Finite Group Databasebeta

← C2×C3×#64265 #13181383

Subgroups — C2×C3×#64265 #13181383

Order 1728.

831 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 831 conjugacy classes of subgroups — too many to draw. Only its 114 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×C32×#41⋊C3C211yesmaximal, characteristic
2C3×#64265C214yesmaximal, direct product factor
3C2×C3×C3⋊#226C212yesmaximal, complemented
4C2×#64265C311yesmaximal, characteristic, direct product factor
5C2×C3×#984—33nomaximal
6C2×C3×#951—31nomaximal
7C2×C3×#634—41nomaximal
8C32×#41⋊C3C2212yes—
9C2×C32×SL(2,𝔽3)C2211yescharacteristic
10C3×C3⋊#226C2214yescomplemented
11C2×C32×#41D311yescharacteristic
12C22×C3×Dic6—31no—
13C2×C3×#41⋊C3—21no—
14C2×C3×#41⋊C3—23no—
15C2×C3×#41⋊C3D313yes—
16C2×C3×#41⋊C3C2×C311yescharacteristic
17#64265C2×C314yesdirect product factor
18C2×C3⋊#226C2×C312yescomplemented
19C3×#984—312no—
20C2×C3×#226—36no—
21C3×#951—34no—
22C3×#951—34no—
23C2×C3×#216—32no—
24C2×C3×#214—32no—
25C2×C3×#208—31no—
26C2×C33×C4—41no—
27C2×C3×C3⋊Dic3—42no—
28C3×#634—44no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30C2×#984—33no—
31C2×C3×#137—91no—
32C2×#951—31no—
33C32×#41—32no—
34C32×#41C2×D312yes—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
37C22×C32×C4—31no—
38C2×#634—41no—
39C22×C3×Dic3—61no—
40C2×C3×Dic6—32no—
41C2×C3×Dic6—34no—
42C2×C3×Dic6—62no—
43C2×C3×Dic6—61no—
44C2×C3×Dic6—123no—
45C3×#41⋊C3—22no—
46C3×#41⋊C3—26no—
47C3×#41⋊C3C2×D316yes—
48C3×#41⋊C3C22×C312yes—
49C2×C3×SL(2,𝔽3)—21no—
50C2×C3×SL(2,𝔽3)—23no—
51C2×C3×SL(2,𝔽3)C2×D313yes—
52C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
53C3⋊#226C22×C314yescomplemented
54C3×#226—312no—
55C3×#216—34no—
56C3×#216—34no—
57C3×#214—38no—
58C3×#208—34no—
59C2×C3×C3⋊C8—32no—
60C22×C33—41no—
61C33×C4—42no—
62C3×C3⋊Dic3—44no—
63C2×C3×#41—21no—
64C2×C3×#41C3×D311yescharacteristic
65C2×C3×#41C3⋊D311yescharacteristic
66C22×C3×Q8—91no—
67C22×Dic6—31no—
68C2×#41⋊C3—23no—
69C2×#41⋊C3C3×D313yes—
70#984—312no—
71C2×#226—36no—
72C3×#137—94no—
73C3×#137—94no—
74C2×C3×Q16—92no—
75C2×C3×SD16—92no—
76C2×C3×M16—91no—
77#951—34no—
78#951—34no—
79C2×#216—32no—
80C2×#214—32no—
81C2×#208—31no—
82C23×C32—31no—
83C32×Q8—31no—
84C32×Q8C22×D311yescharacteristic
85C32×D4—32no—
86C32×D4—61no—
87C2×C32×C4—31no—
88C2×C32×C4—32no—
89C2×C32×C4—32no—
90C2×C32×C4—41no—
91C2×C32×C4—43no—
92C2×C32×C4—83no—
93C2×C32×C4—81no—
94C2×C32×C4S411yescharacteristic
95C2×C3⋊Dic3—42no—
96#634—44no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—126no—
100C3×Dic6—34no—
101C3×Dic6—62no—
102C3×Dic6—64no—
103C3×Dic6—1212no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—23no—
106C3×SL(2,𝔽3)C22×D313yes—
107C3×SL(2,𝔽3)C23×C311yescharacteristic
108C3×C3⋊C8—34no—
109C2×#137—91noSylow
110C2×C33—42no—
111C2×C33—41no—
112C3×#41—22no—
113C3×#41—32no—
114C3×#41—32no—
115C3×#41—62no—
116C3×#41C2×C3×D312yes—
117C3×#41C2×C3⋊D312yes—
118C2×C3×Q8—21no—
119C2×C3×Q8—92no—
120C2×C3×Q8—94no—
121C2×C3×Q8—182no—
122C2×C3×Q8—181no—
123C2×C3×Q8C2×C3×D311yescharacteristic
124C2×C3×Q8C2×C3⋊D311yescharacteristic
125C2×C3×D4—31no—
126C2×C3×D4—31no—
127C2×C3×D4—61no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—61no—
131C22×C3×C4—181no—
132C22×Dic3—61no—
133C2×Dic6—32no—
134C2×Dic6—34no—
135C2×Dic6—62no—
136C2×Dic6—61no—
137C2×Dic6—123no—
138#41⋊C3—26no—
139#41⋊C3C2×C3×D316yes—
140C2×SL(2,𝔽3)—23no—
141C2×SL(2,𝔽3)C2×C3×D313yes—
142#226—312no—
143C3×Q16—94no—
144C3×Q16—94no—
145C3×SD16—98no—
146C3×M16—94no—
147C2×C3×C8—92no—
148#216—34no—
149#216—34no—
150#214—38no—
151#208—34no—
152C2×C3⋊C8—32no—
153C22×C32—32no—
154C22×C32—41no—
155C22×C32—43no—
156C22×C32—62no—
157C22×C32—83no—
158C22×C32—81no—
159C22×C32C2×S411yescharacteristic
160C32×C4—31no—
161C32×C4—31no—
162C32×C4—42no—
163C32×C4—46no—
164C32×C4—86no—
165C32×C4—82no—
166C32×C4C2×S412yes—
167C3⋊Dic3—44no—
168C3×Dic3—64no—
169C3×Dic3—1212no—
170C2×#41C3×C3⋊D311yescharacteristic
171C22×Q8—91no—
172#137—94no—
173#137—94no—
174C2×Q16—92no—
175C2×SD16—92no—
176C2×M16—91no—
177C33—41noSylow
178C23×C3—31no—
179C23×C3—31no—
180C23×C3—61no—
181C3×Q8—21no—
182C3×Q8—31no—
183C3×Q8—31no—
184C3×Q8—61no—
185C3×Q8—94no—
186C3×Q8—182no—
187C3×Q8—184no—
188C3×Q8C22×C3×D311yescharacteristic
189C3×Q8C22×C3⋊D311yescharacteristic
190C3×D4—32no—
191C3×D4—32no—
192C3×D4—62no—
193C3×D4—61no—
194C3×D4—61no—
195C3×D4—62no—
196C2×C3×C4—21no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—32no—
203C2×C3×C4—43no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—61no—
207C2×C3×C4—83no—
208C2×C3×C4—184no—
209C2×C3×C4—182no—
210C2×C3×C4C3×S411yescharacteristic
211C2×C3×C4C3⋊S411yescharacteristic
212C2×Dic3—64no—
213C2×Dic3—62no—
214C2×Dic3—126no—
215Dic6—34no—
216Dic6—62no—
217Dic6—64no—
218Dic6—1212no—
219SL(2,𝔽3)—23no—
220SL(2,𝔽3)C22×C3×D313yes—
221C3×C8—94no—
222C3⋊C8—34no—
223C2×C32—41no—
224C2×C32—42no—
225C2×C32—46no—
226C2×C32—43no—
227C2×C32—62no—
228C2×C32—83no—
229C2×C32—86no—
230C2×C32—81no—
231C2×C32—82no—
232C2×C32#98412yescomplemented
233C2×C32C22×S411yescharacteristic
234#41—32no—
235#41C2×C3×C3⋊D312yes—
236C2×Q8—92no—
237C2×Q8—94no—
238C2×Q8—182no—
239C2×Q8—181no—
240C2×Q8C2×C3×C3⋊D311yescharacteristic
241C2×D4—31no—
242C22×C4—31no—
243C22×C4—181no—
244Q16—94no—
245Q16—94no—
246SD16—98no—
247M16—94no—
248C2×C8—92no—
249C22×C3—21no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—43no—
253C22×C3—62no—
254C22×C3—62no—
255C22×C3—62no—
256C22×C3—83no—
257C22×C3—122no—
258C22×C3C2×C3×S411yescharacteristic
259C22×C3C2×C3⋊S411yescharacteristic
260C3×C4—22no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—46no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—86no—
269C3×C4—184no—
270C3×C4C2×C3×S412yes—
271C3×C4C2×C3⋊S412yes—
272Dic3—64no—
273Dic3—1212no—
274C32—41no—
275C32—43no—
276C32—83no—
277C32—81no—
278C32C2×#98411yescharacteristic, complemented
279C23—31no—
280Q8—31no—
281Q8—94no—
282Q8—182no—
283Q8—184no—
284Q8C22×C3×C3⋊D311yescharacteristic
285D4—32no—
286D4—61no—
287C2×C4—31no—
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—184no—
291C2×C4—182no—
292C2×C4C3×C3⋊S411yescharacteristic
293C8—94no—
294C2×C3—22no—
295C2×C3—21no—
296C2×C3—46no—
297C2×C3—43no—
298C2×C3—62no—
299C2×C3—62no—
300C2×C3—86no—
301C2×C3—83no—
302C2×C3—122no—
303C2×C3C3×#98412yescomplemented
304C2×C3#6426512yesdirect product factor
305C2×C3C22×C3×S411yescharacteristic
306C2×C3C22×C3⋊S411yescharacteristic
307C22—32no—
308C22—62no—
309C22C2×C3×C3⋊S411yescharacteristic
310C4—31no—
311C4—31no—
312C4—184no—
313C4C2×C3×C3⋊S412yes—
314C3—21no—
315C3—43no—
316C3—83no—
317C3C2×C3×#98411yescharacteristic, complemented
318C3C2×#6426511yescharacteristic, direct product factor
319C2—62no—
320C2C3×#6426512yesdirect product factor
321C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
prime radicalC2×C32×#41⋊C3C2
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64265C2direct (N × H)
3. C2×C3×C3⋊#226C2semidirect
4. C2×#64265C3direct (N × H)
10. C3×C3⋊#226C22semidirect
17. #64265C2×C3direct (N × H)
18. C2×C3⋊#226C2×C3semidirect
53. C3⋊#226C22×C3semidirect
232. C2×C32#984semidirect
278. C32C2×#984semidirect
303. C2×C3C3×#984semidirect
304. C2×C3#64265direct (N × H)
317. C3C2×C3×#984semidirect
318. C3C2×#64265direct (N × H)
320. C2C3×#64265direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×#642652
C2×C3×C3⋊#2262
C2×#642653
C2×C3×#9843
C2×C3×#9513
C2×C3×#6344