Finite Group Databasebeta

← C3×#10592667 #13163025

Subgroups — C3×#10592667 #13163025

Order 1728.

1052 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 214 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×C33×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic, complemented
4C3×#63636C212yesmaximal, complemented
5C3×#63635C212yesmaximal, complemented
6#10592667C311yesmaximal, characteristic, direct product factor
7C3×#4968—34nomaximal
8C22×C33×C4—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C33×C4⋊C4—21no—
13C33×C4⋊C4C2212yes—
14C3×#3752—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C2×C32×C4⋊C4—24no—
18C2×C32×C4⋊C4D314yescomplemented
19C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
20#64139C2×C311yescharacteristic
21C3×#930—34no—
22#63657C2×C311yescharacteristic, complemented
23#63636C2×C312yescomplemented
24#63635C2×C312yescomplemented
25C3×#830—34no—
26C3×#809—38no—
27C3×#808—38no—
28C23×C33D411yescharacteristic
29C22×C3×C3⋊D3—21no—
30C2×C33×C4—22no—
31C2×C33×C4—22no—
32C2×C33×C4—42no—
33C2×C33×C4D411yescharacteristic
34C2×C33×C4C2311yescharacteristic
35C3×#638—42no—
36C2×C3×C3⋊Dic3—21no—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C3⋊(C3⋊C8)—22no—
39C3×#481—91no—
40#4968—34no—
41C22×C32×C4—24no—
42C22×C32×C4—28no—
43C22×C32×C4—24no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D314yes—
46C22×C32×C4C22×C311yescharacteristic
47C2×#638—21no—
48C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
49C2×C3×#67—64no—
50C2×C3×D12—34no—
51C32×C4⋊C4—28no—
52C32×C4⋊C4—28no—
53C32×C4⋊C4—24no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4C2×D318yes—
56C32×C4⋊C4C22×C312yes—
57#3752—21no—
58C4⋊(C3⋊Dic3)C22×C311yescharacteristic
59C2×C3⋊(C3⋊C8)C22×C312yes—
60C3×(C2×D3)⋊C4—64no—
61C3×C4⋊Dic3—34no—
62C2×C3×C3⋊C8—38no—
63C22×C33—21no—
64C22×C33—21no—
65C22×C33SD1612yes—
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3—41no—
68C2×C3×C3⋊D3—41no—
69C2×C3×C3⋊D3—41no—
70C33×C4—21no—
71C33×C4—42no—
72C33×C4#4112yes—
73C3×C3⋊Dic3—41no—
74C3×#121—91no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4C3×D314yescomplemented
77C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
78#930—34no—
79C3×#106—92no—
80C3×#102—92no—
81C3×C22⋊C8—91no—
82#830—34no—
83#809—38no—
84#808—38no—
85C23×C32—24no—
86C23×C32#6714yes—
87C23×C32C3×D411yescharacteristic
88C22×C3⋊D3—21no—
89C22×C3×D3—64no—
90C2×C32×C4—216no—
91C2×C32×C4—28no—
92C2×C32×C4—22no—
93C2×C32×C4—216no—
94C2×C32×C4—28no—
95C2×C32×C4—22no—
96C2×C32×C4—24no—
97C2×C32×C4—24no—
98C2×C32×C4—416no—
99C2×C32×C4—48no—
100C2×C32×C4—42no—
101C2×C32×C4#6714yes—
102C2×C32×C4C3×D411yescharacteristic
103C2×C32×C4C22×D314yes—
104C2×C32×C4C23×C311yescharacteristic
105#638—42no—
106C2×C3⋊Dic3—21no—
107C4⋊(C3⋊D3)—22no—
108C3×#67—128no—
109C2×C3×Dic3—64no—
110C3×D12—68no—
111C3⋊(C3⋊C8)—22no—
112C3×C3⋊C8—68no—
113#481—91noSylow
114C2×C33—22no—
115C2×C33#12311yescharacteristic
116C2×C33C2×SD1611yescharacteristic
117C2×C33AGL(1,ℤ/8)11yescharacteristic
118C3×C3⋊D3—81no—
119C2×C3×D4—91no—
120C2×C3×D4—181no—
121C22×C3×C4—28no—
122C22×C3×C4—24no—
123C22×C3×C4—21no—
124C22×C3×C4—24no—
125C22×C3×C4C2×C3×D314yes—
126C22×C3×C4C2×C3⋊D311yescharacteristic
127C2×#67—64no—
128C2×D12—34no—
129C2×C3×C8—92no—
130C3×C4⋊C4—28no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—28no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4C2×C3×D318yes—
136C3×C4⋊C4C2×C3⋊D312yes—
137C3×C22⋊C4—181no—
138(C2×D3)⋊C4—64no—
139C4⋊Dic3—34no—
140C2×C3⋊C8—38no—
141C22×C32—28no—
142C22×C32—28no—
143C22×C32—24no—
144C22×C32—24no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—28no—
148C22×C32—24no—
149C22×C32#21518yes—
150C22×C32C3×SD1612yes—
151C22×C32C2×#6714yes—
152C22×C32C2×C3×D411yescharacteristic
153C2×C3⋊D3—41no—
154C2×C3⋊D3—41no—
155C2×C3⋊D3—41no—
156C2×C3×D3—124no—
157C2×C3×D3—124no—
158C2×C3×D3—124no—
159C32×C4—28no—
160C32×C4—24no—
161C32×C4—21no—
162C32×C4—28no—
163C32×C4—416no—
164C32×C4—48no—
165C32×C4—42no—
166C32×C4#23518yes—
167C32×C4C3×#4112yes—
168C3⋊Dic3—41no—
169C3×Dic3—124no—
170#121—91no—
171C2×C4⋊C4C3×C3⋊D311yescharacteristic, complemented
172#106—92no—
173#102—92no—
174C22⋊C8—91no—
175C33#48111yescharacteristic, complemented, Sylow
176C23×C3—24no—
177C23×C3—181no—
178C23×C3C3×#6714yes—
179C23×C3#63811yescharacteristic
180C22×D3—64no—
181C3×D4—182no—
182C3×D4—362no—
183C2×C3×C4—216no—
184C2×C3×C4—28no—
185C2×C3×C4—22no—
186C2×C3×C4—216no—
187C2×C3×C4—28no—
188C2×C3×C4—22no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—416no—
192C2×C3×C4—48no—
193C2×C3×C4—42no—
194C2×C3×C4—181no—
195C2×C3×C4C3×#6714yes—
196C2×C3×C4#63811yescharacteristic
197C2×C3×C4C22×C3×D314yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199#67—128no—
200C2×Dic3—64no—
201D12—68no—
202C3×C8—182no—
203C3⋊C8—68no—
204C2×C32—216no—
205C2×C32—28no—
206C2×C32—22no—
207C2×C32—24no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32#92914yes—
211C2×C32#93214yes—
212C2×C32C2×#21514yes—
213C2×C32C3×#12311yescharacteristic
214C2×C32C2×C3×SD1611yescharacteristic
215C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
216C3⋊D3—81no—
217C3×D3—244no—
218C2×D4—91no—
219C2×D4—181no—
220C22×C4—21no—
221C22×C4C2×C3×C3⋊D311yescharacteristic
222C2×C8—92no—
223C4⋊C4—21no—
224C4⋊C4—91no—
225C4⋊C4C2×C3×C3⋊D312yes—
226C22⋊C4—181no—
227C22×C3—28no—
228C22×C3—28no—
229C22×C3—24no—
230C22×C3—24no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—28no—
234C22×C3—24no—
235C22×C3—361no—
236C22×C3—361no—
237C22×C3—361no—
238C22×C3C3×#21518yes—
239C22×C3#375512yes—
240C22×C3C2×C3×#6714yes—
241C22×C3C2×#63811yescharacteristic
242C2×D3—124no—
243C2×D3—124no—
244C2×D3—124no—
245C3×C4—28no—
246C3×C4—24no—
247C3×C4—21no—
248C3×C4—28no—
249C3×C4—416no—
250C3×C4—48no—
251C3×C4—42no—
252C3×C4—361no—
253C3×C4C3×#23518yes—
254C3×C4#382812yes—
255Dic3—124no—
256C32—24no—
257C32#496814yescomplemented
258C32C3×#48111yescharacteristic, complemented
259C23—181no—
260C23C3×#63811yescharacteristic
261D4—182no—
262D4—362no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—181no—
267C2×C4C3×#63811yescharacteristic
268C2×C4C22×C3×C3⋊D311yescharacteristic
269C8—182no—
270C2×C3—216no—
271C2×C3—28no—
272C2×C3—22no—
273C2×C3—24no—
274C2×C3—24no—
275C2×C3—24no—
276C2×C3—721no—
277C2×C3C3×#92914yes—
278C2×C3C3×#93214yes—
279C2×C3C2×C3×#21514yes—
280C2×C3#6413811yescharacteristic
281C2×C3#6414111yescharacteristic
282C2×C3C2×#375511yescharacteristic
283D3—244no—
284C22—21no—
285C22—21no—
286C22—361no—
287C22—361no—
288C22—361no—
289C22C3×#375512yes—
290C22C2×C3×#63811yescharacteristic
291C4—21no—
292C4—42no—
293C4—361no—
294C4C3×#382812yes—
295C3—24no—
296C3C3×#496814yescomplemented
297C3#1059266711yescharacteristic, direct product factor
298C2—22no—
299C2—721no—
300C2C3×#6413811yescharacteristic
301C2C3×#6414111yescharacteristic
302C2C2×C3×#375511yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC22×C33C2×D4
prime radicalC3×#64139C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#481
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C33×C4⋊C4C2semidirect
3. C3×#63657C2semidirect
4. C3×#63636C2semidirect
5. C3×#63635C2semidirect
6. #10592667C3direct (N × H)
18. C2×C32×C4⋊C4D3semidirect
19. C2×C32×C4⋊C4C2×C3semidirect
22. #63657C2×C3semidirect
23. #63636C2×C3semidirect
24. #63635C2×C3semidirect
76. C2×C3×C4⋊C4C3×D3semidirect
77. C2×C3×C4⋊C4C3⋊D3semidirect
171. C2×C4⋊C4C3×C3⋊D3semidirect
175. C33#481semidirect
257. C32#4968semidirect
258. C32C3×#481semidirect
296. C3C3×#4968semidirect
297. C3#10592667direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C3×#641392
C3×#636572
C3×#636362
C3×#636352
#105926673
C3×#49683