Subgroups — #13152718

Order 1728.

1162 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88722	C₂	1	2	yes	maximal, complemented
2	C₄×Q₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#64104⋊C₃	C₂	1	6	yes	maximal, complemented
4	#11721385	C₂	1	3	yes	maximal, complemented
5	C₄×C₄⋊((C₃⋊D₃)⋊C₃)	C₂	1	3	yes	maximal, complemented
6	#10596048	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#5577	—	3	1	no	maximal
8	C₂×Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	#88722	—	2	8	no	—
10	C₂×C₄⋊((C₃⋊D₃)⋊C₃)	C₂²	1	6	yes	—
11	C₂×C₄⋊((C₃⋊D₃)⋊C₃)	C₂²	1	3	yes	—
12	C₂×C₄⋊((C₃⋊D₃)⋊C₃)	C₂²	1	6	yes	—
13	C₂×C₄×(C₃⋊D₃)⋊C₃	C₂²	1	6	yes	—
14	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
15	C₄²×H(3,𝔽₃)	C₂²	1	3	yes	—
16	#3752⋊C₃	C₂²	1	6	yes	—
17	C₄⋊((C₃⋊Dic₃)⋊C₃)	C₂²	1	1	yes	characteristic
18	C₂×#3832	C₂×C₃	1	2	yes	complemented
19	C₂×C₃×#239	—	3	2	no	—
20	C₃²×C₄×Q₈	—	2	1	no	—
21	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic, complemented
22	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
23	#64104	C₂×C₃	1	6	yes	complemented
24	#64083	C₂×C₃	1	3	yes	complemented
25	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	3	yes	complemented
26	C₃×#895	—	3	6	no	—
27	C₃×#874	—	3	3	no	—
28	C₃×C₄×D₁₂	—	3	3	no	—
29	C₂²×(C₃⋊D₃)⋊C₃	C₂³	1	6	yes	—
30	Q₈×H(3,𝔽₃)	D₄	1	4	yes	complemented
31	C₂×C₄×H(3,𝔽₃)	C₂³	1	3	yes	—
32	C₂×C₄×H(3,𝔽₃)	C₂³	1	3	yes	—
33	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
34	C₂×(C₃⋊Dic₃)⋊C₃	C₂³	1	2	yes	—
35	C₄⋊((C₃⋊D₃)⋊C₃)	—	2	12	no	—
36	C₄⋊((C₃⋊D₃)⋊C₃)	—	2	6	no	—
37	C₄⋊((C₃⋊D₃)⋊C₃)	—	4	6	no	—
38	C₄×(C₃⋊D₃)⋊C₃	—	2	12	no	—
39	C₃×#550	—	9	1	no	—
40	#5577	—	3	1	no	—
41	#5577	—	9	1	no	—
42	C₂×C₃²×Q₈	—	2	1	no	—
43	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
45	#3832	—	2	8	no	—
46	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	3	yes	—
47	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	6	yes	—
48	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	6	yes	—
49	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	6	yes	—
50	C₃×#239	—	6	8	no	—
51	C₂×C₃×D₁₂	—	3	6	no	—
52	C₂×C₃×D₁₂	—	3	3	no	—
53	C₂×C₃×D₁₂	—	3	6	no	—
54	C₂×C₃×C₄×D₃	—	3	6	no	—
55	C₃²×C₄⋊C₄	—	2	3	no	—
56	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
57	C₃²×C₄⋊C₄	C₂²×C₃	1	3	yes	—
58	C₃²×C₄²	—	2	3	no	—
59	C₃²×C₄²	C₂×D₃	1	3	yes	—
60	C₃²×C₄²	C₂²×C₃	1	3	yes	—
61	#3752	C₂²×C₃	1	6	yes	—
62	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
63	C₃×(C₂×D₃)⋊C₄	—	3	6	no	—
64	C₃×C₄⋊Dic₃	—	3	1	no	—
65	C₂²×H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic
66	C₂×(C₃⋊D₃)⋊C₃	—	2	6	no	—
67	C₂×(C₃⋊D₃)⋊C₃	—	2	6	no	—
68	C₂×(C₃⋊D₃)⋊C₃	—	2	6	no	—
69	C₄×H(3,𝔽₃)	—	2	3	no	—
70	C₄×H(3,𝔽₃)	C₂×D₄	1	6	yes	—
71	C₄×H(3,𝔽₃)	#41	1	2	yes	complemented
72	(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
73	C₂×C₃×#41	—	9	2	no	—
74	C₂×#239	—	3	2	no	—
75	C₂×#239	—	9	2	no	—
76	C₃×#127	—	9	3	no	—
77	C₃×#121	—	9	6	no	—
78	C₃×C₄×Q₈	—	3	1	no	—
79	C₃×C₄×Q₈	—	3	1	no	—
80	C₃×C₄×Q₈	—	6	1	no	—
81	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
82	C₃×C₄×D₄	—	9	3	no	—
83	#895	—	3	6	no	—
84	#895	—	9	6	no	—
85	#874	—	3	3	no	—
86	#874	—	9	3	no	—
87	C₄×D₁₂	—	3	3	no	—
88	C₄×D₁₂	—	9	3	no	—
89	C₂²×C₃⋊D₃	C₂³×C₃	1	6	yes	—
90	C₂²×C₃×D₃	—	3	6	no	—
91	C₃²×Q₈	—	2	4	no	—
92	C₃²×Q₈	D₁₂	1	4	yes	complemented
93	C₃²×Q₈	C₃×D₄	1	4	yes	complemented
94	C₂×C₃²×C₄	—	2	1	no	—
95	C₂×C₃²×C₄	—	2	3	no	—
96	C₂×C₃²×C₄	—	2	3	no	—
97	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
98	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
99	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
101	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
102	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
104	C₄⋊(C₃⋊D₃)	—	2	12	no	—
105	C₄⋊(C₃⋊D₃)	—	2	6	no	—
106	C₄⋊(C₃⋊D₃)	—	4	6	no	—
107	C₄×C₃⋊D₃	—	2	12	no	—
108	C₂×C₃×Dic₃	—	3	2	no	—
109	C₃×D₁₂	—	6	12	no	—
110	C₃×D₁₂	—	6	6	no	—
111	C₃×D₁₂	—	12	6	no	—
112	C₃×C₄×D₃	—	6	12	no	—
113	#550	—	9	1	no	Sylow
114	C₂×H(3,𝔽₃)	C₂²×D₄	1	1	yes	characteristic
115	C₂×H(3,𝔽₃)	C₂×#41	1	1	yes	characteristic
116	C₂×H(3,𝔽₃)	H(5,𝔽₂)	1	1	yes	characteristic
117	(C₃⋊D₃)⋊C₃	—	4	6	no	—
118	C₃×#41	—	18	8	no	—
119	C₂×C₃×Q₈	—	3	1	no	—
120	C₂×C₃×Q₈	—	3	1	no	—
121	C₂×C₃×Q₈	—	6	1	no	—
122	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂×C₃×D₄	—	9	3	no	—
124	C₂×C₃×D₄	—	9	6	no	—
125	C₂×C₃×D₄	—	9	6	no	—
126	C₂²×C₃×C₄	—	9	6	no	—
127	#239	—	6	8	no	—
128	#239	—	18	8	no	—
129	C₂×D₁₂	—	3	3	no	—
130	C₂×D₁₂	—	3	6	no	—
131	C₂×D₁₂	—	3	6	no	—
132	C₂×D₁₂	—	9	3	no	—
133	C₂×D₁₂	—	9	6	no	—
134	C₂×D₁₂	—	9	6	no	—
135	C₂×C₄×D₃	—	3	6	no	—
136	C₂×C₄×D₃	—	9	6	no	—
137	C₃×C₄⋊C₄	—	3	3	no	—
138	C₃×C₄⋊C₄	—	3	3	no	—
139	C₃×C₄⋊C₄	—	6	3	no	—
140	C₃×C₄⋊C₄	—	9	1	no	—
141	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
142	C₃×C₂²⋊C₄	—	9	6	no	—
143	C₃×C₄²	—	3	3	no	—
144	C₃×C₄²	—	3	3	no	—
145	C₃×C₄²	—	6	3	no	—
146	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
147	(C₂×D₃)⋊C₄	—	3	6	no	—
148	(C₂×D₃)⋊C₄	—	9	6	no	—
149	C₄⋊Dic₃	—	3	1	no	—
150	C₄⋊Dic₃	—	9	1	no	—
151	C₂²×C₃²	—	2	1	no	—
152	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
153	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
154	C₂×C₃⋊D₃	—	2	6	no	—
155	C₂×C₃⋊D₃	—	2	6	no	—
156	C₂×C₃⋊D₃	—	2	6	no	—
157	C₂×C₃×D₃	—	6	6	no	—
158	C₂×C₃×D₃	—	6	6	no	—
159	C₂×C₃×D₃	—	6	6	no	—
160	C₃²×C₄	—	2	3	no	—
161	C₃²×C₄	—	2	3	no	—
162	C₃²×C₄	—	2	6	no	—
163	C₃²×C₄	—	2	2	no	—
164	C₃²×C₄	—	4	3	no	—
165	C₃²×C₄	C₂×D₁₂	1	6	yes	—
166	C₃²×C₄	#239	1	2	yes	complemented
167	C₃²×C₄	C₂×C₃×D₄	1	6	yes	—
168	C₃²×C₄	C₃×#41	1	2	yes	complemented
169	C₃⋊Dic₃	—	2	2	no	—
170	C₃×Dic₃	—	6	2	no	—
171	C₂×#41	—	9	2	no	—
172	#127	—	9	3	no	—
173	#121	—	9	6	no	—
174	C₄×Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
175	C₄×D₄	—	9	3	no	—
176	H(3,𝔽₃)	#550	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₃	—	9	6	no	—
178	C₂²×D₃	—	3	6	no	—
179	C₂²×D₃	—	9	6	no	—
180	C₃×Q₈	—	3	4	no	—
181	C₃×Q₈	—	3	4	no	—
182	C₃×Q₈	—	6	4	no	—
183	C₃×Q₈	C₃×D₁₂	1	4	yes	—
184	C₃×D₄	—	18	6	no	—
185	C₃×D₄	—	18	12	no	—
186	C₃×D₄	—	36	6	no	—
187	C₂×C₃×C₄	—	3	1	no	—
188	C₂×C₃×C₄	—	3	3	no	—
189	C₂×C₃×C₄	—	3	1	no	—
190	C₂×C₃×C₄	—	3	3	no	—
191	C₂×C₃×C₄	—	3	3	no	—
192	C₂×C₃×C₄	—	3	3	no	—
193	C₂×C₃×C₄	—	6	1	no	—
194	C₂×C₃×C₄	—	6	3	no	—
195	C₂×C₃×C₄	—	6	3	no	—
196	C₂×C₃×C₄	—	9	2	no	—
197	C₂×C₃×C₄	—	18	12	no	—
198	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
199	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
200	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
201	C₂×Dic₃	—	3	2	no	—
202	C₂×Dic₃	—	9	2	no	—
203	D₁₂	—	6	12	no	—
204	D₁₂	—	6	6	no	—
205	D₁₂	—	12	6	no	—
206	D₁₂	—	18	6	no	—
207	D₁₂	—	18	12	no	—
208	D₁₂	—	36	6	no	—
209	C₄×D₃	—	6	12	no	—
210	C₄×D₃	—	18	12	no	—
211	C₂×C₃²	—	2	1	no	—
212	C₂×C₃²	—	2	1	no	—
213	C₂×C₃²	—	2	1	no	—
214	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
215	C₂×C₃²	C₂×#239	1	1	yes	characteristic
216	C₂×C₃²	#1009	1	1	yes	characteristic
217	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
218	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
219	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
220	C₃⋊D₃	—	4	6	no	—
221	C₃×D₃	—	12	6	no	—
222	#41	—	18	8	no	—
223	C₂×Q₈	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
224	C₂×D₄	—	9	3	no	—
225	C₂×D₄	—	9	6	no	—
226	C₂×D₄	—	9	6	no	—
227	C₂²×C₄	—	9	6	no	—
228	C₄⋊C₄	—	9	1	no	—
229	C₄⋊C₄	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	—
230	C₂²⋊C₄	—	9	6	no	—
231	C₄²	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	—
232	C₂²×C₃	—	3	1	no	—
233	C₂²×C₃	—	3	1	no	—
234	C₂²×C₃	—	6	1	no	—
235	C₂²×C₃	—	18	6	no	—
236	C₂²×C₃	—	18	6	no	—
237	C₂²×C₃	—	18	6	no	—
238	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
239	C₂×D₃	—	6	6	no	—
240	C₂×D₃	—	6	6	no	—
241	C₂×D₃	—	6	6	no	—
242	C₂×D₃	—	18	6	no	—
243	C₂×D₃	—	18	6	no	—
244	C₂×D₃	—	18	6	no	—
245	C₃×C₄	—	2	3	no	—
246	C₃×C₄	—	3	6	no	—
247	C₃×C₄	—	3	6	no	—
248	C₃×C₄	—	3	2	no	—
249	C₃×C₄	—	3	2	no	—
250	C₃×C₄	—	6	3	no	—
251	C₃×C₄	—	6	3	no	—
252	C₃×C₄	—	6	6	no	—
253	C₃×C₄	—	6	2	no	—
254	C₃×C₄	—	12	3	no	—
255	C₃×C₄	—	18	2	no	—
256	C₃×C₄	C₂×C₃×D₁₂	1	6	yes	—
257	C₃×C₄	C₃×#239	1	2	yes	—
258	Dic₃	—	6	2	no	—
259	Dic₃	—	18	2	no	—
260	C₃²	—	2	1	no	—
261	C₃²	#5577	1	1	yes	characteristic, complemented
262	C₃²	C₃×#550	1	1	yes	characteristic, complemented
263	C₂³	—	9	6	no	—
264	Q₈	C₄⋊((C₃⋊D₃)⋊C₃)	1	4	yes	complemented
265	D₄	—	18	12	no	—
266	D₄	—	18	6	no	—
267	D₄	—	36	6	no	—
268	C₂×C₄	—	9	2	no	—
269	C₂×C₄	—	18	12	no	—
270	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃	1	3	yes	—
271	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃	1	3	yes	—
272	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
273	C₂×C₃	—	3	1	no	—
274	C₂×C₃	—	3	1	no	—
275	C₂×C₃	—	3	1	no	—
276	C₂×C₃	—	3	1	no	—
277	C₂×C₃	—	3	1	no	—
278	C₂×C₃	—	3	1	no	—
279	C₂×C₃	—	6	1	no	—
280	C₂×C₃	—	6	1	no	—
281	C₂×C₃	—	6	1	no	—
282	C₂×C₃	—	36	6	no	—
283	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
284	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
285	C₂×C₃	C₃×#1009	1	1	yes	characteristic
286	D₃	—	12	6	no	—
287	D₃	—	36	6	no	—
288	C₂²	—	18	6	no	—
289	C₂²	—	18	6	no	—
290	C₂²	—	18	6	no	—
291	C₂²	C₂³×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
292	C₄	—	2	3	no	—
293	C₄	—	18	2	no	—
294	C₄	C₂×C₄⋊((C₃⋊D₃)⋊C₃)	1	6	yes	—
295	C₄	#88722	1	2	yes	complemented
296	C₃	—	3	1	no	—
297	C₃	—	3	1	no	—
298	C₃	—	6	1	no	—
299	C₃	C₃×#5577	1	1	yes	characteristic
300	C₂	—	36	6	no	—
301	C₂	C₂²×C₄⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
302	C₂	C₂×#88722	1	1	yes	characteristic
303	C₂	#11722638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₄×Q₈×H(3,𝔽₃)	C₂
hypercenter	C₄×Q₈	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×#550
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#88722	C₂	semidirect
2. C₄×Q₈×H(3,𝔽₃)	C₂	semidirect
3. #64104⋊C₃	C₂	semidirect
4. #11721385	C₂	semidirect
5. C₄×C₄⋊((C₃⋊D₃)⋊C₃)	C₂	semidirect
6. #10596048	C₃	semidirect
18. C₂×#3832	C₂×C₃	semidirect
21. C₃²×C₄×Q₈	D₃	semidirect
22. C₃²×C₄×Q₈	C₂×C₃	semidirect
23. #64104	C₂×C₃	semidirect
24. #64083	C₂×C₃	semidirect
25. C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
30. Q₈×H(3,𝔽₃)	D₄	semidirect
71. C₄×H(3,𝔽₃)	#41	semidirect
92. C₃²×Q₈	D₁₂	semidirect
93. C₃²×Q₈	C₃×D₄	semidirect
166. C₃²×C₄	#239	semidirect
168. C₃²×C₄	C₃×#41	semidirect
174. C₄×Q₈	(C₃⋊D₃)⋊C₃	semidirect
176. H(3,𝔽₃)	#550	semidirect
261. C₃²	#5577	semidirect
262. C₃²	C₃×#550	semidirect
264. Q₈	C₄⋊((C₃⋊D₃)⋊C₃)	semidirect
295. C₄	#88722	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88722	2
C₄×Q₈×H(3,𝔽₃)	2
#64104⋊C₃	2
#11721385	2
C₄×C₄⋊((C₃⋊D₃)⋊C₃)	2
#10596048	3
C₃×#5577	3