Subgroups — #11790336

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63644	C₂	1	2	yes	maximal
3	#10592634	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
5	#10589417	C₂	1	2	yes	maximal
6	#70454	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
10	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
11	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃⋊Dic₃	—	2	1	no	—
15	C₃²×C₄⋊C₈	C₄	1	4	yes	complemented
16	#63658	—	2	1	no	—
17	#63644	—	2	4	no	—
18	C₂×C₃×C₄⋊C₈	D₃	1	4	yes	—
19	C₂×#817	—	3	8	no	—
20	#4935	—	3	4	no	—
21	C₄×C₄⋊Dic₃	—	3	4	no	—
22	#4551	—	3	8	no	—
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₂²×C₃⋊Dic₃	—	2	1	no	—
27	C₂²×C₃⋊Dic₃	—	2	1	no	—
28	C₂²×C₃⋊Dic₃	—	2	2	no	—
29	C₂×C₃²×C₈	—	2	2	no	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
32	C₃²×C₄²	C₂×C₄	1	2	yes	—
33	C₃²×C₄²	Q₈	1	2	yes	—
34	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
35	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
39	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
40	C₄⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
41	C₄⋊(C₃⋊Dic₃)	Q₈	1	2	yes	—
42	C₄×C₃⋊Dic₃	—	4	2	no	—
43	#1948	—	9	1	no	Sylow
44	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
45	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
46	C₂×C₄⋊Dic₃	—	3	4	no	—
47	C₂×C₄⋊Dic₃	—	3	4	no	—
48	C₂×C₄⋊Dic₃	—	3	8	no	—
49	C₂×C₄⋊Dic₃	—	6	4	no	—
50	C₂×C₄⋊Dic₃	—	6	4	no	—
51	C₂×C₄×Dic₃	—	6	4	no	—
52	C₃×C₄⋊C₈	Dic₃	1	16	yes	complemented
53	C₃⋊#95	—	6	4	no	—
54	#817	—	6	16	no	—
55	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	—	2	1	no	—
57	C₂×C₃²×C₄	—	2	1	no	—
58	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
60	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
61	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
62	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃²×C₄	#41	1	1	yes	characteristic
67	C₂×C₃²×C₄	#41	1	1	yes	characteristic
68	C₂×C₃⋊Dic₃	—	2	2	no	—
69	C₂×C₃⋊Dic₃	—	2	2	no	—
70	C₂×C₃⋊Dic₃	—	4	1	no	—
71	C₂×C₃⋊Dic₃	—	4	1	no	—
72	C₂×C₃⋊Dic₃	—	4	1	no	—
73	C₂×C₃⋊Dic₃	—	4	1	no	—
74	C₂×C₃⋊Dic₃	—	4	2	no	—
75	C₂×C₃⋊Dic₃	—	4	2	no	—
76	C₂×C₃⋊Dic₃	—	4	2	no	—
77	C₃²×C₈	—	2	4	no	—
78	C₂×#106	—	9	2	no	—
79	C₂×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
80	#384	—	9	1	no	—
81	C₄×C₄⋊C₄	—	9	1	no	—
82	#340	—	9	2	no	—
83	C₂²×C₃×C₄	D₁₂	1	4	yes	—
84	C₂²×C₃×C₄	D₁₂	1	4	yes	—
85	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
86	C₂²×Dic₃	—	6	4	no	—
87	C₂²×Dic₃	—	6	4	no	—
88	C₂²×Dic₃	—	6	8	no	—
89	C₂×C₃×C₈	—	2	8	no	—
90	C₂×C₃×C₈	—	2	8	no	—
91	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
92	C₃×C₄²	Dic₆	1	8	yes	—
93	C₃×C₄²	C₂×Dic₃	1	8	yes	—
94	C₄⋊Dic₃	—	3	8	no	—
95	C₄⋊Dic₃	—	3	8	no	—
96	C₄⋊Dic₃	—	6	4	no	—
97	C₄⋊Dic₃	—	6	4	no	—
98	C₄⋊Dic₃	—	6	8	no	—
99	C₄⋊Dic₃	—	6	8	no	—
100	C₄⋊Dic₃	—	12	8	no	—
101	C₄⋊Dic₃	—	12	8	no	—
102	C₄×Dic₃	—	12	8	no	—
103	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃²	#121	1	1	yes	characteristic
105	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
108	C₂²×C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂²×C₃²	#137	1	1	yes	characteristic
110	C₃²×C₄	—	2	2	no	—
111	C₃²×C₄	#106	1	4	yes	complemented
112	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
113	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
114	C₃²×C₄	#125	1	1	yes	characteristic
115	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
116	C₃⋊Dic₃	—	4	2	no	—
117	C₃⋊Dic₃	—	4	2	no	—
118	C₃⋊Dic₃	—	8	2	no	—
119	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
120	C₂×C₄⋊C₄	—	9	1	no	—
121	C₂×C₄⋊C₄	—	9	1	no	—
122	C₂×C₄⋊C₄	—	9	2	no	—
123	C₂×C₄⋊C₄	—	18	1	no	—
124	C₂×C₄⋊C₄	—	18	1	no	—
125	C₂×C₄²	—	18	1	no	—
126	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
127	#106	—	18	4	no	—
128	C₄⋊C₈	C₃⋊Dic₃	1	4	yes	complemented
129	#95	—	18	1	no	—
130	C₂³×C₃	C₂×D₁₂	1	4	yes	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	#204	1	8	yes	—
134	C₂×C₃×C₄	#204	1	8	yes	—
135	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
136	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
137	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
138	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
139	C₂×C₃×C₄	#237	1	4	yes	—
140	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
141	C₂×C₃×C₄	#239	1	4	yes	—
142	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
143	C₂×Dic₃	—	6	8	no	—
144	C₂×Dic₃	—	6	8	no	—
145	C₂×Dic₃	—	12	4	no	—
146	C₂×Dic₃	—	12	4	no	—
147	C₂×Dic₃	—	12	4	no	—
148	C₂×Dic₃	—	12	4	no	—
149	C₂×Dic₃	—	12	8	no	—
150	C₂×Dic₃	—	12	8	no	—
151	C₂×Dic₃	—	12	8	no	—
152	C₃×C₈	—	2	16	no	—
153	C₂×C₃²	#389	1	1	yes	characteristic
154	C₂×C₃²	C₂×#106	1	1	yes	characteristic
155	C₂×C₃²	#428	1	1	yes	characteristic
156	C₂×C₃²	#460	1	1	yes	characteristic
157	C₂×C₃²	#461	1	1	yes	characteristic
158	C₂×C₃²	#475	1	1	yes	characteristic
159	C₂×C₃²	#476	1	1	yes	characteristic
160	C₂²×C₄	—	18	1	no	—
161	C₂²×C₄	—	18	1	no	—
162	C₂²×C₄	—	18	2	no	—
163	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
169	C₄⋊C₄	—	9	2	no	—
170	C₄⋊C₄	—	9	2	no	—
171	C₄⋊C₄	—	18	1	no	—
172	C₄⋊C₄	—	18	1	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₄⋊C₄	—	18	2	no	—
175	C₄⋊C₄	—	36	2	no	—
176	C₄⋊C₄	—	36	2	no	—
177	C₄²	—	36	2	no	—
178	C₄²	#634	1	2	yes	—
179	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
180	C₂²×C₃	#895	1	4	yes	—
181	C₂²×C₃	#897	1	4	yes	—
182	C₂²×C₃	C₂×#204	1	4	yes	—
183	C₂²×C₃	C₂×#204	1	4	yes	—
184	C₂²×C₃	#908	1	4	yes	—
185	C₂²×C₃	#909	1	4	yes	—
186	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
187	C₃×C₄	—	2	8	no	—
188	C₃×C₄	#817	1	16	yes	complemented
189	C₃×C₄	#888	1	4	yes	—
190	C₃×C₄	#890	1	4	yes	—
191	C₃×C₄	D₄×Dic₃	1	4	yes	—
192	C₃×C₄	Q₈×Dic₃	1	4	yes	—
193	Dic₃	—	12	8	no	—
194	Dic₃	—	12	8	no	—
195	Dic₃	—	24	8	no	—
196	C₃²	#1948	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	18	2	no	—
201	C₂×C₄	—	18	2	no	—
202	C₂×C₄	—	36	1	no	—
203	C₂×C₄	—	36	1	no	—
204	C₂×C₄	—	36	1	no	—
205	C₂×C₄	—	36	1	no	—
206	C₂×C₄	—	36	2	no	—
207	C₂×C₄	—	36	2	no	—
208	C₂×C₄	—	36	2	no	—
209	C₂×C₄	#3744	1	2	yes	—
210	C₂×C₄	#3744	1	2	yes	—
211	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
212	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
213	C₂×C₄	C₂×#634	1	1	yes	characteristic
214	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
215	C₂×C₄	#3830	1	1	yes	characteristic
216	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
217	C₂×C₄	#3832	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₂×C₃	#4841	1	4	yes	—
221	C₂×C₃	#4842	1	4	yes	—
222	C₂×C₃	#4848	1	4	yes	—
223	C₂×C₃	#4851	1	4	yes	—
224	C₂×C₃	#4984	1	4	yes	—
225	C₂×C₃	C₂×#817	1	4	yes	—
226	C₂×C₃	#5121	1	4	yes	—
227	C₂²	#64104	1	1	yes	characteristic
228	C₂²	#64106	1	1	yes	characteristic
229	C₂²	C₂×#3744	1	1	yes	characteristic
230	C₂²	C₂×#3744	1	1	yes	characteristic
231	C₂²	#64117	1	1	yes	characteristic
232	C₂²	#64118	1	1	yes	characteristic
233	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	36	2	no	—
236	C₄	—	36	2	no	—
237	C₄	—	72	2	no	—
238	C₄	#63644	1	4	yes	complemented
239	C₄	#64097	1	1	yes	characteristic
240	C₄	#64099	1	1	yes	characteristic
241	C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
242	C₄	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
243	C₃	#70454	1	4	yes	complemented
244	C₂	#10592540	1	1	yes	characteristic
245	C₂	#10592541	1	1	yes	characteristic
246	C₂	#10592547	1	1	yes	characteristic
247	C₂	#10592550	1	1	yes	characteristic
248	C₂	#10592683	1	1	yes	characteristic
249	C₂	C₂×#63644	1	1	yes	characteristic
250	C₂	#10592820	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	#1948
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₃²×C₄⋊C₈	C₄	semidirect
52. C₃×C₄⋊C₈	Dic₃	semidirect
111. C₃²×C₄	#106	semidirect
128. C₄⋊C₈	C₃⋊Dic₃	semidirect
188. C₃×C₄	#817	semidirect
196. C₃²	#1948	semidirect
238. C₄	#63644	semidirect
243. C₃	#70454	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊C₈	2
C₂×#63644	2
#10592634	2
C₄×C₄⋊(C₃⋊Dic₃)	2
#10589417	2
#70454	3