Finite Group Databasebeta

← C3×#10593340 #13166776

Subgroups — C3×#10593340 #13166776

Order 1728.

536 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 90 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×C3×C4⋊#157C211yesmaximal, characteristic
2C2×C3×#3773C211yesmaximal, characteristic
3C3×#63883C212yesmaximal
4C3×#63737C212yesmaximal
5C3×#63737C211yesmaximal, characteristic
6#10593340C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2211yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#624C2212yes—
11C2×C3×#624C2212yes—
12C3×C4⋊#157—22no—
13C3×#3773—22no—
14C32×Dic3⋊C4—22no—
15C3×Dic3⋊Dic3—22no—
16C2×C4⋊#157C2×C311yescharacteristic
17C2×#3773C2×C311yescharacteristic
18C3×D3×C4⋊C4—62no—
19#63883C2×C312yes—
20#63737C2×C312yes—
21#63737C2×C311yescharacteristic
22C22×C3×C3⋊D3C2311yescharacteristic
23C2×C3×#157—22no—
24C2×C3×#157—21no—
25C2×C3×#157—22no—
26C2×C3×#157D412yes—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3D411yescharacteristic
29C3×C4×C3⋊D3—21no—
30C3×C4×C3⋊D3—21no—
31C2×C32×Dic3—22no—
32C2×C32×Dic3—22no—
33C3×#624—22no—
34C3×#624—22no—
35C3×#624—22no—
36C3×#624Q814yes—
37C3×#398—91nomaximal
38C22×#157C22×C311yescharacteristic
39C22×#157C22×C311yescharacteristic
40C2×C4×C3⋊D3C22×C311yescharacteristic
41C2×C3×C4×D3—62no—
42C2×C3×C4×D3—62no—
43C2×C3×C4×D3—62no—
44C2×#624C22×C312yes—
45C2×#624C22×C312yes—
46C4⋊#157—22no—
47#3773—22no—
48C32×C4⋊C4—62no—
49C3×C4⋊Dic3—62no—
50C3×Dic3⋊C4—22no—
51C3×Dic3⋊C4—22no—
52C3×Dic3⋊C4—42no—
53C3×Dic3⋊C4—62no—
54Dic3⋊Dic3—22no—
55C22×C33C2×D411yescharacteristic
56C2×C3×C3⋊D3C2×D411yescharacteristic
57C2×C3×C3⋊D3C2×Q812yes—
58C2×C3×C3⋊D3#4112yes—
59C2×C3×C3⋊D3#4111yescharacteristic
60C3×#157—22no—
61C3×#157—41no—
62C33×C4—21no—
63C3×C3⋊Dic3—21no—
64C32×Dic3—24no—
65C32×Dic3—42no—
66C2×C3×C4⋊C4—92no—
67C2×C3×C4⋊C4—91no—
68C2×C3×C4⋊C4—91no—
69D3×C4⋊C4—62no—
70C3×#95—91no—
71C3×#95—92no—
72C22×C3⋊D3C23×C311yescharacteristic
73C22×C3×D3—62no—
74C2×#157—22no—
75C2×#157—21no—
76C2×#157—22no—
77C2×#157C3×D412yes—
78C2×C32×C4—22no—
79C2×C32×C4—42no—
80C2×C32×C4—62no—
81C2×C32×C4—62no—
82C2×C32×C4C3×D411yescharacteristic
83C2×C3⋊Dic3C3×D411yescharacteristic
84C4×C3⋊D3—21no—
85C4×C3⋊D3—21no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—42no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—62no—
93C3×C4×D3—64no—
94C3×C4×D3—64no—
95C3×C4×D3—122no—
96C3×C4×D3—122no—
97C3×C4×D3—122no—
98C3×C4×D3—122no—
99#624—22no—
100#624—22no—
101#624—22no—
102#624C3×Q814yes—
103#398—91noSylow
104C2×C33#12111yescharacteristic
105C2×C33C22⋊Q812yes—
106C3×C3⋊D3#12311yescharacteristic
107C3×C3⋊D3#12512yes—
108C3×C3⋊D3C4⋊Q811yescharacteristic
109C22×C3×C4—92no—
110C22×C3×C4—92no—
111C22×C3×C4—91no—
112C22×C3×C4—91no—
113C22×C3×C4—91no—
114C2×C4×D3—62no—
115C2×C4×D3—62no—
116C2×C4×D3—62no—
117C3×C4⋊C4—62no—
118C3×C4⋊C4—62no—
119C3×C4⋊C4—122no—
120C3×C4⋊C4—182no—
121C3×C4⋊C4—182no—
122C3×C4⋊C4—182no—
123C4⋊Dic3—62no—
124Dic3⋊C4—22no—
125Dic3⋊C4—62no—
126C22×C32—22no—
127C22×C32—42no—
128C22×C32C2×C3×D411yescharacteristic
129C2×C3⋊D3C2×C3×D411yescharacteristic
130C2×C3⋊D3C2×C3×Q812yes—
131C2×C3⋊D3C3×#4112yes—
132C2×C3⋊D3C3×#4111yescharacteristic
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—62no—
137C2×C3×D3—62no—
138C2×C3×D3—62no—
139#157—22no—
140#157—41no—
141C32×C4—21no—
142C32×C4—42no—
143C32×C4—64no—
144C32×C4—82no—
145C32×C4—122no—
146C3⋊Dic3—21no—
147C3×Dic3—24no—
148C3×Dic3—24no—
149C3×Dic3—44no—
150C3×Dic3—42no—
151C3×Dic3—42no—
152C3×Dic3—82no—
153C3×Dic3—122no—
154C2×C4⋊C4—92no—
155C2×C4⋊C4—91no—
156C2×C4⋊C4—91no—
157#95—91no—
158#95—92no—
159C33#39811yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C22×D3—62no—
162C2×C3×C4—22no—
163C2×C3×C4—42no—
164C2×C3×C4—62no—
165C2×C3×C4—62no—
166C2×C3×C4—62no—
167C2×C3×C4—62no—
168C2×C3×C4—94no—
169C2×C3×C4—92no—
170C2×C3×C4—91no—
171C2×C3×C4—122no—
172C2×C3×C4—122no—
173C2×C3×C4—181no—
174C2×C3×C4—181no—
175C2×C3×C4—182no—
176C2×C3×C4—182no—
177C2×C3×C4—182no—
178C2×C3×C4—182no—
179C2×C3×C4—181no—
180C2×C3×C4—182no—
181C2×C3×C4GO+(4,𝔽2)11yescharacteristic
182C2×Dic3—22no—
183C2×Dic3—22no—
184C2×Dic3—62no—
185C4×D3—64no—
186C4×D3—64no—
187C4×D3—122no—
188C4×D3—122no—
189C4×D3—122no—
190C4×D3—122no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—22no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—42no—
197C2×C32C3×#12111yescharacteristic
198C2×C32C3×C22⋊Q812yes—
199C3⋊D3C3×#12311yescharacteristic
200C3⋊D3C3×#12512yes—
201C3⋊D3C3×C4⋊Q811yescharacteristic
202C3×D3—62no—
203C3×D3—62no—
204C3×D3—62no—
205C3×D3—62no—
206C22×C4—92no—
207C22×C4—92no—
208C22×C4—91no—
209C22×C4—91no—
210C22×C4—91no—
211C4⋊C4—62no—
212C4⋊C4—182no—
213C4⋊C4—182no—
214C4⋊C4—182no—
215C22×C3—22no—
216C22×C3—42no—
217C22×C3—92no—
218C22×C3—92no—
219C22×C3—91no—
220C22×C3—91no—
221C22×C3C2×GO+(4,𝔽2)11yescharacteristic
222C2×D3—62no—
223C2×D3—62no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—62no—
228C3×C4—21no—
229C3×C4—42no—
230C3×C4—64no—
231C3×C4—64no—
232C3×C4—82no—
233C3×C4—124no—
234C3×C4—122no—
235C3×C4—122no—
236C3×C4—182no—
237C3×C4—181no—
238C3×C4—242no—
239C3×C4—361no—
240Dic3—24no—
241Dic3—42no—
242Dic3—122no—
243C32—22no—
244C32—42no—
245C32C3×#39811yescharacteristic, complemented
246C23—91no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—94no—
250C2×C4—92no—
251C2×C4—91no—
252C2×C4—181no—
253C2×C4—181no—
254C2×C4—182no—
255C2×C4—182no—
256C2×C4—182no—
257C2×C4—182no—
258C2×C4—181no—
259C2×C4—182no—
260C2×C4C3×GO+(4,𝔽2)11yescharacteristic
261C2×C3—22no—
262C2×C3—22no—
263C2×C3—22no—
264C2×C3—42no—
265C2×C3—42no—
266C2×C3—42no—
267C2×C3—91no—
268C2×C3—91no—
269C2×C3—92no—
270C2×C3D32⋊Q812yes—
271C2×C3#6424211yescharacteristic
272D3—62no—
273D3—62no—
274D3—62no—
275D3—62no—
276C22—91no—
277C22—91no—
278C22—92no—
279C22—92no—
280C22C2×C3×GO+(4,𝔽2)11yescharacteristic
281C4—21no—
282C4—64no—
283C4—122no—
284C4—182no—
285C4—181no—
286C4—361no—
287C3—22no—
288C3—42no—
289C3#1059334011yescharacteristic, direct product factor
290C2—91no—
291C2—91no—
292C2—92no—
293C2C3×D32⋊Q812yes—
294C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#398
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593340C3direct (N × H)
159. C33#398semidirect
245. C32C3×#398semidirect
289. C3#10593340direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×#37732
C3×#638832
C3×#637372
#105933403
C3×#3989