Subgroups — #13179707

Order 1728.

3024 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11723988	C₂	1	1	yes	maximal, characteristic, complemented
3	#11723987	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723937	C₂	1	2	yes	maximal, complemented
5	#11723936	C₂	1	1	yes	maximal, characteristic
6	#11723935	C₂	1	2	yes	maximal, complemented
7	#11723929	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723917	C₂	1	2	yes	maximal, complemented
9	#11723916	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
11	#11723913	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596053	—	3	13	no	maximal
13	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	2	yes	—
16	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
18	C₂×#6308	C₂²	1	2	yes	—
19	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃³×C₄²	C₂²	1	1	yes	characteristic
22	C₃³×C₄²	C₂²	1	2	yes	—
23	#88859	C₂²	1	1	yes	characteristic
24	#88859	C₂²	1	1	yes	characteristic
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	#88859	C₂²	1	2	yes	—
30	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
31	#88857	C₂²	1	1	yes	characteristic
32	#88857	C₂²	1	1	yes	characteristic
33	#88857	C₂²	1	2	yes	—
34	#88857	C₂²	1	2	yes	—
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
36	C₃²×C₄×Q₈	D₃	1	13	yes	complemented
37	#64156	—	3	13	no	—
38	#64155	—	3	13	no	—
39	#64105	—	3	26	no	—
40	#64104	—	3	13	no	—
41	#64103	—	3	26	no	—
42	#64097	—	3	13	no	—
43	#64085	—	3	26	no	—
44	#64084	—	3	26	no	—
45	C₄×C₄⋊(C₃⋊D₃)	—	3	13	no	—
46	#64081	—	3	13	no	—
47	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
48	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	2	yes	—
49	C₃³×Q₈	—	2	2	no	—
50	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃³×C₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	C₂³	1	2	yes	—
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
55	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
56	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
57	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
58	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
59	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
60	C₄⋊(C₃⋊(C₃⋊D₃))	—	4	2	no	—
61	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
62	C₄×C₃⋊(C₃⋊D₃)	—	4	2	no	—
63	#6308	—	4	2	no	—
64	#5582	—	9	13	no	—
65	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
66	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
67	C₂×C₄⋊(C₃⋊D₃)	—	3	26	no	—
68	C₂×C₄×C₃⋊D₃	—	3	13	no	—
69	C₂×C₄×C₃⋊D₃	—	3	26	no	—
70	C₂×#634	—	3	26	no	—
71	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
72	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
73	C₃²×C₄²	C₂×D₃	1	13	yes	—
74	C₃²×C₄²	C₂×D₃	1	26	yes	—
75	#3752	—	3	13	no	—
76	#3752	—	3	13	no	—
77	#3752	—	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	#3750	—	3	13	no	—
84	#3750	—	3	13	no	—
85	#3750	—	3	26	no	—
86	#3750	—	3	26	no	—
87	C₄×C₃⋊Dic₃	—	3	13	no	—
88	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
89	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
90	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
91	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
92	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
93	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
94	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
95	C₃³×C₄	—	2	2	no	—
96	C₃³×C₄	—	2	2	no	—
97	C₃³×C₄	—	2	1	no	—
98	C₃³×C₄	—	2	1	no	—
99	C₃³×C₄	#41	1	2	yes	—
100	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
101	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
102	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
103	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
104	C₃×C₄×Q₈	C₃⋊D₃	1	13	yes	complemented
105	#947	—	9	13	no	—
106	#946	—	9	13	no	—
107	#896	—	9	26	no	—
108	#895	—	9	13	no	—
109	#894	—	9	26	no	—
110	#888	—	9	13	no	—
111	#876	—	9	26	no	—
112	#875	—	9	26	no	—
113	C₄×D₁₂	—	9	13	no	—
114	#872	—	9	13	no	—
115	C₂²×C₃⋊D₃	—	3	13	no	—
116	C₂²×C₃⋊D₃	—	3	26	no	—
117	C₃²×Q₈	—	2	26	no	—
118	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
119	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
120	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
121	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
123	C₂×C₃⋊Dic₃	—	3	13	no	—
124	C₂×C₃⋊Dic₃	—	3	13	no	—
125	C₂×C₃⋊Dic₃	—	3	13	no	—
126	C₂×C₃⋊Dic₃	—	3	26	no	—
127	C₄⋊(C₃⋊D₃)	—	6	26	no	—
128	C₄⋊(C₃⋊D₃)	—	12	26	no	—
129	C₄×C₃⋊D₃	—	6	26	no	—
130	C₄×C₃⋊D₃	—	12	26	no	—
131	#634	—	12	26	no	—
132	#542	—	27	1	no	Sylow
133	C₂×C₃³	C₂×#41	1	1	yes	characteristic
134	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
136	C₃⋊(C₃⋊D₃)	—	4	1	no	—
137	C₃⋊(C₃⋊D₃)	—	4	2	no	—
138	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
139	C₂×D₁₂	—	9	13	no	—
140	C₂×D₁₂	—	9	26	no	—
141	C₂×C₄×D₃	—	9	13	no	—
142	C₂×C₄×D₃	—	9	26	no	—
143	C₂×Dic₆	—	9	26	no	—
144	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
145	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
146	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
147	C₃×C₄²	C₂×C₃⋊D₃	1	26	yes	—
148	(C₂×D₃)⋊C₄	—	9	13	no	—
149	(C₂×D₃)⋊C₄	—	9	13	no	—
150	(C₂×D₃)⋊C₄	—	9	26	no	—
151	(C₂×D₃)⋊C₄	—	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	C₄⋊Dic₃	—	9	13	no	—
156	Dic₃⋊C₄	—	9	13	no	—
157	Dic₃⋊C₄	—	9	13	no	—
158	Dic₃⋊C₄	—	9	26	no	—
159	Dic₃⋊C₄	—	9	26	no	—
160	C₄×Dic₃	—	9	13	no	—
161	C₂²×C₃²	C₂³×D₃	1	13	yes	—
162	C₂×C₃⋊D₃	—	6	13	no	—
163	C₂×C₃⋊D₃	—	6	26	no	—
164	C₂×C₃⋊D₃	—	6	13	no	—
165	C₂×C₃⋊D₃	—	6	26	no	—
166	C₂×C₃⋊D₃	—	6	26	no	—
167	C₂×C₃⋊D₃	—	6	13	no	—
168	C₃²×C₄	—	2	26	no	—
169	C₃²×C₄	—	2	13	no	—
170	C₃²×C₄	—	2	26	no	—
171	C₃²×C₄	—	2	13	no	—
172	C₃²×C₄	#235	1	26	yes	—
173	C₃⋊Dic₃	—	6	13	no	—
174	C₃⋊Dic₃	—	6	13	no	—
175	C₃⋊Dic₃	—	6	13	no	—
176	C₃⋊Dic₃	—	6	26	no	—
177	C₄⋊Q₈	—	27	1	no	—
178	#126	—	27	2	no	—
179	#124	—	27	1	no	—
180	#124	—	27	2	no	—
181	#123	—	27	2	no	—
182	C₂²⋊Q₈	—	27	1	no	—
183	C₂²⋊Q₈	—	27	2	no	—
184	#121	—	27	1	no	—
185	C₄×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
186	C₄×D₄	—	27	1	no	—
187	#117	—	27	1	no	—
188	C₃³	#542	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₃	—	9	13	no	—
190	C₂²×D₃	—	9	26	no	—
191	C₃×Q₈	—	2	26	no	—
192	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
193	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
194	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
195	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
196	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
197	C₂×Dic₃	—	9	13	no	—
198	C₂×Dic₃	—	9	13	no	—
199	C₂×Dic₃	—	9	13	no	—
200	C₂×Dic₃	—	9	26	no	—
201	D₁₂	—	18	26	no	—
202	D₁₂	—	36	26	no	—
203	C₄×D₃	—	18	26	no	—
204	C₄×D₃	—	36	26	no	—
205	Dic₆	—	36	26	no	—
206	C₂×C₃²	C₂×#235	1	13	yes	—
207	C₂×C₃²	#1007	1	13	yes	—
208	C₂×C₃²	#1009	1	13	yes	—
209	C₃⋊D₃	—	12	13	no	—
210	C₃⋊D₃	—	12	26	no	—
211	C₂×Q₈	—	27	2	no	—
212	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
213	C₂×D₄	—	27	1	no	—
214	C₂×D₄	—	27	2	no	—
215	C₂²×C₄	—	27	1	no	—
216	C₂²×C₄	—	27	2	no	—
217	C₄⋊C₄	—	27	1	no	—
218	C₄⋊C₄	—	27	1	no	—
219	C₄⋊C₄	—	27	1	no	—
220	C₄⋊C₄	—	27	2	no	—
221	C₄⋊C₄	—	27	2	no	—
222	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
223	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
224	C₂²⋊C₄	—	27	1	no	—
225	C₂²⋊C₄	—	27	1	no	—
226	C₂²⋊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	C₄²	—	27	1	no	—
232	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
233	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
234	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
235	C₂×D₃	—	18	13	no	—
236	C₂×D₃	—	18	26	no	—
237	C₂×D₃	—	18	13	no	—
238	C₂×D₃	—	18	26	no	—
239	C₂×D₃	—	18	26	no	—
240	C₂×D₃	—	18	13	no	—
241	C₃×C₄	—	2	26	no	—
242	C₃×C₄	—	2	13	no	—
243	C₃×C₄	—	2	26	no	—
244	C₃×C₄	—	2	13	no	—
245	C₃×C₄	#3828	1	26	yes	—
246	Dic₃	—	18	13	no	—
247	Dic₃	—	18	13	no	—
248	Dic₃	—	18	13	no	—
249	Dic₃	—	18	26	no	—
250	C₃²	#5582	1	13	yes	complemented
251	C₂³	—	27	1	no	—
252	C₂³	—	27	2	no	—
253	Q₈	—	2	2	no	—
254	Q₈	—	108	2	no	—
255	D₄	—	54	2	no	—
256	D₄	—	108	2	no	—
257	C₂×C₄	—	27	1	no	—
258	C₂×C₄	—	27	1	no	—
259	C₂×C₄	—	27	1	no	—
260	C₂×C₄	—	27	2	no	—
261	C₂×C₄	—	54	2	no	—
262	C₂×C₄	—	108	2	no	—
263	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
265	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
267	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
268	C₂×C₃	C₂×#3828	1	13	yes	—
269	C₂×C₃	#64364	1	13	yes	—
270	C₂×C₃	#64366	1	13	yes	—
271	D₃	—	36	13	no	—
272	D₃	—	36	26	no	—
273	C₂²	—	54	1	no	—
274	C₂²	—	54	2	no	—
275	C₂²	—	54	1	no	—
276	C₂²	—	54	2	no	—
277	C₂²	—	54	2	no	—
278	C₂²	—	54	1	no	—
279	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	54	1	no	—
285	C₄	—	54	1	no	—
286	C₄	—	54	2	no	—
287	C₄	—	54	1	no	—
288	C₄	#89076	1	2	yes	—
289	C₃	#10596053	1	13	yes	complemented
290	C₂	—	108	1	no	—
291	C₂	—	108	2	no	—
292	C₂	C₂×#89076	1	1	yes	characteristic
293	C₂	#11724682	1	1	yes	characteristic
294	C₂	#11724684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₃³×C₄×Q₈	C₂
prime radical	#11723936	C₂
hypercenter	C₄×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#542
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄×Q₈	C₂	semidirect
2. #11723988	C₂	semidirect
3. #11723987	C₂	semidirect
4. #11723937	C₂	semidirect
6. #11723935	C₂	semidirect
7. #11723929	C₂	semidirect
8. #11723917	C₂	semidirect
9. #11723916	C₂	semidirect
10. C₄×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
11. #11723913	C₂	semidirect
36. C₃²×C₄×Q₈	D₃	semidirect
104. C₃×C₄×Q₈	C₃⋊D₃	semidirect
185. C₄×Q₈	C₃⋊(C₃⋊D₃)	semidirect
188. C₃³	#542	semidirect
250. C₃²	#5582	semidirect
289. C₃	#10596053	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
#11723988	2
#11723987	2
#11723937	2
#11723936	2
#11723935	2
#11723929	2
#11723917	2
#11723916	2
C₄×C₄⋊(C₃⋊(C₃⋊D₃))	2
#11723913	2
#10596053	3