Finite Group Databasebeta

← C3×#10593776 #13173247

Subgroups — C3×#10593776 #13173247

Order 1728.

380 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 96 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×#64283C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4C3×#63771C212yesmaximal, complemented
5C3×#63770C212yesmaximal, complemented
6#10593776C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C3×#3792—21no—
9C2×C3×#622—21no—
10C3×#3788—21no—
11C3×#3788C2211yescharacteristic
12C3×#3788C2212yes—
13C3×#3788C2211yescharacteristic
14C3×#3787—21no—
15C33×M16C411yescharacteristic
16C3×#3748C411yescharacteristic
17C3×#3743C412yes—
18C3×C8×C3⋊D3C2212yes—
19#64283C2×C311yescharacteristic
20C2×#3788C2×C311yescharacteristic
21M16×C3⋊D3C2×C311yescharacteristic
22C3×D3×M16—62no—
23#63771C2×C312yescomplemented
24#63770C2×C312yescomplemented
25C22×C3×C3⋊D3Q811yescharacteristic
26C2×C33×C4C2×C411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C3×C4×C3⋊D3C2×C411yescharacteristic
29C3×C4×C3⋊D3D411yescharacteristic
30C3×C4×C3⋊D3Q811yescharacteristic
31C3×C4×C3⋊D3C2311yescharacteristic
32C3×#622—21no—
33C3×#622—21no—
34C3×#622—22no—
35C33×C8C2×C412yes—
36C3×C3⋊(C3⋊C8)C2×C412yes—
37C3×#430—91nomaximal
38C2×C4×C3⋊D3C22×C311yescharacteristic
39C2×C3×C4×D3—62no—
40#3792—21no—
41C2×#622—21no—
42#3788—21no—
43#3788C22×C311yescharacteristic
44#3788C22×C312yes—
45#3788C22×C311yescharacteristic
46#3787—21no—
47C32×M16—22no—
48C32×M16—42no—
49C32×M16C3×C411yescharacteristic
50#3748C3×C411yescharacteristic
51#3743C3×C412yes—
52C8×C3⋊D3C22×C312yes—
53C3×#208—62no—
54C3×ΓL(1,𝔽25)—64no—
55C3×D3×C8—64no—
56C22×C33C4⋊C411yescharacteristic
57C2×C3×C3⋊D3—41no—
58C2×C3×C3⋊D3C4⋊C411yescharacteristic
59C2×C3×C3⋊D3C2×Q811yescharacteristic
60C33×C4C4⋊C411yescharacteristic
61C33×C4C22×C411yescharacteristic
62C3×C3⋊Dic3C4⋊C411yescharacteristic
63C3×C3⋊Dic3C2×D411yescharacteristic
64C2×C3×M16—91no—
65C2×C3×M16—91no—
66C2×C3×M16—91no—
67D3×M16—62no—
68C3×#108—92no—
69C3×#108—92no—
70C22×C3⋊D3C3×Q811yescharacteristic
71C22×C3×D3—62no—
72C2×C32×C4—22no—
73C2×C32×C4—42no—
74C2×C32×C4C2×C3×C411yescharacteristic
75C2×C3⋊Dic3C3×D411yescharacteristic
76C4×C3⋊D3C2×C3×C411yescharacteristic
77C4×C3⋊D3C3×D411yescharacteristic
78C4×C3⋊D3C3×Q811yescharacteristic
79C4×C3⋊D3C23×C311yescharacteristic
80C2×C3×Dic3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85#622—21no—
86#622—21no—
87#622—22no—
88C32×C8—24no—
89C32×C8—44no—
90C32×C8C2×C3×C412yes—
91C3⋊(C3⋊C8)C2×C3×C412yes—
92C3×C3⋊C8—64no—
93#430—91noSylow
94C2×C33—21no—
95C2×C33C2×C4⋊C411yescharacteristic
96C3×C3⋊D3—21no—
97C3×C3⋊D3—21no—
98C22×C3×C4—91no—
99C2×C4×D3—62no—
100C3×M16—22no—
101C3×M16—42no—
102C3×M16—91no—
103C3×M16—92no—
104C3×M16—91no—
105C3×M16—92no—
106C3×M16—91no—
107C3×M16—181no—
108C3×M16—181no—
109C3×M16#15711yescharacteristic
110C2×C3×C8—92no—
111C2×C3×C8—181no—
112C2×C3×C8—181no—
113#208—62no—
114ΓL(1,𝔽25)—64no—
115D3×C8—64no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32C3×C4⋊C411yescharacteristic
119C2×C3⋊D3—41no—
120C2×C3⋊D3C3×C4⋊C411yescharacteristic
121C2×C3⋊D3C2×C3×Q811yescharacteristic
122C2×C3×D3—62no—
123C2×C3×D3—62no—
124C2×C3×D3—124no—
125C32×C4—22no—
126C32×C4—22no—
127C32×C4—42no—
128C32×C4—42no—
129C32×C4C3×C4⋊C411yescharacteristic
130C32×C4C22×C3×C411yescharacteristic
131C3⋊Dic3C3×C4⋊C411yescharacteristic
132C3⋊Dic3C2×C3×D411yescharacteristic
133C3×Dic3—62no—
134C3×Dic3—62no—
135C2×M16—91no—
136C2×M16—91no—
137C2×M16—91no—
138#108—92no—
139#108—92no—
140C33#43011yescharacteristic, complemented, Sylow
141C23×C3—91no—
142C22×D3—62no—
143C2×C3×C4—22no—
144C2×C3×C4—42no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4C2×#15711yescharacteristic
151C2×Dic3—62no—
152C4×D3—62no—
153C4×D3—62no—
154C4×D3—62no—
155C4×D3—62no—
156C3×C8—24no—
157C3×C8—44no—
158C3×C8—92no—
159C3×C8—181no—
160C3×C8—181no—
161C3×C8—182no—
162C3×C8C2×#15712yes—
163C3⋊C8—64no—
164C2×C32—22no—
165C2×C32—21no—
166C2×C32—42no—
167C2×C32—42no—
168C2×C32—82no—
169C2×C32C2×C3×C4⋊C411yescharacteristic
170C3⋊D3—21no—
171C3⋊D3—21no—
172C3×D3—64no—
173C3×D3—122no—
174C22×C4—91no—
175M16—91no—
176M16—92no—
177M16—91no—
178M16—92no—
179M16—91no—
180M16—181no—
181M16—181no—
182M16C3×#15711yescharacteristic
183C2×C8—92no—
184C2×C8—181no—
185C2×C8—181no—
186C22×C3—22no—
187C22×C3—42no—
188C22×C3—91no—
189C22×C3—91no—
190C22×C3—361no—
191C22×C3C4⋊#15711yescharacteristic
192C2×D3—62no—
193C2×D3—62no—
194C2×D3—124no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—42no—
198C3×C4—42no—
199C3×C4—91no—
200C3×C4—91no—
201C3×C4C4⋊#15711yescharacteristic
202C3×C4C22×#15711yescharacteristic
203Dic3—62no—
204Dic3—62no—
205C32—22no—
206C32—42no—
207C32C3×#43011yescharacteristic, complemented
208C23—91no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4C2×C3×#15711yescharacteristic
215C8—92no—
216C8—181no—
217C8—181no—
218C8—182no—
219C8C2×C3×#15712yes—
220C2×C3—21no—
221C2×C3—22no—
222C2×C3—42no—
223C2×C3—42no—
224C2×C3—82no—
225C2×C3—181no—
226C2×C3—181no—
227C2×C3C2×C4⋊#15711yescharacteristic
228D3—64no—
229D3—122no—
230C22—91no—
231C22—91no—
232C22—361no—
233C22C3×C4⋊#15711yescharacteristic
234C4—91no—
235C4—91no—
236C4C3×C4⋊#15711yescharacteristic
237C4C22×C3×#15711yescharacteristic
238C3—22no—
239C3—42no—
240C3#1059377611yescharacteristic, direct product factor
241C2—21no—
242C2—181no—
243C2—181no—
244C2C2×C3×C4⋊#15711yescharacteristic
subgroupG/H
centerC2×C3C2×C4⋊#157
commutatorC32×C4C22×C3×C4
FrattiniC4C22×C3×#157
FittingC33×M16C4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC4×C3⋊D3C23×C3
hypercenterC3×M16#157
hypocenterC32C3×#430
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
4. C3×#63771C2semidirect
5. C3×#63770C2semidirect
6. #10593776C3direct (N × H)
23. #63771C2×C3semidirect
24. #63770C2×C3semidirect
140. C33#430semidirect
207. C32C3×#430semidirect
240. C3#10593776direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642832
C2×C3×#37882
C3×M16×C3⋊D32
C3×#637712
C3×#637702
#105937763
C3×#4309