Finite Group Databasebeta

← C3×#10593464 #13166833

Subgroups — C3×#10593464 #13166833

Order 1728.

476 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C3×D4×#157C211yesmaximal, characteristic, complemented
2C3×#64231C211yesmaximal, characteristic, complemented
3C3×#63926C212yesmaximal, complemented
4C3×#63732C211yesmaximal, characteristic, complemented
5C3×#63729C212yesmaximal, complemented
6#10593464C311yesmaximal, characteristic, direct product factor
7C22×C3×#157—21no—
8C2×C3×GO+(4,𝔽2)—21no—
9C2×C3×GO+(4,𝔽2)—21no—
10C3×D4×C3⋊D3C2211yescharacteristic
11C3×#3802C2212yes—
12C3×C22⋊#157—21no—
13C3×C4⋊#157C2211yescharacteristic
14C3×C4×#157C2211yescharacteristic
15C32×#213—22no—
16C3×#3714—22no—
17C3×#3710C2212yes—
18D4×#157C2×C311yescharacteristic, complemented
19#64231C2×C311yescharacteristic, complemented
20C3×D3×D8—62no—
21#63926C2×C312yescomplemented
22#63732C2×C311yescharacteristic, complemented
23#63729C2×C312yescomplemented
24C22×C3×C3⋊D3—21no—
25C2×C3×D32—22no—
26C2×C3×#157—21no—
27C2×C3×#157—42no—
28C2×C3×#157D411yescharacteristic
29C2×C3×#157D411yescharacteristic
30C3×GO+(4,𝔽2)—42no—
31C3×GO+(4,𝔽2)—42no—
32C33×D4D411yescharacteristic, complemented
33C3×#638—21no—
34C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
35C3×C4×C3⋊D3C2311yescharacteristic
36C32×D12—22no—
37C3×#625—22no—
38C32×C3⋊C8—22no—
39C3×#459—91nomaximal
40C22×#157—21no—
41C2×GO+(4,𝔽2)—21no—
42C2×GO+(4,𝔽2)—21no—
43D4×C3⋊D3C22×C311yescharacteristic
44C3×D3×D4—62no—
45C3×D3×D4—62no—
46#3802C22×C312yes—
47C22⋊#157—21no—
48C4⋊#157C22×C311yescharacteristic
49C4×#157C22×C311yescharacteristic
50C32×D8—62no—
51C3×#213—22no—
52C3×#213—22no—
53C3×#213—42no—
54C3×#213—62no—
55C3×D24—62no—
56C3×D3×C8—62no—
57#3714—22no—
58#3710C22×C312yes—
59C22×C33—21no—
60C2×C3×C3⋊D3—21no—
61C2×C3×C3⋊D3—41no—
62C2×C3×C3⋊D3—41no—
63C2×C3×C3⋊D3C2×D411yescharacteristic
64C2×C32×D3—42no—
65C3×D32—42no—
66C3×D32—42no—
67C3×#157—21no—
68C3×#157—41no—
69C3×#157D812yescomplemented
70C33×C4C2×D411yescharacteristic
71C3×C3⋊Dic3#4111yescharacteristic
72C2×C3×D8—92no—
73C3×#127—91no—
74C3×C4×D4—91no—
75D3×D8—62no—
76C3×C4⋊C8—91no—
77C3×#102—92no—
78C22×C3⋊D3—21no—
79C22×C3×D3—122no—
80C22×C3×D3—122no—
81C2×D32—22no—
82C2×#157—21no—
83C2×#157—42no—
84C2×#157C3×D411yescharacteristic
85C2×#157C3×D411yescharacteristic
86GO+(4,𝔽2)—42no—
87GO+(4,𝔽2)—42no—
88C32×D4—22no—
89C32×D4—42no—
90C32×D4—62no—
91C32×D4C3×D411yescharacteristic, complemented
92#638—21no—
93C4⋊(C3⋊D3)C3×D411yescharacteristic, complemented
94C4×C3⋊D3C23×C311yescharacteristic
95C3×#67—122no—
96C3×#67—122no—
97C3×D12—22no—
98C3×D12—22no—
99C3×D12—42no—
100C3×D12—62no—
101C3×C4×D3—62no—
102#625—22no—
103C32×C8—62no—
104C3×C3⋊C8—22no—
105C3×C3⋊C8—22no—
106C3×C3⋊C8—42no—
107#459—91noSylow
108C2×C33—41no—
109C2×C33#12111yescharacteristic
110C3×C3⋊D3—41no—
111C3×C3⋊D3C2×D811yescharacteristic
112C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
113C32×D3—82no—
114C2×C3×D4—92no—
115C2×C3×D4—91no—
116C2×C3×D4—181no—
117C2×C3×D4—181no—
118C22×C3×C4—181no—
119D3×D4—62no—
120D3×D4—62no—
121C3×D8—62no—
122C3×D8—62no—
123C3×D8—122no—
124C3×D8—182no—
125C2×C3×C8—92no—
126C3×C4⋊C4—91no—
127C3×C22⋊C4—181no—
128C3×C42—91no—
129#213—22no—
130#213—62no—
131D24—62no—
132D3×C8—62no—
133C22×C32—21no—
134C22×C32—42no—
135C22×C32—82no—
136C22×C32—122no—
137C2×C3⋊D3—21no—
138C2×C3⋊D3—41no—
139C2×C3⋊D3—41no—
140C2×C3⋊D3C2×C3×D411yescharacteristic
141C2×C3×D3—42no—
142C2×C3×D3—42no—
143C2×C3×D3—62no—
144C2×C3×D3—82no—
145C2×C3×D3—122no—
146C2×C3×D3—242no—
147C2×C3×D3—242no—
148C2×C3×D3—242no—
149C2×C3×D3—242no—
150D32—42no—
151D32—42no—
152#157—21no—
153#157—41no—
154#157C3×D812yescomplemented
155C32×C4—22no—
156C32×C4—42no—
157C32×C4C2×C3×D411yescharacteristic
158C3⋊Dic3C3×#4111yescharacteristic
159C3×Dic3—62no—
160C2×D8—92no—
161#127—91no—
162C4×D4—91no—
163C4⋊C8—91no—
164#102—92no—
165C33#45911yescharacteristic, complemented, Sylow
166C23×C3—181no—
167C23×C3—182no—
168C22×D3—122no—
169C22×D3—122no—
170C3×D4—22no—
171C3×D4—42no—
172C3×D4—62no—
173C3×D4—62no—
174C3×D4—91no—
175C3×D4—122no—
176C3×D4—181no—
177C3×D4—182no—
178C3×D4—362no—
179C3×D4—362no—
180C3×D4GO+(4,𝔽2)11yescharacteristic, complemented
181C2×C3×C4—91no—
182C2×C3×C4—91no—
183C2×C3×C4—91no—
184C2×C3×C4—181no—
185C2×C3×C4—362no—
186#67—122no—
187#67—122no—
188D12—22no—
189D12—62no—
190C4×D3—62no—
191C3×C8—62no—
192C3×C8—62no—
193C3×C8—122no—
194C3⋊C8—22no—
195C2×C32—22no—
196C2×C32—42no—
197C2×C32—41no—
198C2×C32—82no—
199C2×C32—162no—
200C2×C32—242no—
201C2×C32C3×#12111yescharacteristic
202C3⋊D3—41no—
203C3⋊D3C2×C3×D811yescharacteristic
204C3⋊D3C3×AGL(1,ℤ/8)11yescharacteristic
205C3×D3—62no—
206C3×D3—62no—
207C3×D3—82no—
208C3×D3—82no—
209C3×D3—162no—
210C3×D3—242no—
211C2×D4—92no—
212C2×D4—91no—
213C2×D4—181no—
214C2×D4—181no—
215C22×C4—181no—
216D8—62no—
217D8—182no—
218C2×C8—92no—
219C4⋊C4—91no—
220C22⋊C4—181no—
221C42—91no—
222C22×C3—21no—
223C22×C3—42no—
224C22×C3—82no—
225C22×C3—91no—
226C22×C3—122no—
227C22×C3—122no—
228C22×C3—181no—
229C22×C3—242no—
230C22×C3—361no—
231C22×C3—361no—
232C22×C3—362no—
233C22×C3—362no—
234C2×D3—42no—
235C2×D3—62no—
236C2×D3—122no—
237C2×D3—242no—
238C2×D3—242no—
239C2×D3—242no—
240C2×D3—242no—
241C3×C4—22no—
242C3×C4—42no—
243C3×C4—92no—
244C3×C4—91no—
245C3×C4—181no—
246C3×C4—361no—
247C3×C4C2×GO+(4,𝔽2)11yescharacteristic
248Dic3—62no—
249C32—22no—
250C32—42no—
251C32C3×#45911yescharacteristic, complemented
252C23—181no—
253C23—182no—
254D4—62no—
255D4—91no—
256D4—181no—
257D4—182no—
258D4—362no—
259D4—362no—
260D4C3×GO+(4,𝔽2)11yescharacteristic, complemented
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—181no—
265C2×C4—362no—
266C8—62no—
267C2×C3—22no—
268C2×C3—41no—
269C2×C3—42no—
270C2×C3—82no—
271C2×C3—91no—
272C2×C3—91no—
273C2×C3—162no—
274C2×C3—242no—
275C2×C3—242no—
276C2×C3—361no—
277C2×C3—482no—
278C2×C3#6424211yescharacteristic
279D3—62no—
280D3—62no—
281D3—82no—
282D3—242no—
283C22—21no—
284C22—91no—
285C22—122no—
286C22—181no—
287C22—361no—
288C22—361no—
289C22—362no—
290C22—362no—
291C4—92no—
292C4—91no—
293C4—181no—
294C4—361no—
295C4C2×C3×GO+(4,𝔽2)11yescharacteristic
296C3—22no—
297C3—42no—
298C3#1059346411yescharacteristic, direct product factor
299C2—41no—
300C2—91no—
301C2—91no—
302C2—242no—
303C2—361no—
304C2C3×#6424211yescharacteristic
subgroupG/H
centerC2×C3#64242
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×D4D4
socleC2×C33#121
hypercenterC3×D4GO+(4,𝔽2)
hypocenterC32C3×#459
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×#157C2semidirect
2. C3×#64231C2semidirect
3. C3×#63926C2semidirect
4. C3×#63732C2semidirect
5. C3×#63729C2semidirect
6. #10593464C3direct (N × H)
18. D4×#157C2×C3semidirect
19. #64231C2×C3semidirect
21. #63926C2×C3semidirect
22. #63732C2×C3semidirect
23. #63729C2×C3semidirect
32. C33×D4D4semidirect
34. C3×C4⋊(C3⋊D3)D4semidirect
69. C3×#157D8semidirect
91. C32×D4C3×D4semidirect
93. C4⋊(C3⋊D3)C3×D4semidirect
154. #157C3×D8semidirect
165. C33#459semidirect
180. C3×D4GO+(4,𝔽2)semidirect
251. C32C3×#459semidirect
260. D4C3×GO+(4,𝔽2)semidirect
298. C3#10593464direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×#1572
C3×#642312
C3×#639262
C3×#637322
C3×#637292
#105934643
C3×#4599