Finite Group Databasebeta

← C3×#33761 #10652368

Subgroups — C3×#33761 #10652368

Order 768.

584 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 202 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×#467C212yesmaximal
2C3×#2235C214yesmaximal, complemented
3C3×#2219C214yesmaximal, complemented
4C3×#2188C214yesmaximal, complemented
5C3×#2112C211yesmaximal, characteristic
6#33761C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q16C2212yes—
8C23×C3×C8C2211yescharacteristic
9C2×C3×C22⋊Q8—22no—
10C2×C3×Q32C2214yescomplemented
11C2×C3×M32C2212yes—
12C3×#467—22no—
13C3×#467—22no—
14C3×#467C2214yes—
15C3×#467C2214yes—
16C2×C3×#107C2212yes—
17C2×C3×#103—22no—
18C3×#367C2214yescomplemented
19C3×#358C2218yescomplemented
20C3×C22⋊C16C2214yes—
21C2×#467C2×C312yes—
22#2235C2×C314yescomplemented
23#2219C2×C314yescomplemented
24#2188C2×C314yescomplemented
25#2112C2×C311yescharacteristic
26C22×C3×Q8—22no—
27C23×C3×C4D411yescharacteristic
28C2×C3×Q16—22no—
29C2×C3×Q16—44no—
30C2×C3×Q16C2314yes—
31C22×C3×C8—42no—
32C22×C3×C8D412yes—
33C22×C3×C8D411yescharacteristic
34C22×C3×C8C2311yescharacteristic
35C22×C3×C8C2312yes—
36C3×C22⋊Q8—24no—
37C3×C22⋊Q8—24no—
38C3×C22⋊Q8—42no—
39C3×C22⋊Q8—42no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—42no—
42C2×C3×C22⋊C4—42no—
43C3×Q32—44no—
44C3×M32—42no—
45C2×C3×C16C2314yes—
46C3×#107—22no—
47C3×#107C2314yes—
48C3×#103—24no—
49C3×#103—24no—
50C22×Q16C22×C312yes—
51C23×C8C22×C311yescharacteristic
52C2×C22⋊Q8—22no—
53C2×Q32C22×C314yescomplemented
54C2×M32C22×C312yes—
55#467—22no—
56#467—22no—
57#467C22×C314yes—
58#467C22×C314yes—
59C2×#107C22×C312yes—
60C2×#103—22no—
61#367C22×C314yescomplemented
62#358C22×C318yescomplemented
63C22⋊C16C22×C314yes—
64C24×C3D811yescharacteristic
65C2×C3×Q8—24no—
66C2×C3×Q8—44no—
67C2×C3×Q8—44no—
68C2×C3×Q8—42no—
69C22×C3×C4—41no—
70C22×C3×C4—41no—
71C22×C3×C4—42no—
72C22×C3×C4—42no—
73C22×C3×C4D812yes—
74C22×C3×C4D811yescharacteristic
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C3×Q16—28no—
78C3×Q16—44no—
79C2×C3×C8—22no—
80C2×C3×C8—24no—
81C2×C3×C8—42no—
82C2×C3×C8—41no—
83C2×C3×C8C2×D411yescharacteristic
84C2×C3×C8C2×D412yes—
85C2×C3×C8C2411yescharacteristic
86C3×C4⋊C4—24no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—44no—
89C3×C4⋊C4—44no—
90C3×C22⋊C4—44no—
91C3×C22⋊C4—44no—
92C3×C16—24no—
93C22×Q8—22no—
94C23×C4C3×D411yescharacteristic
95C2×Q16—22no—
96C2×Q16—44no—
97C2×Q16C23×C314yes—
98C22×C8—42no—
99C22×C8C3×D412yes—
100C22×C8C3×D411yescharacteristic
101C22×C8C23×C312yes—
102C22×C8C23×C311yescharacteristic
103C22⋊Q8—24no—
104C22⋊Q8—24no—
105C22⋊Q8—42no—
106C22⋊Q8—42no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—42no—
109C2×C22⋊C4—42no—
110Q32—44no—
111M32—42no—
112C2×C16C23×C314yes—
113#107—22no—
114#107C23×C314yes—
115#103—24no—
116#103—24no—
117C23×C3—24no—
118C23×C3—41no—
119C23×C3C2×D811yescharacteristic
120C23×C3C2×D812yes—
121C3×Q8—44no—
122C3×Q8—44no—
123C3×Q8—84no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—42no—
129C2×C3×C4—41no—
130C2×C3×C4—44no—
131C2×C3×C4—44no—
132C2×C3×C4—84no—
133C2×C3×C4—84no—
134C2×C3×C4C2×D811yescharacteristic
135C2×C3×C4C2×D812yes—
136C2×C3×C4C22×D411yescharacteristic
137C3×C8—21no—
138C3×C8—22no—
139C3×C8H(5,𝔽2)12yes—
140C24C3×D811yescharacteristic
141C2×Q8—24no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×Q8—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4C3×D812yes—
150C22×C4C3×D811yescharacteristic
151C22×C4C2×C3×D412yes—
152C22×C4C2×C3×D411yescharacteristic
153Q16—28no—
154Q16—44no—
155C2×C8—22no—
156C2×C8—24no—
157C2×C8—41no—
158C2×C8—42no—
159C2×C8C2×C3×D411yescharacteristic
160C2×C8C2×C3×D412yes—
161C2×C8C24×C311yescharacteristic
162C4⋊C4—24no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C22⋊C4—44no—
167C22⋊C4—44no—
168C16—24no—
169C22×C3—22no—
170C22×C3—22no—
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—44no—
174C22×C3#51014yes—
175C22×C3C22×D811yescharacteristic
176C3×C4—21no—
177C3×C4—22no—
178C3×C4—84no—
179C3×C4—84no—
180C3×C4#53411yescharacteristic
181C3×C4#57511yescharacteristic
182C23—24no—
183C23—41no—
184C23C2×C3×D811yescharacteristic
185C23C2×C3×D812yes—
186Q8—44no—
187Q8—44no—
188Q8—84no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—84no—
198C2×C4—84no—
199C2×C4C2×C3×D811yescharacteristic
200C2×C4C2×C3×D812yes—
201C2×C4C22×C3×D411yescharacteristic
202C8—22no—
203C8—21no—
204C8C3×H(5,𝔽2)12yes—
205C2×C3—24no—
206C2×C3—41no—
207C2×C3#318611yescharacteristic
208C2×C3C2×#51012yes—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—21no—
213C22—44no—
214C22C3×#51014yes—
215C22C22×C3×D811yescharacteristic
216C4—22no—
217C4—21no—
218C4—84no—
219C4—84no—
220C4C3×#53411yescharacteristic
221C4C3×#57511yescharacteristic
222C3#3376111yescharacteristic, direct product factor, Sylow
223C2—24no—
224C2—41no—
225C2C3×#318611yescharacteristic
226C2C2×C3×#51012yes—
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC24×C3D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2235C2semidirect
3. C3×#2219C2semidirect
4. C3×#2188C2semidirect
6. #33761C3direct (N × H)
10. C2×C3×Q32C22semidirect
18. C3×#367C22semidirect
19. C3×#358C22semidirect
22. #2235C2×C3semidirect
23. #2219C2×C3semidirect
24. #2188C2×C3semidirect
53. C2×Q32C22×C3semidirect
61. #367C22×C3semidirect
62. #358C22×C3semidirect
222. C3#33761direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4672
C3×#22352
C3×#22192
C3×#21882
C3×#21122
#337613