Subgroups — C₂×D₄×#63216 #17978523

Order 4352.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63216	C₂	1	2	yes	maximal, complemented
2	D₄×#63216	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#63216	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×#3622	C₂	1	1	yes	maximal, characteristic
7	C₂²×#63216	—	2	8	no	—
8	C₂²×#63216	C₂²	1	2	yes	—
9	C₂²×#63216	C₂²	1	1	yes	characteristic
10	C₂²×#63216	C₂²	1	1	yes	characteristic
11	C₂²×#63216	C₂²	1	8	yes	complemented
12	C₄⋊#63216	C₂²	1	4	yes	complemented
13	C₂²⋊#63216	C₂²	1	8	yes	complemented
14	C₄×#63216	C₂²	1	4	yes	complemented
15	C₂³×#3622	C₂²	1	2	yes	—
16	D₄×#3622	C₄	1	4	yes	—
17	D₄×#3622	C₂²	1	4	yes	—
18	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊#3622	C₄	1	2	yes	—
20	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₇	C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₇	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₇	C₂×C₄	1	2	yes	—
24	C₄×D₄×C₁₇	C₈	1	4	yes	—
25	C₄×D₄×C₁₇	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₇	C₈	1	2	yes	—
28	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3622	—	2	4	no	—
30	C₂²×#3622	—	2	4	no	—
31	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3622	C₂×C₄	1	2	yes	—
33	C₂²×#3622	C₂×C₄	1	4	yes	—
34	C₂²×#3622	C₂³	1	4	yes	—
35	C₂²×#3622	C₂³	1	1	yes	characteristic
36	C₂×#63216	—	2	4	no	—
37	C₂×#63216	—	2	2	no	—
38	C₂×#63216	—	2	16	no	—
39	C₂×#63216	D₄	1	4	yes	direct product factor
40	C₂×#63216	C₂³	1	4	yes	—
41	C₂×#63216	C₂³	1	2	yes	—
42	C₂×#63216	C₂³	1	2	yes	—
43	C₂²⋊#3622	C₂×C₄	1	8	yes	—
44	C₄⋊#3622	C₂×C₄	1	4	yes	—
45	C₄×#3622	C₂×C₄	1	2	yes	—
46	C₄×#3622	C₂³	1	2	yes	—
47	C₂⁴×C₁₇	C₂×C₈	1	2	yes	—
48	C₂×D₄×C₁₇	C₁₆	1	8	yes	complemented
49	C₂×D₄×C₁₇	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₁₇	—	2	4	no	—
51	C₂²×C₄×C₁₇	—	2	4	no	—
52	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
59	C₂²⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
60	C₄²×C₁₇	C₂×C₈	1	2	yes	—
61	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
62	C₂×#3622	—	2	4	no	—
63	C₂×#3622	—	2	2	no	—
64	C₂×#3622	—	2	8	no	—
65	C₂×#3622	—	2	4	no	—
66	C₂×#3622	—	2	4	no	—
67	C₂×#3622	C₂²×C₄	1	2	yes	—
68	C₂×#3622	C₂²×C₄	1	4	yes	—
69	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
70	C₂×#3622	C₂×D₄	1	2	yes	—
71	C₂×#3622	#41	1	2	yes	—
72	C₂×#3622	C₂⁴	1	1	yes	characteristic
73	#63216	—	2	4	no	—
74	#63216	—	2	2	no	—
75	#63216	C₂×D₄	1	4	yes	direct product factor
76	C₂×D₄×C₁₆	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	—	2	4	no	—
78	C₂³×C₁₇	—	2	4	no	—
79	C₂³×C₁₇	C₂×C₁₆	1	8	yes	complemented
80	C₂³×C₁₇	C₂²×C₈	1	4	yes	—
81	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
82	D₄×C₁₇	C₂×C₁₆	1	16	yes	complemented
83	C₂×C₄×C₁₇	—	2	4	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	4	no	—
86	C₂×C₄×C₁₇	—	2	4	no	—
87	C₂×C₄×C₁₇	—	2	8	no	—
88	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	complemented
89	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
92	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₇	#131	1	2	yes	—
95	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
96	#3622	—	2	4	no	—
97	#3622	—	2	2	no	—
98	#3622	C₄×D₄	1	2	yes	—
99	#3622	C₂²×D₄	1	1	yes	characteristic
100	#3622	C₂×#41	1	1	yes	characteristic
101	C₂³×C₁₆	—	17	2	no	—
102	C₂×C₈×D₄	—	17	1	no	—
103	D₄×C₁₆	—	17	8	no	—
104	C₂×C₄⋊C₁₆	—	17	1	no	—
105	C₂×C₂²⋊C₁₆	—	17	2	no	—
106	C₂×C₄×C₁₆	—	17	1	no	—
107	C₂²×C₁₇	—	2	4	no	—
108	C₂²×C₁₇	—	2	4	no	—
109	C₂²×C₁₇	—	2	16	no	—
110	C₂²×C₁₇	C₈×D₄	1	2	yes	—
111	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
112	C₂²×C₁₇	C₂²×C₁₆	1	8	yes	complemented
113	C₂²×C₁₇	#504	1	2	yes	—
114	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₁₇	—	2	2	no	—
116	C₄×C₁₇	—	2	4	no	—
117	C₄×C₁₇	C₈×D₄	1	2	yes	—
118	C₄×C₁₇	C₂²×C₁₆	1	4	yes	complemented
119	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
121	C₂³×C₈	—	17	2	no	—
122	C₂×C₄×D₄	#595	1	1	yes	characteristic
123	C₂²×C₁₆	—	17	1	no	—
124	C₂²×C₁₆	—	17	1	no	—
125	C₂²×C₁₆	—	17	2	no	—
126	C₂²×C₁₆	—	17	8	no	—
127	C₂²×C₁₆	—	34	8	no	—
128	C₈×D₄	—	17	4	no	—
129	C₈×D₄	—	17	4	no	—
130	C₂×C₄⋊C₈	—	17	1	no	—
131	C₂×C₂²⋊C₈	—	17	2	no	—
132	C₂×C₄×C₈	—	17	1	no	—
133	C₄⋊C₁₆	—	17	4	no	—
134	C₂²⋊C₁₆	—	17	8	no	—
135	C₄×C₁₆	—	17	4	no	—
136	C₂×C₁₇	—	2	8	no	—
137	C₂×C₁₇	D₄×C₁₆	1	4	yes	complemented
138	C₂×C₁₇	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
140	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
141	C₂²×D₄	#3624	1	1	yes	characteristic
142	C₂³×C₄	C₂×#595	1	2	yes	—
143	C₂²×C₈	—	17	1	no	—
144	C₂²×C₈	—	17	4	no	—
145	C₂²×C₈	—	17	4	no	—
146	C₂²×C₈	—	17	2	no	—
147	C₂²×C₈	—	17	1	no	—
148	C₂²×C₈	—	34	4	no	—
149	C₂²×C₈	—	34	4	no	—
150	C₄×D₄	#3624	1	4	yes	—
151	C₄×D₄	C₂×#595	1	4	yes	—
152	C₂×C₄⋊C₄	#3624	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	#3624	1	2	yes	—
154	C₂×C₄²	C₂×#595	1	1	yes	characteristic
155	C₂×C₁₆	—	17	2	no	—
156	C₂×C₁₆	—	17	4	no	—
157	C₂×C₁₆	—	17	2	no	—
158	C₂×C₁₆	—	17	4	no	—
159	C₂×C₁₆	—	34	16	no	—
160	C₂×C₁₆	—	34	2	no	—
161	C₂×C₁₆	—	34	4	no	—
162	C₄⋊C₈	—	17	4	no	—
163	C₂²⋊C₈	—	17	8	no	—
164	C₄×C₈	—	17	2	no	—
165	C₄×C₈	—	17	2	no	—
166	C₁₇	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
167	C₂⁴	C₂×#3624	1	2	yes	—
168	C₂×D₄	#63216	1	8	yes	direct product factor
169	C₂×D₄	C₂×#3624	1	4	yes	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
173	C₂²×C₄	C₂×#3624	1	2	yes	—
174	C₂²×C₄	C₂×#3624	1	4	yes	—
175	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
176	C₂²×C₄	C₂²×#595	1	4	yes	—
177	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
178	C₂×C₈	—	17	2	no	—
179	C₂×C₈	—	17	2	no	—
180	C₂×C₈	—	17	1	no	—
181	C₂×C₈	—	17	1	no	—
182	C₂×C₈	—	17	2	no	—
183	C₂×C₈	—	17	4	no	—
184	C₂×C₈	—	34	4	no	—
185	C₂×C₈	—	34	4	no	—
186	C₂×C₈	—	34	8	no	—
187	C₂×C₈	—	34	2	no	—
188	C₂×C₈	—	34	4	no	—
189	C₄⋊C₄	C₂×#3624	1	4	yes	—
190	C₂²⋊C₄	C₂×#3624	1	8	yes	—
191	C₄²	C₂×#3624	1	2	yes	—
192	C₄²	C₂²×#595	1	2	yes	—
193	C₁₆	—	17	4	no	—
194	C₁₆	—	34	4	no	—
195	C₁₆	—	34	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	4	no	—
198	C₂³	C₂×#63216	1	8	yes	complemented
199	C₂³	C₂²×#3624	1	4	yes	—
200	C₂³	C₂²×#3624	1	1	yes	characteristic
201	D₄	C₂×#63216	1	16	yes	direct product factor
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	C₂×#63216	1	4	yes	complemented
208	C₂×C₄	#10587678	1	2	yes	—
209	C₂×C₄	D₄×#595	1	2	yes	—
210	C₂×C₄	C₂²×#3624	1	2	yes	—
211	C₂×C₄	C₂²×#3624	1	4	yes	—
212	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
213	C₂×C₄	C₂²×#3624	1	2	yes	—
214	C₂×C₄	C₂³×#595	1	1	yes	characteristic
215	C₈	—	17	1	no	—
216	C₈	—	17	1	no	—
217	C₈	—	17	2	no	—
218	C₈	—	34	4	no	—
219	C₈	—	34	2	no	—
220	C₂²	—	2	16	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	D₄×#3624	1	2	yes	—
224	C₂²	C₂²×#63216	1	2	yes	—
225	C₂²	C₂²×#63216	1	8	yes	complemented
226	C₂²	#11763867	1	2	yes	—
227	C₂²	C₂³×#3624	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	D₄×#3624	1	2	yes	—
231	C₄	C₂²×#63216	1	4	yes	complemented
232	C₄	C₂×D₄×#595	1	1	yes	characteristic
233	C₄	C₂×#10587678	1	1	yes	characteristic
234	C₂	—	2	8	no	—
235	C₂	D₄×#63216	1	4	yes	direct product factor
236	C₂	C₂×D₄×#3624	1	1	yes	characteristic
237	C₂	C₂³×#63216	1	1	yes	characteristic
238	C₂	C₂×#11763867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#595
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂×C₄×D₄×C₁₇	C₄
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂²×D₄×C₁₇	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂×C₄×D₄	#595
hypocenter	C₁₇	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#63216	C₂	semidirect
2. D₄×#63216	C₂	direct (N × H)
3. C₂×C₄⋊#63216	C₂	semidirect
4. C₂×C₂²⋊#63216	C₂	semidirect
5. C₂×C₄×#63216	C₂	semidirect
11. C₂²×#63216	C₂²	semidirect
12. C₄⋊#63216	C₂²	semidirect
13. C₂²⋊#63216	C₂²	semidirect
14. C₄×#63216	C₂²	semidirect
39. C₂×#63216	D₄	direct (N × H)
48. C₂×D₄×C₁₇	C₁₆	semidirect
75. #63216	C₂×D₄	direct (N × H)
79. C₂³×C₁₇	C₂×C₁₆	semidirect
82. D₄×C₁₇	C₂×C₁₆	semidirect
88. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
112. C₂²×C₁₇	C₂²×C₁₆	semidirect
118. C₄×C₁₇	C₂²×C₁₆	semidirect
137. C₂×C₁₇	D₄×C₁₆	semidirect
166. C₁₇	C₂×D₄×C₁₆	semidirect
168. C₂×D₄	#63216	direct (N × H)
198. C₂³	C₂×#63216	semidirect
201. D₄	C₂×#63216	direct (N × H)
207. C₂×C₄	C₂×#63216	semidirect
225. C₂²	C₂²×#63216	semidirect
231. C₄	C₂²×#63216	semidirect
235. C₂	D₄×#63216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#63216	2
D₄×#63216	2
C₂×C₄⋊#63216	2
C₂×C₂²⋊#63216	2
C₂×C₄×#63216	2
C₂×D₄×#3622	2
C₂×D₄×C₁₆	17

Subgroups — C2×D4×#63216 #17978523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#63216 #17978523