Subgroups — C₂²×#5078 #11664783

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#869	C₂	1	1	yes	maximal, characteristic
3	C₂×#5078	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#807	C₂	1	4	yes	maximal
5	C₂²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×#500	—	3	1	no	maximal, Sylow
7	C₂³×C₃×Q₈	—	2	1	no	—
8	C₂³×Dic₆	—	2	1	no	—
9	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
10	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	—
11	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
12	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊Dic₃	—	2	1	no	—
15	C₂²×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂³×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×#869	—	2	6	no	—
18	C₂×#869	C₂²	1	12	yes	—
19	#5078	C₂²	1	64	yes	direct product factor
20	C₂×#807	C₂²	1	48	yes	—
21	C₂×C₄×C₃⋊C₈	C₂²	1	12	yes	—
22	C₂²×C₄⋊Q₈	—	3	1	no	—
23	C₂²×C₄⋊Q₈	D₃	1	1	yes	characteristic
24	C₂×#500	—	3	24	no	—
25	C₂²×#107	—	3	4	no	—
26	C₂²×C₄×C₈	—	3	1	no	—
27	C₂²×C₃×Q₈	—	2	12	no	—
28	C₂²×C₃×Q₈	—	4	8	no	—
29	C₂³×C₃×C₄	—	2	2	no	—
30	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
31	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
32	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂³×Dic₃	—	2	2	no	—
34	C₂²×Dic₆	—	2	12	no	—
35	C₂²×Dic₆	—	4	8	no	—
36	C₃×C₄⋊Q₈	—	2	24	no	—
37	C₃×C₄⋊Q₈	C₂³	1	16	yes	—
38	C₂×C₃×C₄⋊C₄	—	2	12	no	—
39	C₂×C₃×C₄⋊C₄	—	2	6	no	—
40	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
41	C₂×C₃×C₄²	D₄	1	6	yes	—
42	C₂×C₃×C₄²	C₂³	1	6	yes	—
43	C₂×C₄⋊Dic₃	—	2	6	no	—
44	C₂×C₄⋊Dic₃	—	2	12	no	—
45	C₂×C₄⋊Dic₃	C₂³	1	12	yes	—
46	C₂²×C₃⋊C₈	Q₈	1	16	yes	complemented
47	C₂²×C₃⋊C₈	C₂³	1	12	yes	—
48	#869	—	2	24	no	—
49	#869	C₂³	1	16	yes	—
50	#807	C₂³	1	64	yes	—
51	C₄×C₃⋊C₈	C₂³	1	16	yes	—
52	C₂³×Q₈	—	2	1	no	—
53	C₂³×Q₈	—	6	1	no	—
54	C₂³×C₈	—	3	2	no	—
55	C₂×C₄⋊Q₈	—	2	6	no	—
56	C₂×C₄⋊Q₈	—	3	12	no	—
57	C₂×C₄⋊Q₈	—	6	6	no	—
58	C₂×C₄⋊Q₈	C₂×D₃	1	12	yes	—
59	C₂²×C₄⋊C₄	—	2	1	no	—
60	C₂²×C₄⋊C₄	—	3	2	no	—
61	C₂²×C₄⋊C₄	—	6	1	no	—
62	C₂²×C₄⋊C₄	C₂×D₃	1	2	yes	—
63	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
64	#500	—	3	64	no	—
65	C₂×#107	—	3	48	no	—
66	C₂×C₄×C₈	—	3	12	no	—
67	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×Q₈	—	2	16	no	—
69	C₂×C₃×Q₈	—	4	48	no	—
70	C₂²×C₃×C₄	—	2	12	no	—
71	C₂²×C₃×C₄	—	4	8	no	—
72	C₂²×C₃×C₄	D₈	1	8	yes	—
73	C₂²×C₃×C₄	Q₁₆	1	8	yes	—
74	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
75	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
76	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
77	C₂²×C₃×C₄	C₂×Q₈	1	8	yes	—
78	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
79	C₂²×Dic₃	—	2	12	no	—
80	C₂²×Dic₃	—	4	8	no	—
81	C₂×Dic₆	—	2	16	no	—
82	C₂×Dic₆	—	4	48	no	—
83	C₃×C₄⋊C₄	—	2	16	no	—
84	C₃×C₄⋊C₄	—	2	12	no	—
85	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
86	C₃×C₄²	C₂×D₄	1	12	yes	—
87	C₃×C₄²	C₂⁴	1	4	yes	—
88	C₄⋊Dic₃	—	2	12	no	—
89	C₄⋊Dic₃	—	2	16	no	—
90	C₄⋊Dic₃	C₂⁴	1	8	yes	—
91	C₂×C₃⋊C₈	C₂×Q₈	1	48	yes	complemented
92	C₂×C₃⋊C₈	C₂⁴	1	8	yes	—
93	C₂²×Q₈	—	2	12	no	—
94	C₂²×Q₈	—	4	8	no	—
95	C₂²×Q₈	—	6	12	no	—
96	C₂²×Q₈	—	12	8	no	—
97	C₂³×C₄	—	2	2	no	—
98	C₂³×C₄	—	6	2	no	—
99	C₂³×C₄	#67	1	1	yes	characteristic
100	C₂³×C₄	#67	1	1	yes	characteristic
101	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₈	—	3	16	no	—
103	C₂²×C₈	—	3	12	no	—
104	C₄⋊Q₈	—	2	24	no	—
105	C₄⋊Q₈	—	3	16	no	—
106	C₄⋊Q₈	—	6	24	no	—
107	C₄⋊Q₈	C₂²×D₃	1	16	yes	—
108	C₂×C₄⋊C₄	—	2	12	no	—
109	C₂×C₄⋊C₄	—	2	6	no	—
110	C₂×C₄⋊C₄	—	3	12	no	—
111	C₂×C₄⋊C₄	—	6	6	no	—
112	C₂×C₄⋊C₄	—	6	12	no	—
113	C₂×C₄⋊C₄	C₂²×D₃	1	12	yes	—
114	C₂×C₄²	#67	1	6	yes	—
115	C₂×C₄²	C₂²×D₃	1	6	yes	—
116	#107	—	3	64	no	—
117	C₄×C₈	—	3	16	no	—
118	C₂³×C₃	C₄⋊Q₈	1	4	yes	—
119	C₂³×C₃	C₂×D₈	1	4	yes	—
120	C₂³×C₃	C₂×Q₁₆	1	4	yes	—
121	C₂³×C₃	C₂²×D₄	1	3	yes	—
122	C₃×Q₈	—	4	32	no	—
123	C₂×C₃×C₄	—	2	8	no	—
124	C₂×C₃×C₄	—	4	24	no	—
125	C₂×C₃×C₄	C₄⋊Q₈	1	12	yes	—
126	C₂×C₃×C₄	C₂×D₈	1	24	yes	—
127	C₂×C₃×C₄	C₂×Q₁₆	1	24	yes	—
128	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
129	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
130	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
131	C₂×C₃×C₄	C₂²×Q₈	1	12	yes	—
132	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
133	C₂×Dic₃	—	2	8	no	—
134	C₂×Dic₃	—	4	24	no	—
135	Dic₆	—	4	32	no	—
136	C₃⋊C₈	C₂²×Q₈	1	16	yes	complemented
137	C₂⁴	C₂×#67	1	1	yes	characteristic
138	C₂×Q₈	—	2	16	no	—
139	C₂×Q₈	—	4	48	no	—
140	C₂×Q₈	—	6	16	no	—
141	C₂×Q₈	—	12	48	no	—
142	C₂²×C₄	—	2	12	no	—
143	C₂²×C₄	—	4	8	no	—
144	C₂²×C₄	—	6	12	no	—
145	C₂²×C₄	—	12	8	no	—
146	C₂²×C₄	#213	1	8	yes	—
147	C₂²×C₄	#216	1	8	yes	—
148	C₂²×C₄	D₃×Q₈	1	8	yes	—
149	C₂²×C₄	C₂×#67	1	6	yes	—
150	C₂²×C₄	C₂×#67	1	6	yes	—
151	C₂²×C₄	C₂×#67	1	3	yes	—
152	C₂²×C₄	C₂³×D₃	1	3	yes	—
153	C₂×C₈	—	3	48	no	—
154	C₂×C₈	—	3	8	no	—
155	C₄⋊C₄	—	2	16	no	—
156	C₄⋊C₄	—	2	12	no	—
157	C₄⋊C₄	—	3	8	no	—
158	C₄⋊C₄	—	6	12	no	—
159	C₄⋊C₄	—	6	16	no	—
160	C₄⋊C₄	C₂³×D₃	1	8	yes	—
161	C₄²	C₂×#67	1	12	yes	—
162	C₄²	C₂³×D₃	1	4	yes	—
163	C₂²×C₃	#500	1	16	yes	complemented
164	C₂²×C₃	C₂×C₄⋊Q₈	1	6	yes	—
165	C₂²×C₃	C₂²×D₈	1	6	yes	—
166	C₂²×C₃	C₂²×Q₁₆	1	6	yes	—
167	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
168	C₃×C₄	—	4	8	no	—
169	C₃×C₄	C₂×C₄⋊Q₈	1	6	yes	—
170	C₃×C₄	C₂²×D₈	1	8	yes	—
171	C₃×C₄	C₂²×Q₁₆	1	8	yes	—
172	C₃×C₄	C₂³×Q₈	1	2	yes	—
173	Dic₃	—	4	8	no	—
174	C₂³	C₂×#213	1	4	yes	—
175	C₂³	C₂×#216	1	4	yes	—
176	C₂³	#944	1	4	yes	—
177	C₂³	C₂²×#67	1	3	yes	—
178	Q₈	—	4	32	no	—
179	Q₈	—	12	32	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	4	24	no	—
182	C₂×C₄	—	6	8	no	—
183	C₂×C₄	—	12	24	no	—
184	C₂×C₄	C₂×#213	1	24	yes	—
185	C₂×C₄	C₂×#216	1	24	yes	—
186	C₂×C₄	#944	1	12	yes	—
187	C₂×C₄	C₂×D₃×Q₈	1	12	yes	—
188	C₂×C₄	C₂²×#67	1	4	yes	—
189	C₂×C₄	C₂²×#67	1	4	yes	—
190	C₂×C₄	C₂²×#67	1	3	yes	—
191	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
192	C₈	—	3	16	no	—
193	C₂×C₃	C₂×#500	1	12	yes	complemented
194	C₂×C₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×C₃	C₂³×D₈	1	1	yes	characteristic
196	C₂×C₃	C₂³×Q₁₆	1	1	yes	characteristic
197	C₂²	#5078	1	16	yes	direct product factor
198	C₂²	C₂²×#213	1	6	yes	—
199	C₂²	C₂²×#216	1	6	yes	—
200	C₂²	C₂×#944	1	6	yes	—
201	C₂²	C₂³×#67	1	1	yes	characteristic
202	C₄	—	4	8	no	—
203	C₄	—	12	8	no	—
204	C₄	C₂²×#213	1	8	yes	—
205	C₄	C₂²×#216	1	8	yes	—
206	C₄	C₂×#944	1	6	yes	—
207	C₄	C₂²×D₃×Q₈	1	2	yes	—
208	C₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
209	C₂	C₂×#5078	1	12	yes	direct product factor
210	C₂	C₂³×#213	1	1	yes	characteristic
211	C₂	C₂³×#216	1	1	yes	characteristic
212	C₂	C₂²×#944	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×C₄⋊Q₈	C₂
socle, prime radical	C₂⁴×C₃	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₃
hypocenter	C₃	C₂²×#500
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#5078	C₂	direct (N × H)
19. #5078	C₂²	direct (N × H)
46. C₂²×C₃⋊C₈	Q₈	semidirect
91. C₂×C₃⋊C₈	C₂×Q₈	semidirect
136. C₃⋊C₈	C₂²×Q₈	semidirect
163. C₂²×C₃	#500	semidirect
193. C₂×C₃	C₂×#500	semidirect
197. C₂²	#5078	direct (N × H)
208. C₃	C₂²×#500	semidirect
209. C₂	C₂×#5078	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×C₃×C₄⋊Q₈	2
C₂²×#869	2
C₂×#5078	2
C₂²×#807	2
C₂²×C₄×C₃⋊C₈	2
C₂²×#500	3

Subgroups — C22×#5078 #11664783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#5078 #11664783