Finite Group Databasebeta

← C2×#64265⋊C3 #13164427

Subgroups — C2×#64265⋊C3 #13164427

Order 1728.

578 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 71 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2#64265⋊C3C214yesmaximal, direct product factor
3C2×(C3⋊#226)⋊C3C212yesmaximal, complemented
4C2×#64265C311yesmaximal, characteristic, complemented
5C2×C3×#951—31nomaximal
6C2×#634⋊C3—41nomaximal
7#41⋊H(3,𝔽3)C2212yes—
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9(C3⋊#226)⋊C3C2214yescomplemented
10C2×C32×#41D311yescharacteristic
11C22×C3×Dic6—31no—
12C2×C3×#41⋊C3—21no—
13C2×C3×#41⋊C3C2×C311yescharacteristic
14#64265C2×C314yescomplemented
15C2×C3⋊#226C2×C312yescomplemented
16C3×#951—34no—
17C3×#951—34no—
18C2×C3×#216—32no—
19C2×C3×#214—32no—
20C2×C3×#208—31no—
21C2×C4×H(3,𝔽3)—41no—
22C2×(C3⋊Dic3)⋊C3—42no—
23#634⋊C3—44no—
24Q8⋊H(3,𝔽3)C2311yescharacteristic
25C2×#984—91no—
26C2×C3×#137—91no—
27C2×#951—31no—
28C32×#41—32no—
29C32×#41C2×D312yes—
30C2×C32×Q8C2×D311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×#634—41no—
34C22×C3×Dic3—61no—
35C2×C3×Dic6—32no—
36C2×C3×Dic6—34no—
37C2×C3×Dic6—62no—
38C2×C3×Dic6—61no—
39C3×#41⋊C3—22no—
40C3×#41⋊C3C22×C312yes—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
43C3⋊#226C22×C314yescomplemented
44C3×#216—34no—
45C3×#216—34no—
46C3×#214—38no—
47C3×#208—34no—
48C2×C3×C3⋊C8—32no—
49C22×H(3,𝔽3)—41no—
50C4×H(3,𝔽3)—42no—
51(C3⋊Dic3)⋊C3—44no—
52C2×C3×#41—31no—
53C2×C3×#41C3×D311yescharacteristic
54C22×C3×Q8—91no—
55C22×Dic6—31no—
56C2×#41⋊C3—31no—
57C2×#41⋊C3—61no—
58#984—94no—
59C2×#226—92no—
60C3×#137—94no—
61C3×#137—94no—
62C2×C3×Q16—92no—
63C2×C3×SD16—92no—
64C2×C3×M16—91no—
65#951—34no—
66#951—34no—
67C2×#216—32no—
68C2×#214—32no—
69C2×#208—31no—
70C23×C32—31no—
71C32×Q8—31no—
72C32×Q8C22×D311yescharacteristic
73C32×D4—32no—
74C32×D4—61no—
75C2×C32×C4—31no—
76C2×C32×C4—32no—
77C2×C32×C4—32no—
78C2×C32×C4—41no—
79C2×C32×C4—81no—
80C2×C32×C4S411yescharacteristic
81C2×C3⋊Dic3—42no—
82#634—44no—
83C2×C3×Dic3—64no—
84C2×C3×Dic3—62no—
85C3×Dic6—34no—
86C3×Dic6—62no—
87C3×Dic6—64no—
88C3×SL(2,𝔽3)—21no—
89C3×SL(2,𝔽3)C23×C311yescharacteristic
90C3×C3⋊C8—34no—
91C2×#137—91noSylow
92C2×H(3,𝔽3)—42no—
93C2×H(3,𝔽3)—41no—
94C3×#41—32no—
95C3×#41—32no—
96C3×#41—32no—
97C3×#41—62no—
98C3×#41C2×C3×D312yes—
99C2×C3×Q8—31no—
100C2×C3×Q8—92no—
101C2×C3×Q8—94no—
102C2×C3×Q8—182no—
103C2×C3×Q8—181no—
104C2×C3×Q8C2×C3×D311yescharacteristic
105C2×C3×D4—31no—
106C2×C3×D4—31no—
107C2×C3×D4—61no—
108C22×C3×C4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—61no—
111C22×C3×C4—181no—
112C22×Dic3—61no—
113C2×Dic6—32no—
114C2×Dic6—34no—
115C2×Dic6—62no—
116C2×Dic6—61no—
117C2×Dic6—361no—
118#41⋊C3—32no—
119#41⋊C3—62no—
120C2×SL(2,𝔽3)—31no—
121C2×SL(2,𝔽3)—61no—
122#226—94no—
123C3×Q16—94no—
124C3×Q16—94no—
125C3×SD16—98no—
126C3×M16—94no—
127C2×C3×C8—92no—
128#216—34no—
129#216—34no—
130#214—38no—
131#208—34no—
132C2×C3⋊C8—32no—
133C22×C32—32no—
134C22×C32—41no—
135C22×C32—62no—
136C22×C32—81no—
137C22×C32C2×S411yescharacteristic
138C32×C4—31no—
139C32×C4—31no—
140C32×C4—42no—
141C32×C4—82no—
142C32×C4C2×S412yes—
143C3⋊Dic3—44no—
144C3×Dic3—64no—
145C2×#41(C3⋊D3)⋊C311yescharacteristic
146C22×Q8—91no—
147#137—94no—
148#137—94no—
149C2×Q16—92no—
150C2×SD16—92no—
151C2×M16—91no—
152H(3,𝔽3)—41noSylow
153C23×C3—31no—
154C23×C3—31no—
155C23×C3—61no—
156C3×Q8—31no—
157C3×Q8—31no—
158C3×Q8—31no—
159C3×Q8—61no—
160C3×Q8—94no—
161C3×Q8—182no—
162C3×Q8—184no—
163C3×Q8C22×C3×D311yescharacteristic
164C3×D4—32no—
165C3×D4—32no—
166C3×D4—62no—
167C3×D4—61no—
168C3×D4—61no—
169C3×D4—62no—
170C2×C3×C4—31no—
171C2×C3×C4—32no—
172C2×C3×C4—32no—
173C2×C3×C4—31no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—31no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—61no—
180C2×C3×C4—121no—
181C2×C3×C4—184no—
182C2×C3×C4—182no—
183C2×C3×C4—241no—
184C2×C3×C4C3×S411yescharacteristic
185C2×Dic3—64no—
186C2×Dic3—62no—
187C2×Dic3—362no—
188Dic6—34no—
189Dic6—62no—
190Dic6—64no—
191Dic6—364no—
192SL(2,𝔽3)—31no—
193SL(2,𝔽3)—61no—
194C3×C8—94no—
195C3⋊C8—34no—
196C2×C32—41no—
197C2×C32—42no—
198C2×C32—62no—
199C2×C32—81no—
200C2×C32—82no—
201C2×C32#98412yescomplemented
202C2×C32C22×S411yescharacteristic
203#41—32no—
204#41C2×(C3⋊D3)⋊C312yes—
205C2×Q8—92no—
206C2×Q8—94no—
207C2×Q8—182no—
208C2×Q8—181no—
209C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
210C2×D4—31no—
211C22×C4—31no—
212C22×C4—181no—
213Q16—94no—
214Q16—94no—
215SD16—98no—
216M16—94no—
217C2×C8—92no—
218C22×C3—32no—
219C22×C3—32no—
220C22×C3—31no—
221C22×C3—62no—
222C22×C3—62no—
223C22×C3—62no—
224C22×C3—122no—
225C22×C3—121no—
226C22×C3—241no—
227C22×C3C2×C3×S411yescharacteristic
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—32no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—122no—
236C3×C4—184no—
237C3×C4—242no—
238C3×C4C2×C3×S412yes—
239Dic3—64no—
240Dic3—364no—
241C32—41no—
242C32—81no—
243C32C2×#98411yescharacteristic, complemented
244C23—31no—
245Q8—31no—
246Q8—94no—
247Q8—182no—
248Q8—184no—
249Q8C22×(C3⋊D3)⋊C311yescharacteristic
250D4—32no—
251D4—61no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—184no—
256C2×C4—182no—
257C2×C4(C3⋊S4)⋊C311yescharacteristic
258C8—94no—
259C2×C3—31no—
260C2×C3—32no—
261C2×C3—62no—
262C2×C3—62no—
263C2×C3—122no—
264C2×C3—121no—
265C2×C3—122no—
266C2×C3—241no—
267C2×C3—242no—
268C2×C3C3×#98412yes—
269C2×C3C22×C3×S411yescharacteristic
270C22—32no—
271C22—62no—
272C22C2×(C3⋊S4)⋊C311yescharacteristic
273C4—31no—
274C4—31no—
275C4—184no—
276C4C2×(C3⋊S4)⋊C312yes—
277C3—31no—
278C3—121no—
279C3—241no—
280C3C2×C3×#98411yescharacteristic
281C2—62no—
282C2#64265⋊C312yesdirect product factor
283C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2×C3C22×C3×S4
FittingC2×C32×#41D3
socleC22×C3C2×C3×S4
prime radicalC2×#41⋊H(3,𝔽3)C2
hypercenterC2×C4(C3⋊S4)⋊C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #64265⋊C3C2direct (N × H)
3. C2×(C3⋊#226)⋊C3C2semidirect
4. C2×#64265C3semidirect
9. (C3⋊#226)⋊C3C22semidirect
14. #64265C2×C3semidirect
15. C2×C3⋊#226C2×C3semidirect
43. C3⋊#226C22×C3semidirect
201. C2×C32#984semidirect
243. C32C2×#984semidirect
282. C2#64265⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#64265⋊C32
C2×(C3⋊#226)⋊C32
C2×#642653
C2×C3×#9513
C2×#634⋊C34