Finite Group Databasebeta

← C2×#11722445 #13164667

Subgroups — C2×#11722445 #13164667

Order 1728.

919 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 919 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88631C212yesmaximal, complemented
3#11722445C218yesmaximal, direct product factor
4C2×#88627C212yesmaximal, complemented
5C2×#88626C212yesmaximal, complemented
6C23×#272C411yescharacteristic
7C22×C4×H(3,𝔽3)C411yescharacteristic
8C22×#1084C2211yescharacteristic
9C2×C4×#272C414yes—
10C2×C4×#272C411yescharacteristic
11C2×C4×#272C411yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2214yes—
15#88631C2218yescomplemented
16C2×#6188C2214yescomplemented
17#88627C2218yescomplemented
18#88626C2218yescomplemented
19C22×C3×C4×D3—62no—
20C23×H(3,𝔽3)C2×C411yescharacteristic
21C22×#272—24no—
22C22×#272C2×C411yescharacteristic
23C22×#272C2×C414yes—
24C22×#272C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C414yes—
26C2×C4×H(3,𝔽3)C2×C411yescharacteristic
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28C2×#1084C2×C412yes—
29C2×#1084C2×C411yescharacteristic
30C2×#1084C2311yescharacteristic
31C2×#1084C2312yes—
32C4×#272C2×C412yes—
33C4×#272C2×C414yes—
34C4×#272C2×C412yes—
35C4×#272C2×C414yes—
36C4×#272C2312yes—
37C4×#272C2312yes—
38#6188C2318yescomplemented
39C22×C3×M16—91nomaximal
40C23×C3×D3—62no—
41C22×C32×C4—22no—
42C22×C3×Dic3—62no—
43C2×C3×C4×D3—62no—
44C2×C3×C4×D3—68no—
45C2×C3×C4×D3—62no—
46C2×C3×C4×D3—62no—
47C2×C3×C4×D3—68no—
48C2×C3×C4×D3—62no—
49C22×H(3,𝔽3)—22no—
50C22×H(3,𝔽3)C22×C412yes—
51C22×H(3,𝔽3)C22×C411yescharacteristic
52C2×#272—22no—
53C2×#272—28no—
54C2×#272M1614yescomplemented
55C2×#272C22×C412yes—
56C2×#272C22×C412yes—
57C4×H(3,𝔽3)C22×C412yes—
58C4×H(3,𝔽3)C22×C412yes—
59#1084C22×C411yescharacteristic
60#1084C22×C412yes—
61#1084C2411yescharacteristic
62C23×C3×C4—91no—
63C22×C4×D3—62no—
64C2×C3×M16—92no—
65C2×C3×M16—98no—
66C2×C3×M16—92no—
67C22×C3×C8—92no—
68C23×C32—22no—
69C22×C3×D3—62no—
70C22×C3×D3—68no—
71C22×C3×D3—62no—
72C22×C3×D3—616no—
73C2×C32×C4—22no—
74C2×C32×C4—28no—
75C2×C32×C4—22no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—64no—
80C3×C4×D3—68no—
81C3×C4×D3—64no—
82C3×C4×D3—64no—
83C3×C4×D3—68no—
84C3×C4×D3—64no—
85C3×C4×D3—64no—
86C22×M16—91noSylow
87C2×H(3,𝔽3)—22no—
88C2×H(3,𝔽3)C2×M1612yescomplemented
89C2×H(3,𝔽3)C23×C411yescharacteristic
90#272—22no—
91#272C2×M1614yescomplemented
92C24×C3—91no—
93C23×D3—62no—
94C22×C3×C4—62no—
95C22×C3×C4—91no—
96C22×C3×C4—91no—
97C22×C3×C4—94no—
98C22×C3×C4—91no—
99C22×C3×C4—94no—
100C22×C3×C4—91no—
101C22×C3×C4—91no—
102C22×C3×C4#15711yescharacteristic
103C22×Dic3—62no—
104C2×C4×D3—62no—
105C2×C4×D3—62no—
106C2×C4×D3—68no—
107C2×C4×D3—62no—
108C2×C4×D3—68no—
109C2×C4×D3—62no—
110C3×M16—98no—
111C3×M16—98no—
112C2×C3×C8—98no—
113C2×C3×C8—94no—
114C22×C32—28no—
115C22×C32—24no—
116C22×C32—22no—
117C2×C3×D3—68no—
118C2×C3×D3—64no—
119C2×C3×D3—64no—
120C2×C3×D3—632no—
121C2×C3×D3—68no—
122C32×C4—24no—
123C32×C4—24no—
124C3×Dic3—62no—
125C3×Dic3—64no—
126C3×Dic3—62no—
127C23×C4—91no—
128C2×M16—92no—
129C2×M16—92no—
130C2×M16—98no—
131C22×C8—92no—
132H(3,𝔽3)C22×M1611yescharacteristic, complemented, Sylow
133C23×C3—62no—
134C23×C3—91no—
135C23×C3—94no—
136C23×C3—91no—
137C23×C3—184no—
138C23×C3C2×#15711yescharacteristic
139C22×D3—62no—
140C22×D3—68no—
141C22×D3—62no—
142C22×D3—616no—
143C2×C3×C4—68no—
144C2×C3×C4—62no—
145C2×C3×C4—62no—
146C2×C3×C4—91no—
147C2×C3×C4—92no—
148C2×C3×C4—92no—
149C2×C3×C4—91no—
150C2×C3×C4—92no—
151C2×C3×C4—92no—
152C2×C3×C4—94no—
153C2×C3×C4—92no—
154C2×C3×C4—92no—
155C2×C3×C4—94no—
156C2×C3×C4C2×#15711yescharacteristic
157C2×C3×C4C2×#15714yes—
158C2×C3×C4C2×#15711yescharacteristic
159C2×Dic3—62no—
160C2×Dic3—64no—
161C2×Dic3—64no—
162C2×Dic3—62no—
163C4×D3—64no—
164C4×D3—64no—
165C4×D3—64no—
166C4×D3—68no—
167C4×D3—64no—
168C4×D3—68no—
169C3×C8—98no—
170C2×C32—28no—
171C2×C32—24no—
172C2×C32—22no—
173C3×D3—68no—
174C3×D3—68no—
175C24—91no—
176C22×C4—91no—
177C22×C4—91no—
178C22×C4—91no—
179C22×C4—94no—
180C22×C4—91no—
181C22×C4—94no—
182C22×C4—91no—
183C22×C4#108811yescharacteristic
184M16—98no—
185M16—98no—
186C2×C8—94no—
187C2×C8—98no—
188C22×C3—22no—
189C22×C3—68no—
190C22×C3—64no—
191C22×C3—62no—
192C22×C3—94no—
193C22×C3—92no—
194C22×C3—92no—
195C22×C3—188no—
196C22×C3—182no—
197C22×C3C22×#15712yes—
198C22×C3C22×#15711yescharacteristic
199C2×D3—64no—
200C2×D3—64no—
201C2×D3—68no—
202C2×D3—632no—
203C2×D3—68no—
204C3×C4—64no—
205C3×C4—64no—
206C3×C4—91no—
207C3×C4—91no—
208C3×C4—92no—
209C3×C4C22×#15712yes—
210C3×C4C22×#15712yes—
211Dic3—62no—
212Dic3—62no—
213Dic3—64no—
214C32—22no—
215C23—91no—
216C23—94no—
217C23—91no—
218C23—184no—
219C23C2×#108811yescharacteristic
220C2×C4—91no—
221C2×C4—92no—
222C2×C4—92no—
223C2×C4—91no—
224C2×C4—92no—
225C2×C4—92no—
226C2×C4—92no—
227C2×C4—94no—
228C2×C4—92no—
229C2×C4—94no—
230C2×C4C2×#108814yes—
231C2×C4C2×#108811yescharacteristic
232C2×C4C2×#108811yescharacteristic
233C8—98no—
234C2×C3—22no—
235C2×C3—68no—
236C2×C3—62no—
237C2×C3—64no—
238C2×C3—94no—
239C2×C3—182no—
240C2×C3#6428312yes—
241C2×C3C23×#15711yescharacteristic
242D3—68no—
243D3—68no—
244C22—22no—
245C22—92no—
246C22—94no—
247C22—92no—
248C22—188no—
249C22—182no—
250C22C22×#108812yes—
251C22C22×#108811yescharacteristic
252C4—91no—
253C4—91no—
254C4—92no—
255C4C22×#108812yes—
256C4C22×#108812yes—
257C3—62no—
258C3C2×#6428311yescharacteristic
259C2—22no—
260C2—94no—
261C2—182no—
262C2#1172244512yesdirect product factor
263C2C23×#108811yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC2×C3C23×#157
FittingC22×C4×H(3,𝔽3)C4
socleC22×C3C22×#157
prime radicalC23×#272C4
cosocle#1084C24
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)C22×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88631C2semidirect
3. #11722445C2direct (N × H)
4. C2×#88627C2semidirect
5. C2×#88626C2semidirect
15. #88631C22semidirect
16. C2×#6188C22semidirect
17. #88627C22semidirect
18. #88626C22semidirect
38. #6188C23semidirect
54. C2×#272M16semidirect
88. C2×H(3,𝔽3)C2×M16semidirect
91. #272C2×M16semidirect
132. H(3,𝔽3)C22×M16semidirect
262. C2#11722445direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#886312
#117224452
C2×#886272
C2×#886262
C22×C3×M169