Finite Group Databasebeta

← C2×#11724344 #13181144

Subgroups — C2×#11724344 #13181144

Order 1728.

1576 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 175 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×#157⋊D3C214yesmaximal, complemented
2C2×C4⋊#1113C212yesmaximal, complemented
3C2×C3×C4×#157C211yesmaximal, characteristic, complemented
4#11724344C218yesmaximal, direct product factor
5C2×#64231—31nomaximal
6C22×#1113C2214yes—
7C22×C3×#157C2212yes—
8C2×#157⋊D3—216no—
9C2×#157⋊D3C22116yescomplemented
10C2×C3×C4⋊(C3⋊D3)—22no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6295—22no—
13C4⋊#1113—24no—
14C4⋊#1113C2218yescomplemented
15C3×C4×#157C2214yescomplemented
16C22×GO+(4,𝔽2)—34no—
17C2×#3802—32no—
18C2×D3×D12—62no—
19C2×C4×#157D311yescharacteristic, complemented
20#64231—38no—
21C22×C3×C3⋊D3—24no—
22C22×C3×C3⋊D3C2311yescharacteristic
23C2×#1113—216no—
24C2×#1113—24no—
25C2×#1113C2318yes—
26C2×C3×#157D418yescomplemented
27C2×C3×#157C2314yes—
28#157⋊D3—232no—
29C2×C33×C4D411yescharacteristic, complemented
30C2×C3×C3⋊Dic3D411yescharacteristic, complemented
31C3×C4⋊(C3⋊D3)—28no—
32C3×C4×C3⋊D3D412yescomplemented
33C3×C4×C3⋊D3C2312yes—
34#6295—28no—
35C2×#874—91nomaximal
36C22×D32—34no—
37C22×D32—64no—
38C22×#157C2×D312yes—
39C2×GO+(4,𝔽2)—316no—
40C2×GO+(4,𝔽2)—616no—
41C2×C4⋊(C3⋊D3)—22no—
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×C3×D12—62no—
44C2×C3×D12—62no—
45C2×C3×D12—122no—
46C2×C3×C4×D3—62no—
47C2×#626—64no—
48C2×#625—62no—
49#3802—38no—
50#3802—64no—
51D3×D12—68no—
52D3×D12—68no—
53C4×#157C2×D314yescomplemented
54C22×C33C2×D411yescharacteristic
55C2×C3×C3⋊D3—28no—
56C2×C3×C3⋊D3—48no—
57C2×C3×C3⋊D3C2×D411yescharacteristic
58C2×C3×C3⋊D3C2×D414yes—
59C2×C3×C3⋊D3C2411yescharacteristic
60#1113—216no—
61#1113—48no—
62C3×#157C2×D418yescomplemented
63C33×C4C2×D412yescomplemented
64C3×C3⋊Dic3C2×D412yescomplemented
65C2×D3×D4—182no—
66C22×D12—92no—
67C22×D12—94no—
68C2×C3×C42—91no—
69#874—98no—
70C22×C3⋊D3—24no—
71C22×C3⋊D3C22×D311yescharacteristic
72C22×C3×D3—62no—
73C22×C3×D3—64no—
74C22×C3×D3—64no—
75C22×C3×D3—124no—
76C2×D32—38no—
77C2×D32—68no—
78C2×D32—68no—
79C2×D32—64no—
80C2×D32—616no—
81C2×D32—1216no—
82C2×#157D1218yescomplemented
83C2×#157C22×D314yes—
84GO+(4,𝔽2)—632no—
85C2×C32×C4—22no—
86C2×C32×C4—42no—
87C2×C32×C4D1211yescharacteristic, complemented
88C2×C3⋊Dic3D1211yescharacteristic, complemented
89C4⋊(C3⋊D3)—28no—
90C4×C3⋊D3D1212yescomplemented
91C4×C3⋊D3C22×D312yes—
92C2×C3×Dic3—62no—
93C3×D12—68no—
94C3×D12—68no—
95C3×D12—128no—
96C3×C4×D3—64no—
97C3×C4×D3—64no—
98#626—616no—
99#625—68no—
100C2×#127—271noSylow
101C2×C33#12712yescomplemented
102C2×C33C22×D411yescharacteristic
103C3×C3⋊D3—48no—
104C3×C3⋊D3#12712yescomplemented
105C3×C3⋊D3C22×D412yes—
106C23×D3—94no—
107C23×D3—184no—
108C2×C3×D4—182no—
109C22×C3×C4—92no—
110C22×C3×C4—91no—
111C2×#67—184no—
112D3×D4—188no—
113D3×D4—188no—
114C2×D12—62no—
115C2×D12—62no—
116C2×D12—98no—
117C2×D12—916no—
118C2×D12—122no—
119C2×D12—184no—
120C2×D12—182no—
121C2×D12—1816no—
122C2×C4×D3—62no—
123C3×C42—94no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32C2×D1211yescharacteristic
127C2×C3⋊D3—28no—
128C2×C3⋊D3—48no—
129C2×C3⋊D3C2×D1211yescharacteristic
130C2×C3⋊D3C2×D1214yes—
131C2×C3⋊D3C23×D311yescharacteristic
132C2×C3×D3—62no—
133C2×C3×D3—68no—
134C2×C3×D3—62no—
135C2×C3×D3—68no—
136C2×C3×D3—68no—
137C2×C3×D3—128no—
138C2×C3×D3—128no—
139C2×C3×D3—128no—
140C2×C3×D3—248no—
141D32—616no—
142D32—1216no—
143D32—1216no—
144D32—128no—
145#157C2×D1218yescomplemented
146C32×C4—24no—
147C32×C4—44no—
148C32×C4C2×D1212yescomplemented
149C3⋊Dic3C2×D1212yescomplemented
150C3×Dic3—64no—
151C22×D4—272no—
152C22×D4—274no—
153#127—278no—
154C2×C42—91no—
155C33C2×#12711yescharacteristic, complemented, Sylow
156C23×C3—91no—
157C23×C3—184no—
158C22×D3—62no—
159C22×D3—64no—
160C22×D3—64no—
161C22×D3—98no—
162C22×D3—124no—
163C22×D3—188no—
164C22×D3—184no—
165C22×D3—188no—
166C22×D3—1816no—
167C22×D3—3616no—
168C3×D4—188no—
169C2×C3×C4—22no—
170C2×C3×C4—42no—
171C2×C3×C4—98no—
172C2×C3×C4—94no—
173C2×C3×C4—91no—
174C2×C3×C4—92no—
175C2×C3×C4—92no—
176C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
177#67—1816no—
178C2×Dic3—62no—
179D12—68no—
180D12—68no—
181D12—128no—
182D12—188no—
183D12—1832no—
184C4×D3—64no—
185C4×D3—64no—
186C2×C32—22no—
187C2×C32—24no—
188C2×C32—44no—
189C2×C32—42no—
190C2×C32#87412yescomplemented
191C2×C32C22×D1211yescharacteristic
192C3⋊D3—48no—
193C3⋊D3#87412yescomplemented
194C3⋊D3C22×D1212yes—
195C3×D3—64no—
196C3×D3—64no—
197C3×D3—128no—
198C3×D3—128no—
199C3×D3—248no—
200C24—274no—
201C2×D4—182no—
202C2×D4—278no—
203C2×D4—2716no—
204C2×D4—544no—
205C2×D4—542no—
206C2×D4—5416no—
207C22×C4—92no—
208C22×C4—91no—
209C42—94no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—91no—
213C22×C3—91no—
214C22×C3—94no—
215C22×C3—188no—
216C22×C3—368no—
217C22×C3C2×GO+(4,𝔽2)11yescharacteristic
218C2×D3—62no—
219C2×D3—62no—
220C2×D3—68no—
221C2×D3—68no—
222C2×D3—68no—
223C2×D3—128no—
224C2×D3—128no—
225C2×D3—128no—
226C2×D3—1816no—
227C2×D3—248no—
228C2×D3—3616no—
229C2×D3—3616no—
230C2×D3—368no—
231C3×C4—24no—
232C3×C4—44no—
233C3×C4—98no—
234C3×C4—92no—
235C3×C4C2×GO+(4,𝔽2)12yescomplemented
236Dic3—64no—
237C32—22no—
238C32—42no—
239C32C2×#87411yescharacteristic, complemented
240C23—91no—
241C23—184no—
242C23—278no—
243C23—544no—
244C23—5416no—
245D4—188no—
246D4—548no—
247D4—5432no—
248C2×C4—94no—
249C2×C4—98no—
250C2×C4—91no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4#157⋊D311yescharacteristic, complemented
254C2×C3—24no—
255C2×C3—22no—
256C2×C3—44no—
257C2×C3—42no—
258C2×C3—92no—
259C2×C3—92no—
260C2×C3—368no—
261C2×C3#6423112yescomplemented
262C2×C3C22×GO+(4,𝔽2)11yescharacteristic
263D3—64no—
264D3—64no—
265D3—128no—
266D3—128no—
267D3—248no—
268C22—94no—
269C22—91no—
270C22—91no—
271C22—188no—
272C22—368no—
273C22—5416no—
274C22—1088no—
275C22C2×#157⋊D311yescharacteristic
276C4—98no—
277C4—92no—
278C4C2×#157⋊D312yescomplemented
279C3—22no—
280C3—42no—
281C3C2×#6423111yescharacteristic, complemented
282C2—92no—
283C2—92no—
284C2—368no—
285C2#1172434412yesdirect product factor
286C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#127
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×#157⋊D3C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C2×C3×C4×#157C2semidirect
4. #11724344C2direct (N × H)
9. C2×#157⋊D3C22semidirect
14. C4⋊#1113C22semidirect
15. C3×C4×#157C22semidirect
19. C2×C4×#157D3semidirect
26. C2×C3×#157D4semidirect
29. C2×C33×C4D4semidirect
30. C2×C3×C3⋊Dic3D4semidirect
32. C3×C4×C3⋊D3D4semidirect
53. C4×#157C2×D3semidirect
62. C3×#157C2×D4semidirect
63. C33×C4C2×D4semidirect
64. C3×C3⋊Dic3C2×D4semidirect
82. C2×#157D12semidirect
87. C2×C32×C4D12semidirect
88. C2×C3⋊Dic3D12semidirect
90. C4×C3⋊D3D12semidirect
101. C2×C33#127semidirect
104. C3×C3⋊D3#127semidirect
145. #157C2×D12semidirect
148. C32×C4C2×D12semidirect
149. C3⋊Dic3C2×D12semidirect
155. C33C2×#127semidirect
176. C2×C3×C4GO+(4,𝔽2)semidirect
190. C2×C32#874semidirect
193. C3⋊D3#874semidirect
235. C3×C4C2×GO+(4,𝔽2)semidirect
239. C32C2×#874semidirect
253. C2×C4#157⋊D3semidirect
261. C2×C3#64231semidirect
278. C4C2×#157⋊D3semidirect
281. C3C2×#64231semidirect
285. C2#11724344direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
C2×C4⋊#11132
C2×C3×C4×#1572
#117243442
C2×#642313
C2×#8749