Finite Group Databasebeta

← C2×#64267⋊C3 #13164429

Subgroups — C2×#64267⋊C3 #13164429

Order 1728.

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

normal subgroups only

This group has 682 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, complemented
2#64267⋊C3C214yesmaximal, direct product factor
3C2×(C3⋊GL(2,𝔽3))⋊C3C212yesmaximal, complemented
4C2×#64267C311yesmaximal, characteristic, complemented
5C2×C3×#949—31nomaximal
6C2×C4⋊((C3⋊D3)⋊C3)—41nomaximal
7#41⋊H(3,𝔽3)C2212yescomplemented
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic, complemented
9(C3⋊GL(2,𝔽3))⋊C3C2214yescomplemented
10C2×C32×#41D311yescharacteristic, complemented
11C22×C3×D12—31no—
12C2×C3×#41⋊C3—21no—
13C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
14#64267C2×C314yescomplemented
15C2×C3⋊GL(2,𝔽3)C2×C312yescomplemented
16C3×#949—34no—
17C3×#949—34no—
18C2×C3×#215—32no—
19C2×C3×#213—32no—
20C2×C3×#208—31no—
21C22×(C3⋊D3)⋊C3—42no—
22C2×C4×H(3,𝔽3)—41no—
23C4⋊((C3⋊D3)⋊C3)—44no—
24Q8⋊H(3,𝔽3)C2311yescharacteristic, complemented
25C2×#986—91no—
26C2×C3×AGL(1,ℤ/8)—91no—
27C2×#949—31no—
28C23×C3×D3—61no—
29C32×#41—32no—
30C32×#41C2×D312yescomplemented
31C2×C32×Q8C2×D311yescharacteristic
32C2×C32×D4—31no—
33C22×C32×C4—31no—
34C2×C4⋊(C3⋊D3)—41no—
35C2×C3×D12—32no—
36C2×C3×D12—34no—
37C2×C3×D12—62no—
38C2×C3×D12—61no—
39C3×#41⋊C3—22no—
40C3×#41⋊C3C22×C312yescomplemented
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)C22×C311yescharacteristic, complemented
43C3⋊GL(2,𝔽3)C22×C314yescomplemented
44C3×#215—34no—
45C3×#215—34no—
46C3×#213—38no—
47C3×#208—34no—
48C2×C3×C3⋊C8—32no—
49C22×H(3,𝔽3)—41no—
50C2×(C3⋊D3)⋊C3—44no—
51C2×(C3⋊D3)⋊C3—84no—
52C4×H(3,𝔽3)—42no—
53C2×C3×#41—31no—
54C2×C3×#41C3×D311yescharacteristic
55C22×C3×D4—91no—
56C22×D12—31no—
57C2×#41⋊C3—31no—
58C2×#41⋊C3—61no—
59#986—94no—
60C2×GL(2,𝔽3)—92no—
61C3×AGL(1,ℤ/8)—94no—
62C3×AGL(1,ℤ/8)—94no—
63C2×C3×SD16—92no—
64C2×C3×D8—92no—
65C2×C3×M16—91no—
66#949—34no—
67#949—34no—
68C2×#215—32no—
69C2×#213—32no—
70C2×#208—31no—
71C23×C32—31no—
72C22×C3⋊D3—42no—
73C22×C3×D3—64no—
74C22×C3×D3—62no—
75C22×C3×D3—124no—
76C32×Q8—31no—
77C32×Q8C22×D311yescharacteristic
78C32×D4—32no—
79C32×D4—61no—
80C2×C32×C4—31no—
81C2×C32×C4—32no—
82C2×C32×C4—32no—
83C2×C32×C4—41no—
84C2×C32×C4—81no—
85C2×C32×C4S411yescharacteristic
86C4⋊(C3⋊D3)—44no—
87C3×D12—34no—
88C3×D12—62no—
89C3×D12—64no—
90C3×SL(2,𝔽3)—21no—
91C3×SL(2,𝔽3)C23×C311yescharacteristic, complemented
92C3×C3⋊C8—34no—
93C2×AGL(1,ℤ/8)—91noSylow
94C2×H(3,𝔽3)—42no—
95C2×H(3,𝔽3)—41no—
96(C3⋊D3)⋊C3—84no—
97C24×C3—181no—
98C23×D3—61no—
99C3×#41—32no—
100C3×#41—32no—
101C3×#41—32no—
102C3×#41—62no—
103C3×#41C2×C3×D312yes—
104C2×C3×Q8—31no—
105C2×C3×Q8C2×C3×D311yescharacteristic
106C2×C3×D4—31no—
107C2×C3×D4—31no—
108C2×C3×D4—61no—
109C2×C3×D4—92no—
110C2×C3×D4—94no—
111C2×C3×D4—182no—
112C2×C3×D4—181no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—61no—
116C2×D12—32no—
117C2×D12—34no—
118C2×D12—62no—
119C2×D12—61no—
120C2×D12—361no—
121#41⋊C3—32no—
122#41⋊C3—62no—
123C2×SL(2,𝔽3)—31no—
124C2×SL(2,𝔽3)—61no—
125GL(2,𝔽3)—94no—
126C3×SD16—94no—
127C3×SD16—94no—
128C3×D8—98no—
129C3×M16—94no—
130C2×C3×C8—92no—
131#215—34no—
132#215—34no—
133#213—38no—
134#208—34no—
135C2×C3⋊C8—32no—
136C22×C32—32no—
137C22×C32—41no—
138C22×C32—62no—
139C22×C32—81no—
140C22×C32C2×S411yescharacteristic
141C2×C3⋊D3—44no—
142C2×C3⋊D3—84no—
143C2×C3×D3—64no—
144C2×C3×D3—128no—
145C2×C3×D3—124no—
146C32×C4—31no—
147C32×C4—31no—
148C32×C4—42no—
149C32×C4—82no—
150C32×C4C2×S412yes—
151C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
152C22×D4—91no—
153AGL(1,ℤ/8)—94no—
154AGL(1,ℤ/8)—94no—
155C2×SD16—92no—
156C2×D8—92no—
157C2×M16—91no—
158H(3,𝔽3)—41noSylow
159C23×C3—31no—
160C23×C3—31no—
161C23×C3—61no—
162C23×C3—184no—
163C23×C3—182no—
164C23×C3—364no—
165C22×D3—64no—
166C22×D3—62no—
167C22×D3—124no—
168C22×D3—362no—
169C3×Q8—31no—
170C3×Q8—31no—
171C3×Q8—31no—
172C3×Q8—61no—
173C3×Q8C22×C3×D311yescharacteristic
174C3×D4—32no—
175C3×D4—32no—
176C3×D4—62no—
177C3×D4—61no—
178C3×D4—61no—
179C3×D4—62no—
180C3×D4—94no—
181C3×D4—182no—
182C3×D4—184no—
183C2×C3×C4—31no—
184C2×C3×C4—32no—
185C2×C3×C4—32no—
186C2×C3×C4—31no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—31no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—61no—
193C2×C3×C4—121no—
194C2×C3×C4—241no—
195C2×C3×C4C3×S411yescharacteristic
196D12—34no—
197D12—62no—
198D12—64no—
199D12—364no—
200SL(2,𝔽3)—31no—
201SL(2,𝔽3)—61no—
202C3×C8—94no—
203C3⋊C8—34no—
204C2×C32—42no—
205C2×C32—41no—
206C2×C32—62no—
207C2×C32—82no—
208C2×C32—81no—
209C2×C32#98612yescomplemented
210C2×C32C22×S411yescharacteristic
211C3⋊D3—84no—
212C3×D3—124no—
213C24—181no—
214#41—32no—
215#41C2×(C3⋊D3)⋊C312yescomplemented
216C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
217C2×D4—31no—
218C2×D4—92no—
219C2×D4—94no—
220C2×D4—182no—
221C2×D4—181no—
222C22×C4—31no—
223SD16—94no—
224SD16—94no—
225D8—98no—
226M16—94no—
227C2×C8—92no—
228C22×C3—32no—
229C22×C3—32no—
230C22×C3—31no—
231C22×C3—62no—
232C22×C3—62no—
233C22×C3—62no—
234C22×C3—122no—
235C22×C3—121no—
236C22×C3—184no—
237C22×C3—241no—
238C22×C3—364no—
239C22×C3—368no—
240C22×C3C2×C3×S411yescharacteristic
241C2×D3—64no—
242C2×D3—128no—
243C2×D3—124no—
244C2×D3—364no—
245C2×D3—724no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—32no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—122no—
254C3×C4—242no—
255C3×C4C2×C3×S412yes—
256C32—41no—
257C32—81no—
258C32C2×#98611yescharacteristic, complemented
259C23—31no—
260C23—184no—
261C23—182no—
262C23—364no—
263Q8—31no—
264Q8C22×(C3⋊D3)⋊C311yescharacteristic
265D4—32no—
266D4—61no—
267D4—94no—
268D4—182no—
269D4—184no—
270C2×C4—31no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4(C3⋊S4)⋊C311yescharacteristic
274C8—94no—
275C2×C3—31no—
276C2×C3—32no—
277C2×C3—62no—
278C2×C3—62no—
279C2×C3—122no—
280C2×C3—121no—
281C2×C3—122no—
282C2×C3—241no—
283C2×C3—242no—
284C2×C3—364no—
285C2×C3C3×#98612yes—
286C2×C3C22×C3×S411yescharacteristic
287D3—124no—
288D3—724no—
289C22—32no—
290C22—62no—
291C22—184no—
292C22—368no—
293C22—364no—
294C22C2×(C3⋊S4)⋊C311yescharacteristic
295C4—31no—
296C4—31no—
297C4C2×(C3⋊S4)⋊C312yes—
298C3—31no—
299C3—121no—
300C3—241no—
301C3C2×C3×#98611yescharacteristic
302C2—62no—
303C2—364no—
304C2#64267⋊C312yesdirect product factor
305C2C22×(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
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
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. #64267⋊C3C2direct (N × H)
3. C2×(C3⋊GL(2,𝔽3))⋊C3C2semidirect
4. C2×#64267C3semidirect
7. #41⋊H(3,𝔽3)C22semidirect
8. C2×Q8⋊H(3,𝔽3)C22semidirect
9. (C3⋊GL(2,𝔽3))⋊C3C22semidirect
10. C2×C32×#41D3semidirect
13. C2×C3×#41⋊C3C2×C3semidirect
14. #64267C2×C3semidirect
15. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
24. Q8⋊H(3,𝔽3)C23semidirect
30. C32×#41C2×D3semidirect
40. C3×#41⋊C3C22×C3semidirect
42. C2×C3×SL(2,𝔽3)C22×C3semidirect
43. C3⋊GL(2,𝔽3)C22×C3semidirect
91. C3×SL(2,𝔽3)C23×C3semidirect
151. C2×#41(C3⋊D3)⋊C3semidirect
209. C2×C32#986semidirect
215. #41C2×(C3⋊D3)⋊C3semidirect
258. C32C2×#986semidirect
304. C2#64267⋊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
#64267⋊C32
C2×(C3⋊GL(2,𝔽3))⋊C32
C2×#642673
C2×C3×#9493
C2×C4⋊((C3⋊D3)⋊C3)4