Subgroups — #13179604

Order 1728.

3276 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
2	#11723971	C₂	1	1	yes	maximal, characteristic
3	#11723970	C₂	1	2	yes	maximal, complemented
4	C₄×#6312	C₂	1	2	yes	maximal, complemented
5	#11723965	C₂	1	1	yes	maximal, characteristic, complemented
6	#11723917	C₂	1	2	yes	maximal, complemented
7	#11723916	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
9	#11723913	C₂	1	2	yes	maximal, complemented
10	#11723911	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6308	C₂	1	1	yes	maximal, characteristic, complemented
12	#10595950	—	3	13	no	maximal
13	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
16	C₂×#6312	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
19	C₂×#6308	C₂²	1	1	yes	characteristic
20	C₃³×C₄²	C₂²	1	1	yes	characteristic
21	C₃³×C₄²	C₂²	1	1	yes	characteristic
22	C₃³×C₄²	C₂²	1	1	yes	characteristic
23	C₃³×C₄²	C₂²	1	1	yes	characteristic
24	#88864	C₂²	1	2	yes	—
25	#88859	C₂²	1	2	yes	—
26	#88859	C₂²	1	2	yes	—
27	#88859	C₂²	1	2	yes	—
28	#88859	C₂²	1	2	yes	—
29	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
30	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
31	#88857	C₂²	1	2	yes	—
32	#88857	C₂²	1	2	yes	—
33	#88857	C₂²	1	2	yes	—
34	#88857	C₂²	1	2	yes	—
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
36	C₂×C₃²×C₄²	D₃	1	13	yes	complemented
37	#64139	—	3	13	no	—
38	#64138	—	3	26	no	—
39	C₄×#638	—	3	26	no	—
40	#64133	—	3	13	no	—
41	#64085	—	3	26	no	—
42	#64084	—	3	13	no	—
43	C₄×C₄⋊(C₃⋊D₃)	—	3	13	no	—
44	#64081	—	3	26	no	—
45	#64079	—	3	13	no	—
46	C₄×#634	—	3	13	no	—
47	C₂³×C₃³	C₂³	1	1	yes	characteristic
48	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	2	yes	—
49	C₂×C₃³×C₄	—	2	2	no	—
50	C₂×C₃³×C₄	—	2	2	no	—
51	C₂×C₃³×C₄	—	2	2	no	—
52	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
55	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
58	#6312	—	2	4	no	—
59	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
60	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
61	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
62	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
63	C₄×C₃⋊(C₃⋊D₃)	—	2	4	no	—
64	#6308	—	2	2	no	—
65	#5479	—	9	13	no	—
66	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
67	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
68	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
69	C₂×#638	—	3	26	no	—
70	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
71	C₂×C₄×C₃⋊D₃	—	3	26	no	—
72	C₂×#634	—	3	13	no	—
73	C₃²×C₄²	C₂×D₃	1	13	yes	—
74	C₃²×C₄²	C₂×D₃	1	13	yes	—
75	C₃²×C₄²	C₂×D₃	1	13	yes	—
76	C₃²×C₄²	C₂×D₃	1	13	yes	—
77	#3757	—	3	26	no	—
78	#3752	—	3	26	no	—
79	#3752	—	3	26	no	—
80	#3752	—	3	26	no	—
81	#3752	—	3	26	no	—
82	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
83	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
84	#3750	—	3	26	no	—
85	#3750	—	3	26	no	—
86	#3750	—	3	26	no	—
87	#3750	—	3	26	no	—
88	C₄×C₃⋊Dic₃	—	3	26	no	—
89	C₂²×C₃³	—	2	1	no	—
90	C₂²×C₃³	—	2	1	no	—
91	C₂²×C₃³	#41	1	2	yes	—
92	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
93	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
94	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
95	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
96	C₃³×C₄	—	2	1	no	—
97	C₃³×C₄	—	2	1	no	—
98	C₃³×C₄	—	2	1	no	—
99	C₃³×C₄	—	2	1	no	—
100	C₃³×C₄	#41	1	2	yes	—
101	C₃³×C₄	#41	1	2	yes	—
102	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
103	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
104	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
105	C₂×C₃×C₄²	C₃⋊D₃	1	13	yes	complemented
106	#930	—	9	13	no	—
107	#929	—	9	26	no	—
108	C₄×#67	—	9	26	no	—
109	#924	—	9	13	no	—
110	#876	—	9	26	no	—
111	#875	—	9	13	no	—
112	C₄×D₁₂	—	9	13	no	—
113	#872	—	9	26	no	—
114	#870	—	9	13	no	—
115	C₄×Dic₆	—	9	13	no	—
116	C₂³×C₃²	C₂²×D₃	1	13	yes	—
117	C₂²×C₃⋊D₃	—	3	26	no	—
118	C₂×C₃²×C₄	—	2	26	no	—
119	C₂×C₃²×C₄	—	2	26	no	—
120	C₂×C₃²×C₄	—	2	26	no	—
121	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
123	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
124	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
125	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
126	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
127	#638	—	6	52	no	—
128	C₂×C₃⋊Dic₃	—	3	26	no	—
129	C₂×C₃⋊Dic₃	—	3	26	no	—
130	C₂×C₃⋊Dic₃	—	3	26	no	—
131	C₄⋊(C₃⋊D₃)	—	6	26	no	—
132	C₄×C₃⋊D₃	—	6	52	no	—
133	#634	—	6	26	no	—
134	#529	—	27	1	no	Sylow
135	C₂×C₃³	—	2	2	no	—
136	C₂×C₃³	C₂×#41	1	1	yes	characteristic
137	C₂×C₃³	C₂×#41	1	1	yes	characteristic
138	C₂×C₃³	C₂×#41	1	1	yes	characteristic
139	C₃⋊(C₃⋊D₃)	—	4	2	no	—
140	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
141	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
142	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
143	C₂×#67	—	9	26	no	—
144	C₂×D₁₂	—	9	13	no	—
145	C₂×C₄×D₃	—	9	26	no	—
146	C₂×Dic₆	—	9	13	no	—
147	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
148	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
149	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
150	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
151	#217	—	9	26	no	—
152	(C₂×D₃)⋊C₄	—	9	26	no	—
153	(C₂×D₃)⋊C₄	—	9	26	no	—
154	(C₂×D₃)⋊C₄	—	9	26	no	—
155	(C₂×D₃)⋊C₄	—	9	26	no	—
156	C₄⋊Dic₃	—	9	13	no	—
157	C₄⋊Dic₃	—	9	13	no	—
158	Dic₃⋊C₄	—	9	26	no	—
159	Dic₃⋊C₄	—	9	26	no	—
160	Dic₃⋊C₄	—	9	26	no	—
161	Dic₃⋊C₄	—	9	26	no	—
162	C₄×Dic₃	—	9	26	no	—
163	C₂²×C₃²	—	2	13	no	—
164	C₂²×C₃²	—	2	13	no	—
165	C₂²×C₃²	#235	1	26	yes	—
166	C₂²×C₃²	C₂³×D₃	1	13	yes	—
167	C₂×C₃⋊D₃	—	6	26	no	—
168	C₂×C₃⋊D₃	—	6	26	no	—
169	C₂×C₃⋊D₃	—	6	26	no	—
170	C₃²×C₄	—	2	13	no	—
171	C₃²×C₄	—	2	13	no	—
172	C₃²×C₄	—	2	13	no	—
173	C₃²×C₄	—	2	13	no	—
174	C₃²×C₄	#235	1	26	yes	—
175	C₃²×C₄	#235	1	26	yes	—
176	C₃⋊Dic₃	—	6	26	no	—
177	C₃⋊Dic₃	—	6	26	no	—
178	C₃⋊Dic₃	—	6	26	no	—
179	#126	—	27	2	no	—
180	#125	—	27	1	no	—
181	#124	—	27	1	no	—
182	#123	—	27	2	no	—
183	C₂²⋊Q₈	—	27	1	no	—
184	#121	—	27	1	no	—
185	C₄×Q₈	—	27	1	no	—
186	C₄×D₄	—	27	2	no	—
187	C₄×D₄	—	27	1	no	—
188	#117	—	27	2	no	—
189	C₂×C₄²	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
190	C₃³	#529	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
192	C₂²×D₃	—	9	26	no	—
193	C₂×C₃×C₄	—	2	26	no	—
194	C₂×C₃×C₄	—	2	26	no	—
195	C₂×C₃×C₄	—	2	26	no	—
196	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
197	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
198	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
199	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
200	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
201	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
202	#67	—	18	52	no	—
203	C₂×Dic₃	—	9	26	no	—
204	C₂×Dic₃	—	9	26	no	—
205	C₂×Dic₃	—	9	26	no	—
206	D₁₂	—	18	26	no	—
207	C₄×D₃	—	18	52	no	—
208	Dic₆	—	18	26	no	—
209	C₂×C₃²	—	2	26	no	—
210	C₂×C₃²	C₂×#235	1	13	yes	—
211	C₂×C₃²	C₂×#235	1	13	yes	—
212	C₂×C₃²	C₂×#235	1	13	yes	—
213	C₃⋊D₃	—	12	26	no	—
214	C₂×Q₈	—	27	1	no	—
215	C₂×D₄	—	27	2	no	—
216	C₂×D₄	—	27	1	no	—
217	C₂²×C₄	—	27	2	no	—
218	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
221	C₄⋊C₄	—	27	2	no	—
222	C₄⋊C₄	—	27	2	no	—
223	C₄⋊C₄	—	27	1	no	—
224	C₄⋊C₄	—	27	2	no	—
225	C₄⋊C₄	—	27	1	no	—
226	C₄⋊C₄	—	27	2	no	—
227	C₂²⋊C₄	—	27	2	no	—
228	C₂²⋊C₄	—	27	2	no	—
229	C₂²⋊C₄	—	27	2	no	—
230	C₂²⋊C₄	—	27	2	no	—
231	C₂²⋊C₄	—	27	2	no	—
232	C₄²	—	27	2	no	—
233	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
234	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
235	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
236	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
237	C₂²×C₃	—	2	13	no	—
238	C₂²×C₃	—	2	13	no	—
239	C₂²×C₃	#3828	1	26	yes	—
240	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
241	C₂×D₃	—	18	26	no	—
242	C₂×D₃	—	18	26	no	—
243	C₂×D₃	—	18	26	no	—
244	C₃×C₄	—	2	13	no	—
245	C₃×C₄	—	2	13	no	—
246	C₃×C₄	—	2	13	no	—
247	C₃×C₄	—	2	13	no	—
248	C₃×C₄	#3828	1	26	yes	—
249	C₃×C₄	#3828	1	26	yes	—
250	Dic₃	—	18	26	no	—
251	Dic₃	—	18	26	no	—
252	Dic₃	—	18	26	no	—
253	C₃²	#5479	1	13	yes	complemented
254	C₂³	—	27	2	no	—
255	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
256	Q₈	—	54	2	no	—
257	D₄	—	54	4	no	—
258	D₄	—	54	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	27	2	no	—
263	C₂×C₄	—	27	2	no	—
264	C₂×C₄	—	27	2	no	—
265	C₂×C₄	—	54	4	no	—
266	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
272	C₂×C₃	—	2	26	no	—
273	C₂×C₃	C₂×#3828	1	13	yes	—
274	C₂×C₃	C₂×#3828	1	13	yes	—
275	C₂×C₃	C₂×#3828	1	13	yes	—
276	D₃	—	36	26	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	54	2	no	—
280	C₂²	—	54	2	no	—
281	C₂²	—	54	2	no	—
282	C₂²	#89076	1	2	yes	—
283	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	54	2	no	—
289	C₄	—	54	2	no	—
290	C₄	—	54	2	no	—
291	C₄	#89076	1	2	yes	—
292	C₄	#89076	1	2	yes	—
293	C₃	#10595950	1	13	yes	complemented
294	C₂	—	2	2	no	—
295	C₂	—	108	2	no	—
296	C₂	C₂×#89076	1	1	yes	characteristic
297	C₂	C₂×#89076	1	1	yes	characteristic
298	C₂	C₂×#89076	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₄²	C₂
prime radical	#11723971	C₂
hypercenter	C₂×C₄²	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#529
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×C₄²	C₂	semidirect
3. #11723970	C₂	semidirect
4. C₄×#6312	C₂	semidirect
5. #11723965	C₂	semidirect
6. #11723917	C₂	semidirect
7. #11723916	C₂	semidirect
8. C₄×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
9. #11723913	C₂	semidirect
10. #11723911	C₂	semidirect
11. C₄×#6308	C₂	semidirect
36. C₂×C₃²×C₄²	D₃	semidirect
105. C₂×C₃×C₄²	C₃⋊D₃	semidirect
189. C₂×C₄²	C₃⋊(C₃⋊D₃)	semidirect
190. C₃³	#529	semidirect
253. C₃²	#5479	semidirect
293. C₃	#10595950	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×C₄²	2
#11723971	2
#11723970	2
C₄×#6312	2
#11723965	2
#11723917	2
#11723916	2
C₄×C₄⋊(C₃⋊(C₃⋊D₃))	2
#11723913	2
#11723911	2
C₄×#6308	2
#10595950	3