Finite Group Databasebeta

← C3×#10216 #10628823

Subgroups — C3×#10216 #10628823

Order 768.

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

normal subgroups only

This group has 468 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×#3255C211yesmaximal, characteristic, complemented
2C3×#3235C212yesmaximal
3C3×#1963C211yesmaximal, characteristic
4C3×#1506C212yesmaximal, complemented
5C3×#1307C211yesmaximal, characteristic, complemented
6#10216C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#533C2212yes—
8C3×#532C2211yescharacteristic
9C3×#501—22no—
10C3×#490—22no—
11C3×#482—24no—
12C3×#476—22no—
13C3×#475—22no—
14C3×#474—22no—
15C3×#470—22no—
16C3×#462—22no—
17C3×#432C2212yes—
18C3×#423C2211yescharacteristic
19C2×C3×#104—22no—
20C3×#376C2211yescharacteristic
21C3×#328—24no—
22#3255C2×C311yescharacteristic, complemented
23#3235C2×C312yes—
24#1963C2×C311yescharacteristic
25#1506C2×C312yescomplemented
26#1307C2×C311yescharacteristic, complemented
27C2×C3×#41—21no—
28C2×C3×Q16—22no—
29C2×C3×M16D412yes—
30C3×C4⋊Q8D412yes—
31C3×C4⋊Q8D412yes—
32C3×C4⋊Q8D411yescharacteristic
33C3×#127D411yescharacteristic
34C3×#125—22no—
35C3×#124—21no—
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×#121—22no—
39C3×C4×Q8—22no—
40C3×C4×Q8—22no—
41C3×C4×D4—22no—
42C3×#117—22no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45C3×#107—24no—
46C3×#106—22no—
47C3×#106—24no—
48C3×C4⋊C8—22no—
49C3×C4⋊C8—22no—
50C3×#104—44no—
51C3×#103—24no—
52C3×#102—24no—
53C3×#102—24no—
54C3×C22⋊C8D414yes—
55C3×#97—22no—
56C3×#95—22no—
57#533C22×C312yes—
58#532C22×C311yescharacteristic
59#501—22no—
60#490—22no—
61#482—24no—
62#476—22no—
63#475—22no—
64#474—22no—
65#470—22no—
66#462—22no—
67#432C22×C312yes—
68#423C22×C311yescharacteristic
69C2×#104—22no—
70#376C22×C311yescharacteristic
71#328—24no—
72C3×#41—42no—
73C3×#41—42no—
74C2×C3×Q8—22no—
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—41no—
80C22×C3×C4—41no—
81C22×C3×C4—41no—
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4C2×D412yes—
84C3×Q16—82no—
85C3×M16—42no—
86C2×C3×C8—24no—
87C2×C3×C8—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—42no—
94C3×C4⋊C4—42no—
95C3×C22⋊C4—42no—
96C3×C22⋊C4—42no—
97C3×C42—22no—
98C3×C42—41no—
99C3×C42—42no—
100C3×C42C2×D411yescharacteristic
101C3×C42C2×D412yes—
102C3×C42#4111yescharacteristic
103C2×#41—21no—
104C2×Q16—22no—
105C2×M16C3×D412yes—
106C4⋊Q8C3×D411yescharacteristic
107C4⋊Q8C3×D412yes—
108C4⋊Q8C3×D412yes—
109#127C3×D411yescharacteristic
110#125—22no—
111#124—21no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114#121—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×D4—22no—
118#117—22no—
119C2×C42—21no—
120C2×C42C23×C311yescharacteristic
121#107—24no—
122#106—22no—
123#106—24no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126#104—44no—
127#103—24no—
128#102—24no—
129#102—24no—
130C22⋊C8C3×D414yes—
131#97—22no—
132#95—22no—
133C23×C3—41no—
134C23×C3#12711yescharacteristic
135C3×Q8—42no—
136C3×Q8—42no—
137C3×Q8—42no—
138C3×Q8—82no—
139C3×D4—44no—
140C3×D4—42no—
141C3×D4—82no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—82no—
150C2×C3×C4—81no—
151C2×C3×C4—81no—
152C2×C3×C4#12012yes—
153C2×C3×C4#12011yescharacteristic
154C2×C3×C4#12112yes—
155C2×C3×C4#12111yescharacteristic
156C3×C8—44no—
157C3×C8—42no—
158#41—42no—
159#41—42no—
160C2×Q8—22no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4C2×C3×D411yescharacteristic
169C22×C4C2×C3×D412yes—
170Q16—82no—
171M16—42no—
172C2×C8—24no—
173C2×C8—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C42—22no—
184C42—41no—
185C42—42no—
186C42C2×C3×D411yescharacteristic
187C42C2×C3×D412yes—
188C42C3×#4111yescharacteristic
189C22×C3—21no—
190C22×C3—22no—
191C22×C3—81no—
192C22×C3—82no—
193C22×C3#39211yescharacteristic
194C3×C4—21no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—82no—
198C3×C4—81no—
199C3×C4—82no—
200C3×C4—82no—
201C3×C4#45311yescharacteristic
202C3×C4#45411yescharacteristic
203C23—41no—
204C23C3×#12711yescharacteristic
205Q8—42no—
206Q8—42no—
207Q8—42no—
208Q8—82no—
209D4—42no—
210D4—44no—
211D4—82no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4C3×#12012yes—
223C2×C4C3×#12011yescharacteristic
224C2×C4C3×#12112yes—
225C2×C4C3×#12111yescharacteristic
226C8—44no—
227C8—42no—
228C2×C3—41no—
229C2×C3—161no—
230C2×C3#200611yescharacteristic
231C2×C3#200912yes—
232C22—21no—
233C22—22no—
234C22—81no—
235C22—82no—
236C22C3×#39211yescharacteristic
237C4—21no—
238C4—22no—
239C4—22no—
240C4—82no—
241C4—81no—
242C4—82no—
243C4—82no—
244C4C3×#45311yescharacteristic
245C4C3×#45411yescharacteristic
246C3#1021611yescharacteristic, direct product factor, Sylow
247C2—41no—
248C2—161no—
249C2C3×#200611yescharacteristic
250C2C3×#200912yes—
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#532C22
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
1. C3×#3255C2semidirect
4. C3×#1506C2semidirect
5. C3×#1307C2semidirect
6. #10216C3direct (N × H)
22. #3255C2×C3semidirect
25. #1506C2×C3semidirect
26. #1307C2×C3semidirect
246. C3#10216direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32552
C3×#32352
C3×#19632
C3×#15062
C3×#13072
#102163