Finite Group Databasebeta

← C2×#10620157 #13049718

Subgroups — C2×#10620157 #13049718

Order 1296.

1268 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1268 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×C3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C22×C32×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4#10620157C214yesmaximal, direct product factor
5C2×#6295—34nomaximal
6C2×#6291—31nomaximal
7C2×#6290—31nomaximal
8C22×C34C2211yescharacteristic
9C2×C3×C3⋊(C3⋊D3)—22no—
10C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
11C2×C32×C3⋊D3—22no—
12C2×C32×C3⋊D3C2212yescomplemented
13C3×C3⋊(C3⋊Dic3)C2212yescomplemented
14C22×C3⋊(C3⋊D3)D311yescharacteristic, complemented
15C22×C3×C3⋊D3—31no—
16C22×C3×C3⋊D3—34no—
17C22×C3×C3⋊D3—64no—
18C22×C3×C3⋊D3D314yescomplemented
19C22×C32×D3—31no—
20C22×C32×D3—31no—
21C22×C32×D3—61no—
22C2×C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
23C2×C3×C3⋊Dic3—31no—
24C2×C3×C3⋊Dic3—34no—
25C2×C3×C3⋊Dic3—64no—
26#6295—316no—
27#6291—34no—
28#6290—34no—
29C2×C34D412yescomplemented
30C2×C34C2311yescharacteristic
31C3×C3⋊(C3⋊D3)—22no—
32C32×C3⋊D3—22no—
33C2×#638—91no—
34C2×#626—94no—
35C2×#626—91no—
36C2×#625—94no—
37C22×C33—24no—
38C22×C33—21no—
39C22×C33—24no—
40C22×C33—44no—
41C22×C33C2×D311yescharacteristic
42C22×C33C2×D311yescharacteristic
43C22×C33C2×D314yes—
44C2×C3⋊(C3⋊D3)—22no—
45C2×C3⋊(C3⋊D3)C2×D312yescomplemented
46C2×C3×C3⋊D3—28no—
47C2×C3×C3⋊D3—32no—
48C2×C3×C3⋊D3—38no—
49C2×C3×C3⋊D3—68no—
50C2×C3×C3⋊D3—616no—
51C2×C3×C3⋊D3—62no—
52C2×C3×C3⋊D3—68no—
53C2×C3×C3⋊D3C2×D318yescomplemented
54C2×C32×D3—32no—
55C2×C32×D3—32no—
56C2×C32×D3—62no—
57C2×C32×D3—64no—
58C2×C32×D3—62no—
59C2×C32×D3—62no—
60C3⋊(C3⋊Dic3)C2×D312yescomplemented
61C3×C3⋊Dic3—32no—
62C3×C3⋊Dic3—38no—
63C3×C3⋊Dic3—68no—
64C34C2×D411yescharacteristic, complemented, Sylow
65C23×C32—91no—
66C22×C3⋊D3—31no—
67C22×C3⋊D3—34no—
68C22×C3⋊D3—64no—
69C22×C3⋊D3C3⋊D311yescharacteristic, complemented
70C22×C3×D3—34no—
71C22×C3×D3—34no—
72C22×C3×D3—64no—
73C22×C3×D3—94no—
74C22×C3×D3—91no—
75C22×C3×D3—184no—
76#638—94no—
77C2×C3⋊Dic3—31no—
78C2×C3⋊Dic3—34no—
79C2×C3⋊Dic3—64no—
80C2×C3×Dic3—94no—
81C2×C3×Dic3—91no—
82C2×C3×Dic3—184no—
83#626—916no—
84#626—94no—
85#625—916no—
86C2×C33—28no—
87C2×C33—22no—
88C2×C33—28no—
89C2×C33—24no—
90C2×C33—21no—
91C2×C33—24no—
92C2×C33—48no—
93C2×C33—44no—
94C2×C33D1212yescomplemented
95C2×C33#6712yescomplemented
96C2×C33#6718yescomplemented
97C2×C33C22×D311yescharacteristic
98C2×C33C22×D311yescharacteristic
99C2×C33C22×D314yes—
100C3⋊(C3⋊D3)—22no—
101C3×C3⋊D3—28no—
102C3×C3⋊D3—616no—
103C3×C3⋊D3—62no—
104C3×C3⋊D3—68no—
105C32×D3—64no—
106C32×D3—62no—
107C32×D3—62no—
108C2×#67—271no—
109C2×#67—274no—
110C2×D12—271no—
111C22×C32—24no—
112C22×C32—24no—
113C22×C32—24no—
114C22×C32—24no—
115C22×C32—24no—
116C22×C32—44no—
117C22×C32—412no—
118C22×C32—44no—
119C22×C32—92no—
120C22×C32—182no—
121C22×C32D3214yes—
122C22×C32D3214yes—
123C22×C32D3211yescharacteristic
124C22×C32C2×C3⋊D311yescharacteristic
125C2×C3⋊D3—22no—
126C2×C3⋊D3—32no—
127C2×C3⋊D3—38no—
128C2×C3⋊D3—68no—
129C2×C3⋊D3—616no—
130C2×C3⋊D3—68no—
131C2×C3⋊D3—62no—
132C2×C3⋊D3C2×C3⋊D312yescomplemented
133C2×C3×D3—38no—
134C2×C3×D3—38no—
135C2×C3×D3—68no—
136C2×C3×D3—616no—
137C2×C3×D3—68no—
138C2×C3×D3—68no—
139C2×C3×D3—98no—
140C2×C3×D3—92no—
141C2×C3×D3—188no—
142C2×C3×D3—1816no—
143C2×C3×D3—188no—
144C2×C3×D3—182no—
145C3⋊Dic3—32no—
146C3⋊Dic3—38no—
147C3⋊Dic3—68no—
148C3×Dic3—98no—
149C3×Dic3—92no—
150C3×Dic3—188no—
151C33—24no—
152C33—21no—
153C33—24no—
154C33—44no—
155C33C2×D1211yescharacteristic, complemented
156C33C2×#6711yescharacteristic, complemented
157C33C2×#6714yescomplemented
158C23×C3—94no—
159C23×C3—271no—
160C22×D3—31no—
161C22×D3—31no—
162C22×D3—61no—
163C22×D3—91no—
164C22×D3—94no—
165C22×D3—184no—
166C2×C3×C4—271no—
167#67—274no—
168#67—2716no—
169C2×Dic3—91no—
170C2×Dic3—94no—
171C2×Dic3—184no—
172D12—274no—
173C2×C32—28no—
174C2×C32—28no—
175C2×C32—24no—
176C2×C32—24no—
177C2×C32—28no—
178C2×C32—28no—
179C2×C32—28no—
180C2×C32—24no—
181C2×C32—24no—
182C2×C32—24no—
183C2×C32—48no—
184C2×C32—44no—
185C2×C32—412no—
186C2×C32—424no—
187C2×C32—48no—
188C2×C32—44no—
189C2×C32—182no—
190C2×C32#62518yescomplemented
191C2×C32#62618yescomplemented
192C2×C32#62612yescomplemented
193C2×C32#63812yescomplemented
194C2×C32C2×D3214yes—
195C2×C32C2×D3214yes—
196C2×C32C2×D3211yescharacteristic
197C2×C32C22×C3⋊D311yescharacteristic
198C3⋊D3—22no—
199C3⋊D3—616no—
200C3⋊D3—68no—
201C3⋊D3—62no—
202C3×D3—616no—
203C3×D3—68no—
204C3×D3—68no—
205C3×D3—1816no—
206C3×D3—188no—
207C3×D3—182no—
208C2×D4—811noSylow
209C22×C3—24no—
210C22×C3—21no—
211C22×C3—24no—
212C22×C3—44no—
213C22×C3—98no—
214C22×C3—188no—
215C22×C3—272no—
216C22×C3—542no—
217C22×C3D3×C3⋊D311yescharacteristic
218C22×C3D3×C3⋊D311yescharacteristic
219C22×C3#111314yes—
220C2×D3—32no—
221C2×D3—32no—
222C2×D3—62no—
223C2×D3—64no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—92no—
227C2×D3—98no—
228C2×D3—188no—
229C2×D3—1816no—
230C2×D3—182no—
231C2×D3—188no—
232C3×C4—272no—
233Dic3—92no—
234Dic3—98no—
235Dic3—188no—
236C32—24no—
237C32—24no—
238C32—24no—
239C32—24no—
240C32—24no—
241C32—44no—
242C32—412no—
243C32—44no—
244C32C2×#62514yescomplemented
245C32C2×#62614yescomplemented
246C32C2×#62611yescharacteristic, complemented
247C32C2×#63811yescharacteristic, complemented
248C23—91no—
249C23—271no—
250D4—814no—
251C2×C4—271no—
252C2×C3—24no—
253C2×C3—21no—
254C2×C3—24no—
255C2×C3—22no—
256C2×C3—28no—
257C2×C3—28no—
258C2×C3—44no—
259C2×C3—48no—
260C2×C3—188no—
261C2×C3—542no—
262C2×C3#629012yescomplemented
263C2×C3#629112yescomplemented
264C2×C3#629518yescomplemented
265C2×C3C2×D3×C3⋊D311yescharacteristic
266C2×C3C2×D3×C3⋊D311yescharacteristic
267C2×C3C2×#111314yes—
268D3—64no—
269D3—62no—
270D3—62no—
271D3—1816no—
272D3—182no—
273D3—188no—
274C22—92no—
275C22—182no—
276C22—272no—
277C22—542no—
278C22C3⋊#111311yescharacteristic
279C4—272no—
280C3—21no—
281C3—24no—
282C3—24no—
283C3—44no—
284C3C2×#629011yescharacteristic, complemented
285C3C2×#629111yescharacteristic, complemented
286C3C2×#629514yescomplemented
287C2—182no—
288C2—542no—
289C2#1062015712yesdirect product factor
290C2C2×C3⋊#111311yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊#1113
commutator, cosocleC2×C34C23
FrattiniC2C2×C3⋊#1113
Fitting, socleC22×C34C22
hypocenterC34C2×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊(C3⋊D3)C2semidirect
2. C22×C32×C3⋊D3C2semidirect
3. C2×C3×C3⋊(C3⋊Dic3)C2semidirect
4. #10620157C2direct (N × H)
10. C2×C3×C3⋊(C3⋊D3)C22semidirect
12. C2×C32×C3⋊D3C22semidirect
13. C3×C3⋊(C3⋊Dic3)C22semidirect
14. C22×C3⋊(C3⋊D3)D3semidirect
18. C22×C3×C3⋊D3D3semidirect
22. C2×C3⋊(C3⋊Dic3)D3semidirect
29. C2×C34D4semidirect
45. C2×C3⋊(C3⋊D3)C2×D3semidirect
53. C2×C3×C3⋊D3C2×D3semidirect
60. C3⋊(C3⋊Dic3)C2×D3semidirect
64. C34C2×D4semidirect
69. C22×C3⋊D3C3⋊D3semidirect
94. C2×C33D12semidirect
95. C2×C33#67semidirect
96. C2×C33#67semidirect
132. C2×C3⋊D3C2×C3⋊D3semidirect
155. C33C2×D12semidirect
156. C33C2×#67semidirect
157. C33C2×#67semidirect
190. C2×C32#625semidirect
191. C2×C32#626semidirect
192. C2×C32#626semidirect
193. C2×C32#638semidirect
244. C32C2×#625semidirect
245. C32C2×#626semidirect
246. C32C2×#626semidirect
247. C32C2×#638semidirect
262. C2×C3#6290semidirect
263. C2×C3#6291semidirect
264. C2×C3#6295semidirect
284. C3C2×#6290semidirect
285. C3C2×#6291semidirect
286. C3C2×#6295semidirect
289. C2#10620157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊D3)2
C22×C32×C3⋊D32
C2×C3×C3⋊(C3⋊Dic3)2
#106201572
C2×#62953
C2×#62913
C2×#62903