Subgroups — C₂×#69864 #10948309

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#4553	C₂	1	2	yes	maximal
3	#69864	C₂	1	8	yes	maximal, direct product factor
4	C₂×#4466	C₂	1	2	yes	maximal, complemented
5	C₂×#4463	C₂	1	2	yes	maximal, complemented
6	C₂×#2165	—	3	1	no	maximal, Sylow
7	C₂²×C₃×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
10	C₃×C₄×M₁₆	C₄	1	4	yes	—
11	C₃×C₄×M₁₆	C₂²	1	4	yes	—
12	C₂×C₃×#97	C₄	1	1	yes	characteristic
13	C₂×C₃×#97	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×C₈	C₄	1	1	yes	characteristic
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×#812	C₂²	1	4	yes	—
17	#4553	C₂²	1	8	yes	—
18	#4466	C₂²	1	8	yes	complemented
19	#4463	C₂²	1	8	yes	complemented
20	C₂×C₄×M₁₆	D₃	1	1	yes	characteristic
21	#2165	—	3	8	no	—
22	C₂×#364	—	3	2	no	—
23	C₂×#349	—	3	2	no	—
24	C₂×#347	—	3	2	no	—
25	C₂³×C₃×C₄	—	2	1	no	—
26	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×M₁₆	—	2	2	no	—
28	C₂×C₃×M₁₆	—	2	2	no	—
29	C₂×C₃×M₁₆	—	2	2	no	—
30	C₂×C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₃×M₁₆	C₂³	1	2	yes	—
34	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄²	—	2	1	no	—
39	C₂×C₃×C₄²	—	2	2	no	—
40	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
43	C₃×#97	C₂×C₄	1	4	yes	—
44	C₃×#97	C₂×C₄	1	2	yes	—
45	C₃×#97	C₂³	1	2	yes	—
46	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
47	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
48	C₃×C₄×C₈	C₂³	1	2	yes	—
49	#812	—	2	4	no	—
50	#812	C₂³	1	8	yes	—
51	C₂×C₃⋊C₁₆	—	2	4	no	—
52	C₂²×M₁₆	Dic₃	1	1	yes	characteristic
53	C₂²×M₁₆	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₄²	Dic₃	1	1	yes	characteristic
55	C₂×M₃₂	—	3	4	no	—
56	C₄×M₁₆	Dic₃	1	4	yes	—
57	C₄×M₁₆	C₂×D₃	1	4	yes	—
58	C₂×#97	Dic₃	1	1	yes	characteristic
59	C₂×#97	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄×C₈	Dic₃	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	#364	—	3	8	no	—
63	#349	—	3	8	no	—
64	#347	—	3	8	no	—
65	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	—	2	4	no	—
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	—	4	4	no	—
74	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
75	C₂²×C₃×C₄	M₁₆	1	2	yes	—
76	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
77	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₃×M₁₆	—	2	4	no	—
79	C₃×M₁₆	—	2	8	no	—
80	C₃×M₁₆	—	2	4	no	—
81	C₂×C₃×C₈	—	2	1	no	—
82	C₂×C₃×C₈	—	2	1	no	—
83	C₂×C₃×C₈	—	2	1	no	—
84	C₂×C₃×C₈	—	2	1	no	—
85	C₂×C₃×C₈	M₁₆	1	2	yes	—
86	C₂×C₃×C₈	M₁₆	1	2	yes	—
87	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₃×C₈	#41	1	2	yes	—
92	C₂×C₃×C₈	#41	1	2	yes	—
93	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	—	2	4	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	C₂²×C₄	1	2	yes	—
98	C₃×C₄²	C₂²×C₄	1	2	yes	—
99	C₃⋊C₁₆	—	2	8	no	—
100	C₂³×C₄	—	2	1	no	—
101	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	C₂×Dic₃	1	4	yes	—
107	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
108	C₂×M₁₆	C₂²×D₃	1	2	yes	—
109	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₄²	—	2	1	no	—
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₃	1	4	yes	—
118	M₃₂	—	3	8	no	—
119	M₃₂	—	6	4	no	—
120	C₂×C₁₆	—	6	4	no	—
121	#97	C₂×Dic₃	1	4	yes	—
122	#97	C₂×Dic₃	1	2	yes	—
123	#97	C₂²×D₃	1	2	yes	—
124	C₄×C₈	C₂×Dic₃	1	4	yes	—
125	C₄×C₈	C₂×Dic₃	1	2	yes	—
126	C₄×C₈	C₂²×D₃	1	2	yes	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	—	2	1	no	—
129	C₂³×C₃	—	4	2	no	—
130	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂³×C₃	C₂×M₁₆	1	2	yes	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	4	8	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	#117	1	1	yes	characteristic
144	C₂×C₃×C₄	#117	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
146	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	#117	1	2	yes	—
153	C₃×C₈	C₂×M₁₆	1	4	yes	—
154	C₃×C₈	C₂×#41	1	2	yes	—
155	C₂⁴	#208	1	1	yes	characteristic
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	#208	1	1	yes	characteristic
165	C₂²×C₄	#208	1	2	yes	—
166	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
167	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	M₁₆	—	2	4	no	—
170	M₁₆	—	2	4	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	#208	1	2	yes	—
176	C₂×C₈	#208	1	2	yes	—
177	C₂×C₈	#235	1	2	yes	—
178	C₂×C₈	#235	1	2	yes	—
179	C₂×C₈	C₂²×Dic₃	1	2	yes	—
180	C₂×C₈	C₂²×Dic₃	1	2	yes	—
181	C₂×C₈	C₂²×Dic₃	1	2	yes	—
182	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
183	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
184	C₄²	—	2	1	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	1	no	—
187	C₄²	C₂²×Dic₃	1	2	yes	—
188	C₄²	C₂²×Dic₃	1	2	yes	—
189	C₁₆	—	6	8	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	—	4	2	no	—
195	C₂²×C₃	#432	1	1	yes	characteristic
196	C₂²×C₃	#432	1	1	yes	characteristic
197	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	#432	1	1	yes	characteristic
202	C₃×C₄	#432	1	1	yes	characteristic
203	C₃×C₄	C₂×#117	1	1	yes	characteristic
204	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	4	2	no	—
208	C₂³	C₂×#208	1	1	yes	characteristic
209	C₂³	C₂×#208	1	2	yes	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂×#208	1	2	yes	—
222	C₂×C₄	C₂×#208	1	1	yes	characteristic
223	C₂×C₄	C₂×#208	1	1	yes	characteristic
224	C₂×C₄	C₂×#208	1	1	yes	characteristic
225	C₂×C₄	#926	1	1	yes	characteristic
226	C₂×C₄	#926	1	1	yes	characteristic
227	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	C₂×#208	1	4	yes	—
231	C₈	#926	1	2	yes	—
232	C₈	C₂×#235	1	2	yes	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	4	2	no	—
236	C₂×C₃	#2165	1	2	yes	complemented
237	C₂×C₃	C₂×#432	1	1	yes	characteristic
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	#4926	1	1	yes	characteristic
244	C₂²	#4926	1	1	yes	characteristic
245	C₂²	C₂²×#208	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	2	no	—
249	C₄	#4926	1	1	yes	characteristic
250	C₄	#4926	1	1	yes	characteristic
251	C₄	C₂²×#208	1	1	yes	characteristic
252	C₄	C₂×#926	1	1	yes	characteristic
253	C₃	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#69864	1	2	yes	direct product factor
258	C₂	C₂×#4926	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#926
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×C₄×M₁₆	C₂
socle	C₂²×C₃	#432
prime radical	C₂⁴×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₃
hypocenter	C₃	C₂×#2165
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #69864	C₂	direct (N × H)
4. C₂×#4466	C₂	semidirect
5. C₂×#4463	C₂	semidirect
18. #4466	C₂²	semidirect
19. #4463	C₂²	semidirect
236. C₂×C₃	#2165	semidirect
253. C₃	C₂×#2165	semidirect
257. C₂	#69864	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×M₁₆	2
C₂×#4553	2
#69864	2
C₂×#4466	2
C₂×#4463	2
C₂×#2165	3

Subgroups — C2×#69864 #10948309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69864 #10948309