Finite Group Databasebeta

← C2×C3×#64222 #13181014

Subgroups — C2×C3×#64222 #13181014

Order 1728.

968 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 226 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×#3797C212yesmaximal
2C2×C3×C4×#157C211yesmaximal, characteristic
3C2×C3×#3773C214yesmaximal
4C3×#64222C218yesmaximal, direct product factor
5C2×#64222C311yesmaximal, characteristic, direct product factor
6C22×C3×#157C2212yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6—22no—
9C2×C3×#627—22no—
10C2×C3×#624C2214yes—
11C3×#3797—24no—
12C3×#3797C2218yes—
13C3×C4×#157C2214yes—
14C3×#3773C22116yes—
15C2×C3×D3×Q8—62no—
16C2×#3797C2×C312yes—
17C2×C4×#157C2×C311yescharacteristic
18C2×#3773C2×C314yes—
19#64222C2×C318yesdirect product factor
20C22×C3×C3⋊D3C2311yescharacteristic
21C2×C3×#157Q818yescomplemented
22C2×C3×#157C2314yes—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C3×C4×C3⋊D3D412yes—
26C3×C4×C3⋊D3C2312yes—
27C2×C32×Dic3—24no—
28C32×Dic6—28no—
29C3×#627—28no—
30C3×#624—24no—
31C3×#624C2318yes—
32C2×C3×C4⋊Q8—91nomaximal
33C22×#157C22×C312yes—
34C2×C32×Q8—62no—
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C3×D3×Q8—68no—
37C3×D3×Q8—68no—
38C2×C3×C4×D3—64no—
39C2×C3×C4×D3—62no—
40C2×C3×Dic6—22no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6—42no—
43C2×C3×Dic6—64no—
44C2×#627—22no—
45C2×#624C22×C314yes—
46#3797—24no—
47#3797C22×C318yes—
48C4×#157C22×C314yes—
49#3773C22×C3116yes—
50C22×C33C2×D411yescharacteristic
51C2×C3×C3⋊D3C2×D411yescharacteristic
52C2×C3×C3⋊D3C2×Q814yes—
53C2×C3×C3⋊D3C2411yescharacteristic
54C3×#157C2×Q818yescomplemented
55C33×C4C2×D412yes—
56C3×C3⋊Dic3C2×D412yes—
57C32×Dic3—28no—
58C22×C3×Q8—92no—
59C2×D3×Q8—62no—
60C3×C4⋊Q8—98no—
61C2×C3×C4⋊C4—94no—
62C2×C3×C42—91no—
63C22×C3⋊D3C23×C311yescharacteristic
64C22×C3×D3—62no—
65C2×#157C3×Q818yescomplemented
66C2×#157C23×C314yes—
67C32×Q8—68no—
68C2×C32×C4—22no—
69C2×C32×C4—42no—
70C2×C32×C4—64no—
71C2×C32×C4C3×D411yescharacteristic
72C2×C3⋊Dic3C3×D411yescharacteristic
73C4×C3⋊D3C3×D412yes—
74C4×C3⋊D3C23×C312yes—
75C2×C3×Dic3—24no—
76C2×C3×Dic3—24no—
77C2×C3×Dic3—44no—
78C2×C3×Dic3—62no—
79C3×C4×D3—68no—
80C3×C4×D3—64no—
81C3×C4×D3—64no—
82C3×C4×D3—68no—
83C3×Dic6—28no—
84C3×Dic6—28no—
85C3×Dic6—48no—
86C3×Dic6—616no—
87#627—28no—
88#624—24no—
89#624C23×C318yes—
90C2×C4⋊Q8—91noSylow
91C2×C33C4⋊Q812yescomplemented
92C2×C33C22×D411yescharacteristic
93C3×C3⋊D3C4⋊Q812yescomplemented
94C3×C3⋊D3C22×Q812yes—
95C2×C3×Q8—62no—
96C2×C3×Q8—62no—
97C2×C3×Q8—98no—
98C2×C3×Q8—122no—
99C2×C3×Q8—182no—
100C2×C3×Q8—184no—
101C22×C3×C4—94no—
102C22×C3×C4—92no—
103C22×C3×C4—91no—
104D3×Q8—68no—
105D3×Q8—68no—
106C2×C4×D3—64no—
107C2×C4×D3—62no—
108C2×Dic6—22no—
109C2×Dic6—64no—
110C3×C4⋊C4—916no—
111C3×C42—94no—
112C22×C32—22no—
113C22×C32—42no—
114C22×C32C2×C3×D411yescharacteristic
115C2×C3⋊D3C2×C3×D411yescharacteristic
116C2×C3⋊D3C2×C3×Q814yes—
117C2×C3⋊D3C24×C311yescharacteristic
118C2×C3×D3—62no—
119C2×C3×D3—68no—
120C2×C3×D3—62no—
121#157C2×C3×Q818yescomplemented
122C32×C4—24no—
123C32×C4—44no—
124C32×C4—68no—
125C32×C4C2×C3×D412yes—
126C3⋊Dic3C2×C3×D412yes—
127C3×Dic3—28no—
128C3×Dic3—28no—
129C3×Dic3—48no—
130C3×Dic3—64no—
131C22×Q8—92no—
132C4⋊Q8—98no—
133C2×C4⋊C4—94no—
134C2×C42—91no—
135C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
136C23×C3—91no—
137C22×D3—62no—
138C3×Q8—68no—
139C3×Q8—68no—
140C3×Q8—128no—
141C3×Q8—188no—
142C2×C3×C4—22no—
143C2×C3×C4—42no—
144C2×C3×C4—64no—
145C2×C3×C4—64no—
146C2×C3×C4—98no—
147C2×C3×C4—98no—
148C2×C3×C4—94no—
149C2×C3×C4—91no—
150C2×C3×C4—92no—
151C2×C3×C4—92no—
152C2×C3×C4—124no—
153C2×C3×C4—184no—
154C2×C3×C4GO+(4,𝔽2)11yescharacteristic
155C2×Dic3—24no—
156C2×Dic3—62no—
157C4×D3—68no—
158C4×D3—64no—
159C4×D3—64no—
160C4×D3—68no—
161Dic6—28no—
162Dic6—616no—
163C2×C32—22no—
164C2×C32—24no—
165C2×C32—44no—
166C2×C32—42no—
167C2×C32C3×C4⋊Q812yescomplemented
168C2×C32C22×C3×D411yescharacteristic
169C3⋊D3C3×C4⋊Q812yescomplemented
170C3⋊D3C22×C3×Q812yes—
171C3×D3—64no—
172C3×D3—64no—
173C2×Q8—62no—
174C2×Q8—98no—
175C2×Q8—182no—
176C2×Q8—184no—
177C22×C4—94no—
178C22×C4—92no—
179C22×C4—91no—
180C4⋊C4—916no—
181C42—94no—
182C22×C3—22no—
183C22×C3—42no—
184C22×C3—91no—
185C22×C3—91no—
186C22×C3—94no—
187C22×C3C2×GO+(4,𝔽2)11yescharacteristic
188C2×D3—62no—
189C2×D3—62no—
190C2×D3—68no—
191C3×C4—24no—
192C3×C4—44no—
193C3×C4—68no—
194C3×C4—68no—
195C3×C4—98no—
196C3×C4—92no—
197C3×C4—128no—
198C3×C4C2×GO+(4,𝔽2)12yes—
199Dic3—28no—
200Dic3—64no—
201C32—22no—
202C32—42no—
203C32C2×C3×C4⋊Q811yescharacteristic, complemented
204C23—91no—
205Q8—68no—
206Q8—188no—
207C2×C4—64no—
208C2×C4—98no—
209C2×C4—94no—
210C2×C4—98no—
211C2×C4—91no—
212C2×C4—92no—
213C2×C4—92no—
214C2×C4—184no—
215C2×C4C3×GO+(4,𝔽2)11yescharacteristic
216C2×C3—22no—
217C2×C3—24no—
218C2×C3—42no—
219C2×C3—44no—
220C2×C3—92no—
221C2×C3—92no—
222C2×C3#6422212yesdirect product factor
223C2×C3C22×GO+(4,𝔽2)11yescharacteristic
224D3—64no—
225D3—64no—
226C22—91no—
227C22—91no—
228C22—94no—
229C22C2×C3×GO+(4,𝔽2)11yescharacteristic
230C4—68no—
231C4—98no—
232C4—92no—
233C4C2×C3×GO+(4,𝔽2)12yes—
234C3—22no—
235C3—42no—
236C3C2×#6422211yescharacteristic, direct product factor
237C2—92no—
238C2—92no—
239C2C3×#6422212yesdirect product factor
240C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×C4⋊Q8
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#64222C2direct (N × H)
5. C2×#64222C3direct (N × H)
19. #64222C2×C3direct (N × H)
21. C2×C3×#157Q8semidirect
54. C3×#157C2×Q8semidirect
65. C2×#157C3×Q8semidirect
91. C2×C33C4⋊Q8semidirect
93. C3×C3⋊D3C4⋊Q8semidirect
121. #157C2×C3×Q8semidirect
135. C33C2×C4⋊Q8semidirect
167. C2×C32C3×C4⋊Q8semidirect
169. C3⋊D3C3×C4⋊Q8semidirect
203. C32C2×C3×C4⋊Q8semidirect
222. C2×C3#64222direct (N × H)
236. C3C2×#64222direct (N × H)
239. C2C3×#64222direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×C3×C4×#1572
C2×C3×#37732
C3×#642222
C2×#642223
C2×C3×C4⋊Q89