22×C311yescharacteristic
82C22⋊C8C22×C311yescharacteristic
83#95—21no—
84#95C22×C311yescharacteristic
85C23×C3—21no—
86C23×C3—21no—
87C23×C3—21no—
88C23×C3—21no—
89C23×C3—41no—
90C23×C3—41no—
91C23×C3C2×D411yescharacteristic
92C3×Q8—22no—
93C3×Q8—22no—
94C3×Q8—41no—
95C3×Q8—41no—
96C3×Q8—41no—
97C3×D4—41no—
98C3×D4—41no—
99C3×D4—41no—
100C3×D4—81no—
101C3×D4—81no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—41no—
108C2×C3×C4—41no—
109C2×C3×C4—41no—
110C2×C3×C4—41no—
111C2×C3×C4—41no—
112C2×C3×C4—41no—
113C2×C3×C4—41no—
114C2×C3×C4—81no—
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D411yescharacteristic
117C3×C8—41no—
118C3×C8—41no—
119C24C3×D411yescharacteristic
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—42no—
123C2×Q8C3×D411yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C3×D411yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4C3×D411yescharacteristic
132C22×C4C23×C311yescharacteristic
133Q16—42no—
134SD16—42no—
135C2×C8—21no—
136C2×C8—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4C3×D411yescharacteristic
140C4⋊C4C3×D411yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C22⋊C4—21no—
148C22⋊C4—42no—
149C42—21no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—42no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3—81no—
160C22×C3#12011yescharacteristic
161C22×C3#13511yescharacteristic
162C22×C3#13711yescharacteristic
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—41no—
168C3×C4—42no—
169C3×C4—41no—
170C3×C4—81no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×C3×D411yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—41no—
181Q8—41no—
182Q8—41no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—81no—
187D4—81no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—81no—
201C2×C4C2×C3×D411yescharacteristic
202C2×C4C2×C3×D411yescharacteristic
203C8—41no—
204C8—41no—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3—81no—
208C2×C3—81no—
209C2×C3#45211yescharacteristic
210C2×C3#45511yescharacteristic
211C2×C3B(4,𝔽2)11yescharacteristic
212C22—21no—
213C22—21no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—81no—
222C22C3×#12011yescharacteristic
223C22C3×#13511yescharacteristic
224C22C3×#13711yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—81no—
233C3#160611yescharacteristic, direct product factor, Sylow
234C2—21no—
235C2—21no—
236C2—81no—
237C2—81no—
238C2C3×#45211yescharacteristic
239C2C3×#45511yescharacteristic
240C2C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC3×#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#451C2semidirect
3. C3×#448C2semidirect
4. C3×#393C2semidirect
5. C3×#328C2semidirect
6. C3×#327C2semidirect
7. C3×#323C2semidirect
8. #1606C3direct (N × H)
29. #451C2×C3semidirect
30. #448C2×C3semidirect
31. #393C2×C3semidirect
32. #328C2×C3semidirect
33. #327C2×C3semidirect
34. #323C2×C3semidirect
233. C3#1606direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5382
C3×#4512
C3×#4482
C3×#3932
C3×#3282
C3×#3272
C3×#3232
#16063