Finite Group Databasebeta

← C3×#10593292 #13166661

Subgroups — C3×#10593292 #13166661

Order 1728.

460 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 74 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×#64231C211yesmaximal, characteristic
2C3×#64227C212yesmaximal, complemented
3C3×#64222C211yesmaximal, characteristic, complemented
4C3×#63796C212yesmaximal, complemented
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#10593292C311yesmaximal, characteristic, direct product factor
7C2×C3×GO+(4,𝔽2)—22no—
8C3×#3802C2212yes—
9C3×#3797C2212yes—
10C3×C4×#157C2211yescharacteristic
11C3×#3787C2211yescharacteristic
12C3×#3775—24no—
13C3×#3773—22no—
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×#204—22no—
16C3×#3715—22no—
17#64231C2×C311yescharacteristic
18#64227C2×C312yescomplemented
19#64222C2×C311yescharacteristic, complemented
20C3×D3×SD16—62no—
21#63796C2×C312yescomplemented
22C8×#157C2×C311yescharacteristic, complemented
23C2×C3×D32—22no—
24C2×C3×#157D412yes—
25C3×GO+(4,𝔽2)—44no—
26C3×C4×C3⋊D3C2311yescharacteristic
27C32×D12—22no—
28C32×Dic6—22no—
29C3×#627—22no—
30C3×#625—22no—
31C3×#624—22no—
32C3×#622D412yescomplemented
33C33×C8D411yescharacteristic, complemented
34C3×C3⋊(C3⋊C8)D411yescharacteristic, complemented
35C3×#492—91nomaximal
36C2×GO+(4,𝔽2)—22no—
37C3×D3×Q8—62no—
38C3×D3×D4—62no—
39#3802C22×C312yes—
40#3797C22×C312yes—
41C4×#157C22×C311yescharacteristic
42#3787C22×C311yescharacteristic
43#3775—24no—
44#3773—22no—
45C32×SD16—62no—
46C8×C3⋊D3C22×C311yescharacteristic
47C3×#215—62no—
48C3×#214—62no—
49C3×#204—22no—
50C3×#204—22no—
51C3×#204—42no—
52C3×D3×C8—62no—
53#3715—22no—
54C2×C3×C3⋊D3C2×D411yescharacteristic
55C2×C32×D3—42no—
56C3×D32—44no—
57C3×#157SD1614yescomplemented
58C33×C4C2×D411yescharacteristic
59C3×C3⋊Dic3C2×D411yescharacteristic
60C32×Dic3—42no—
61C2×C3×SD16—92no—
62C2×C3×SD16—92no—
63C3×C4⋊Q8—91no—
64C3×#127—91no—
65D3×SD16—62no—
66C3×C4×C8—91no—
67C22×C3×D3—122no—
68C2×D32—22no—
69C2×#157C3×D412yes—
70GO+(4,𝔽2)—44no—
71C32×Q8—62no—
72C32×D4—62no—
73C4×C3⋊D3C23×C311yescharacteristic
74C3×#67—122no—
75C3×D12—22no—
76C3×D12—22no—
77C3×D12—42no—
78C3×C4×D3—62no—
79C3×C4×D3—122no—
80C3×Dic6—22no—
81C3×Dic6—22no—
82C3×Dic6—42no—
83C3×Dic6—122no—
84#627—22no—
85#625—22no—
86#624—22no—
87#622C3×D412yescomplemented
88C32×C8—22no—
89C32×C8—42no—
90C32×C8C3×D411yescharacteristic, complemented
91C3⋊(C3⋊C8)C3×D411yescharacteristic, complemented
92C3×C3⋊C8—62no—
93#492—91noSylow
94C2×C33#12711yescharacteristic
95C3×C3⋊D3C2×SD1612yes—
96C32×D3—82no—
97C2×C3×Q8—92no—
98C2×C3×D4—92no—
99C2×C3×D4—182no—
100D3×Q8—62no—
101D3×D4—62no—
102C3×SD16—62no—
103C3×SD16—62no—
104C3×SD16—122no—
105C3×SD16—182no—
106C3×SD16—184no—
107C2×C3×C8—91no—
108C2×C3×C8—91no—
109C3×C4⋊C4—182no—
110C3×C42—91no—
111#215—62no—
112#214—62no—
113#204—22no—
114D3×C8—62no—
115C22×C32—122no—
116C2×C3⋊D3C2×C3×D411yescharacteristic
117C2×C3×D3—42no—
118C2×C3×D3—42no—
119C2×C3×D3—62no—
120C2×C3×D3—82no—
121C2×C3×D3—244no—
122D32—44no—
123#157C3×SD1614yescomplemented
124C32×C4—22no—
125C32×C4—42no—
126C32×C4—122no—
127C32×C4C2×C3×D411yescharacteristic
128C3⋊Dic3C2×C3×D411yescharacteristic
129C3×Dic3—42no—
130C3×Dic3—42no—
131C3×Dic3—62no—
132C3×Dic3—82no—
133C2×SD16—92no—
134C2×SD16—92no—
135C4⋊Q8—91no—
136#127—91no—
137C4×C8—91no—
138C33#49211yescharacteristic, complemented, Sylow
139C23×C3—182no—
140C22×D3—122no—
141C3×Q8—62no—
142C3×Q8—62no—
143C3×Q8—122no—
144C3×Q8—182no—
145C3×D4—62no—
146C3×D4—62no—
147C3×D4—122no—
148C3×D4—182no—
149C3×D4—364no—
150C2×C3×C4—92no—
151C2×C3×C4—91no—
152C2×C3×C4—182no—
153#67—122no—
154D12—22no—
155C4×D3—62no—
156C4×D3—122no—
157Dic6—22no—
158Dic6—122no—
159C3×C8—22no—
160C3×C8—42no—
161C3×C8—92no—
162C3×C8—91no—
163C3×C8GO+(4,𝔽2)11yescharacteristic, complemented
164C3⋊C8—62no—
165C2×C32—22no—
166C2×C32—42no—
167C2×C32—242no—
168C2×C32C3×#12711yescharacteristic
169C3⋊D3C2×C3×SD1612yes—
170C3×D3—64no—
171C3×D3—82no—
172C3×D3—82no—
173C3×D3—162no—
174C2×Q8—92no—
175C2×D4—92no—
176C2×D4—182no—
177SD16—62no—
178SD16—182no—
179SD16—184no—
180C2×C8—91no—
181C2×C8—91no—
182C4⋊C4—182no—
183C42—91no—
184C22×C3—91no—
185C22×C3—122no—
186C22×C3—122no—
187C22×C3—242no—
188C22×C3—364no—
189C2×D3—42no—
190C2×D3—62no—
191C2×D3—244no—
192C3×C4—22no—
193C3×C4—42no—
194C3×C4—94no—
195C3×C4—91no—
196C3×C4—122no—
197C3×C4—122no—
198C3×C4—242no—
199C3×C4C2×GO+(4,𝔽2)11yescharacteristic
200Dic3—42no—
201Dic3—62no—
202C32—22no—
203C32—42no—
204C32C3×#49211yescharacteristic, complemented
205C23—182no—
206Q8—62no—
207Q8—182no—
208D4—62no—
209D4—182no—
210D4—364no—
211C2×C4—92no—
212C2×C4—91no—
213C2×C4—182no—
214C8—92no—
215C8—91no—
216C8C3×GO+(4,𝔽2)11yescharacteristic, complemented
217C2×C3—22no—
218C2×C3—42no—
219C2×C3—92no—
220C2×C3—242no—
221C2×C3—242no—
222C2×C3—482no—
223C2×C3#6423111yescharacteristic
224D3—64no—
225D3—82no—
226C22—91no—
227C22—122no—
228C22—364no—
229C4—94no—
230C4—91no—
231C4—122no—
232C4C2×C3×GO+(4,𝔽2)11yescharacteristic
233C3—22no—
234C3—42no—
235C3#1059329211yescharacteristic, direct product factor
236C2—92no—
237C2—242no—
238C2C3×#6423111yescharacteristic
subgroupG/H
centerC2×C3#64231
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#127
prime radicalC3×#64231C2
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#492
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#64227C2semidirect
3. C3×#64222C2semidirect
4. C3×#63796C2semidirect
5. C3×C8×#157C2semidirect
6. #10593292C3direct (N × H)
18. #64227C2×C3semidirect
19. #64222C2×C3semidirect
21. #63796C2×C3semidirect
22. C8×#157C2×C3semidirect
32. C3×#622D4semidirect
33. C33×C8D4semidirect
34. C3×C3⋊(C3⋊C8)D4semidirect
57. C3×#157SD16semidirect
87. #622C3×D4semidirect
90. C32×C8C3×D4semidirect
91. C3⋊(C3⋊C8)C3×D4semidirect
123. #157C3×SD16semidirect
138. C33#492semidirect
163. C3×C8GO+(4,𝔽2)semidirect
204. C32C3×#492semidirect
216. C8C3×GO+(4,𝔽2)semidirect
235. C3#10593292direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642312
C3×#642272
C3×#642222
C3×#637962
C3×C8×#1572
#105932923
C3×#4929