Subgroups — C₂×#4824 #78978

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#950	C₂	1	1	yes	maximal, characteristic
2	C₂×#948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#907	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#822	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#805	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#4824	C₂	1	8	yes	maximal, direct product factor
9	C₂×#443	—	3	1	no	maximal, Sylow
10	C₂×C₃×#41	C₂²	1	1	yes	characteristic
11	C₂×#235	C₂²	1	1	yes	characteristic
12	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	#950	C₄	1	4	yes	complemented
15	#950	C₂²	1	4	yes	—
16	#948	—	2	2	no	—
17	#948	C₂²	1	4	yes	complemented
18	C₂×#216	C₄	1	1	yes	characteristic, complemented
19	C₂×#215	C₄	1	1	yes	characteristic, complemented
20	C₂×#214	C₄	1	1	yes	characteristic, complemented
21	C₂×#213	C₄	1	1	yes	characteristic, complemented
22	C₂×#208	—	2	1	no	—
23	C₂×#208	C₂²	1	1	yes	characteristic
24	C₂²×C₃⋊C₈	—	2	1	no	—
25	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
27	#907	—	2	2	no	—
28	#907	C₂²	1	4	yes	complemented
29	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
30	C₂×D₃×C₈	—	2	1	no	—
31	C₃×#104	C₂²	1	4	yes	complemented
32	#822	C₂²	1	4	yes	complemented
33	#805	C₂²	1	4	yes	complemented
34	C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
35	C₂×#135	—	3	1	no	—
36	C₂×#131	—	3	1	no	—
37	C₂×#131	—	3	1	no	—
38	#443	—	3	8	no	—
39	C₂×#108	—	3	1	no	—
40	C₂×#104	—	3	1	no	—
41	C₂×#104	D₃	1	1	yes	characteristic, complemented
42	C₂×C₄×C₈	—	3	1	no	—
43	C₃×#41	—	2	1	no	—
44	C₃×#41	—	2	1	no	—
45	C₃×#41	C₂×C₄	1	2	yes	—
46	C₃×#41	C₂³	1	2	yes	—
47	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×D₄	—	2	1	no	—
49	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₃×C₄	—	2	1	no	—
51	C₂²×C₃×C₄	—	2	1	no	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂×#67	—	2	1	no	—
54	#235	—	2	1	no	—
55	#235	—	2	1	no	—
56	#235	C₂×C₄	1	2	yes	—
57	#235	C₂³	1	2	yes	—
58	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₃	—	2	1	no	—
60	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
61	C₃×M₁₆	—	2	1	no	—
62	C₃×M₁₆	C₂³	1	2	yes	—
63	C₂×C₃×C₈	—	2	1	no	—
64	C₃×C₄²	—	2	1	no	—
65	C₃×C₄²	C₂³	1	2	yes	—
66	#216	C₂×C₄	1	4	yes	complemented
67	#215	C₂×C₄	1	4	yes	complemented
68	#214	C₂×C₄	1	4	yes	complemented
69	#213	C₂×C₄	1	4	yes	complemented
70	#208	—	2	1	no	—
71	#208	—	2	4	no	—
72	#208	C₂³	1	2	yes	—
73	C₂×C₃⋊C₈	—	2	1	no	—
74	C₂×C₃⋊C₈	—	2	4	no	—
75	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
77	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
78	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
79	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
80	C₂×C₃⋊C₈	D₄	1	2	yes	—
81	C₂×C₃⋊C₈	C₂³	1	2	yes	—
82	C₂×C₃⋊C₈	C₂³	1	2	yes	—
83	ΓL(1,𝔽₂₅)	—	2	4	no	—
84	D₃×C₈	—	2	4	no	—
85	C₂×#41	—	3	1	no	—
86	C₂×#41	C₂×D₃	1	1	yes	characteristic
87	#135	—	3	4	no	—
88	#135	—	3	4	no	—
89	C₂×Q₁₆	—	3	1	no	—
90	C₂×SD₁₆	—	3	1	no	—
91	C₂×SD₁₆	—	3	1	no	—
92	C₂×D₈	—	3	1	no	—
93	#131	—	3	4	no	—
94	#131	—	3	4	no	—
95	#131	—	6	2	no	—
96	#131	—	6	2	no	—
97	C₂×M₁₆	—	3	1	no	—
98	C₂×M₁₆	—	6	1	no	—
99	C₂×M₁₆	—	6	1	no	—
100	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
101	C₂²×C₈	—	3	1	no	—
102	C₂²×C₈	—	3	1	no	—
103	C₂²×C₈	—	6	1	no	—
104	C₂²×C₈	—	6	1	no	—
105	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
106	#108	—	3	4	no	—
107	#104	—	3	4	no	—
108	#104	C₂×D₃	1	4	yes	complemented
109	C₄×C₈	—	3	4	no	—
110	C₂³×C₃	—	2	1	no	—
111	C₂³×C₃	#41	1	1	yes	characteristic
112	C₂²×D₃	—	2	1	no	—
113	C₃×Q₈	—	2	1	no	—
114	C₃×Q₈	C₂²×C₄	1	2	yes	—
115	C₃×D₄	—	2	2	no	—
116	C₃×D₄	—	2	2	no	—
117	C₃×D₄	—	2	1	no	—
118	C₃×D₄	C₂²×C₄	1	2	yes	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	#41	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
130	#67	—	2	2	no	—
131	#67	—	2	2	no	—
132	C₂×Dic₃	—	2	1	no	—
133	D₁₂	—	2	1	no	—
134	D₁₂	C₂²×C₄	1	2	yes	—
135	C₄×D₃	—	2	2	no	—
136	C₄×D₃	—	2	2	no	—
137	Dic₆	—	2	1	no	—
138	Dic₆	C₂²×C₄	1	2	yes	—
139	C₃×C₈	—	2	2	no	—
140	C₃⋊C₈	—	2	2	no	—
141	C₃⋊C₈	C₂²×C₄	1	2	yes	—
142	C₃⋊C₈	C₂²×C₄	1	2	yes	—
143	C₃⋊C₈	C₂×D₄	1	2	yes	—
144	C₃⋊C₈	C₂×D₄	1	2	yes	—
145	#41	—	2	1	no	—
146	#41	—	2	1	no	—
147	#41	—	3	2	no	—
148	#41	—	3	2	no	—
149	#41	—	6	1	no	—
150	#41	—	6	1	no	—
151	#41	C₄×D₃	1	2	yes	—
152	#41	C₂²×D₃	1	2	yes	—
153	C₂×Q₈	—	3	1	no	—
154	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	3	1	no	—
157	C₂×D₄	—	6	1	no	—
158	C₂×D₄	C₄×D₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	6	1	no	—
162	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
163	Q₁₆	—	3	4	no	—
164	SD₁₆	—	3	4	no	—
165	SD₁₆	—	3	4	no	—
166	D₈	—	3	4	no	—
167	M₁₆	—	2	1	no	—
168	M₁₆	—	3	2	no	—
169	M₁₆	—	6	4	no	—
170	M₁₆	—	6	4	no	—
171	M₁₆	—	6	1	no	—
172	M₁₆	C₂²×D₃	1	2	yes	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	3	1	no	—
175	C₂×C₈	—	3	1	no	—
176	C₂×C₈	—	3	2	no	—
177	C₂×C₈	—	3	2	no	—
178	C₂×C₈	—	3	1	no	—
179	C₂×C₈	—	3	1	no	—
180	C₂×C₈	—	3	2	no	—
181	C₂×C₈	—	3	2	no	—
182	C₂×C₈	—	6	4	no	—
183	C₂×C₈	—	6	4	no	—
184	C₂×C₈	—	6	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₂²×D₃	1	2	yes	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	4	2	no	—
190	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
191	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
192	C₂²×C₃	C₂×#41	1	1	yes	characteristic
193	C₂×D₃	—	2	2	no	—
194	C₂×D₃	—	4	2	no	—
195	C₃×C₄	—	2	4	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	C₄×D₄	1	1	yes	characteristic
198	C₃×C₄	C₄×D₄	1	1	yes	characteristic
199	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
200	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
201	Dic₃	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	6	1	no	—
204	C₂³	#237	1	1	yes	characteristic
205	Q₈	—	2	1	no	—
206	Q₈	—	3	2	no	—
207	Q₈	—	6	1	no	—
208	Q₈	C₂×C₄×D₃	1	2	yes	—
209	D₄	—	2	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	3	2	no	—
213	D₄	—	6	2	no	—
214	D₄	—	6	2	no	—
215	D₄	—	6	1	no	—
216	D₄	C₂×C₄×D₃	1	2	yes	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	6	1	no	—
223	C₂×C₄	—	6	2	no	—
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
227	C₂×C₄	D₃×D₄	1	1	yes	characteristic
228	C₂×C₄	D₃×D₄	1	1	yes	characteristic
229	C₂×C₄	#237	1	1	yes	characteristic
230	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	3	2	no	—
233	C₈	—	3	2	no	—
234	C₈	—	3	2	no	—
235	C₈	—	3	2	no	—
236	C₈	—	6	2	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	4	2	no	—
240	C₂×C₃	#443	1	2	yes	complemented
241	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
242	D₃	—	4	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	6	2	no	—
247	C₂²	—	12	2	no	—
248	C₂²	#67⋊C₄	1	1	yes	characteristic
249	C₂²	#67⋊C₄	1	1	yes	characteristic
250	C₂²	C₂×#237	1	1	yes	characteristic
251	C₄	—	2	4	no	—
252	C₄	—	2	2	no	—
253	C₄	—	6	2	no	—
254	C₄	#67⋊C₄	1	1	yes	characteristic
255	C₄	#67⋊C₄	1	1	yes	characteristic
256	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
257	C₄	C₂×D₃×D₄	1	1	yes	characteristic
258	C₃	C₂×#443	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	12	2	no	—
263	C₂	#4824	1	2	yes	direct product factor
264	C₂	C₂×#67⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#104	C₂
socle	C₂²×C₃	C₄×D₄
prime radical	C₂×#950	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#104	D₃
hypocenter	C₃	C₂×#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. C₂×#948	C₂	semidirect
3. C₂×#907	C₂	semidirect
4. C₂×C₃×#104	C₂	semidirect
5. C₂×#822	C₂	semidirect
6. C₂×#805	C₂	semidirect
7. C₂×C₄×C₃⋊C₈	C₂	semidirect
8. #4824	C₂	direct (N × H)
14. #950	C₄	semidirect
17. #948	C₂²	semidirect
18. C₂×#216	C₄	semidirect
19. C₂×#215	C₄	semidirect
20. C₂×#214	C₄	semidirect
21. C₂×#213	C₄	semidirect
28. #907	C₂²	semidirect
31. C₃×#104	C₂²	semidirect
32. #822	C₂²	semidirect
33. #805	C₂²	semidirect
34. C₄×C₃⋊C₈	C₂²	semidirect
41. C₂×#104	D₃	semidirect
66. #216	C₂×C₄	semidirect
67. #215	C₂×C₄	semidirect
68. #214	C₂×C₄	semidirect
69. #213	C₂×C₄	semidirect
108. #104	C₂×D₃	semidirect
240. C₂×C₃	#443	semidirect
258. C₃	C₂×#443	semidirect
263. C₂	#4824	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×#950	2
C₂×#948	2
C₂×#907	2
C₂×C₃×#104	2
C₂×#822	2
C₂×#805	2
C₂×C₄×C₃⋊C₈	2
#4824	2
C₂×#443	3

Subgroups — C2×#4824 #78978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4824 #78978