Finite Group Databasebeta

← C3×#10213 #10628820

Subgroups — C3×#10213 #10628820

Order 768.

498 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3253C211yesmaximal, characteristic
2C3×#3216C212yesmaximal, complemented
3C3×#1995C211yesmaximal, characteristic, complemented
4C3×#1505C212yesmaximal, complemented
5C3×#1307C211yesmaximal, characteristic, complemented
6#10213C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#533C2211yescharacteristic
8C3×#532C2212yes—
9C3×#497—22no—
10C3×#489—22no—
11C3×#476—22no—
12C3×#469—22no—
13C3×#462—22no—
14C3×#460—22no—
15C3×#459—22no—
16C3×#452—24no—
17C3×#432C2212yes—
18C3×#423C2211yescharacteristic
19C3×#376C2211yescharacteristic
20C3×#339—22no—
21C3×#327—24no—
22#3253C2×C311yescharacteristic
23#3216C2×C312yescomplemented
24#1995C2×C311yescharacteristic, complemented
25#1505C2×C312yescomplemented
26#1307C2×C311yescharacteristic, complemented
27C2×C3×#41—22no—
28C2×C3×Q16—24no—
29C2×C3×SD16—24no—
30C2×C3×D8—22no—
31C2×C3×M16D412yes—
32C3×C4⋊Q8D412yes—
33C3×C4⋊Q8D412yes—
34C3×#127D412yes—
35C3×#125—21no—
36C3×#124—22no—
37C3×C22⋊Q8—22no—
38C3×#121—22no—
39C3×#121—22no—
40C3×C4×Q8—22no—
41C3×C4×D4—22no—
42C3×C4×D4—22no—
43C3×#117—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C3×#106—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×#103—24no—
50C3×#102—24no—
51C3×#102—24no—
52C3×C22⋊C8D414yes—
53C3×#97—22no—
54C3×#95—22no—
55#533C22×C311yescharacteristic
56#532C22×C312yes—
57#497—22no—
58#489—22no—
59#476—22no—
60#469—22no—
61#462—22no—
62#460—22no—
63#459—22no—
64#452—24no—
65#432C22×C312yes—
66#423C22×C311yescharacteristic
67#376C22×C311yescharacteristic
68#339—22no—
69#327—24no—
70C3×#41—84no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—42no—
77C22×C3×C4—41no—
78C22×C3×C4—42no—
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D412yes—
81C3×Q16—84no—
82C3×SD16—84no—
83C3×D8—82no—
84C3×M16—42no—
85C2×C3×C8—24no—
86C2×C3×C8—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—42no—
92C3×C22⋊C4—41no—
93C3×C22⋊C4—42no—
94C3×C22⋊C4—42no—
95C3×C42—22no—
96C3×C42—41no—
97C3×C42—41no—
98C3×C42C2×D411yescharacteristic
99C3×C42C2×D412yes—
100C3×C42#4111yescharacteristic
101C2×#41—22no—
102C2×Q16—24no—
103C2×SD16—24no—
104C2×D8—22no—
105C2×M16C3×D412yes—
106C4⋊Q8C3×D412yes—
107C4⋊Q8C3×D412yes—
108#127C3×D412yes—
109#125—21no—
110#124—22no—
111C22⋊Q8—22no—
112#121—22no—
113#121—22no—
114C4×Q8—22no—
115C4×D4—22no—
116C4×D4—22no—
117#117—21no—
118C2×C42—21no—
119C2×C42C23×C311yescharacteristic
120#106—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123#103—24no—
124#102—24no—
125#102—24no—
126C22⋊C8C3×D414yes—
127#97—22no—
128#95—22no—
129C23×C3—42no—
130C23×C3#12711yescharacteristic
131C3×Q8—42no—
132C3×Q8—42no—
133C3×Q8—44no—
134C3×D4—44no—
135C3×D4—42no—
136C3×D4—42no—
137C3×D4—82no—
138C3×D4—84no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—84no—
146C2×C3×C4—82no—
147C2×C3×C4#12012yes—
148C2×C3×C4#12011yescharacteristic
149C2×C3×C4#12111yescharacteristic
150C2×C3×C4#12112yes—
151C3×C8—44no—
152C3×C8—42no—
153#41—84no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—42no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4C2×C3×D411yescharacteristic
163C22×C4C2×C3×D412yes—
164Q16—84no—
165SD16—84no—
166D8—82no—
167M16—42no—
168C2×C8—24no—
169C2×C8—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C22⋊C4—41no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—22no—
179C42—41no—
180C42—41no—
181C42C2×C3×D411yescharacteristic
182C42C2×C3×D412yes—
183C42C3×#4111yescharacteristic
184C22×C3—21no—
185C22×C3—22no—
186C22×C3—82no—
187C22×C3—82no—
188C22×C3—82no—
189C22×C3#39211yescharacteristic
190C3×C4—21no—
191C3×C4—22no—
192C3×C4—22no—
193C3×C4—82no—
194C3×C4—82no—
195C3×C4—82no—
196C3×C4#45511yescharacteristic
197C3×C4#45611yescharacteristic
198C23—42no—
199C23C3×#12711yescharacteristic
200Q8—42no—
201Q8—42no—
202Q8—44no—
203D4—44no—
204D4—42no—
205D4—42no—
206D4—82no—
207D4—84no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—84no—
215C2×C4—82no—
216C2×C4C3×#12012yes—
217C2×C4C3×#12011yescharacteristic
218C2×C4C3×#12111yescharacteristic
219C2×C4C3×#12112yes—
220C8—44no—
221C8—42no—
222C2×C3—41no—
223C2×C3—162no—
224C2×C3#200711yescharacteristic
225C2×C3#200812yes—
226C22—21no—
227C22—22no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22C3×#39211yescharacteristic
232C4—21no—
233C4—22no—
234C4—22no—
235C4—82no—
236C4—82no—
237C4—82no—
238C4C3×#45511yescharacteristic
239C4C3×#45611yescharacteristic
240C3#1021311yescharacteristic, direct product factor, Sylow
241C2—41no—
242C2—162no—
243C2C3×#200711yescharacteristic
244C2C3×#200812yes—
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#3253C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3216C2semidirect
3. C3×#1995C2semidirect
4. C3×#1505C2semidirect
5. C3×#1307C2semidirect
6. #10213C3direct (N × H)
23. #3216C2×C3semidirect
24. #1995C2×C3semidirect
25. #1505C2×C3semidirect
26. #1307C2×C3semidirect
240. C3#10213direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32532
C3×#32162
C3×#19952
C3×#15052
C3×#13072
#102133