Finite Group Databasebeta

← C3×#10593527 #13166895

Subgroups — C3×#10593527 #13166895

Order 1728.

444 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 74 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×#622C211yesmaximal, characteristic
2C2×C3×#3776C211yesmaximal, characteristic, complemented
3C3×#63982C212yesmaximal
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593527C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3×#627C2212yes—
9C2×C3×#622—22no—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3776—22no—
13C32×#217—22no—
14C3×#3724C2212yes—
15C3×Dic3⋊Dic3—22no—
16C22×#622C2×C311yescharacteristic
17C2×#3776C2×C311yescharacteristic, complemented
18C3×#878—62no—
19#63982C2×C312yes—
20#63743C2×C311yescharacteristic, complemented
21#63740C2×C312yescomplemented
22C23×C33D411yescharacteristic
23C2×C3×C3⋊Dic3—21no—
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—22no—
28C2×C32×Dic3—22no—
29C3×#627—22no—
30C3×#627—22no—
31C3×#622—21no—
32C3×#622D412yes—
33C3×#467—91nomaximal
34C22×C3⋊Dic3C22×C311yescharacteristic
35C22×C3×Dic3—62no—
36C2×C3×Dic6—62no—
37C2×#627C22×C312yes—
38C2×#622—22no—
39C2×#622C22×C311yescharacteristic
40C2×#622C22×C311yescharacteristic
41#3776—22no—
42C32×C22⋊C4—62no—
43C3×#217—22no—
44C3×#217—22no—
45C3×#217—42no—
46C3×C4⋊Dic3—62no—
47C3×Dic3⋊C4—62no—
48C3×Dic3⋊C4—62no—
49#3724C22×C312yes—
50Dic3⋊Dic3—22no—
51C22×C33—21no—
52C22×C33—21no—
53C22×C33Q1612yescomplemented
54C22×C33C2×D411yescharacteristic
55C3×C3⋊Dic3—21no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3#4111yescharacteristic
58C32×Dic3—42no—
59C32×Dic3—42no—
60C2×C3×Q16—91no—
61C22×C3×C8—91no—
62C3×C22⋊Q8—92no—
63#878—62no—
64C3×#107—91no—
65C3×#103—92no—
66C23×C32—22no—
67C23×C32—42no—
68C23×C32C3×D411yescharacteristic
69C2×C32×C4—62no—
70C2×C32×C4—62no—
71C2×C3⋊Dic3—21no—
72C2×C3⋊Dic3—21no—
73C2×C3⋊Dic3C3×D411yescharacteristic
74C2×C3⋊Dic3C23×C311yescharacteristic
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—22no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—122no—
84C2×C3×Dic3—122no—
85C3×Dic6—122no—
86C3×Dic6—122no—
87#627—22no—
88#627—22no—
89#622—21no—
90#622C3×D412yes—
91#467—91noSylow
92C2×C33—22no—
93C2×C33#12111yescharacteristic
94C2×C33C2×Q1611yescharacteristic
95C2×C33#13511yescharacteristic
96C2×C3×Q8—92no—
97C22×C3×C4—91no—
98C22×Dic3—62no—
99C2×Dic6—62no—
100C3×Q16—182no—
101C2×C3×C8—91no—
102C2×C3×C8—91no—
103C2×C3×C8—182no—
104C3×C4⋊C4—92no—
105C3×C4⋊C4—182no—
106C3×C22⋊C4—62no—
107C3×C22⋊C4—62no—
108C3×C22⋊C4—122no—
109#217—22no—
110C4⋊Dic3—62no—
111Dic3⋊C4—62no—
112Dic3⋊C4—62no—
113C22×C32—22no—
114C22×C32—24no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—42no—
118C22×C32—44no—
119C22×C32—42no—
120C22×C32—42no—
121C22×C32—82no—
122C22×C32—82no—
123C22×C32C3×Q1612yescomplemented
124C22×C32C2×C3×D411yescharacteristic
125C32×C4—122no—
126C32×C4—122no—
127C3⋊Dic3—21no—
128C3⋊Dic3C2×C3×D411yescharacteristic
129C3⋊Dic3C3×#4111yescharacteristic
130C3×Dic3—42no—
131C3×Dic3—42no—
132C3×Dic3—42no—
133C3×Dic3—42no—
134C3×Dic3—62no—
135C3×Dic3—62no—
136C3×Dic3—82no—
137C3×Dic3—82no—
138C3×Dic3—122no—
139C2×Q16—91no—
140C22×C8—91no—
141C22⋊Q8—92no—
142#107—91no—
143#103—92no—
144C33#46711yescharacteristic, complemented, Sylow
145C23×C3—22no—
146C23×C3—42no—
147C23×C3GO+(4,𝔽2)11yescharacteristic
148C3×Q8—182no—
149C3×Q8—182no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—62no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—122no—
157C2×C3×C4—122no—
158C2×C3×C4—181no—
159C2×C3×C4—181no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—62no—
163C2×Dic3—62no—
164C2×Dic3—122no—
165C2×Dic3—122no—
166Dic6—122no—
167Dic6—122no—
168C3×C8—92no—
169C3×C8—181no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—22no—
173C2×C32—22no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32—42no—
177C2×C32—44no—
178C2×C32—84no—
179C2×C32C3×#12111yescharacteristic
180C2×C32C2×C3×Q1611yescharacteristic
181C2×C32C3×#13511yescharacteristic
182C2×Q8—92no—
183C22×C4—91no—
184Q16—182no—
185C2×C8—91no—
186C2×C8—91no—
187C2×C8—182no—
188C4⋊C4—92no—
189C4⋊C4—182no—
190C22⋊C4—62no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—22no—
194C22×C3—24no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—44no—
199C22×C3—82no—
200C22×C3—82no—
201C22×C3#377612yescomplemented
202C22×C3C2×GO+(4,𝔽2)11yescharacteristic
203C3×C4—91no—
204C3×C4—91no—
205C3×C4—122no—
206C3×C4—122no—
207C3×C4—122no—
208C3×C4—122no—
209C3×C4—181no—
210C3×C4—242no—
211C3×C4—242no—
212Dic3—42no—
213Dic3—42no—
214Dic3—62no—
215Dic3—62no—
216Dic3—122no—
217C32—22no—
218C32—42no—
219C32C3×#46711yescharacteristic, complemented
220C23C3×GO+(4,𝔽2)11yescharacteristic
221Q8—182no—
222Q8—182no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—181no—
228C2×C4—181no—
229C8—92no—
230C8—181no—
231C2×C3—22no—
232C2×C3—22no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—44no—
236C2×C3—42no—
237C2×C3—42no—
238C2×C3—42no—
239C2×C3—84no—
240C2×C3#6423711yescharacteristic
241C2×C3C2×#377611yescharacteristic
242C2×C3#6424211yescharacteristic
243C22—21no—
244C22—21no—
245C22C3×#377612yescomplemented
246C22C2×C3×GO+(4,𝔽2)11yescharacteristic
247C4—91no—
248C4—91no—
249C4—122no—
250C4—122no—
251C4—181no—
252C3—22no—
253C3—42no—
254C3#1059352711yescharacteristic, direct product factor
255C2—22no—
256C2C3×#6423711yescharacteristic
257C2C2×C3×#377611yescharacteristic
258C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#467
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3776C2semidirect
4. C3×#63743C2semidirect
5. C3×#63740C2semidirect
6. #10593527C3direct (N × H)
17. C2×#3776C2×C3semidirect
20. #63743C2×C3semidirect
21. #63740C2×C3semidirect
53. C22×C33Q16semidirect
123. C22×C32C3×Q16semidirect
144. C33#467semidirect
201. C22×C3#3776semidirect
219. C32C3×#467semidirect
245. C22C3×#3776semidirect
254. C3#10593527direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6222
C2×C3×#37762
C3×#639822
C3×#637432
C3×#637402
#105935273
C3×#4679