Subgroups — C₂×C₄×#88106 #16806717

Order 3328.

918 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	C₄×#88106	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718231	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#6092	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	1	yes	characteristic
9	C₂×#88106	C₄	1	16	yes	direct product factor
10	C₂×#88106	C₂²	1	8	yes	—
11	C₂²×#6092	C₂²	1	4	yes	—
12	C₂×C₄×#1055	C₄	1	1	yes	characteristic
13	C₂×C₄×#1055	C₄	1	4	yes	—
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	4	yes	—
18	#11718231	C₂²	1	8	yes	complemented
19	C₄×#6092	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₃	C₈	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₈	1	4	yes	—
23	C₂×C₄²×C₁₃	C₈	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₈	1	8	yes	—
28	C₂²×#1055	C₂×C₄	1	2	yes	—
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	8	yes	—
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	4	yes	—
34	C₂²×#1055	C₂³	1	1	yes	characteristic
35	C₂²×#1055	C₂³	1	2	yes	—
36	#88106	C₂×C₄	1	32	yes	direct product factor
37	C₂×#6092	C₂×C₄	1	16	yes	complemented
38	C₂×#6092	C₂³	1	8	yes	—
39	C₄×#1055	C₂×C₄	1	2	yes	—
40	C₄×#1055	C₂×C₄	1	2	yes	—
41	C₄×#1055	C₂×C₄	1	4	yes	—
42	C₄×#1055	C₂×C₄	1	4	yes	—
43	C₄×#1055	C₂³	1	2	yes	—
44	C₄×#1055	C₂³	1	2	yes	—
45	C₂×C₄×M₃₂	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	8	no	—
48	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×C₈	1	2	yes	—
58	C₄²×C₁₃	C₂×C₈	1	8	yes	—
59	C₄²×C₁₃	C₂×C₈	1	2	yes	—
60	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂×#1055	C₄²	1	8	yes	—
63	C₂×#1055	C₄²	1	4	yes	—
64	C₂×#1055	C₂×C₈	1	16	yes	—
65	C₂×#1055	C₂×C₈	1	4	yes	—
66	C₂×#1055	C₂×C₈	1	4	yes	—
67	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
68	C₂×#1055	C₂²×C₄	1	2	yes	—
69	C₂×#1055	C₂²×C₄	1	4	yes	—
70	C₂×#1055	C₂²×C₄	1	8	yes	—
71	C₂×#1055	C₂²×C₄	1	2	yes	—
72	C₂×#1055	C₂²×C₄	1	2	yes	—
73	C₂×#1055	C₂⁴	1	1	yes	characteristic
74	#6092	C₂²×C₄	1	16	yes	complemented
75	C₂²×M₃₂	—	13	2	no	—
76	C₂²×C₄×C₈	—	13	1	no	—
77	C₄×M₃₂	—	13	8	no	—
78	C₂×#346	—	13	2	no	—
79	C₂×C₄×C₁₆	—	13	2	no	—
80	C₂³×C₁₃	—	2	4	no	—
81	C₂³×C₁₃	C₄×C₈	1	4	yes	—
82	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
83	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	16	no	—
86	C₂×C₄×C₁₃	C₄×C₈	1	8	yes	—
87	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
88	C₂×C₄×C₁₃	M₃₂	1	8	yes	complemented
89	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
90	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
92	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	#1055	C₂×C₄²	1	2	yes	—
99	#1055	C₂×C₄²	1	4	yes	—
100	#1055	C₂²×C₈	1	4	yes	—
101	#1055	C₂²×C₈	1	4	yes	—
102	#1055	C₂³×C₄	1	2	yes	—
103	C₂³×C₈	—	13	1	no	—
104	C₂³×C₈	—	13	1	no	—
105	C₂²×C₄²	#261	1	1	yes	characteristic
106	C₂×M₃₂	—	13	8	no	—
107	C₂×M₃₂	—	13	16	no	—
108	C₂²×C₁₆	—	13	4	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₂×C₄×C₈	—	13	4	no	—
111	C₂×C₄×C₈	—	13	4	no	—
112	C₂×C₄×C₈	—	13	1	no	—
113	C₂×C₄×C₈	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	#346	—	13	8	no	—
116	C₄×C₁₆	—	13	8	no	—
117	C₂²×C₁₃	—	2	8	no	—
118	C₂²×C₁₃	—	2	4	no	—
119	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
120	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
121	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
122	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
123	C₄×C₁₃	—	2	4	no	—
124	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
125	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
126	C₄×C₁₃	C₂×M₃₂	1	8	yes	complemented
127	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
128	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
129	C₂³×C₄	#1057	1	2	yes	—
130	C₂³×C₄	C₂×#261	1	1	yes	characteristic
131	C₂²×C₈	—	13	8	no	—
132	C₂²×C₈	—	13	8	no	—
133	C₂²×C₈	—	13	4	no	—
134	C₂²×C₈	—	13	1	no	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	—	13	2	no	—
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	1	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₂×C₄²	#1057	1	2	yes	—
141	C₂×C₄²	#1057	1	4	yes	—
142	C₂×C₄²	C₂×#261	1	1	yes	characteristic
143	C₂×C₄²	C₂×#261	1	4	yes	—
144	C₂×C₄²	C₂×#261	1	1	yes	characteristic
145	M₃₂	—	13	32	no	—
146	C₂×C₁₆	—	13	16	no	—
147	C₂×C₁₆	—	13	8	no	—
148	C₄×C₈	—	13	2	no	—
149	C₄×C₈	—	13	2	no	—
150	C₄×C₈	—	13	2	no	—
151	C₄×C₈	—	13	4	no	—
152	C₄×C₈	—	13	4	no	—
153	C₄×C₈	—	13	2	no	—
154	C₂×C₁₃	—	2	4	no	—
155	C₂×C₁₃	C₄×M₃₂	1	4	yes	complemented
156	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
157	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
158	C₂⁴	C₂×#1057	1	1	yes	characteristic
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	D₁₃⋊C₈	1	4	yes	—
161	C₂²×C₄	C₄×#261	1	4	yes	—
162	C₂²×C₄	C₂×#1057	1	2	yes	—
163	C₂²×C₄	C₂×#1057	1	8	yes	—
164	C₂²×C₄	C₂×#1057	1	2	yes	—
165	C₂²×C₄	C₂×#1057	1	2	yes	—
166	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
167	C₂²×C₄	C₂²×#261	1	2	yes	—
168	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
169	C₂×C₈	—	13	16	no	—
170	C₂×C₈	—	13	4	no	—
171	C₂×C₈	—	13	4	no	—
172	C₂×C₈	—	13	4	no	—
173	C₂×C₈	—	13	8	no	—
174	C₂×C₈	—	13	8	no	—
175	C₂×C₈	—	13	4	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	2	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	1	no	—
180	C₂×C₈	—	13	2	no	—
181	C₄²	C₂×#1057	1	2	yes	—
182	C₄²	C₂×#1057	1	8	yes	—
183	C₄²	C₂×#1057	1	2	yes	—
184	C₄²	C₂²×#261	1	2	yes	—
185	C₄²	C₂²×#261	1	2	yes	—
186	C₁₆	—	13	16	no	—
187	C₁₃	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	2	4	no	—
189	C₂³	C₄×#1057	1	4	yes	—
190	C₂³	C₂²×#1057	1	2	yes	—
191	C₂³	C₂²×#1057	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	#88106	1	8	yes	direct product factor
195	C₂×C₄	C₄×#1057	1	8	yes	—
196	C₂×C₄	C₄×#1057	1	4	yes	—
197	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
198	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
199	C₂×C₄	C₂×C₄×#261	1	4	yes	—
200	C₂×C₄	C₂×C₄×#261	1	2	yes	—
201	C₂×C₄	C₂²×#1057	1	4	yes	—
202	C₂×C₄	C₂²×#1057	1	4	yes	—
203	C₂×C₄	C₂²×#1057	1	2	yes	—
204	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
205	C₂×C₄	C₂³×#261	1	1	yes	characteristic
206	C₈	—	13	4	no	—
207	C₈	—	13	4	no	—
208	C₈	—	13	2	no	—
209	C₈	—	13	2	no	—
210	C₈	—	13	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	8	no	—
213	C₂²	C₂×#88106	1	4	yes	—
214	C₂²	C₂×C₄×#1057	1	4	yes	—
215	C₂²	C₂×C₄×#1057	1	2	yes	—
216	C₂²	C₂³×#1057	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	C₂×#88106	1	8	yes	direct product factor
219	C₄	C₂×C₄×#1057	1	2	yes	—
220	C₄	C₂×C₄×#1057	1	4	yes	—
221	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
222	C₄	C₂²×C₄×#261	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₄×#88106	1	4	yes	direct product factor
225	C₂	C₂²×C₄×#1057	1	1	yes	characteristic
226	C₂	C₂²×#88106	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#88106	C₂	direct (N × H)
3. C₂×#11718231	C₂	semidirect
4. C₂×C₄×#6092	C₂	semidirect
9. C₂×#88106	C₄	direct (N × H)
18. #11718231	C₂²	semidirect
19. C₄×#6092	C₂²	semidirect
36. #88106	C₂×C₄	direct (N × H)
37. C₂×#6092	C₂×C₄	semidirect
74. #6092	C₂²×C₄	semidirect
88. C₂×C₄×C₁₃	M₃₂	semidirect
126. C₄×C₁₃	C₂×M₃₂	semidirect
155. C₂×C₁₃	C₄×M₃₂	semidirect
187. C₁₃	C₂×C₄×M₃₂	semidirect
194. C₂×C₄	#88106	direct (N × H)
218. C₄	C₂×#88106	direct (N × H)
224. C₂	C₄×#88106	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
C₄×#88106	2
C₂×#11718231	2
C₂×C₄×#6092	2
C₂²×C₄×#1055	2
C₂×C₄×M₃₂	13

Subgroups — C2×C4×#88106 #16806717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88106 #16806717