Finite Group Databasebeta

← C2×C3×#352 #72834

Subgroups — C2×C3×#352 #72834

Order 384.

278 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×C3×#123C211yesmaximal, characteristic
2C2×C3×#100C211yesmaximal, characteristic
3C2×C3×#99C211yesmaximal, characteristic
4C3×#352C214yesmaximal, direct product factor
5C2×#352C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×D4C2211yescharacteristic
7C23×C3×C4C411yescharacteristic, complemented
8C2×C3×M16—21no—
9C3×#123—21no—
10C3×#123—21no—
11C3×#123C412yescomplemented
12C3×#123C2212yes—
13C2×C3×C4⋊C4—21no—
14C2×C3×C22⋊C4—21no—
15C2×C3×C22⋊C4—21no—
16C2×C3×C22⋊C4C411yescharacteristic, complemented
17C3×#100—21no—
18C3×#100C2212yes—
19C3×#99—21no—
20C3×#99C2212yes—
21C2×#123C2×C311yescharacteristic
22C2×#100C2×C311yescharacteristic
23C2×#99C2×C311yescharacteristic
24#352C2×C314yesdirect product factor
25C24×C3C2×C411yescharacteristic
26C24×C3D411yescharacteristic
27C2×C3×D4—41no—
28C2×C3×D4—41no—
29C2×C3×D4C2×C411yescharacteristic
30C2×C3×D4D411yescharacteristic
31C2×C3×D4D411yescharacteristic
32C2×C3×D4C2311yescharacteristic
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4—41no—
37C22×C3×C4—41no—
38C22×C3×C4—41no—
39C22×C3×C4C2×C412yescomplemented
40C22×C3×C4D411yescharacteristic
41C3×M16—22no—
42C3×M16—41no—
43C2×C3×C8—41no—
44C3×C4⋊C4—22no—
45C3×C4⋊C4—22no—
46C3×C22⋊C4—22no—
47C3×C22⋊C4—21no—
48C3×C22⋊C4—22no—
49C3×C22⋊C4—22no—
50C3×C22⋊C4—41no—
51C3×C22⋊C4C2×C412yescomplemented
52C22×D4C22×C311yescharacteristic
53C23×C4C3×C411yescharacteristic, complemented
54C2×M16—21no—
55#123—21no—
56#123—21no—
57#123C3×C412yescomplemented
58#123C22×C312yes—
59C2×C4⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4C3×C411yescharacteristic, complemented
63#100—21no—
64#100C22×C312yes—
65#99—21no—
66#99C22×C312yes—
67C23×C3—21no—
68C23×C3—21no—
69C23×C3—21no—
70C23×C3—21no—
71C23×C3—42no—
72C23×C3—42no—
73C23×C3C22⋊C411yescharacteristic
74C23×C3C22⋊C411yescharacteristic
75C23×C3C22⋊C411yescharacteristic
76C23×C3C22×C411yescharacteristic
77C23×C3C2×D411yescharacteristic
78C3×D4—41no—
79C3×D4—41no—
80C3×D4—41no—
81C3×D4—41no—
82C2×C3×C4—21no—
83C2×C3×C4—21no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—42no—
87C2×C3×C4—42no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—42no—
91C2×C3×C4—42no—
92C2×C3×C4—82no—
93C2×C3×C4C22⋊C411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95C3×C8—42no—
96C24C2×C3×C411yescharacteristic
97C24C3×D411yescharacteristic
98C2×D4—41no—
99C2×D4—41no—
100C2×D4C2×C3×C411yescharacteristic
101C2×D4C3×D411yescharacteristic
102C2×D4C3×D411yescharacteristic
103C2×D4C23×C311yescharacteristic
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4C2×C3×C412yescomplemented
111C22×C4C3×D411yescharacteristic
112M16—22no—
113M16—41no—
114C2×C8—41no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—21no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—41no—
122C22⋊C4C2×C3×C412yescomplemented
123C22×C3—22no—
124C22×C3—21no—
125C22×C3—21no—
126C22×C3—21no—
127C22×C3—21no—
128C22×C3—41no—
129C22×C3—41no—
130C22×C3—41no—
131C22×C3—41no—
132C22×C3—42no—
133C22×C3—42no—
134C22×C3—42no—
135C22×C3—81no—
136C22×C3#9911yescharacteristic
137C22×C3#9911yescharacteristic
138C22×C3C2×C22⋊C411yescharacteristic
139C3×C4—21no—
140C3×C4—21no—
141C3×C4—42no—
142C3×C4—42no—
143C3×C4—82no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—42no—
149C23—42no—
150C23C3×C22⋊C411yescharacteristic
151C23C3×C22⋊C411yescharacteristic
152C23C3×C22⋊C411yescharacteristic
153C23C22×C3×C411yescharacteristic
154C23C2×C3×D411yescharacteristic
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—82no—
170C2×C4C3×C22⋊C411yescharacteristic
171C2×C4C2×C3×D411yescharacteristic
172C8—42no—
173C2×C3—21no—
174C2×C3—21no—
175C2×C3—41no—
176C2×C3—41no—
177C2×C3—41no—
178C2×C3—41no—
179C2×C3#35212yesdirect product factor
180C2×C3C2×#9911yescharacteristic
181C22—22no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—81no—
194C22C3×#9911yescharacteristic
195C22C3×#9911yescharacteristic
196C22C2×C3×C22⋊C411yescharacteristic
197C4—21no—
198C4—21no—
199C4—42no—
200C4—42no—
201C4—82no—
202C3C2×#35211yescharacteristic, direct product factor, Sylow
203C2—21no—
204C2—21no—
205C2—41no—
206C2—41no—
207C2—41no—
208C2—41no—
209C2C3×#35212yesdirect product factor
210C2C2×C3×#9911yescharacteristic
subgroupG/H
center, socleC22×C3#99
commutatorC23C22×C3×C4
Frattini, cosocleC2×D4C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#352C2direct (N × H)
5. C2×#352C3direct (N × H)
7. C23×C3×C4C4semidirect
11. C3×#123C4semidirect
16. C2×C3×C22⋊C4C4semidirect
24. #352C2×C3direct (N × H)
39. C22×C3×C4C2×C4semidirect
51. C3×C22⋊C4C2×C4semidirect
53. C23×C4C3×C4semidirect
57. #123C3×C4semidirect
62. C2×C22⋊C4C3×C4semidirect
110. C22×C4C2×C3×C4semidirect
122. C22⋊C4C2×C3×C4semidirect
179. C2×C3#352direct (N × H)
202. C3C2×#352direct (N × H)
209. C2C3×#352direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×C3×#1002
C2×C3×#992
C3×#3522
C2×#3523