Subgroups — #11860674

Order 1152.

1584 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₈	C₂	1	1	yes	maximal, characteristic
2	#10596269	C₂	1	2	yes	maximal, complemented
3	#10592914	C₂	1	2	yes	maximal, complemented
4	#10592851	C₂	1	4	yes	maximal, complemented
5	D₈×C₃⋊Dic₃	C₂	1	4	yes	maximal, complemented
6	#10592805	C₂	1	1	yes	maximal, characteristic, complemented
7	#10592800	C₂	1	1	yes	maximal, characteristic, complemented
8	#83823	—	3	4	no	maximal
9	C₂²×C₃²×D₄	C₂²	1	2	yes	—
10	C₂×C₃²×D₈	C₄	1	8	yes	complemented
11	C₂×C₃²×D₈	C₂²	1	2	yes	—
12	C₂×C₃²×D₈	C₂²	1	2	yes	—
13	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₂×#3757	—	2	4	no	—
15	D₄×C₃⋊Dic₃	—	2	4	no	—
16	D₄×C₃⋊Dic₃	C₂²	1	8	yes	complemented
17	#64135	C₂²	1	2	yes	—
18	#64135	C₂²	1	1	yes	characteristic
19	C₂×#3748	C₂²	1	1	yes	characteristic
20	#63659	C₂²	1	8	yes	complemented
21	#63645	C₂²	1	2	yes	complemented
22	#63644	C₂²	1	2	yes	complemented
23	#63642	C₂²	1	2	yes	complemented
24	C₈×C₃⋊Dic₃	C₂²	1	2	yes	complemented
25	C₂²×C₃×D₈	D₃	1	4	yes	—
26	#5798	—	3	8	no	—
27	#5215	—	3	8	no	—
28	#5152	—	3	16	no	—
29	Dic₃×D₈	—	3	16	no	—
30	#5106	—	3	4	no	—
31	#5101	—	3	4	no	—
32	C₂⁴×C₃²	—	2	2	no	—
33	C₂×C₃²×D₄	—	2	2	no	—
34	C₂×C₃²×D₄	—	2	4	no	—
35	C₂×C₃²×D₄	C₂×C₄	1	8	yes	complemented
36	C₂×C₃²×D₄	C₂³	1	4	yes	—
37	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	2	8	no	—
39	C₃²×D₈	C₂×C₄	1	8	yes	complemented
40	C₃²×D₈	C₂×C₄	1	8	yes	complemented
41	C₂×C₃²×C₈	C₂×C₄	1	4	yes	complemented
42	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
44	#3757	—	2	2	no	—
45	#3757	—	2	8	no	—
46	#3757	D₄	1	2	yes	—
47	C₄⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
48	C₄⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
49	C₄×C₃⋊Dic₃	—	2	2	no	—
50	C₄×C₃⋊Dic₃	C₂³	1	2	yes	—
51	#3748	—	2	2	no	—
52	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
53	#2948	—	9	1	no	Sylow
54	C₂²×C₃×D₄	C₂×D₃	1	8	yes	—
55	C₂×C₃×D₈	Dic₃	1	32	yes	complemented
56	C₂×C₃×D₈	C₂×D₃	1	8	yes	—
57	C₂×C₃×D₈	C₂×D₃	1	8	yes	—
58	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
59	C₂×#217	—	6	16	no	—
60	D₄×Dic₃	—	3	32	no	—
61	D₄×Dic₃	—	6	16	no	—
62	#926	—	3	4	no	—
63	#926	—	3	8	no	—
64	C₂×#208	—	3	4	no	—
65	#832	—	3	32	no	—
66	#818	—	3	8	no	—
67	#817	—	3	8	no	—
68	#815	—	3	8	no	—
69	C₈×Dic₃	—	3	8	no	—
70	C₂³×C₃²	—	2	4	no	—
71	C₂³×C₃²	—	2	8	no	—
72	C₂³×C₃²	—	8	4	no	—
73	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
74	C₃²×D₄	—	2	8	no	—
75	C₃²×D₄	—	2	4	no	—
76	C₃²×D₄	C₂²×C₄	1	8	yes	—
77	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	2	yes	—
80	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
81	C₂×C₃⋊Dic₃	—	2	4	no	—
82	C₂×C₃⋊Dic₃	—	8	8	no	—
83	C₂×C₃⋊Dic₃	C₂×D₄	1	4	yes	—
84	C₃²×C₈	C₂²×C₄	1	2	yes	—
85	C₃²×C₈	C₂²×C₄	1	2	yes	—
86	C₃⋊(C₃⋊C₈)	—	2	2	no	—
87	C₂²×D₈	C₃⋊D₃	1	1	yes	characteristic
88	#518	—	9	2	no	—
89	#442	—	9	4	no	—
90	C₄×D₈	—	9	4	no	—
91	#427	—	9	1	no	—
92	#418	—	9	2	no	—
93	#405	—	9	1	no	—
94	C₂⁴×C₃	—	2	8	no	—
95	C₂×C₃×D₄	—	2	8	no	—
96	C₂×C₃×D₄	—	2	16	no	—
97	C₂×C₃×D₄	C₂×Dic₃	1	32	yes	complemented
98	C₂×C₃×D₄	C₂²×D₃	1	16	yes	—
99	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
100	C₂²×Dic₃	—	6	32	no	—
101	C₃×D₈	C₂×Dic₃	1	32	yes	complemented
102	C₃×D₈	C₂×Dic₃	1	32	yes	complemented
103	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	complemented
104	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
105	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
106	#217	—	3	8	no	—
107	#217	—	6	32	no	—
108	#217	—	6	8	no	—
109	C₄⋊Dic₃	—	3	8	no	—
110	C₄⋊Dic₃	—	3	16	no	—
111	C₄×Dic₃	—	3	8	no	—
112	C₄×Dic₃	—	6	8	no	—
113	#208	—	6	8	no	—
114	C₂×C₃⋊C₈	—	3	8	no	—
115	C₂²×C₃²	—	2	2	no	—
116	C₂²×C₃²	—	2	8	no	—
117	C₂²×C₃²	—	4	8	no	—
118	C₂²×C₃²	—	8	8	no	—
119	C₂²×C₃²	C₄×D₄	1	2	yes	—
120	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
121	C₃²×C₄	C₄×D₄	1	2	yes	—
122	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
123	C₃²×C₄	C₂×#41	1	1	yes	characteristic
124	C₃⋊Dic₃	—	2	4	no	—
125	C₃⋊Dic₃	—	4	4	no	—
126	C₂²×D₄	C₂×C₃⋊D₃	1	2	yes	—
127	C₂×D₈	C₃⋊Dic₃	1	8	yes	complemented
128	C₂×D₈	C₂×C₃⋊D₃	1	2	yes	—
129	C₂×D₈	C₂×C₃⋊D₃	1	2	yes	—
130	C₂×M₁₆	—	9	1	no	—
131	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
132	C₄×D₄	—	9	8	no	—
133	C₄×D₄	—	18	4	no	—
134	#117	—	9	1	no	—
135	#117	—	9	2	no	—
136	C₂×C₂²⋊C₄	—	18	4	no	—
137	#107	—	9	2	no	—
138	#106	—	9	2	no	—
139	#102	—	9	8	no	—
140	#97	—	9	2	no	—
141	C₄×C₈	—	9	2	no	—
142	C₂³×C₃	—	2	16	no	—
143	C₂³×C₃	—	2	32	no	—
144	C₂³×C₃	—	8	16	no	—
145	C₂³×C₃	D₃×D₄	1	4	yes	—
146	C₃×D₄	—	2	32	no	—
147	C₃×D₄	—	2	16	no	—
148	C₃×D₄	C₂²×Dic₃	1	32	yes	—
149	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
150	C₂×C₃×C₄	#237	1	8	yes	—
151	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
152	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
153	C₂×Dic₃	—	3	16	no	—
154	C₂×Dic₃	—	6	16	no	—
155	C₂×Dic₃	—	24	32	no	—
156	C₃×C₈	C₂²×Dic₃	1	8	yes	—
157	C₃×C₈	C₂²×Dic₃	1	8	yes	—
158	C₃⋊C₈	—	6	8	no	—
159	C₂×C₃²	—	2	2	no	—
160	C₂×C₃²	—	4	8	no	—
161	C₂×C₃²	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×C₃²	#576	1	2	yes	—
163	C₂⁴	—	2	2	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	C₂×C₃⋊Dic₃	1	8	yes	complemented
167	C₂×D₄	C₂²×C₃⋊D₃	1	4	yes	—
168	C₂²×C₄	—	18	8	no	—
169	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
170	D₈	C₂×C₃⋊Dic₃	1	8	yes	complemented
171	D₈	C₂×C₃⋊Dic₃	1	8	yes	complemented
172	M₁₆	—	18	2	no	—
173	C₂×C₈	—	9	2	no	—
174	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	complemented
175	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
177	C₄⋊C₄	—	9	2	no	—
178	C₄⋊C₄	—	9	4	no	—
179	C₂²⋊C₄	—	9	2	no	—
180	C₂²⋊C₄	—	18	8	no	—
181	C₂²⋊C₄	—	18	2	no	—
182	C₄²	—	9	2	no	—
183	C₄²	—	18	2	no	—
184	C₂²×C₃	—	2	8	no	—
185	C₂²×C₃	—	2	32	no	—
186	C₂²×C₃	—	4	32	no	—
187	C₂²×C₃	—	8	32	no	—
188	C₂²×C₃	D₄×Dic₃	1	8	yes	—
189	C₂²×C₃	C₂×D₃×D₄	1	4	yes	—
190	C₃×C₄	D₄×Dic₃	1	8	yes	—
191	C₃×C₄	C₂×#237	1	4	yes	—
192	C₃×C₄	C₂³×Dic₃	1	4	yes	—
193	Dic₃	—	6	16	no	—
194	Dic₃	—	12	16	no	—
195	C₃²	#2948	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	2	4	no	—
197	C₂³	—	2	8	no	—
198	C₂³	—	8	4	no	—
199	C₂³	D₄×C₃⋊D₃	1	1	yes	characteristic
200	D₄	—	2	4	no	—
201	D₄	—	2	8	no	—
202	D₄	C₂²×C₃⋊Dic₃	1	8	yes	—
203	C₂×C₄	—	9	4	no	—
204	C₂×C₄	—	18	4	no	—
205	C₂×C₄	—	72	8	no	—
206	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
207	C₂×C₄	#3830	1	2	yes	—
208	C₂×C₄	C₂²×C₃⋊Dic₃	1	2	yes	—
209	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
210	C₈	—	18	2	no	—
211	C₈	C₂²×C₃⋊Dic₃	1	2	yes	—
212	C₈	C₂²×C₃⋊Dic₃	1	2	yes	—
213	C₂×C₃	—	2	8	no	—
214	C₂×C₃	—	4	32	no	—
215	C₂×C₃	#5757	1	8	yes	—
216	C₂×C₃	C₂×D₄×Dic₃	1	4	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	4	8	no	—
220	C₂²	—	8	8	no	—
221	C₂²	D₄×C₃⋊Dic₃	1	2	yes	—
222	C₂²	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
223	C₄	—	18	4	no	—
224	C₄	—	36	4	no	—
225	C₄	D₄×C₃⋊Dic₃	1	2	yes	—
226	C₄	C₂×#3830	1	1	yes	characteristic
227	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
228	C₃	#83823	1	4	yes	complemented
229	C₂	—	2	2	no	—
230	C₂	—	4	8	no	—
231	C₂	#10596228	1	2	yes	—
232	C₂	C₂×D₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting, prime radical	C₂²×C₃²×D₈	C₂
socle	C₂²×C₃²	C₂²×D₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂²×D₈	C₃⋊D₃
hypocenter	C₃²	#2948
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
2. #10596269	C₂	semidirect
3. #10592914	C₂	semidirect
4. #10592851	C₂	semidirect
5. D₈×C₃⋊Dic₃	C₂	semidirect
6. #10592805	C₂	semidirect
7. #10592800	C₂	semidirect
10. C₂×C₃²×D₈	C₄	semidirect
16. D₄×C₃⋊Dic₃	C₂²	semidirect
20. #63659	C₂²	semidirect
21. #63645	C₂²	semidirect
22. #63644	C₂²	semidirect
23. #63642	C₂²	semidirect
24. C₈×C₃⋊Dic₃	C₂²	semidirect
35. C₂×C₃²×D₄	C₂×C₄	semidirect
39. C₃²×D₈	C₂×C₄	semidirect
40. C₃²×D₈	C₂×C₄	semidirect
41. C₂×C₃²×C₈	C₂×C₄	semidirect
55. C₂×C₃×D₈	Dic₃	semidirect
97. C₂×C₃×D₄	C₂×Dic₃	semidirect
101. C₃×D₈	C₂×Dic₃	semidirect
102. C₃×D₈	C₂×Dic₃	semidirect
103. C₂×C₃×C₈	C₂×Dic₃	semidirect
127. C₂×D₈	C₃⋊Dic₃	semidirect
166. C₂×D₄	C₂×C₃⋊Dic₃	semidirect
170. D₈	C₂×C₃⋊Dic₃	semidirect
171. D₈	C₂×C₃⋊Dic₃	semidirect
174. C₂×C₈	C₂×C₃⋊Dic₃	semidirect
195. C₃²	#2948	semidirect
228. C₃	#83823	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×D₈	2
#10596269	2
#10592914	2
#10592851	2
D₈×C₃⋊Dic₃	2
#10592805	2
#10592800	2
#83823	3