Finite Group Databasebeta

← C3×#13457 #10632064

Subgroups — C3×#13457 #10632064

Order 768.

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

normal subgroups only

This group has 356 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
1C2×C3×#498C211yesmaximal, characteristic
2C2×C3×#365C212yesmaximal
3C3×#1945C211yesmaximal, characteristic
4C3×#1386C212yesmaximal
5C3×#1376C211yesmaximal, characteristic
6#13457C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×C4⋊Q8—21no—
8C2×C3×M32C2212yes—
9C3×#498—24no—
10C2×C3×#108C2211yescharacteristic
11C3×#427C2211yescharacteristic
12C2×C3×#106—21no—
13C2×C3×#106C2212yes—
14C3×#424—21no—
15C2×C3×C4×C8C2211yescharacteristic
16C3×#365—24no—
17C3×#339—21no—
18C2×#498C2×C311yescharacteristic
19C2×#365C2×C312yes—
20#1945C2×C311yescharacteristic
21#1386C2×C312yes—
22#1376C2×C311yescharacteristic
23C22×C3×Q8—41no—
24C2×C3×M16—21no—
25C22×C3×C8D411yescharacteristic
26C22×C3×C8C2311yescharacteristic
27C3×C4⋊Q8—24no—
28C3×C4⋊Q8—42no—
29C3×#117—21no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—41no—
32C2×C3×C42D411yescharacteristic
33C3×M32—22no—
34C3×M32Q814yescomplemented
35C2×C3×C16—22no—
36C3×#108—22no—
37C3×#107Q812yescomplemented
38C3×#106—24no—
39C3×#106—22no—
40C3×#106—21no—
41C3×#106D414yes—
42C3×C4⋊C8—42no—
43C3×C4×C8—22no—
44C2×C4⋊Q8—21no—
45C2×M32C22×C312yes—
46#498—24no—
47C2×#108C22×C311yescharacteristic
48#427C22×C311yescharacteristic
49C2×#106—21no—
50C2×#106C22×C312yes—
51#424—21no—
52C2×C4×C8C22×C311yescharacteristic
53#365—24no—
54#339—21no—
55C2×C3×Q8—44no—
56C2×C3×Q8—84no—
57C22×C3×C4—21no—
58C22×C3×C4—42no—
59C22×C3×C4C2×D411yescharacteristic
60C3×M16—41no—
61C3×M16—41no—
62C2×C3×C8—21no—
63C2×C3×C8—21no—
64C2×C3×C8—41no—
65C2×C3×C8SD1612yes—
66C2×C3×C8C2×D412yes—
67C2×C3×C8C2×Q812yes—
68C2×C3×C8C2×Q811yescharacteristic
69C2×C3×C8#4111yescharacteristic
70C3×C4⋊C4—24no—
71C3×C4⋊C4—22no—
72C3×C4⋊C4—44no—
73C3×C4⋊C4—42no—
74C3×C22⋊C4—41no—
75C3×C42—21no—
76C3×C42—41no—
77C3×C42SD1612yes—
78C3×C16—24no—
79C22×Q8—41no—
80C2×M16—21no—
81C22×C8C3×D411yescharacteristic
82C22×C8C23×C311yescharacteristic
83C4⋊Q8—24no—
84C4⋊Q8—42no—
85#117—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—41no—
88C2×C42C3×D411yescharacteristic
89M32—22no—
90M32C3×Q814yescomplemented
91C2×C16—22no—
92#108—22no—
93#107C3×Q812yescomplemented
94#106—24no—
95#106—22no—
96#106—21no—
97#106C3×D414yes—
98C4⋊C8—42no—
99C4×C8—22no—
100C23×C3AGL(1,ℤ/8)11yescharacteristic
101C3×Q8—88no—
102C2×C3×C4—24no—
103C2×C3×C4—22no—
104C2×C3×C4—44no—
105C2×C3×C4—42no—
106C2×C3×C4—84no—
107C2×C3×C4#12011yescharacteristic
108C2×C3×C4C22⋊Q811yescharacteristic
109C2×C3×C4C4⋊Q811yescharacteristic
110C2×C3×C4C2×SD1611yescharacteristic
111C2×C3×C4C2×SD1611yescharacteristic
112C2×C3×C4#13711yescharacteristic
113C3×C8—22no—
114C3×C8—81no—
115C3×C8C22⋊Q812yes—
116C3×C8C4⋊Q812yes—
117C2×Q8—44no—
118C2×Q8—84no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4C2×C3×D411yescharacteristic
122M16—41no—
123M16—41no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—41no—
127C2×C8C3×SD1612yes—
128C2×C8C2×C3×D412yes—
129C2×C8C2×C3×Q812yes—
130C2×C8C2×C3×Q811yescharacteristic
131C2×C8C3×#4111yescharacteristic
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—44no—
135C4⋊C4—42no—
136C22⋊C4—41no—
137C42—21no—
138C42—41no—
139C42C3×SD1612yes—
140C16—24no—
141C22×C3—22no—
142C22×C3#45011yescharacteristic
143C22×C3#47611yescharacteristic
144C22×C3#50111yescharacteristic
145C3×C4—24no—
146C3×C4—84no—
147C3×C4—82no—
148C3×C4#39511yescharacteristic
149C3×C4#44811yescharacteristic
150C3×C4#47511yescharacteristic
151C3×C4#49811yescharacteristic
152C23C3×AGL(1,ℤ/8)11yescharacteristic
153Q8—88no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—44no—
157C2×C4—42no—
158C2×C4—84no—
159C2×C4C3×#12011yescharacteristic
160C2×C4C3×C22⋊Q811yescharacteristic
161C2×C4C3×C4⋊Q811yescharacteristic
162C2×C4C2×C3×SD1611yescharacteristic
163C2×C4C2×C3×SD1611yescharacteristic
164C2×C4C3×#13711yescharacteristic
165C8—22no—
166C8—81no—
167C8C3×C22⋊Q812yes—
168C8C3×C4⋊Q812yes—
169C2×C3—21no—
170C2×C3—21no—
171C2×C3#205811yescharacteristic
172C2×C3#223012yes—
173C22—22no—
174C22C3×#45011yescharacteristic
175C22C3×#47611yescharacteristic
176C22C3×#50111yescharacteristic
177C4—24no—
178C4—84no—
179C4—82no—
180C4C3×#39511yescharacteristic
181C4C3×#44811yescharacteristic
182C4C3×#47511yescharacteristic
183C4C3×#49811yescharacteristic
184C3#1345711yescharacteristic, direct product factor, Sylow
185C2—21no—
186C2—21no—
187C2C3×#205811yescharacteristic
188C2C3×#223012yes—
subgroupG/H
center, socleC22×C3#476
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC23×C3AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G

Complements

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

NH ≅ G/Ntype
6. #13457C3direct (N × H)
34. C3×M32Q8semidirect
37. C3×#107Q8semidirect
90. M32C3×Q8semidirect
93. #107C3×Q8semidirect
184. C3#13457direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4982
C2×C3×#3652
C3×#19452
C3×#13862
C3×#13762
#134573