Finite Group Databasebeta

← C3×#10590206 #13157131

Subgroups — C3×#10590206 #13157131

Order 1728.

656 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 76 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×C22⋊#157C212yesmaximal
2C2×C3×#3752C211yesmaximal, characteristic
3#10590206C311yesmaximal, characteristic, direct product factor
4C23×C3×C3⋊D3C2211yescharacteristic
5C22×C3×#157—22no—
6C22×C33×C4C411yescharacteristic, complemented
7C22×C3×C3⋊Dic3C411yescharacteristic, complemented
8C3×C22⋊#157—22no—
9C3×C22⋊#157C414yescomplemented
10C3×#3752—22no—
11C2×C22⋊#157C2×C312yes—
12C2×#3752C2×C311yescharacteristic
13C2×C3×(C2×D3)⋊C4—62no—
14C23×C33C2×C411yescharacteristic
15C22×C3×C3⋊D3—41no—
16C22×C3×C3⋊D3—41no—
17C22×C3×C3⋊D3C2×C412yes—
18C22×C3×C3⋊D3D412yes—
19C22×C3×C3⋊D3D411yescharacteristic
20C22×C3×C3⋊D3Q811yescharacteristic
21C2×C3×#157—24no—
22C2×C3×#157—42no—
23C2×C3×#157—42no—
24C2×C33×C4—21no—
25C2×C33×C4—41no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—41no—
28C3×#342—91nomaximal
29C23×C3⋊D3C22×C311yescharacteristic
30C23×C3×D3—62no—
31C22×#157—22no—
32C22×C32×C4—22no—
33C22×C32×C4—42no—
34C22×C32×C4C3×C411yescharacteristic, complemented
35C22×C3⋊Dic3C3×C411yescharacteristic, complemented
36C22×C3×Dic3—62no—
37C22⋊#157—22no—
38C22⋊#157C3×C414yescomplemented
39#3752—22no—
40C3×(C2×D3)⋊C4—68no—
41C22×C33—21no—
42C22×C33—21no—
43C22×C33C4211yescharacteristic
44C22×C33C22⋊C411yescharacteristic
45C22×C33C4⋊C411yescharacteristic
46C2×C3×C3⋊D3—22no—
47C2×C3×C3⋊D3—22no—
48C2×C3×C3⋊D3—42no—
49C2×C3×C3⋊D3—41no—
50C2×C3×C3⋊D3—41no—
51C2×C3×C3⋊D3C22⋊C412yes—
52C2×C3×C3⋊D3C4⋊C412yes—
53C3×#157—44no—
54C33×C4—41no—
55C3×C3⋊Dic3—41no—
56C2×C3×C22⋊C4—92no—
57C2×C3×C22⋊C4—91no—
58C2×(C2×D3)⋊C4—62no—
59C23×C32—22no—
60C23×C32—42no—
61C23×C32C2×C3×C411yescharacteristic
62C22×C3⋊D3—41no—
63C22×C3⋊D3—41no—
64C22×C3⋊D3C2×C3×C412yes—
65C22×C3⋊D3C3×D412yes—
66C22×C3⋊D3C3×D411yescharacteristic
67C22×C3⋊D3C3×Q811yescharacteristic
68C22×C3×D3—64no—
69C22×C3×D3—64no—
70C22×C3×D3—62no—
71C22×C3×D3—62no—
72C22×C3×D3—124no—
73C22×C3×D3—124no—
74C2×#157—24no—
75C2×#157—42no—
76C2×#157—42no—
77C2×C32×C4—21no—
78C2×C32×C4—24no—
79C2×C32×C4—44no—
80C2×C32×C4—44no—
81C2×C32×C4—48no—
82C2×C32×C4—41no—
83C2×C3⋊Dic3—21no—
84C2×C3⋊Dic3—41no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—124no—
87#342—91noSylow
88C2×C33—21no—
89C2×C33—21no—
90C2×C33#9511yescharacteristic
91C2×C33#9911yescharacteristic
92C2×C33#9911yescharacteristic
93C3×C3⋊D3—22no—
94C3×C3⋊D3—22no—
95C24×C3—91no—
96C23×D3—62no—
97C22×C3×C4—22no—
98C22×C3×C4—42no—
99C22×C3×C4—91no—
100C22×C3×C4—182no—
101C22×C3×C4#15711yescharacteristic, complemented
102C22×Dic3—62no—
103C3×C22⋊C4—94no—
104C3×C22⋊C4—182no—
105C3×C22⋊C4—182no—
106(C2×D3)⋊C4—68no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32—24no—
110C22×C32—24no—
111C22×C32—22no—
112C22×C32—22no—
113C22×C32—22no—
114C22×C32—44no—
115C22×C32—44no—
116C22×C32—42no—
117C22×C32—42no—
118C22×C32—42no—
119C22×C32C3×C4211yescharacteristic
120C22×C32C3×C22⋊C411yescharacteristic
121C22×C32C3×C4⋊C411yescharacteristic
122C2×C3⋊D3—22no—
123C2×C3⋊D3—22no—
124C2×C3⋊D3—42no—
125C2×C3⋊D3—41no—
126C2×C3⋊D3—41no—
127C2×C3⋊D3C3×C22⋊C412yes—
128C2×C3⋊D3C3×C4⋊C412yes—
129C2×C3×D3—64no—
130C2×C3×D3—64no—
131C2×C3×D3—124no—
132C2×C3×D3—124no—
133C2×C3×D3—128no—
134C2×C3×D3—124no—
135C2×C3×D3—124no—
136#157—44no—
137C32×C4—44no—
138C32×C4—48no—
139C32×C4—41no—
140C3⋊Dic3—41no—
141C3×Dic3—124no—
142C2×C22⋊C4—92no—
143C2×C22⋊C4—91no—
144C33#34211yescharacteristic, complemented, Sylow
145C23×C3—22no—
146C23×C3—42no—
147C23×C3—92no—
148C23×C3—92no—
149C23×C3—91no—
150C23×C3—91no—
151C23×C3—361no—
152C23×C3—361no—
153C23×C3C2×#15711yescharacteristic
154C22×D3—64no—
155C22×D3—64no—
156C22×D3—62no—
157C22×D3—62no—
158C22×D3—124no—
159C22×D3—124no—
160C2×C3×C4—24no—
161C2×C3×C4—21no—
162C2×C3×C4—44no—
163C2×C3×C4—44no—
164C2×C3×C4—48no—
165C2×C3×C4—41no—
166C2×C3×C4—184no—
167C2×C3×C4—181no—
168C2×C3×C4—362no—
169C2×C3×C4—362no—
170C2×C3×C4—361no—
171C2×Dic3—64no—
172C2×Dic3—124no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—24no—
176C2×C32—24no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—44no—
181C2×C32—44no—
182C2×C32—42no—
183C2×C32—42no—
184C2×C32—42no—
185C2×C32C3×#9511yescharacteristic
186C2×C32C3×#9911yescharacteristic
187C2×C32C3×#9911yescharacteristic
188C3⋊D3—22no—
189C3⋊D3—22no—
190C3×D3—124no—
191C3×D3—124no—
192C24—91no—
193C22×C4—91no—
194C22×C4—182no—
195C22×C4C3×#15711yescharacteristic, complemented
196C22⋊C4—94no—
197C22⋊C4—182no—
198C22⋊C4—182no—
199C22×C3—24no—
200C22×C3—24no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—44no—
207C22×C3—44no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—92no—
212C22×C3—92no—
213C22×C3—182no—
214C22×C3—182no—
215C22×C3—362no—
216C22×C3—361no—
217C22×C3—361no—
218C22×C3C4×#15711yescharacteristic
219C22×C3C4⋊#15711yescharacteristic
220C22×C3C22⋊#15711yescharacteristic
221C2×D3—64no—
222C2×D3—64no—
223C2×D3—124no—
224C2×D3—124no—
225C2×D3—128no—
226C2×D3—124no—
227C2×D3—124no—
228C3×C4—44no—
229C3×C4—48no—
230C3×C4—41no—
231C3×C4—364no—
232C3×C4—361no—
233Dic3—124no—
234C32—22no—
235C32—42no—
236C32C3×#34211yescharacteristic, complemented
237C23—92no—
238C23—92no—
239C23—91no—
240C23—91no—
241C23—361no—
242C23—361no—
243C23C2×C3×#15711yescharacteristic
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—184no—
247C2×C4—181no—
248C2×C4—362no—
249C2×C4—362no—
250C2×C4—361no—
251C2×C3—24no—
252C2×C3—24no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—22no—
256C2×C3—22no—
257C2×C3—22no—
258C2×C3—44no—
259C2×C3—44no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—42no—
263C2×C3—182no—
264C2×C3—182no—
265C2×C3#6377411yescharacteristic
266C2×C3#6378111yescharacteristic
267C2×C3#6378611yescharacteristic
268D3—124no—
269D3—124no—
270C22—21no—
271C22—21no—
272C22—92no—
273C22—92no—
274C22—182no—
275C22—182no—
276C22—362no—
277C22—361no—
278C22—361no—
279C22C3×C4×#15711yescharacteristic
280C22C3×C4⋊#15711yescharacteristic
281C22C3×C22⋊#15711yescharacteristic
282C4—41no—
283C4—364no—
284C4—361no—
285C3—22no—
286C3—42no—
287C3#1059020611yescharacteristic, direct product factor
288C2—21no—
289C2—21no—
290C2—182no—
291C2—182no—
292C2C3×#6377411yescharacteristic
293C2C3×#6378111yescharacteristic
294C2C3×#6378611yescharacteristic
subgroupG/H
centerC22×C3C22⋊#157
commutatorC22×C32C3×C42
FrattiniC23C2×C3×#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C3⋊D3C22×C3
hypercenterC22×C3×C4#157
hypocenterC32C3×#342
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. #10590206C3direct (N × H)
6. C22×C33×C4C4semidirect
7. C22×C3×C3⋊Dic3C4semidirect
9. C3×C22⋊#157C4semidirect
34. C22×C32×C4C3×C4semidirect
35. C22×C3⋊Dic3C3×C4semidirect
38. C22⋊#157C3×C4semidirect
101. C22×C3×C4#157semidirect
144. C33#342semidirect
195. C22×C4C3×#157semidirect
236. C32C3×#342semidirect
287. C3#10590206direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊#1572
C2×C3×#37522
#105902063
C3×#3429