Finite Group Databasebeta

← C3×#10014 #10628621

Subgroups — C3×#10014 #10628621

Order 768.

566 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3103C212yesmaximal
2C3×#2399C211yesmaximal, characteristic, complemented
3C3×#1963C211yesmaximal, characteristic
4C3×#1487C212yesmaximal, complemented
5C3×#1274C211yesmaximal, characteristic, complemented
6#10014C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#532C2211yescharacteristic
8C2×C3×C4×Q8C2212yes—
9C3×#481—22no—
10C3×#475—22no—
11C3×#460—22no—
12C3×#448—22no—
13C3×C4×SD16—24no—
14C3×#431C2212yes—
15C3×#423C2211yescharacteristic
16C2×C3×#104—22no—
17C3×#398—22no—
18C3×#395—22no—
19C3×C4×C4⋊C4—22no—
20C3×#376C2211yescharacteristic
21C3×Q8⋊C8—24no—
22#3103C2×C312yes—
23#2399C2×C311yescharacteristic, complemented
24#1963C2×C311yescharacteristic
25#1487C2×C312yescomplemented
26#1274C2×C311yescharacteristic, complemented
27C2×C3×#41—21no—
28C22×C3×Q8D412yes—
29C2×C3×SD16—24no—
30C2×C3×M16—21no—
31C3×C4⋊Q8D411yescharacteristic
32C3×#127D411yescharacteristic
33C3×#124—21no—
34C3×#121—22no—
35C3×C4×Q8—22no—
36C3×C4×Q8—22no—
37C3×C4×Q8—22no—
38C3×C4×Q8—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4—24no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C42—22no—
44C2×C3×C42D412yes—
45C2×C3×C42C2311yescharacteristic
46C3×#106—24no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×#104—44no—
50C3×#103—24no—
51C3×#102—24no—
52C3×C22⋊C8—22no—
53C3×C4×C8D414yes—
54C3×#95—22no—
55C3×#95—22no—
56C3×#95—22no—
57#532C22×C311yescharacteristic
58C2×C4×Q8C22×C312yes—
59#481—22no—
60#475—22no—
61#460—22no—
62#448—22no—
63C4×SD16—24no—
64#431C22×C312yes—
65#423C22×C311yescharacteristic
66C2×#104—22no—
67#398—22no—
68#395—22no—
69C4×C4⋊C4—22no—
70#376C22×C311yescharacteristic
71Q8⋊C8—24no—
72C3×#41—42no—
73C3×#41—42no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8—21no—
77C2×C3×Q8—44no—
78C2×C3×D4—22no—
79C2×C3×D4—21no—
80C2×C3×D4—41no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—41no—
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C3×SD16—48no—
88C3×M16—42no—
89C2×C3×C8—24no—
90C2×C3×C8—41no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—48no—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—42no—
97C3×C22⋊C4—42no—
98C3×C42—22no—
99C3×C42—42no—
100C3×C42—42no—
101C3×C42SD1614yescomplemented
102C3×C42C2×D411yescharacteristic
103C3×C42C2×D412yes—
104C3×C42#4111yescharacteristic
105C2×#41—21no—
106C22×Q8C3×D412yes—
107C2×SD16—24no—
108C2×M16—21no—
109C4⋊Q8C3×D411yescharacteristic
110#127C3×D411yescharacteristic
111#124—21no—
112#121—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×D4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4C3×D412yes—
121C2×C42—22no—
122C2×C42C3×D412yes—
123C2×C42C23×C311yescharacteristic
124#106—24no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127#104—44no—
128#103—24no—
129#102—24no—
130C22⋊C8—22no—
131C4×C8C3×D414yes—
132#95—22no—
133#95—22no—
134#95—22no—
135C23×C3—41no—
136C23×C3#12011yescharacteristic
137C3×Q8—24no—
138C3×Q8—24no—
139C3×Q8—42no—
140C3×Q8—42no—
141C3×Q8—42no—
142C3×D4—44no—
143C3×D4—42no—
144C3×D4—82no—
145C2×C3×C4—24no—
146C2×C3×C4—24no—
147C2×C3×C4—42no—
148C2×C3×C4—44no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—44no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—81no—
155C2×C3×C4—81no—
156C2×C3×C4#12012yes—
157C2×C3×C4#12112yes—
158C2×C3×C4#12111yescharacteristic
159C2×C3×C4#12711yescharacteristic
160C2×C3×C4C2×SD1612yes—
161C2×C3×C4#13712yes—
162C3×C8—44no—
163C3×C8—81no—
164#41—42no—
165#41—42no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—21no—
169C2×Q8—44no—
170C2×D4—22no—
171C2×D4—21no—
172C2×D4—41no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4C2×C3×D412yes—
178C22×C4C2×C3×D411yescharacteristic
179SD16—48no—
180M16—42no—
181C2×C8—24no—
182C2×C8—41no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—48no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C22⋊C4—42no—
190C42—22no—
191C42—42no—
192C42—42no—
193C42C3×SD1614yescomplemented
194C42C2×C3×D411yescharacteristic
195C42C2×C3×D412yes—
196C42C3×#4111yescharacteristic
197C22×C3—22no—
198C22×C3—82no—
199C22×C3—81no—
200C22×C3#39211yescharacteristic
201C22×C3#44812yes—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—44no—
205C3×C4—44no—
206C3×C4—82no—
207C3×C4—81no—
208C3×C4#46112yes—
209C3×C4#49211yescharacteristic
210C3×C4#49711yescharacteristic
211C23—41no—
212C23C3×#12011yescharacteristic
213Q8—24no—
214Q8—24no—
215Q8—42no—
216Q8—42no—
217Q8—42no—
218D4—44no—
219D4—42no—
220D4—82no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4C3×#12012yes—
233C2×C4C3×#12112yes—
234C2×C4C3×#12111yescharacteristic
235C2×C4C3×#12711yescharacteristic
236C2×C4C2×C3×SD1612yes—
237C2×C4C3×#13712yes—
238C8—44no—
239C8—81no—
240C2×C3—22no—
241C2×C3—161no—
242C2×C3#201012yes—
243C2×C3#201311yescharacteristic
244C22—22no—
245C22—82no—
246C22—81no—
247C22C3×#39211yescharacteristic
248C22C3×#44812yes—
249C4—22no—
250C4—22no—
251C4—44no—
252C4—44no—
253C4—82no—
254C4—81no—
255C4C3×#46112yes—
256C4C3×#49211yescharacteristic
257C4C3×#49711yescharacteristic
258C3#1001411yescharacteristic, direct product factor, Sylow
259C2—22no—
260C2—161no—
261C2C3×#201012yes—
262C2C3×#201311yescharacteristic
subgroupG/H
centerC2×C3×C4#127
commutator, Frattini, cosocleC2×C42C23×C3
socleC22×C3#392
prime radicalC3×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2399C2semidirect
4. C3×#1487C2semidirect
5. C3×#1274C2semidirect
6. #10014C3direct (N × H)
23. #2399C2×C3semidirect
25. #1487C2×C3semidirect
26. #1274C2×C3semidirect
101. C3×C42SD16semidirect
193. C42C3×SD16semidirect
258. C3#10014direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31032
C3×#23992
C3×#19632
C3×#14872
C3×#12742
#100143