Finite Group Databasebeta

← C2×#63422⋊C3 #13146909

Subgroups — C2×#63422⋊C3 #13146909

Order 1728.

322 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 55 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×#88691C211yesmaximal, characteristic
2#63422⋊C3C212yesmaximal, direct product factor
3C2×C3×#837—31nomaximal
4C2×#63422C311yesmaximal, characteristic, complemented
5#88691C411yescharacteristic, complemented
6#88691C2211yescharacteristic
7C2×#6204C411yescharacteristic, complemented
8C2×3−1+2⋊C8—41nomaximal
9C2×C32×#41D311yescharacteristic
10C2×C3×C4×Dic3—31no—
11C2×C3×#208—31no—
12C2×#41⋊C9—21no—
13C2×#41⋊C9C2×C311yescharacteristic
14C3×#837—32no—
15C3×#837—32no—
16#63422C2×C312yescomplemented
17C2×C4×3−1+2—41no—
18#6204C2×C411yescharacteristic, complemented
193−1+2⋊C8—42no—
20C2×C3×#104—91no—
21C2×#837—31no—
22C32×#41—31no—
23C32×#41—31no—
24C32×#41Dic311yescharacteristic
25C32×#41C2×D311yescharacteristic
26C2×C32×Q8Dic311yescharacteristic
27C2×C32×D4—31no—
28C22×C32×C4—31no—
29C22×C3×Dic3—61no—
30C3×C4×Dic3—32no—
31C3×C4×Dic3—61no—
32C3×#208—32no—
33C3×#208—61no—
34C2×C3×C3⋊C8—61no—
35#41⋊C9—21no—
36#41⋊C9—21no—
37#41⋊C9C3×C411yescharacteristic, complemented
38#41⋊C9C22×C311yescharacteristic
39C2×Q8⋊C9—21no—
40C2×Q8⋊C9C3×C411yescharacteristic, complemented
41C2×C9⋊C8—41no—
42C22×3−1+2—41no—
43C4×3−1+2—41no—
44C4×3−1+2—41no—
45C2×C3×#41—31no—
46C2×C3×#41C3×D311yescharacteristic
47C2×C3×M16—91no—
48C2×C3×C42—91no—
49C2×C4×Dic3—31no—
50C2×#208—31no—
51C3×#104—92no—
52C3×#104—92no—
53#837—32no—
54#837—32no—
55C23×C32—31no—
56C32×Q8—31no—
57C32×Q8C2×Dic311yescharacteristic
58C32×D4—31no—
59C32×D4—31no—
60C32×D4—61no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4S411yescharacteristic
67C2×C3×Dic3—62no—
68C2×C3×Dic3—64no—
69C3×C3⋊C8—62no—
70C2×C4×C9—41no—
71C2×C4×C9—81no—
72Q8⋊C9—21no—
73Q8⋊C9C2×C3×C411yescharacteristic, complemented
74C9⋊C8—42no—
75C2×#104—91noSylow
76C2×3−1+2—42no—
77C2×3−1+2—41no—
78C3×#41—31no—
79C3×#41—31no—
80C3×#41—31no—
81C3×#41—31no—
82C3×#41—31no—
83C3×#41—31no—
84C3×#41—61no—
85C3×#41—61no—
86C3×#41C3×Dic311yescharacteristic
87C3×#41C2×C3×D311yescharacteristic
88C2×C3×Q8—31no—
89C2×C3×Q8C3×Dic311yescharacteristic
90C2×C3×D4—31no—
91C2×C3×D4—31no—
92C2×C3×D4—61no—
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4—61no—
96C22×C3×C4—181no—
97C22×Dic3—61no—
98C3×M16—92no—
99C3×M16—181no—
100C2×C3×C8—181no—
101C3×C42—92no—
102C3×C42—181no—
103C4×Dic3—32no—
104C4×Dic3—61no—
105#208—32no—
106#208—61no—
107C2×C3⋊C8—61no—
108C22×C32—31no—
109C22×C32—31no—
110C22×C32—62no—
111C22×C32SL(2,ℤ/4)11yescharacteristic
112C32×C4—31no—
113C32×C4—31no—
114C32×C4SL(2,ℤ/4)11yescharacteristic
115C32×C4C2×S411yescharacteristic
116C3×Dic3—64no—
117C22×C9—41no—
118C22×C9—81no—
119C4×C9—41no—
120C4×C9—41no—
121C4×C9—81no—
122C4×C9—81no—
123C2×#41AGL(1,ℤ/9)11yescharacteristic
124C2×M16—91no—
125C2×C42—91no—
126#104—92no—
127#104—92no—
1283−1+2—41noSylow
129C23×C3—31no—
130C23×C3—31no—
131C23×C3—61no—
132C3×Q8—31no—
133C3×Q8—31no—
134C3×Q8—31no—
135C3×Q8—61no—
136C3×Q8C2×C3×Dic311yescharacteristic
137C3×D4—31no—
138C3×D4—31no—
139C3×D4—31no—
140C3×D4—31no—
141C3×D4—61no—
142C3×D4—61no—
143C3×D4—61no—
144C3×D4—61no—
145C3×D4—61no—
146C3×D4—61no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—61no—
159C2×C3×C4—61no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—61no—
163C2×C3×C4—182no—
164C2×C3×C4—184no—
165C2×C3×C4C3×S411yescharacteristic
166C2×Dic3—62no—
167C2×Dic3—64no—
168C3×C8—182no—
169C3⋊C8—62no—
170C2×C32—61no—
171C2×C32—61no—
172C2×C32GU(2,𝔽3)12yes—
173C2×C32C2×SL(2,ℤ/4)11yescharacteristic
174C2×C9—41no—
175C2×C9—42no—
176C2×C9—81no—
177C2×C9—82no—
178#41—31no—
179#41—31no—
180#413−1+2⋊C411yescharacteristic
181#41C2×AGL(1,ℤ/9)11yescharacteristic
182C2×Q83−1+2⋊C411yescharacteristic
183C2×D4—31no—
184C22×C4—31no—
185C22×C4—181no—
186M16—92no—
187M16—181no—
188C2×C8—181no—
189C42—92no—
190C42—181no—
191C22×C3—31no—
192C22×C3—31no—
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—61no—
197C22×C3—61no—
198C22×C3—62no—
199C22×C3—62no—
200C22×C3—122no—
201C22×C3C3×SL(2,ℤ/4)11yescharacteristic
202C3×C4—31no—
203C3×C4—31no—
204C3×C4—31no—
205C3×C4—31no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—61no—
209C3×C4—61no—
210C3×C4—184no—
211C3×C4C3×SL(2,ℤ/4)11yescharacteristic
212C3×C4C2×C3×S411yescharacteristic
213Dic3—64no—
214C32C2×GU(2,𝔽3)11yescharacteristic
215C9—41no—
216C9—81no—
217C23—31no—
218Q8—31no—
219Q8C2×3−1+2⋊C411yescharacteristic
220D4—31no—
221D4—31no—
222D4—61no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—182no—
229C2×C4—184no—
230C2×C4((C22⋊C9)⋊C2)⋊C311yescharacteristic
231C8—182no—
232C2×C3—32no—
233C2×C3—31no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—61no—
237C2×C3—61no—
238C2×C3—121no—
239C2×C3—121no—
240C2×C3C3×GU(2,𝔽3)12yes—
241C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
242C22—31no—
243C22—31no—
244C22—62no—
245C22#1094⋊C411yescharacteristic
246C4—31no—
247C4—31no—
248C4—184no—
249C4#1094⋊C411yescharacteristic
250C4C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
251C3—31no—
252C3C2×C3×GU(2,𝔽3)11yescharacteristic
253C2—61no—
254C2—61no—
255C2#63422⋊C312yesdirect product factor
256C2C2×#1094⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C4((C22⋊C9)⋊C2)⋊C3
commutator, hypocenterQ8⋊C9C2×C3×C4
FrattiniC3×C4C2×C3×S4
Fitting, prime radicalC2×C32×#41D3
socleC22×C3C3×SL(2,ℤ/4)
cosocle#41⋊C9C22×C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #63422⋊C3C2direct (N × H)
4. C2×#63422C3semidirect
5. #88691C4semidirect
7. C2×#6204C4semidirect
16. #63422C2×C3semidirect
18. #6204C2×C4semidirect
37. #41⋊C9C3×C4semidirect
40. C2×Q8⋊C9C3×C4semidirect
73. Q8⋊C9C2×C3×C4semidirect
255. C2#63422⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886912
#63422⋊C32
C2×C3×#8373
C2×#634223
C2×3−1+2⋊C84