Finite Group Databasebeta

← C2×#11722394 #13164498

Subgroups — C2×#11722394 #13164498

Order 1728.

828 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 70 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#11722394C214yesmaximal, direct product factor
3C2×#88611C212yesmaximal, complemented
4C2×C3×#986—33nomaximal
5C2×C3×#949—31nomaximal
6C2×C4⋊#272—41nomaximal
7#41⋊H(3,𝔽3)C2212yescomplemented
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic, complemented
9#88611C2214yescomplemented
10C2×C32×#41D311yescharacteristic, complemented
11C22×C3×D12—31no—
12C2×C3×#41⋊C3D313yescomplemented
13C3×#986—312no—
14C2×C3×GL(2,𝔽3)—36no—
15C3×#949—34no—
16C3×#949—34no—
17C2×C3×#215—32no—
18C2×C3×#213—32no—
19C2×C3×#208—31no—
20C22×#272—42no—
21C2×C4×H(3,𝔽3)—41no—
22C4⋊#272—44no—
23Q8⋊H(3,𝔽3)C2311yescharacteristic, complemented
24C2×#986—33no—
25C2×C3×AGL(1,ℤ/8)—91no—
26C2×#949—31no—
27C23×C3×D3—61no—
28C32×#41—32no—
29C32×#41C2×D312yescomplemented
30C2×C32×Q8C2×D311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×C3×D12—32no—
34C2×C3×D12—34no—
35C2×C3×D12—62no—
36C2×C3×D12—61no—
37C2×C3×D12—123no—
38C3×#41⋊C3C2×D316yescomplemented
39C2×C3×SL(2,𝔽3)C2×D313yescomplemented
40C3×GL(2,𝔽3)—312no—
41C3×#215—34no—
42C3×#215—34no—
43C3×#213—38no—
44C3×#208—34no—
45C2×C3×C3⋊C8—32no—
46C22×H(3,𝔽3)—41no—
47C2×#272—44no—
48C2×#272—84no—
49C4×H(3,𝔽3)—42no—
50C2×C3×#41—31no—
51C2×C3×#41C3⋊D311yescharacteristic
52C22×C3×D4—91no—
53C22×D12—31no—
54C2×#41⋊C3—33no—
55#986—312no—
56C2×GL(2,𝔽3)—36no—
57C3×AGL(1,ℤ/8)—94no—
58C3×AGL(1,ℤ/8)—94no—
59C2×C3×SD16—92no—
60C2×C3×D8—92no—
61C2×C3×M16—91no—
62#949—34no—
63#949—34no—
64C2×#215—32no—
65C2×#213—32no—
66C2×#208—31no—
67C23×C32—31no—
68C22×C3×D3—64no—
69C22×C3×D3—62no—
70C22×C3×D3—126no—
71C22×C3×D3—124no—
72C32×Q8—31no—
73C32×Q8C22×D311yescharacteristic
74C32×D4—32no—
75C32×D4—61no—
76C2×C32×C4—31no—
77C2×C32×C4—32no—
78C2×C32×C4—32no—
79C2×C32×C4—43no—
80C2×C32×C4S411yescharacteristic
81C3×D12—34no—
82C3×D12—62no—
83C3×D12—64no—
84C3×D12—1212no—
85C3×SL(2,𝔽3)C22×D313yescomplemented
86C3×C3⋊C8—34no—
87C2×AGL(1,ℤ/8)—91noSylow
88C2×H(3,𝔽3)—42no—
89C2×H(3,𝔽3)—41no—
90#272—84no—
91C24×C3—181no—
92C23×D3—61no—
93C3×#41—32no—
94C3×#41—32no—
95C3×#41—32no—
96C3×#41—62no—
97C3×#41C2×C3⋊D312yes—
98C2×C3×Q8—31no—
99C2×C3×Q8C2×C3⋊D311yescharacteristic
100C2×C3×D4—31no—
101C2×C3×D4—31no—
102C2×C3×D4—61no—
103C2×C3×D4—92no—
104C2×C3×D4—94no—
105C2×C3×D4—182no—
106C2×C3×D4—181no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—61no—
110C2×D12—32no—
111C2×D12—34no—
112C2×D12—62no—
113C2×D12—61no—
114C2×D12—123no—
115#41⋊C3—36no—
116C2×SL(2,𝔽3)—33no—
117GL(2,𝔽3)—312no—
118C3×SD16—94no—
119C3×SD16—94no—
120C3×D8—98no—
121C3×M16—94no—
122C2×C3×C8—92no—
123#215—34no—
124#215—34no—
125#213—38no—
126#208—34no—
127C2×C3⋊C8—32no—
128C22×C32—32no—
129C22×C32—43no—
130C22×C32—62no—
131C22×C32C2×S411yescharacteristic
132C2×C3×D3—64no—
133C2×C3×D3—1212no—
134C2×C3×D3—128no—
135C2×C3×D3—124no—
136C2×C3×D3—2412no—
137C32×C4—31no—
138C32×C4—31no—
139C32×C4—46no—
140C32×C4C2×S412yes—
141C2×#41#27211yescharacteristic, complemented
142C22×D4—91no—
143AGL(1,ℤ/8)—94no—
144AGL(1,ℤ/8)—94no—
145C2×SD16—92no—
146C2×D8—92no—
147C2×M16—91no—
148H(3,𝔽3)—41noSylow
149C23×C3—31no—
150C23×C3—31no—
151C23×C3—61no—
152C23×C3—184no—
153C23×C3—182no—
154C23×C3—364no—
155C22×D3—64no—
156C22×D3—62no—
157C22×D3—126no—
158C22×D3—124no—
159C3×Q8—31no—
160C3×Q8—31no—
161C3×Q8—31no—
162C3×Q8—61no—
163C3×Q8C22×C3⋊D311yescharacteristic
164C3×D4—32no—
165C3×D4—32no—
166C3×D4—62no—
167C3×D4—61no—
168C3×D4—61no—
169C3×D4—62no—
170C3×D4—94no—
171C3×D4—182no—
172C3×D4—184no—
173C2×C3×C4—31no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—31no—
177C2×C3×C4—32no—
178C2×C3×C4—32no—
179C2×C3×C4—31no—
180C2×C3×C4—62no—
181C2×C3×C4—62no—
182C2×C3×C4—61no—
183C2×C3×C4—123no—
184C2×C3×C4C3⋊S411yescharacteristic
185D12—34no—
186D12—62no—
187D12—64no—
188D12—1212no—
189SL(2,𝔽3)—33no—
190C3×C8—94no—
191C3⋊C8—34no—
192C2×C32—46no—
193C2×C32—43no—
194C2×C32—62no—
195C2×C32#98612yescomplemented
196C2×C32C22×S411yescharacteristic
197C3×D3—124no—
198C3×D3—2412no—
199C24—181no—
200#41—32no—
201#41C2×#27212yescomplemented
202C2×Q8C2×#27211yescharacteristic
203C2×D4—31no—
204C2×D4—92no—
205C2×D4—94no—
206C2×D4—182no—
207C2×D4—181no—
208C22×C4—31no—
209SD16—94no—
210SD16—94no—
211D8—98no—
212M16—94no—
213C2×C8—92no—
214C22×C3—32no—
215C22×C3—32no—
216C22×C3—31no—
217C22×C3—62no—
218C22×C3—62no—
219C22×C3—62no—
220C22×C3—122no—
221C22×C3—123no—
222C22×C3—184no—
223C22×C3—368no—
224C22×C3—364no—
225C22×C3C2×C3⋊S411yescharacteristic
226C2×D3—64no—
227C2×D3—1212no—
228C2×D3—124no—
229C2×D3—128no—
230C2×D3—2412no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—32no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4—126no—
239C3×C4C2×C3⋊S412yes—
240C32—43no—
241C32C2×#98611yescharacteristic, complemented
242C23—31no—
243C23—184no—
244C23—182no—
245C23—364no—
246Q8—31no—
247Q8C22×#27211yescharacteristic
248D4—32no—
249D4—61no—
250D4—94no—
251D4—182no—
252D4—184no—
253C2×C4—31no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4#625811yescharacteristic
257C8—94no—
258C2×C3—31no—
259C2×C3—32no—
260C2×C3—62no—
261C2×C3—62no—
262C2×C3—122no—
263C2×C3—123no—
264C2×C3—126no—
265C2×C3—364no—
266C2×C3#6426712yes—
267C2×C3C22×C3⋊S411yescharacteristic
268D3—124no—
269D3—2412no—
270C22—32no—
271C22—62no—
272C22—184no—
273C22—368no—
274C22—364no—
275C22C2×#625811yescharacteristic
276C4—31no—
277C4—31no—
278C4C2×#625812yes—
279C3—31no—
280C3—123no—
281C3C2×#6426711yescharacteristic
282C2—62no—
283C2—364no—
284C2#1172239412yesdirect product factor
285C2C22×#625811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC2×C32×#41D3
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. #11722394C2direct (N × H)
3. C2×#88611C2semidirect
7. #41⋊H(3,𝔽3)C22semidirect
8. C2×Q8⋊H(3,𝔽3)C22semidirect
9. #88611C22semidirect
10. C2×C32×#41D3semidirect
12. C2×C3×#41⋊C3D3semidirect
23. Q8⋊H(3,𝔽3)C23semidirect
29. C32×#41C2×D3semidirect
38. C3×#41⋊C3C2×D3semidirect
39. C2×C3×SL(2,𝔽3)C2×D3semidirect
85. C3×SL(2,𝔽3)C22×D3semidirect
141. C2×#41#272semidirect
195. C2×C32#986semidirect
201. #41C2×#272semidirect
241. C32C2×#986semidirect
284. C2#11722394direct (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
#117223942
C2×#886112
C2×C3×#9863
C2×C3×#9493
C2×C4⋊#2724