Subgroups — C₂×#69886 #10948314

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#4553	C₂	1	2	yes	maximal
3	#69886	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×#2166	—	3	1	no	maximal, Sylow
7	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₈	C₄	1	1	yes	characteristic
9	C₂×C₃×#117	C₄	1	1	yes	characteristic
10	C₃×#405	C₄	1	4	yes	—
11	C₃×#405	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×#405	D₃	1	1	yes	characteristic
21	#2166	—	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	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×M₁₆	—	2	2	no	—
27	C₂×C₃×M₁₆	—	2	2	no	—
28	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₃×M₁₆	C₂³	1	2	yes	—
30	C₂²×C₃×C₈	—	2	2	no	—
31	C₂²×C₃×C₈	—	2	2	no	—
32	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂×C₄	1	4	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	C₂³	1	1	yes	characteristic
37	C₃×#117	—	2	2	no	—
38	C₃×#117	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
41	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
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	C₂×D₃	1	1	yes	characteristic
53	C₂³×C₈	Dic₃	1	1	yes	characteristic
54	C₂×#117	Dic₃	1	1	yes	characteristic
55	C₂×M₃₂	—	3	4	no	—
56	#405	Dic₃	1	4	yes	—
57	#405	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	1	no	—
72	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
73	C₂²×C₃×C₄	M₁₆	1	2	yes	—
74	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
75	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₃×M₁₆	—	2	4	no	—
77	C₃×M₁₆	—	2	4	no	—
78	C₂×C₃×C₈	—	2	1	no	—
79	C₂×C₃×C₈	—	2	1	no	—
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	—	2	1	no	—
82	C₂×C₃×C₈	—	2	8	no	—
83	C₂×C₃×C₈	M₁₆	1	2	yes	—
84	C₂×C₃×C₈	M₁₆	1	2	yes	—
85	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₈	#41	1	2	yes	—
90	C₂×C₃×C₈	#41	1	2	yes	—
91	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
92	C₃×C₄⋊C₄	—	2	4	no	—
93	C₃×C₂²⋊C₄	—	2	4	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	C₂²×C₄	1	2	yes	—
97	C₃×C₄²	C₂²×C₄	1	2	yes	—
98	C₃⋊C₁₆	—	2	8	no	—
99	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
103	C₂×M₁₆	C₂²×D₃	1	2	yes	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₃	1	4	yes	—
108	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
111	#117	—	2	2	no	—
112	#117	C₂×Dic₃	1	4	yes	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₂²⋊C₄	—	2	1	no	—
115	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
117	M₃₂	—	3	8	no	—
118	M₃₂	—	6	4	no	—
119	C₂×C₁₆	—	6	4	no	—
120	#97	C₂×Dic₃	1	4	yes	—
121	#97	C₂×Dic₃	1	2	yes	—
122	#97	C₂²×D₃	1	2	yes	—
123	C₄×C₈	C₂×Dic₃	1	4	yes	—
124	C₄×C₈	C₂×Dic₃	1	2	yes	—
125	C₄×C₈	C₂²×D₃	1	2	yes	—
126	C₂³×C₃	—	2	1	no	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	—	4	2	no	—
129	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂³×C₃	C₂×M₁₆	1	2	yes	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	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	—	4	2	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	#117	1	1	yes	characteristic
142	C₂×C₃×C₄	#117	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
144	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	—	2	2	no	—
150	C₃×C₈	#117	1	2	yes	—
151	C₃×C₈	C₂×M₁₆	1	4	yes	—
152	C₃×C₈	C₂×#41	1	2	yes	—
153	C₂⁴	#208	1	1	yes	characteristic
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
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	#208	1	1	yes	characteristic
161	C₂²×C₄	#208	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
164	M₁₆	—	2	4	no	—
165	M₁₆	—	2	4	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	#208	1	2	yes	—
172	C₂×C₈	#208	1	2	yes	—
173	C₂×C₈	#235	1	2	yes	—
174	C₂×C₈	#235	1	2	yes	—
175	C₂×C₈	C₂²×Dic₃	1	2	yes	—
176	C₂×C₈	C₂²×Dic₃	1	2	yes	—
177	C₂×C₈	C₂²×Dic₃	1	2	yes	—
178	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
179	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
180	C₄⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	C₂²×Dic₃	1	2	yes	—
185	C₄²	C₂²×Dic₃	1	2	yes	—
186	C₁₆	—	6	8	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	4	2	no	—
190	C₂²×C₃	—	4	2	no	—
191	C₂²×C₃	—	4	2	no	—
192	C₂²×C₃	#432	1	1	yes	characteristic
193	C₂²×C₃	#432	1	1	yes	characteristic
194	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	#432	1	1	yes	characteristic
199	C₃×C₄	#432	1	1	yes	characteristic
200	C₃×C₄	C₂×#117	1	1	yes	characteristic
201	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	4	2	no	—
205	C₂³	C₂×#208	1	1	yes	characteristic
206	C₂³	C₂×#208	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	C₂×#208	1	2	yes	—
218	C₂×C₄	C₂×#208	1	1	yes	characteristic
219	C₂×C₄	C₂×#208	1	1	yes	characteristic
220	C₂×C₄	C₂×#208	1	1	yes	characteristic
221	C₂×C₄	#926	1	1	yes	characteristic
222	C₂×C₄	#926	1	1	yes	characteristic
223	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	C₂×#208	1	4	yes	—
227	C₈	#926	1	2	yes	—
228	C₈	C₂×#235	1	2	yes	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	4	2	no	—
232	C₂×C₃	#2166	1	2	yes	complemented
233	C₂×C₃	C₂×#432	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#4926	1	1	yes	characteristic
240	C₂²	#4926	1	1	yes	characteristic
241	C₂²	C₂²×#208	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	#4926	1	1	yes	characteristic
246	C₄	#4926	1	1	yes	characteristic
247	C₄	C₂²×#208	1	1	yes	characteristic
248	C₄	C₂×#926	1	1	yes	characteristic
249	C₃	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	#69886	1	2	yes	direct product factor
254	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×#405	C₂
socle	C₂²×C₃	#432
prime radical	C₂⁴×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×#405	D₃
hypocenter	C₃	C₂×#2166
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #69886	C₂	direct (N × H)
4. C₂×#4466	C₂	semidirect
5. C₂×#4463	C₂	semidirect
18. #4466	C₂²	semidirect
19. #4463	C₂²	semidirect
232. C₂×C₃	#2166	semidirect
249. C₃	C₂×#2166	semidirect
253. C₂	#69886	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×#405	2
C₂×#4553	2
#69886	2
C₂×#4466	2
C₂×#4463	2
C₂×#2166	3

Subgroups — C2×#69886 #10948314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69886 #10948314