Finite Group Databasebeta

← C3×#10592669 #13163000

Subgroups — C3×#10592669 #13163000

Order 1728.

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

normal subgroups only

This group has 992 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
2C3×#64133C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic
4C3×#63637C212yesmaximal, complemented
5C3×#63634C212yesmaximal, complemented
6#10592669C311yesmaximal, characteristic, direct product factor
7C3×#4970—34nomaximal
8C22×C33×C4—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C33×C4⋊C4—21no—
12C33×C4⋊C4C2212yes—
13C3×#3757—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#3750—21no—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C2×C32×C4⋊C4—24no—
18C2×C32×C4⋊C4D314yes—
19C2×C32×C4⋊C4C2×C311yescharacteristic
20#64133C2×C311yescharacteristic
21C3×#924—34no—
22#63657C2×C311yescharacteristic
23#63637C2×C312yescomplemented
24#63634C2×C312yescomplemented
25C3×#830—34no—
26C3×#810—38no—
27C3×#807—38no—
28C23×C33D411yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4—22no—
31C2×C33×C4—42no—
32C2×C33×C4D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C3×#634—22no—
37C3×C3⋊(C3⋊C8)—22no—
38C3×#484—91no—
39#4970—34no—
40C22×C32×C4—24no—
41C22×C32×C4—28no—
42C22×C32×C4—24no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D314yes—
45C22×C32×C4C22×C311yescharacteristic
46C2×#634C22×C311yescharacteristic
47C2×C3×Dic6—34no—
48C32×C4⋊C4—28no—
49C32×C4⋊C4—28no—
50C32×C4⋊C4—24no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4C2×D318yes—
53C32×C4⋊C4C22×C312yes—
54#3757—21no—
55C4⋊(C3⋊Dic3)C22×C311yescharacteristic
56#3750—21no—
57C2×C3⋊(C3⋊C8)C22×C312yes—
58C3×#217—64no—
59C3×C4⋊Dic3—34no—
60C3×Dic3⋊C4—64no—
61C2×C3×C3⋊C8—38no—
62C22×C33—21no—
63C22×C33—21no—
64C22×C33Q1612yes—
65C22×C33C2×D411yescharacteristic
66C33×C4—21no—
67C33×C4—42no—
68C33×C4#4112yes—
69C3×C3⋊Dic3—41no—
70C3×C3⋊Dic3—41no—
71C3×C22⋊Q8—91no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4C3×D314yes—
74C2×C3×C4⋊C4C3⋊D311yescharacteristic
75#924—34no—
76C3×#107—92no—
77C3×#103—92no—
78C3×C22⋊C8—91no—
79#830—34no—
80#810—38no—
81#807—38no—
82C23×C32—24no—
83C23×C32#6714yes—
84C23×C32C3×D411yescharacteristic
85C2×C32×C4—216no—
86C2×C32×C4—28no—
87C2×C32×C4—22no—
88C2×C32×C4—216no—
89C2×C32×C4—28no—
90C2×C32×C4—22no—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4—416no—
94C2×C32×C4—48no—
95C2×C32×C4—42no—
96C2×C32×C4#6714yes—
97C2×C32×C4C3×D411yescharacteristic
98C2×C32×C4C22×D314yes—
99C2×C32×C4C23×C311yescharacteristic
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102#634—22no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—64no—
105C3×Dic6—68no—
106C3⋊(C3⋊C8)—22no—
107C3×C3⋊C8—68no—
108#484—91noSylow
109C2×C33—22no—
110C2×C33#12311yescharacteristic
111C2×C33C2×Q1611yescharacteristic
112C2×C33AGL(1,ℤ/8)11yescharacteristic
113C2×C3×Q8—91no—
114C22×C3×C4—28no—
115C22×C3×C4—24no—
116C22×C3×C4—21no—
117C22×C3×C4—24no—
118C22×C3×C4C2×C3×D314yes—
119C22×C3×C4C2×C3⋊D311yescharacteristic
120C2×Dic6—34no—
121C2×C3×C8—92no—
122C3×C4⋊C4—28no—
123C3×C4⋊C4—24no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—28no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—181no—
128C3×C4⋊C4C2×C3×D318yes—
129C3×C4⋊C4C2×C3⋊D312yes—
130C3×C22⋊C4—181no—
131#217—64no—
132C4⋊Dic3—34no—
133Dic3⋊C4—64no—
134C2×C3⋊C8—38no—
135C22×C32—28no—
136C22×C32—28no—
137C22×C32—24no—
138C22×C32—24no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—28no—
142C22×C32—24no—
143C22×C32#21618yes—
144C22×C32C3×Q1612yes—
145C22×C32C2×#6714yes—
146C22×C32C2×C3×D411yescharacteristic
147C32×C4—28no—
148C32×C4—24no—
149C32×C4—21no—
150C32×C4—28no—
151C32×C4—416no—
152C32×C4—48no—
153C32×C4—42no—
154C32×C4#23518yes—
155C32×C4C3×#4112yes—
156C3⋊Dic3—41no—
157C3⋊Dic3—41no—
158C3×Dic3—124no—
159C3×Dic3—124no—
160C22⋊Q8—91no—
161C2×C4⋊C4C3×C3⋊D311yescharacteristic
162#107—92no—
163#103—92no—
164C22⋊C8—91no—
165C33#48411yescharacteristic, complemented, Sylow
166C23×C3—24no—
167C23×C3C3×#6714yes—
168C23×C3#63811yescharacteristic
169C3×Q8—182no—
170C2×C3×C4—216no—
171C2×C3×C4—28no—
172C2×C3×C4—22no—
173C2×C3×C4—216no—
174C2×C3×C4—28no—
175C2×C3×C4—22no—
176C2×C3×C4—24no—
177C2×C3×C4—24no—
178C2×C3×C4—416no—
179C2×C3×C4—48no—
180C2×C3×C4—42no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4C3×#6714yes—
184C2×C3×C4#63811yescharacteristic
185C2×C3×C4C22×C3×D314yes—
186C2×C3×C4C22×C3⋊D311yescharacteristic
187C2×Dic3—64no—
188C2×Dic3—64no—
189Dic6—68no—
190C3×C8—182no—
191C3⋊C8—68no—
192C2×C32—216no—
193C2×C32—28no—
194C2×C32—22no—
195C2×C32—24no—
196C2×C32—24no—
197C2×C32—24no—
198C2×C32#92914yes—
199C2×C32#93214yes—
200C2×C32C2×#21614yes—
201C2×C32C3×#12311yescharacteristic
202C2×C32C2×C3×Q1611yescharacteristic
203C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
204C2×Q8—91no—
205C22×C4—21no—
206C22×C4C2×C3×C3⋊D311yescharacteristic
207C2×C8—92no—
208C4⋊C4—21no—
209C4⋊C4—91no—
210C4⋊C4—181no—
211C4⋊C4C2×C3×C3⋊D312yes—
212C22⋊C4—181no—
213C22×C3—28no—
214C22×C3—28no—
215C22×C3—24no—
216C22×C3—24no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—28no—
220C22×C3—24no—
221C22×C3C3×#21618yes—
222C22×C3#375612yes—
223C22×C3C2×C3×#6714yes—
224C22×C3C2×#63811yescharacteristic
225C3×C4—28no—
226C3×C4—24no—
227C3×C4—21no—
228C3×C4—28no—
229C3×C4—416no—
230C3×C4—48no—
231C3×C4—42no—
232C3×C4—361no—
233C3×C4—361no—
234C3×C4C3×#23518yes—
235C3×C4#382812yes—
236Dic3—124no—
237Dic3—124no—
238C32—24no—
239C32#497014yescomplemented
240C32C3×#48411yescharacteristic, complemented
241C23C3×#63811yescharacteristic
242Q8—182no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—181no—
247C2×C4—181no—
248C2×C4C3×#63811yescharacteristic
249C2×C4C22×C3×C3⋊D311yescharacteristic
250C8—182no—
251C2×C3—216no—
252C2×C3—28no—
253C2×C3—22no—
254C2×C3—24no—
255C2×C3—24no—
256C2×C3—24no—
257C2×C3C3×#92914yes—
258C2×C3C3×#93214yes—
259C2×C3C2×C3×#21614yes—
260C2×C3#6413811yescharacteristic
261C2×C3#6414111yescharacteristic
262C2×C3C2×#375611yescharacteristic
263C22—21no—
264C22—21no—
265C22C3×#375612yes—
266C22C2×C3×#63811yescharacteristic
267C4—21no—
268C4—42no—
269C4—361no—
270C4—361no—
271C4C3×#382812yes—
272C3—24no—
273C3C3×#497014yescomplemented
274C3#1059266911yescharacteristic, direct product factor
275C2—22no—
276C2C3×#6413811yescharacteristic
277C2C3×#6414111yescharacteristic
278C2C2×C3×#375611yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#484
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
4. C3×#63637C2semidirect
5. C3×#63634C2semidirect
6. #10592669C3direct (N × H)
23. #63637C2×C3semidirect
24. #63634C2×C3semidirect
165. C33#484semidirect
239. C32#4970semidirect
240. C32C3×#484semidirect
273. C3C3×#4970semidirect
274. C3#10592669direct (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×#641332
C3×#636572
C3×#636372
C3×#636342
#105926693
C3×#49703