Subgroups — #11772406

Order 1152.

313 conjugacy classes of subgroups.

The subgroup lattice has 257 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₃×#5106	C₂	1	1	yes	maximal, characteristic
4	#67729	—	3	1	no	maximal
5	#67724	—	3	1	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₃×#926	—	2	1	no	—
8	C₂×C₃×#208	—	2	1	no	—
9	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
10	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
11	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
12	C₃×#819	—	2	1	no	—
13	C₃×#819	C₄	1	2	yes	complemented
14	C₃×#818	C₄	1	2	yes	—
15	C₃×#817	—	2	1	no	—
16	C₂×C₃×#108	—	3	1	no	—
17	C₃×#427	—	3	1	no	—
18	C₂×#819	D₃	1	1	yes	characteristic
19	#5106	D₃	1	1	yes	characteristic
20	C₂²×C₃⋊C₁₆	—	3	1	no	—
21	C₂²×C₃⋊C₁₆	—	3	1	no	—
22	C₂²×C₃⋊C₁₆	—	6	1	no	—
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
30	C₃×#217	—	4	1	no	—
31	C₃×C₄⋊Dic₃	—	2	2	no	—
32	C₃×C₄×Dic₃	—	4	1	no	—
33	C₃×#208	—	2	2	no	—
34	C₃×#208	—	4	1	no	—
35	C₂×C₃×C₃⋊C₈	—	4	1	no	—
36	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
37	#1382	—	9	1	no	Sylow
38	C₂×C₃×M₁₆	—	6	1	no	—
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
42	C₃×#117	—	6	1	no	—
43	#926	—	2	1	no	—
44	C₂×#208	—	2	1	no	—
45	C₃×#108	—	3	2	no	—
46	C₃×#108	—	6	1	no	—
47	C₃×#107	—	3	2	no	—
48	C₃×#106	—	6	1	no	—
49	#819	—	2	1	no	—
50	#819	Dic₃	1	2	yes	complemented
51	#818	Dic₃	1	2	yes	—
52	#817	—	2	1	no	—
53	C₂×C₃⋊C₁₆	—	3	2	no	—
54	C₂×C₃⋊C₁₆	—	3	2	no	—
55	C₂×C₃⋊C₁₆	—	6	2	no	—
56	C₂×C₃⋊C₁₆	—	6	2	no	—
57	C₂×C₃⋊C₁₆	—	6	2	no	—
58	C₂×C₃⋊C₁₆	—	6	1	no	—
59	C₂×C₃⋊C₁₆	—	6	1	no	—
60	C₂×C₃⋊C₁₆	—	6	1	no	—
61	C₂×C₃⋊C₁₆	—	6	1	no	—
62	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
68	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
69	C₂×C₃×Dic₃	—	4	2	no	—
70	C₃²×C₈	C₄²	1	1	yes	characteristic
71	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃×C₃⋊C₈	—	4	2	no	—
75	C₂²×C₁₆	—	9	1	no	—
76	C₂×#108	—	3	1	no	—
77	#427	—	3	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	#67	1	1	yes	characteristic
80	C₂²×C₃×C₄	#67	1	1	yes	characteristic
81	C₃×M₁₆	—	6	2	no	—
82	C₃×M₁₆	—	12	1	no	—
83	C₂×C₃×C₈	—	2	1	no	—
84	C₂×C₃×C₈	—	2	1	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	12	1	no	—
90	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
91	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
92	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
96	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
97	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
100	C₂×C₃×C₈	#67	1	1	yes	characteristic
101	C₂×C₃×C₈	#67	1	1	yes	characteristic
102	C₃×C₄⋊C₄	—	6	2	no	—
103	C₃×C₂²⋊C₄	—	12	1	no	—
104	C₃×C₄²	—	12	1	no	—
105	#217	—	4	1	no	—
106	C₄⋊Dic₃	—	2	2	no	—
107	C₄×Dic₃	—	4	1	no	—
108	#208	—	2	2	no	—
109	#208	—	4	1	no	—
110	C₂×C₃⋊C₈	—	4	1	no	—
111	C₃⋊C₁₆	—	6	4	no	—
112	C₃⋊C₁₆	—	6	2	no	—
113	C₃⋊C₁₆	—	6	2	no	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	#102	1	1	yes	characteristic
116	C₂²×C₃²	#103	1	1	yes	characteristic
117	C₂²×C₃²	#106	1	1	yes	characteristic
118	C₃²×C₄	#95	1	1	yes	characteristic
119	C₃²×C₄	#102	1	1	yes	characteristic
120	C₃²×C₄	#103	1	1	yes	characteristic
121	C₃²×C₄	#107	1	1	yes	characteristic
122	C₃×Dic₃	—	8	2	no	—
123	C₂×M₁₆	—	6	1	no	—
124	C₂²×C₈	D₃²	1	1	yes	characteristic
125	#117	—	6	1	no	—
126	C₂×C₁₆	—	9	2	no	—
127	C₂×C₁₆	—	18	1	no	—
128	C₂×C₁₆	—	18	1	no	—
129	#108	—	3	2	no	—
130	#108	—	6	1	no	—
131	#107	—	3	2	no	—
132	#106	—	6	1	no	—
133	C₂³×C₃	—	2	1	no	—
134	C₂³×C₃	#214	1	1	yes	characteristic
135	C₂³×C₃	#215	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	12	2	no	—
143	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
145	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
146	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	#213	1	1	yes	characteristic
148	C₂×C₃×C₄	#213	1	1	yes	characteristic
149	C₂×C₃×C₄	#214	1	1	yes	characteristic
150	C₂×C₃×C₄	#215	1	1	yes	characteristic
151	C₂×C₃×C₄	#216	1	1	yes	characteristic
152	C₂×C₃×C₄	#216	1	1	yes	characteristic
153	C₂×C₃×C₄	#217	1	1	yes	characteristic
154	C₂×C₃×C₄	#217	1	1	yes	characteristic
155	C₂×Dic₃	—	4	2	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	12	2	no	—
161	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
162	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
163	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
164	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
165	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
166	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
167	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
168	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₃⋊C₈	—	4	2	no	—
170	C₂×C₃²	—	2	1	no	—
171	C₂×C₃²	—	2	1	no	—
172	C₂×C₃²	#340	1	1	yes	characteristic
173	C₂×C₃²	#359	1	2	yes	—
174	C₂²×C₄	#625	1	1	yes	characteristic
175	M₁₆	—	6	2	no	—
176	M₁₆	—	12	1	no	—
177	C₂×C₈	—	12	1	no	—
178	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
179	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
180	C₂×C₈	#624	1	1	yes	characteristic
181	C₂×C₈	#626	1	1	yes	characteristic
182	C₂×C₈	#626	1	1	yes	characteristic
183	C₂×C₈	#627	1	1	yes	characteristic
184	C₄⋊C₄	—	6	2	no	—
185	C₂²⋊C₄	—	12	1	no	—
186	C₄²	—	12	1	no	—
187	C₁₆	—	18	2	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	#808	1	1	yes	characteristic
195	C₂²×C₃	#808	1	1	yes	characteristic
196	C₂²×C₃	#809	1	1	yes	characteristic
197	C₂²×C₃	#810	1	1	yes	characteristic
198	C₂²×C₃	#832	1	1	yes	characteristic
199	C₂²×C₃	#835	1	1	yes	characteristic
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	24	2	no	—
205	C₃×C₄	#807	1	1	yes	characteristic
206	C₃×C₄	#807	1	1	yes	characteristic
207	C₃×C₄	#809	1	1	yes	characteristic
208	C₃×C₄	#810	1	1	yes	characteristic
209	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
210	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
211	C₃×C₄	#832	1	1	yes	characteristic
212	C₃×C₄	#835	1	1	yes	characteristic
213	Dic₃	—	8	2	no	—
214	C₃²	#1382	1	1	yes	characteristic, complemented, Sylow
215	C₂³	#3715	1	1	yes	characteristic
216	C₂×C₄	—	12	2	no	—
217	C₂×C₄	#3713	1	1	yes	characteristic
218	C₂×C₄	#3715	1	1	yes	characteristic
219	C₂×C₄	#3718	1	1	yes	characteristic
220	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
221	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
222	C₂×C₄	#3724	1	1	yes	characteristic
223	C₈	—	12	2	no	—
224	C₈	Dic₃²	1	1	yes	characteristic
225	C₈	#3722	1	1	yes	characteristic
226	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
227	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
228	C₂×C₃	—	2	2	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	2	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	#4497	1	2	yes	—
237	C₂×C₃	#4529	1	1	yes	characteristic
238	C₂×C₃	#4529	1	1	yes	characteristic
239	C₂×C₃	#4567	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	#63562	1	1	yes	characteristic
242	C₂²	#63565	1	1	yes	characteristic
243	C₂²	#63574	1	1	yes	characteristic
244	C₄	—	24	2	no	—
245	C₄	#63562	1	1	yes	characteristic
246	C₄	#63565	1	1	yes	characteristic
247	C₄	#63576	1	1	yes	characteristic
248	C₄	#63579	1	1	yes	characteristic
249	C₃	—	2	1	no	—
250	C₃	#67724	1	1	yes	characteristic, complemented
251	C₃	#67729	1	1	yes	characteristic, complemented
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	#10589120	1	2	yes	—
255	C₂	#10589163	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3713
commutator	C₃²×C₈	C₄²
Frattini, hypercenter	C₂²×C₈	D₃²
Fitting, cosocle	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	#102
prime radical	C₂³×C₃²	SD₁₆
hypocenter	C₃²	#1382
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×C₃⋊(C₃⋊C₁₆)	C₄	semidirect
13. C₃×#819	C₄	semidirect
50. #819	Dic₃	semidirect
214. C₃²	#1382	semidirect
250. C₃	#67724	semidirect
251. C₃	#67729	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₂×C₃×#819	2
C₃×#5106	2
#67729	3
#67724	3