Subgroups — #11784314

Order 1152.

816 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#63642	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
6	#10589416	C₂	1	1	yes	maximal, characteristic
7	#10589416	C₂	1	1	yes	maximal, characteristic
8	#69810	—	3	4	no	maximal
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
13	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃²×#97	C₄	1	4	yes	complemented
18	#63658	C₄	1	2	yes	—
19	#63642	—	2	2	no	—
20	C₈×C₃⋊Dic₃	—	2	2	no	—
21	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
22	C₂×C₃×#97	D₃	1	4	yes	—
23	C₂×#815	—	3	4	no	—
24	C₂×C₈×Dic₃	—	3	4	no	—
25	C₄×C₄⋊Dic₃	—	3	4	no	—
26	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
27	#4550	—	3	4	no	—
28	#4550	—	3	4	no	—
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
33	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	—	2	1	no	—
35	C₂×C₃²×C₈	—	2	1	no	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃²×C₈	D₄	1	2	yes	—
39	C₂×C₃²×C₈	Q₈	1	2	yes	—
40	C₃²×C₄²	C₂×C₄	1	2	yes	—
41	C₃²×C₄²	C₂×C₄	1	2	yes	—
42	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
43	C₄×C₃⋊Dic₃	—	2	1	no	—
44	C₄×C₃⋊Dic₃	—	2	1	no	—
45	C₄×C₃⋊Dic₃	—	2	1	no	—
46	C₄×C₃⋊Dic₃	—	2	1	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
51	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
52	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
53	#1771	—	9	1	no	Sylow
54	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
55	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
56	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
57	C₂×C₄⋊Dic₃	—	3	8	no	—
58	C₂×C₄×Dic₃	—	3	4	no	—
59	C₂×C₄×Dic₃	—	3	4	no	—
60	C₂²×C₃⋊C₈	—	3	4	no	—
61	C₂²×C₃⋊C₈	—	3	4	no	—
62	C₃×#97	Dic₃	1	16	yes	complemented
63	C₃⋊#95	—	3	8	no	—
64	#815	—	6	8	no	—
65	C₈×Dic₃	—	6	8	no	—
66	C₄⋊(C₃⋊C₈)	—	3	16	no	—
67	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	—	2	2	no	—
69	C₂×C₃²×C₄	—	2	2	no	—
70	C₂×C₃²×C₄	C₄²	1	2	yes	—
71	C₂×C₃²×C₄	C₄²	1	2	yes	—
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	#41	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	2	2	no	—
79	C₂×C₃⋊Dic₃	—	2	2	no	—
80	C₂×C₃⋊Dic₃	—	2	2	no	—
81	C₂×C₃⋊Dic₃	—	2	2	no	—
82	C₂×C₃⋊Dic₃	—	2	2	no	—
83	C₂×C₃⋊Dic₃	M₁₆	1	4	yes	—
84	C₃²×C₈	—	2	2	no	—
85	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
86	C₃⋊(C₃⋊C₈)	—	2	2	no	—
87	C₃⋊(C₃⋊C₈)	—	2	2	no	—
88	C₂×C₄⋊C₈	—	9	1	no	—
89	C₂×#97	—	9	1	no	—
90	C₂×#97	C₃⋊D₃	1	1	yes	characteristic
91	C₂×C₄×C₈	—	9	1	no	—
92	C₄×C₄⋊C₄	—	9	1	no	—
93	#336	—	9	1	no	—
94	#336	—	9	1	no	—
95	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
96	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
97	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
98	C₂²×Dic₃	—	3	8	no	—
99	C₂²×Dic₃	—	3	8	no	—
100	C₂×C₃×C₈	—	2	4	no	—
101	C₂×C₃×C₈	—	2	4	no	—
102	C₂×C₃×C₈	Dic₆	1	8	yes	—
103	C₂×C₃×C₈	D₁₂	1	8	yes	—
104	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
105	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
106	C₃×C₄²	C₄×D₃	1	8	yes	—
107	C₃×C₄²	C₂×Dic₃	1	8	yes	—
108	C₄⋊Dic₃	—	6	16	no	—
109	C₄×Dic₃	—	6	4	no	—
110	C₄×Dic₃	—	6	4	no	—
111	C₄×Dic₃	—	6	4	no	—
112	C₄×Dic₃	—	6	4	no	—
113	C₂×C₃⋊C₈	—	3	8	no	—
114	C₂×C₃⋊C₈	—	3	8	no	—
115	C₂×C₃⋊C₈	—	6	4	no	—
116	C₂×C₃⋊C₈	—	6	4	no	—
117	C₂×C₃⋊C₈	—	6	4	no	—
118	C₂×C₃⋊C₈	—	6	4	no	—
119	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
120	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
121	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
122	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
123	C₂²×C₃²	#131	1	2	yes	—
124	C₃²×C₄	—	2	2	no	—
125	C₃²×C₄	—	2	2	no	—
126	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₃²×C₄	#117	1	1	yes	characteristic
128	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
129	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
130	C₃⋊Dic₃	—	2	4	no	—
131	C₃⋊Dic₃	—	4	2	no	—
132	C₂²×C₈	—	9	1	no	—
133	C₂²×C₈	—	9	1	no	—
134	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
135	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	9	2	no	—
137	C₂×C₄²	—	9	1	no	—
138	C₂×C₄²	—	9	1	no	—
139	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
140	C₄⋊C₈	—	9	4	no	—
141	#97	—	18	2	no	—
142	#97	C₃⋊Dic₃	1	4	yes	complemented
143	C₄×C₈	—	18	2	no	—
144	#95	—	9	2	no	—
145	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
146	C₂×C₃×C₄	—	2	8	no	—
147	C₂×C₃×C₄	—	2	8	no	—
148	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
149	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
150	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
151	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
152	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
153	C₂×C₃×C₄	#235	1	4	yes	—
154	C₂×C₃×C₄	#235	1	4	yes	—
155	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
156	C₂×Dic₃	—	3	16	no	—
157	C₂×Dic₃	—	6	8	no	—
158	C₂×Dic₃	—	6	8	no	—
159	C₂×Dic₃	—	6	8	no	—
160	C₂×Dic₃	—	6	8	no	—
161	C₂×Dic₃	—	6	8	no	—
162	C₃×C₈	—	2	8	no	—
163	C₃×C₈	C₄⋊Dic₃	1	16	yes	complemented
164	C₃⋊C₈	—	6	8	no	—
165	C₃⋊C₈	—	6	8	no	—
166	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
168	C₂×C₃²	#405	1	1	yes	characteristic
169	C₂×C₃²	#435	1	2	yes	—
170	C₂×C₃²	#446	1	2	yes	—
171	C₂²×C₄	—	9	2	no	—
172	C₂²×C₄	—	9	2	no	—
173	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
174	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	9	2	no	—
179	C₂×C₈	—	9	2	no	—
180	C₂×C₈	—	18	1	no	—
181	C₂×C₈	—	18	1	no	—
182	C₂×C₈	—	18	1	no	—
183	C₂×C₈	—	18	1	no	—
184	C₂×C₈	#634	1	2	yes	—
185	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
186	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
187	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
188	C₄⋊C₄	—	18	4	no	—
189	C₄²	—	18	1	no	—
190	C₄²	—	18	1	no	—
191	C₄²	—	18	1	no	—
192	C₄²	—	18	1	no	—
193	C₄²	C₄×C₃⋊D₃	1	2	yes	—
194	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
195	C₂²×C₃	C₄×Dic₆	1	4	yes	—
196	C₂²×C₃	C₄×D₁₂	1	4	yes	—
197	C₂²×C₃	D₃×M₁₆	1	8	yes	—
198	C₂²×C₃	#907	1	8	yes	—
199	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
200	C₃×C₄	—	2	8	no	—
201	C₃×C₄	—	2	8	no	—
202	C₃×C₄	C₄×Dic₆	1	4	yes	—
203	C₃×C₄	C₄×D₁₂	1	4	yes	—
204	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
205	C₃×C₄	#926	1	4	yes	—
206	Dic₃	—	6	16	no	—
207	Dic₃	—	12	8	no	—
208	C₃²	#1771	1	1	yes	characteristic, complemented, Sylow
209	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	9	4	no	—
213	C₂×C₄	—	18	2	no	—
214	C₂×C₄	—	18	2	no	—
215	C₂×C₄	—	18	2	no	—
216	C₂×C₄	—	18	2	no	—
217	C₂×C₄	—	18	2	no	—
218	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
219	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
220	C₂×C₄	C₂×#634	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
223	C₂×C₄	#3828	1	1	yes	characteristic
224	C₂×C₄	#3828	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	18	2	no	—
228	C₈	—	18	2	no	—
229	C₈	C₄⋊(C₃⋊Dic₃)	1	4	yes	complemented
230	C₂×C₃	#4701	1	8	yes	—
231	C₂×C₃	#4706	1	8	yes	—
232	C₂×C₃	C₄×C₄⋊Dic₃	1	4	yes	—
233	C₂×C₃	Dic₃×M₁₆	1	4	yes	—
234	C₂×C₃	#5122	1	4	yes	—
235	C₂²	C₄×#634	1	1	yes	characteristic
236	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
237	C₂²	M₁₆×C₃⋊D₃	1	2	yes	—
238	C₂²	#64116	1	2	yes	—
239	C₂²	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	18	4	no	—
243	C₄	—	36	2	no	—
244	C₄	C₄×#634	1	1	yes	characteristic
245	C₄	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
246	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
247	C₄	#64135	1	1	yes	characteristic
248	C₃	#69810	1	4	yes	complemented
249	C₂	#10592400	1	2	yes	—
250	C₂	#10592405	1	2	yes	—
251	C₂	C₄×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
252	C₂	M₁₆×C₃⋊Dic₃	1	1	yes	characteristic
253	C₂	#10592821	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂×C₄²
Fitting	C₂×C₃²×#97	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#97	C₃⋊D₃
hypocenter	C₃²	#1771
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
17. C₃²×#97	C₄	semidirect
21. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
62. C₃×#97	Dic₃	semidirect
85. C₃²×C₈	C₄⋊C₄	semidirect
142. #97	C₃⋊Dic₃	semidirect
163. C₃×C₈	C₄⋊Dic₃	semidirect
208. C₃²	#1771	semidirect
229. C₈	C₄⋊(C₃⋊Dic₃)	semidirect
248. C₃	#69810	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#97	2
C₂×#63642	2
C₂×C₈×C₃⋊Dic₃	2
C₄×C₄⋊(C₃⋊Dic₃)	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10589416	2
#69810	3