Finite Group Databasebeta

← C3×#29658 #10648265

Subgroups — C3×#29658 #10648265

Order 768.

1086 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 238 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×#3458C211yesmaximal, characteristic, complemented
2C3×#2482C213yesmaximal
3C3×#2062C214yesmaximal, complemented
4C3×#2029C214yesmaximal
5C3×#1820C213yesmaximal, complemented
6#29658C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#544—23no—
8C3×#544—23no—
9C3×#543C2214yes—
10C3×#533—26no—
11C2×C3×#117C2213yes—
12C22×C3×C22⋊C4C2211yescharacteristic
13C3×#399—23no—
14C3×#397—23no—
15C3×#388—26no—
16C3×#385C2216yes—
17C3×#381C2213yes—
18C3×#379C2216yes—
19C3×#342C22112yes—
20#3458C2×C311yescharacteristic, complemented
21#2482C2×C313yes—
22#2062C2×C314yescomplemented
23#2029C2×C314yes—
24#1820C2×C313yescomplemented
25C25×C3C2311yescharacteristic
26C23×C3×C4C2311yescharacteristic
27C23×C3×C4C2311yescharacteristic
28C3×C4⋊Q8—26no—
29C3×C4⋊Q8—26no—
30C3×#125—212no—
31C3×C22⋊Q8—212no—
32C3×C22⋊Q8—212no—
33C3×C4×Q8—24no—
34C3×C4×Q8—24no—
35C3×#117—26no—
36C3×#117—26no—
37C2×C3×C4⋊C4—26no—
38C2×C3×C4⋊C4D413yes—
39C2×C3×C4⋊C4D413yes—
40C2×C3×C22⋊C4—26no—
41C2×C3×C22⋊C4—23no—
42C2×C3×C22⋊C4—23no—
43C2×C3×C22⋊C4—26no—
44C2×C3×C22⋊C4C2316yes—
45C2×C3×C22⋊C4C2316yes—
46C2×C3×C42D416yes—
47C3×#95—23no—
48C3×#95—23no—
49C3×#95—26no—
50#544—23no—
51#544—23no—
52#543C22×C314yes—
53#533—26no—
54C2×#117C22×C313yes—
55C22×C22⋊C4C22×C311yescharacteristic
56#399—23no—
57#397—23no—
58#388—26no—
59#385C22×C316yes—
60#381C22×C313yes—
61#379C22×C316yes—
62#342C22×C3112yes—
63C24×C3—23no—
64C24×C3—24no—
65C24×C3—42no—
66C24×C3—81no—
67C24×C3C2411yescharacteristic
68C2×C3×Q8—24no—
69C2×C3×Q8—24no—
70C22×C3×C4—24no—
71C22×C3×C4—26no—
72C22×C3×C4—23no—
73C22×C3×C4—81no—
74C22×C3×C4C2×D413yes—
75C22×C3×C4C2×D413yes—
76C22×C3×C4C2×D416yes—
77C22×C3×C4C2×D416yes—
78C3×C4⋊C4—212no—
79C3×C4⋊C4—212no—
80C3×C4⋊C4—26no—
81C3×C4⋊C4—26no—
82C3×C4⋊C4—86no—
83C3×C22⋊C4—212no—
84C3×C22⋊C4—212no—
85C3×C22⋊C4—412no—
86C3×C22⋊C4—41no—
87C3×C22⋊C4—43no—
88C3×C22⋊C4—86no—
89C3×C42—26no—
90C3×C42—26no—
91C25C23×C311yescharacteristic
92C23×C4C23×C311yescharacteristic
93C23×C4C23×C311yescharacteristic
94C4⋊Q8—26no—
95C4⋊Q8—26no—
96#125—212no—
97C22⋊Q8—212no—
98C22⋊Q8—212no—
99C4×Q8—24no—
100C4×Q8—24no—
101#117—26no—
102#117—26no—
103C2×C4⋊C4—26no—
104C2×C4⋊C4C3×D413yes—
105C2×C4⋊C4C3×D413yes—
106C2×C22⋊C4—23no—
107C2×C22⋊C4—23no—
108C2×C22⋊C4—26no—
109C2×C22⋊C4—26no—
110C2×C22⋊C4C23×C316yes—
111C2×C22⋊C4C23×C316yes—
112C2×C42C3×D416yes—
113#95—23no—
114#95—23no—
115#95—26no—
116C23×C3—26no—
117C23×C3—41no—
118C23×C3—41no—
119C23×C3—46no—
120C23×C3—43no—
121C23×C3—41no—
122C23×C3—46no—
123C23×C3—88no—
124C23×C3C22×D413yes—
125C23×C32−1+414yes—
126C3×Q8—84no—
127C3×Q8—84no—
128C2×C3×C4—26no—
129C2×C3×C4—26no—
130C2×C3×C4—26no—
131C2×C3×C4—26no—
132C2×C3×C4—46no—
133C2×C3×C4—46no—
134C2×C3×C4—46no—
135C2×C3×C4—46no—
136C2×C3×C4—46no—
137C2×C3×C4—41no—
138C2×C3×C4—46no—
139C2×C3×C4#12011yescharacteristic
140C2×C3×C4#12013yes—
141C24—23no—
142C24—24no—
143C24—42no—
144C24—81no—
145C24C24×C311yescharacteristic
146C2×Q8—24no—
147C2×Q8—24no—
148C22×C4—24no—
149C22×C4—26no—
150C22×C4—23no—
151C22×C4—81no—
152C22×C4C2×C3×D413yes—
153C22×C4C2×C3×D413yes—
154C22×C4C2×C3×D416yes—
155C22×C4C2×C3×D416yes—
156C4⋊C4—212no—
157C4⋊C4—212no—
158C4⋊C4—26no—
159C4⋊C4—26no—
160C4⋊C4—86no—
161C22⋊C4—212no—
162C22⋊C4—212no—
163C22⋊C4—412no—
164C22⋊C4—41no—
165C22⋊C4—43no—
166C22⋊C4—86no—
167C42—26no—
168C42—26no—
169C22×C3—23no—
170C22×C3—23no—
171C22×C3—44no—
172C22×C3—412no—
173C22×C3—42no—
174C22×C3—86no—
175C22×C3—81no—
176C22×C3—81no—
177C22×C3C2×#12011yescharacteristic
178C22×C3#53616yes—
179C3×C4—21no—
180C3×C4—23no—
181C3×C4—46no—
182C3×C4—46no—
183C3×C4—86no—
184C3×C4—81no—
185C23—26no—
186C23—41no—
187C23—41no—
188C23—46no—
189C23—43no—
190C23—41no—
191C23—46no—
192C23—88no—
193C23C22×C3×D413yes—
194C23C3×2−1+414yes—
195Q8—84no—
196Q8—84no—
197C2×C4—26no—
198C2×C4—26no—
199C2×C4—26no—
200C2×C4—26no—
201C2×C4—46no—
202C2×C4—46no—
203C2×C4—46no—
204C2×C4—46no—
205C2×C4—46no—
206C2×C4—41no—
207C2×C4—46no—
208C2×C4C3×#12011yescharacteristic
209C2×C4C3×#12013yes—
210C2×C3—26no—
211C2×C3—82no—
212C2×C3#261411yescharacteristic
213C2×C3#302511yescharacteristic
214C2×C3#302711yescharacteristic
215C22—23no—
216C22—23no—
217C22—44no—
218C22—412no—
219C22—42no—
220C22—86no—
221C22—81no—
222C22—81no—
223C22C2×C3×#12011yescharacteristic
224C22C3×#53616yes—
225C4—21no—
226C4—23no—
227C4—46no—
228C4—46no—
229C4—86no—
230C4—81no—
231C3#2965811yescharacteristic, direct product factor, Sylow
232C2—26no—
233C2—82no—
234C2C3×#261411yescharacteristic
235C2C3×#302511yescharacteristic
236C2C3×#302711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3458C2semidirect
3. C3×#2062C2semidirect
5. C3×#1820C2semidirect
6. #29658C3direct (N × H)
20. #3458C2×C3semidirect
22. #2062C2×C3semidirect
24. #1820C2×C3semidirect
231. C3#29658direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34582
C3×#24822
C3×#20622
C3×#20292
C3×#18202
#296583