Finite Group Databasebeta

← C2×#11722391 #13164494

Subgroups — C2×#11722391 #13164494

Order 1728.

928 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 109 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×C3×#41⋊C9C211yesmaximal, characteristic, complemented
2#11722391C214yesmaximal, direct product factor
3C2×C3⋊#3689C212yesmaximal, complemented
4C2×#64158—31nomaximal
5C2×#63692—33nomaximal
6C2×C4⋊(C3⋊D9)—41nomaximal
7C3×#41⋊C9C2212yescomplemented
8C2×C3×Q8⋊C9C2211yescharacteristic, complemented
9C3⋊#3689C2214yescomplemented
10C2×C32×#41D311yescharacteristic
11C22×C4⋊(C3⋊D3)—31no—
12#64158—34no—
13#64158—34no—
14C2×#3755—32no—
15C2×#3753—32no—
16C2×#3748—31no—
17C2×#41⋊C9D313yescomplemented
18#63692—312no—
19C2×#3689—36no—
20C22×C3⋊D9—42no—
21C2×C3×C4×C9—41no—
22C4⋊(C3⋊D9)—44no—
23C3×Q8⋊C9C2311yescharacteristic, complemented
24C2×#949—93no—
25C2×#949—91no—
26C23×C3⋊D3—61no—
27C32×#41—32no—
28C32×#41C2×D312yes—
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C4⋊(C3⋊D3)—32no—
33C2×C4⋊(C3⋊D3)—34no—
34C2×C4⋊(C3⋊D3)—62no—
35C2×C4⋊(C3⋊D3)—61no—
36#3755—34no—
37#3755—34no—
38#3753—38no—
39#3748—34no—
40C2×C3⋊(C3⋊C8)—32no—
41C2×D36—123no—
42#41⋊C9C2×D316yescomplemented
43C2×Q8⋊C9C2×D313yescomplemented
44#3689—312no—
45C22×C3×C9—41no—
46C2×C3⋊D9—44no—
47C2×C3⋊D9—84no—
48C3×C4×C9—42no—
49C2×C3×#41D913yescomplemented
50C2×C3×#41C3⋊D311yescharacteristic
51C22×D12—93no—
52C22×D12—91no—
53#949—912no—
54#949—94no—
55#949—912no—
56#949—94no—
57C2×#215—96no—
58C2×#215—92no—
59C2×#213—96no—
60C2×#213—92no—
61C2×#208—93no—
62C2×#208—91no—
63C23×C32—31no—
64C22×C3⋊D3—64no—
65C22×C3⋊D3—62no—
66C22×C3⋊D3—124no—
67C32×Q8—31no—
68C32×Q8C22×D311yescharacteristic
69C32×D4—32no—
70C32×D4—61no—
71C2×C32×C4—31no—
72C2×C32×C4—32no—
73C2×C32×C4—32no—
74C2×C32×C4S411yescharacteristic
75C4⋊(C3⋊D3)—34no—
76C4⋊(C3⋊D3)—62no—
77C4⋊(C3⋊D3)—64no—
78C22×D9—126no—
79C3⋊(C3⋊C8)—34no—
80C2×C4×C9—43no—
81D36—1212no—
82Q8⋊C9C22×D313yescomplemented
83C2×AGL(1,ℤ/8)—271noSylow
84C2×C3×C9—42no—
85C2×C3×C9—41no—
86C3⋊D9—84no—
87C23×D3—181no—
88C23×D3—183no—
89C3×#41—36no—
90C3×#41—32no—
91C3×#41C2×D916yescomplemented
92C3×#41C2×C3⋊D312yes—
93C2×C3×Q8C2×D913yes—
94C2×C3×Q8C2×C3⋊D311yescharacteristic
95C2×C3×D4—33no—
96C2×C3×D4—31no—
97C22×C3×C4—33no—
98C22×C3×C4—31no—
99C2×D12—92no—
100C2×D12—96no—
101C2×D12—94no—
102C2×D12—912no—
103C2×D12—182no—
104C2×D12—186no—
105C2×D12—183no—
106C2×D12—181no—
107#215—912no—
108#215—94no—
109#215—912no—
110#215—94no—
111#213—924no—
112#213—98no—
113#208—94no—
114#208—912no—
115C2×C3⋊C8—92no—
116C2×C3⋊C8—96no—
117C22×C32—32no—
118C22×C32—62no—
119C22×C32C2×S411yescharacteristic
120C2×C3⋊D3—64no—
121C2×C3⋊D3—128no—
122C2×C3⋊D3—124no—
123C32×C4—31no—
124C32×C4—31no—
125C32×C4C2×S412yes—
126C22×C9—43no—
127C2×D9—1212no—
128C2×D9—2412no—
129C4×C9—46no—
130C2×#41C3⋊D911yescharacteristic, complemented
131C22×D4—271no—
132AGL(1,ℤ/8)—274no—
133AGL(1,ℤ/8)—274no—
134C2×SD16—272no—
135C2×D8—272no—
136C2×M16—271no—
137C3×C9—41noSylow
138C23×C3—31no—
139C23×C3—33no—
140C22×D3—184no—
141C22×D3—1812no—
142C22×D3—182no—
143C22×D3—186no—
144C22×D3—364no—
145C22×D3—3612no—
146C3×Q8—33no—
147C3×Q8—31no—
148C3×Q8C22×D913yes—
149C3×Q8C22×C3⋊D311yescharacteristic
150C3×D4—36no—
151C3×D4—32no—
152C3×D4—61no—
153C3×D4—63no—
154C2×C3×C4—31no—
155C2×C3×C4—33no—
156C2×C3×C4—32no—
157C2×C3×C4—32no—
158C2×C3×C4—36no—
159C2×C3×C4—36no—
160C2×C3×C4(C22⋊C9)⋊C213yes—
161C2×C3×C4C3⋊S411yescharacteristic
162D12—94no—
163D12—912no—
164D12—182no—
165D12—186no—
166D12—184no—
167D12—1812no—
168C3⋊C8—94no—
169C3⋊C8—912no—
170C2×C32—62no—
171C2×C32#98612yes—
172C2×C32C22×S411yescharacteristic
173C3⋊D3—124no—
174C2×C9—46no—
175C2×C9—43no—
176D9—2412no—
177C24—541no—
178#41—32no—
179#41C2×C3⋊D912yescomplemented
180C2×Q8C2×C3⋊D911yescharacteristic
181C2×D4—31no—
182C2×D4—272no—
183C2×D4—274no—
184C2×D4—542no—
185C2×D4—541no—
186C22×C4—31no—
187SD16—274no—
188SD16—274no—
189D8—278no—
190M16—274no—
191C2×C8—272no—
192C22×C3—32no—
193C22×C3—36no—
194C22×C3—62no—
195C22×C3—66no—
196C22×C3C2×(C22⋊C9)⋊C213yes—
197C22×C3C2×C3⋊S411yescharacteristic
198C2×D3—184no—
199C2×D3—1812no—
200C2×D3—3624no—
201C2×D3—368no—
202C2×D3—364no—
203C2×D3—3612no—
204C3×C4—31no—
205C3×C4—31no—
206C3×C4—33no—
207C3×C4—33no—
208C3×C4C2×(C22⋊C9)⋊C216yes—
209C3×C4C2×C3⋊S412yes—
210C32C2×#98611yescharacteristic
211C9—43no—
212C23—31no—
213C23—544no—
214C23—542no—
215C23—1084no—
216Q8—31no—
217Q8C22×C3⋊D911yescharacteristic
218D4—32no—
219D4—61no—
220D4—274no—
221D4—542no—
222D4—544no—
223C2×C4—31no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
227C8—274no—
228C2×C3—66no—
229C2×C3—62no—
230C2×C3#6369216yescomplemented
231C2×C3C22×(C22⋊C9)⋊C213yes—
232C2×C3#6426712yes—
233C2×C3C22×C3⋊S411yescharacteristic
234D3—3612no—
235D3—364no—
236C22—32no—
237C22—62no—
238C22—544no—
239C22—1088no—
240C22—1084no—
241C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
242C4—31no—
243C4—31no—
244C4C2×C3⋊((C22⋊C9)⋊C2)12yes—
245C3C2×#6369213yescomplemented
246C3C2×#6426711yescharacteristic
247C2—62no—
248C2—1084no—
249C2#1172239112yesdirect product factor
250C2C22×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×Q8⋊C9C23
FrattiniC2×C3C22×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. #11722391C2direct (N × H)
3. C2×C3⋊#3689C2semidirect
7. C3×#41⋊C9C22semidirect
8. C2×C3×Q8⋊C9C22semidirect
9. C3⋊#3689C22semidirect
17. C2×#41⋊C9D3semidirect
23. C3×Q8⋊C9C23semidirect
42. #41⋊C9C2×D3semidirect
43. C2×Q8⋊C9C2×D3semidirect
49. C2×C3×#41D9semidirect
82. Q8⋊C9C22×D3semidirect
91. C3×#41C2×D9semidirect
130. C2×#41C3⋊D9semidirect
179. #41C2×C3⋊D9semidirect
230. C2×C3#63692semidirect
245. C3C2×#63692semidirect
249. C2#11722391direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
#117223912
C2×C3⋊#36892
C2×#641583
C2×#636923
C2×C4⋊(C3⋊D9)4