Finite Group Databasebeta

← C3×#13078 #10631685

Subgroups — C3×#13078 #10631685

Order 768.

454 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3407C211yesmaximal, characteristic
2C3×#3059C211yesmaximal, characteristic
3C2×C3×#362C212yesmaximal, complemented
4C3×#1813C211yesmaximal, characteristic
5C3×#1389C212yesmaximal, complemented
6#13078C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q16C2211yescharacteristic
8C2×C3×#131C2211yescharacteristic
9C3×#536—21no—
10C3×#504—24no—
11C2×C3×M32—22no—
12C2×C3×M32C2212yes—
13C3×#470C414yes—
14C3×#467—22no—
15C2×C3×#108C2212yes—
16C3×#427C2211yescharacteristic
17C3×#419—21no—
18C3×#413—21no—
19C3×#362—24no—
20C3×#338—21no—
21#3407C2×C311yescharacteristic
22#3059C2×C311yescharacteristic
23C2×#362C2×C312yescomplemented
24#1813C2×C311yescharacteristic
25#1389C2×C312yescomplemented
26C2×C3×#41D411yescharacteristic
27C22×C3×Q8—21no—
28C2×C3×Q16—22no—
29C2×C3×Q16—21no—
30C2×C3×Q16—22no—
31C2×C3×Q16C2×C412yes—
32C3×#131—22no—
33C3×#131—22no—
34C2×C3×M16—22no—
35C2×C3×M16C2×C412yes—
36C2×C3×M16D411yescharacteristic
37C22×C3×C8D412yes—
38C22×C3×C8C2311yescharacteristic
39C3×C4⋊Q8—41no—
40C3×#124—41no—
41C3×#123—42no—
42C3×C22⋊Q8—24no—
43C3×#117—21no—
44C3×M32—24no—
45C3×M32—22no—
46C3×M32D414yes—
47C2×C3×C16—24no—
48C2×C3×C16—22no—
49C3×#108—22no—
50C3×#108—22no—
51C3×#107—21no—
52C3×#106C2×C412yes—
53C3×#103—24no—
54C3×#101—41no—
55C3×#100—41no—
56C22×Q16C22×C311yescharacteristic
57C2×#131C22×C311yescharacteristic
58#536—21no—
59#504—24no—
60C2×M32—22no—
61C2×M32C22×C312yes—
62#470C3×C414yes—
63#467—22no—
64C2×#108C22×C312yes—
65#427C22×C311yescharacteristic
66#419—21no—
67#413—21no—
68#362—24no—
69#338—21no—
70C3×#41—22no—
71C3×#41—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—41no—
74C2×C3×Q8—42no—
75C2×C3×Q8—42no—
76C2×C3×Q8D811yescharacteristic
77C2×C3×D4D811yescharacteristic
78C2×C3×D4SD1612yes—
79C22×C3×C4—41no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C2×D411yescharacteristic
82C3×Q16—24no—
83C3×Q16—41no—
84C3×Q16—41no—
85C3×M16—22no—
86C3×M16—24no—
87C3×M16—42no—
88C3×M16—42no—
89C2×C3×C8—22no—
90C2×C3×C8—22no—
91C2×C3×C8—42no—
92C2×C3×C8C22⋊C412yes—
93C2×C3×C8C22×C411yescharacteristic
94C2×C3×C8C2×D412yes—
95C2×C3×C8C2×D411yescharacteristic
96C2×C3×C8#4112yes—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—44no—
99C3×C22⋊C4—44no—
100C3×C22⋊C4—41no—
101C3×C42—41no—
102C3×C16—24no—
103C2×#41C3×D411yescharacteristic
104C22×Q8—21no—
105C2×Q16—22no—
106C2×Q16—21no—
107C2×Q16—22no—
108C2×Q16C2×C3×C412yes—
109#131—22no—
110#131—22no—
111C2×M16—22no—
112C2×M16C2×C3×C412yes—
113C2×M16C3×D411yescharacteristic
114C22×C8C3×D412yes—
115C22×C8C23×C311yescharacteristic
116C4⋊Q8—41no—
117#124—41no—
118#123—42no—
119C22⋊Q8—24no—
120#117—21no—
121M32—24no—
122M32—22no—
123M32C3×D414yes—
124C2×C16—24no—
125C2×C16—22no—
126#108—22no—
127#108—22no—
128#107—21no—
129#106C2×C3×C412yes—
130#103—24no—
131#101—41no—
132#100—41no—
133C23×C3#10212yes—
134C23×C3C2×SD1611yescharacteristic
135C3×Q8—22no—
136C3×Q8—42no—
137C3×Q8—42no—
138C3×Q8—82no—
139C3×D4—24no—
140C3×D4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—82no—
146C2×C3×C4#10212yes—
147C2×C3×C4C2×C22⋊C411yescharacteristic
148C2×C3×C4#12112yes—
149C2×C3×C4C2×D811yescharacteristic
150C2×C3×C4#13512yes—
151C3×C8—22no—
152C3×C8—82no—
153C3×C8C4×D412yes—
154C3×C8#12011yescharacteristic
155C3×C8#12311yescharacteristic
156#41—22no—
157#41—22no—
158C2×Q8—22no—
159C2×Q8—41no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8C3×D811yescharacteristic
163C2×D4C3×D811yescharacteristic
164C2×D4C3×SD1612yes—
165C22×C4—41no—
166C22×C4C3×C22⋊C412yes—
167C22×C4C2×C3×D411yescharacteristic
168Q16—24no—
169Q16—41no—
170Q16—41no—
171M16—22no—
172M16—24no—
173M16—42no—
174M16—42no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—42no—
178C2×C8C3×C22⋊C412yes—
179C2×C8C22×C3×C411yescharacteristic
180C2×C8C2×C3×D412yes—
181C2×C8C2×C3×D411yescharacteristic
182C2×C8C3×#4112yes—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C22⋊C4—44no—
186C22⋊C4—41no—
187C42—41no—
188C16—24no—
189C22×C3—22no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3C2×#10211yescharacteristic
193C22×C3#46512yes—
194C3×C4—22no—
195C3×C4—82no—
196C3×C4—82no—
197C3×C4#38611yescharacteristic
198C3×C4#41611yescharacteristic
199C3×C4#46612yes—
200C23C3×#10212yes—
201C23C2×C3×SD1611yescharacteristic
202Q8—22no—
203Q8—42no—
204Q8—42no—
205Q8—82no—
206D4—24no—
207D4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4C3×#10212yes—
214C2×C4C2×C3×C22⋊C411yescharacteristic
215C2×C4C3×#12112yes—
216C2×C4C2×C3×D811yescharacteristic
217C2×C4C3×#13512yes—
218C8—22no—
219C8—82no—
220C8C3×C4×D412yes—
221C8C3×#12011yescharacteristic
222C8C3×#12311yescharacteristic
223C2×C3—21no—
224C2×C3—22no—
225C2×C3—42no—
226C2×C3#189311yescharacteristic
227C22—22no—
228C22—41no—
229C22—42no—
230C22C2×C3×#10211yescharacteristic
231C22C3×#46512yes—
232C4—22no—
233C4—82no—
234C4—82no—
235C4C3×#38611yescharacteristic
236C4C3×#41611yescharacteristic
237C4C3×#46612yes—
238C3#1307811yescharacteristic, direct product factor, Sylow
239C2—21no—
240C2—22no—
241C2—42no—
242C2C3×#189311yescharacteristic
subgroupG/H
center, socleC2×C3#1893
commutatorC2×C8C22×C3×C4
Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#41D4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#362C2semidirect
5. C3×#1389C2semidirect
6. #13078C3direct (N × H)
23. C2×#362C2×C3semidirect
25. #1389C2×C3semidirect
238. C3#13078direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34072
C3×#30592
C2×C3×#3622
C3×#18132
C3×#13892
#130783