Finite Group Databasebeta

← C2×#11724251 #13180956

Subgroups — C2×#11724251 #13180956

Order 1728.

936 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 178 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×#89014C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4C2×#88925C212yesmaximal, complemented
5C2×D3×#622C212yesmaximal, complemented
6#11724251C218yesmaximal, direct product factor
7C2×#64287—31nomaximal
8C2×#6308C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C411yescharacteristic
11C2×#6290C412yes—
12C2×D3×C3⋊Dic3C2212yes—
13#89014C414yes—
14#89014C2214yes—
15C2×C3⋊#622C2211yescharacteristic
16C2×C3⋊#622C2211yescharacteristic
17#88988C2214yescomplemented
18C2×C3×#622C2212yes—
19C3×#3788C2214yescomplemented
20#88925C2218yescomplemented
21D3×#622C2218yescomplemented
22C2×#3830—31no—
23C2×#3795—62no—
24C2×#3792—32no—
25C22×#622—32no—
26#64287—38no—
27C2×#3788D311yescharacteristic, complemented
28C2×#3787—31no—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C22×C32×D3C2×C412yes—
31C2×C33×C4C2×C411yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C412yes—
33#6308C2×C414yes—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×C4×C3⋊D3C2×C412yes—
36C3×C4×C3⋊D3C2312yes—
37C32×D12C2×C414yes—
38#6290C2×C418yes—
39D3×C3⋊Dic3C2×C414yes—
40D3×C3⋊Dic3C2314yes—
41C3⋊#622C2312yes—
42C3⋊#622C2312yes—
43C3×#622C2314yes—
44C2×#907—91nomaximal
45C2×C32×D4—31no—
46C2×#638—32no—
47C22×C3⋊Dic3—32no—
48#3830—34no—
49#3830—34no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×#634—31no—
52C2×#634—62no—
53C2×#634—122no—
54C2×C3×D12—22no—
55C2×C3×C4×D3—62no—
56C2×#625—64no—
57C2×D3×Dic3—64no—
58#3795—68no—
59#3795—68no—
60#3792—38no—
61C2×#622—31no—
62C2×#622—31no—
63C2×#622—38no—
64C2×#622C2×D312yes—
65#3788C2×D314yescomplemented
66#3787—34no—
67C22×C33C22×C411yescharacteristic
68C2×C3×C3⋊D3—22no—
69C2×C3×C3⋊D3C22×C412yes—
70C2×C32×D3—24no—
71C2×C32×D3C22×C414yes—
72C33×C4C22×C412yes—
73C3⋊(C3⋊Dic3)C22×C414yes—
74C3×C3⋊Dic3C22×C411yescharacteristic
75C3×C3⋊Dic3C2411yescharacteristic
76C2×#237—182no—
77C2×#235—91no—
78C2×C3×M16—91no—
79C22×C3⋊C8—91no—
80#907—98no—
81C2×ΓL(1,𝔽25)—92no—
82C2×D3×C8—92no—
83C23×C32—32no—
84C22×C3⋊D3C4×D311yescharacteristic
85C22×C3×D3—24no—
86C22×C3×D3—62no—
87C32×D4—34no—
88C2×C32×C4—22no—
89C2×C32×C4—42no—
90C2×C32×C4C4×D311yescharacteristic
91#638—38no—
92C2×C3⋊Dic3—32no—
93C2×C3⋊Dic3—34no—
94C2×C3⋊Dic3—34no—
95C2×C3⋊Dic3—64no—
96C2×C3⋊Dic3—124no—
97C2×C3⋊Dic3C22×D311yescharacteristic
98C4×C3⋊D3C4×D312yes—
99C4×C3⋊D3C22×D312yes—
100#634—34no—
101#634—68no—
102#634—128no—
103C2×C3×Dic3—62no—
104C3×D12—28no—
105C3×C4×D3—64no—
106C3×C4×D3—64no—
107#625—616no—
108D3×Dic3—68no—
109D3×Dic3—68no—
110#622—32no—
111#622—32no—
112#622C22×D314yes—
113C2×#131—271noSylow
114C2×C33#13112yescomplemented
115C2×C33C23×C411yescharacteristic
116C3×C3⋊D3—22no—
117C32×D3—24no—
118C2×C3×D4—62no—
119C22×C3×C4—91no—
120C2×#67—92no—
121C2×#67—184no—
122C22×Dic3—184no—
123#237—188no—
124#237—188no—
125#235—94no—
126#235—94no—
127C2×D12#15711yescharacteristic
128C2×C4×D3—62no—
129C2×C4×D3—92no—
130C2×Dic6—91no—
131C2×Dic6—182no—
132C2×Dic6—362no—
133C3×M16—94no—
134C2×C3×C8—92no—
135C2×C3⋊C8—91no—
136C2×C3⋊C8—91no—
137C2×C3⋊C8—94no—
138ΓL(1,𝔽25)—98no—
139D3×C8—98no—
140C22×C32—22no—
141C22×C32—34no—
142C22×C32—42no—
143C22×C32—64no—
144C22×C32C2×C4×D311yescharacteristic
145C2×C3⋊D3—22no—
146C2×C3⋊D3C2×C4×D312yes—
147C2×C3×D3—28no—
148C2×C3×D3—48no—
149C2×C3×D3—64no—
150C2×C3×D3—124no—
151C32×C4—24no—
152C32×C4—44no—
153C32×C4C2×C4×D312yes—
154C3⋊Dic3—34no—
155C3⋊Dic3—68no—
156C3⋊Dic3—128no—
157C3⋊Dic3C2×C4×D311yescharacteristic
158C3⋊Dic3C23×D311yescharacteristic
159C3×Dic3—62no—
160C3×Dic3—62no—
161C2×#41—271no—
162#131—278no—
163C2×M16—91no—
164C2×M16—272no—
165C22×C8—272no—
166C22×C8—271no—
167C33C2×#13111yescharacteristic, complemented, Sylow
168C23×C3—64no—
169C23×C3—91no—
170C22×D3—62no—
171C22×D3C2×#15712yes—
172C3×D4—68no—
173C2×C3×C4—22no—
174C2×C3×C4—42no—
175C2×C3×C4—91no—
176C2×C3×C4—92no—
177C2×C3×C4—92no—
178C2×C3×C4C2×#15711yescharacteristic
179#67—98no—
180#67—1816no—
181C2×Dic3—62no—
182C2×Dic3—92no—
183C2×Dic3—188no—
184C2×Dic3—188no—
185C2×Dic3—184no—
186C2×Dic3—364no—
187D12C2×#15714yes—
188C4×D3—64no—
189C4×D3—64no—
190C4×D3—94no—
191C4×D3—94no—
192Dic6—94no—
193Dic6—188no—
194Dic6—368no—
195C3×C8—94no—
196C3⋊C8—92no—
197C3⋊C8—92no—
198C2×C32—24no—
199C2×C32—22no—
200C2×C32—44no—
201C2×C32—42no—
202C2×C32—64no—
203C2×C32#90712yescomplemented
204C2×C32C22×C4×D311yescharacteristic
205C3⋊D3—22no—
206C3×D3—48no—
207C3×D3—124no—
208#41—274no—
209#41—274no—
210C2×Q8—271no—
211C2×D4—31no—
212C2×D4—272no—
213C22×C4—91no—
214C22×C4—272no—
215M16—94no—
216M16—278no—
217C2×C8—92no—
218C2×C8—278no—
219C2×C8—271no—
220C2×C8—271no—
221C2×C8—274no—
222C22×C3—22no—
223C22×C3—42no—
224C22×C3—68no—
225C22×C3—92no—
226C22×C3—128no—
227C22×C3—182no—
228C22×C3C22×#15711yescharacteristic
229C2×D3—24no—
230C2×D3—64no—
231C2×D3—124no—
232C2×D3C22×#15714yes—
233C3×C4—24no—
234C3×C4—44no—
235C3×C4—91no—
236C3×C4—91no—
237C3×C4C22×#15712yes—
238Dic3—62no—
239Dic3—62no—
240Dic3—94no—
241Dic3—188no—
242Dic3—368no—
243C32—22no—
244C32—42no—
245C32C2×#90711yescharacteristic, complemented
246C23—32no—
247C23—91no—
248Q8—274no—
249D4—34no—
250D4—278no—
251C2×C4—91no—
252C2×C4—92no—
253C2×C4—92no—
254C2×C4—274no—
255C2×C4—274no—
256C2×C4—272no—
257C2×C4D3×#15711yescharacteristic
258C8—94no—
259C8—272no—
260C8—272no—
261C2×C3—22no—
262C2×C3—24no—
263C2×C3—42no—
264C2×C3—44no—
265C2×C3—128no—
266C2×C3—182no—
267C2×C3#6428712yescomplemented
268C2×C3C23×#15711yescharacteristic
269D3—24no—
270D3—124no—
271C22—34no—
272C22—64no—
273C22—92no—
274C22—182no—
275C22C2×D3×#15711yescharacteristic
276C4—91no—
277C4—91no—
278C4—274no—
279C4C2×D3×#15712yes—
280C3—22no—
281C3—42no—
282C3C2×#6428711yescharacteristic, complemented
283C2—64no—
284C2—182no—
285C2#1172425112yesdirect product factor
286C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC2×#89014C2
cosocleC3×C3⋊Dic3C24
hypercenterC2×C4D3×#157
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88988C2semidirect
3. C2×C3×#3788C2semidirect
4. C2×#88925C2semidirect
5. C2×D3×#622C2semidirect
6. #11724251C2direct (N × H)
17. #88988C22semidirect
19. C3×#3788C22semidirect
20. #88925C22semidirect
21. D3×#622C22semidirect
27. C2×#3788D3semidirect
65. #3788C2×D3semidirect
114. C2×C33#131semidirect
167. C33C2×#131semidirect
203. C2×C32#907semidirect
245. C32C2×#907semidirect
267. C2×C3#64287semidirect
282. C3C2×#64287semidirect
285. C2#11724251direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890142
C2×#889882
C2×C3×#37882
C2×#889252
C2×D3×#6222
#117242512
C2×#642873
C2×#9079