Subgroups — C₂²×C₃×#400 #10653148

Order 768.

3550 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#400	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#95	C₂	1	2	yes	maximal
5	C₂²×C₃×#95	C₂	1	2	yes	maximal
6	C₂²×C₃×#95	C₂	1	1	yes	maximal, characteristic
7	C₂²×#400	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂⁴×C₃×C₄	C₂²	1	2	yes	—
10	C₂⁴×C₃×C₄	C₂²	1	2	yes	—
11	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄⋊C₄	—	2	8	no	—
14	C₂²×C₃×C₄⋊C₄	—	2	8	no	—
15	C₂²×C₃×C₄⋊C₄	C₂²	1	12	yes	—
16	C₂²×C₃×C₄⋊C₄	C₂²	1	12	yes	—
17	C₃×#400	C₂²	1	64	yes	direct product factor
18	C₂×C₃×#95	C₂²	1	12	yes	—
19	C₂×C₃×#95	C₂²	1	24	yes	—
20	C₂×C₃×#95	C₂²	1	24	yes	—
21	C₂³×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₂³×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₂×#400	C₂×C₃	1	24	yes	direct product factor
24	C₂²×#95	C₂×C₃	1	2	yes	—
25	C₂²×#95	C₂×C₃	1	2	yes	—
26	C₂²×#95	C₂×C₃	1	1	yes	characteristic
27	C₂⁵×C₃	C₂³	1	1	yes	characteristic
28	C₂³×C₃×C₄	—	2	4	no	—
29	C₂³×C₃×C₄	—	2	4	no	—
30	C₂³×C₃×C₄	—	2	8	no	—
31	C₂³×C₃×C₄	—	2	4	no	—
32	C₂³×C₃×C₄	—	2	8	no	—
33	C₂³×C₃×C₄	—	2	8	no	—
34	C₂³×C₃×C₄	—	2	16	no	—
35	C₂³×C₃×C₄	—	2	8	no	—
36	C₂³×C₃×C₄	—	2	8	no	—
37	C₂³×C₃×C₄	—	2	8	no	—
38	C₂³×C₃×C₄	D₄	1	8	yes	—
39	C₂³×C₃×C₄	Q₈	1	8	yes	—
40	C₂³×C₃×C₄	C₂³	1	12	yes	—
41	C₂³×C₃×C₄	C₂³	1	6	yes	—
42	C₂³×C₃×C₄	C₂³	1	12	yes	—
43	C₂³×C₃×C₄	C₂³	1	6	yes	—
44	C₂³×C₃×C₄	C₂³	1	6	yes	—
45	C₂×C₃×C₄⋊C₄	—	2	48	no	—
46	C₂×C₃×C₄⋊C₄	—	2	48	no	—
47	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	—
48	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	—
49	C₃×#95	C₂³	1	16	yes	—
50	C₃×#95	C₂³	1	32	yes	—
51	C₃×#95	C₂³	1	32	yes	—
52	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂⁴×C₄	C₂²×C₃	1	2	yes	—
54	C₂⁴×C₄	C₂²×C₃	1	2	yes	—
55	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₄⋊C₄	—	2	8	no	—
58	C₂²×C₄⋊C₄	—	2	8	no	—
59	C₂²×C₄⋊C₄	C₂²×C₃	1	12	yes	—
60	C₂²×C₄⋊C₄	C₂²×C₃	1	12	yes	—
61	#400	C₂²×C₃	1	64	yes	direct product factor
62	C₂×#95	C₂²×C₃	1	24	yes	—
63	C₂×#95	C₂²×C₃	1	24	yes	—
64	C₂×#95	C₂²×C₃	1	12	yes	—
65	C₂⁴×C₃	C₂×D₄	1	4	yes	—
66	C₂⁴×C₃	C₂×Q₈	1	4	yes	—
67	C₂⁴×C₃	#41	1	4	yes	—
68	C₂⁴×C₃	#41	1	8	yes	—
69	C₂⁴×C₃	#41	1	8	yes	—
70	C₂⁴×C₃	C₂⁴	1	3	yes	—
71	C₂²×C₃×C₄	—	2	32	no	—
72	C₂²×C₃×C₄	—	2	32	no	—
73	C₂²×C₃×C₄	—	2	12	no	—
74	C₂²×C₃×C₄	—	2	12	no	—
75	C₂²×C₃×C₄	—	2	24	no	—
76	C₂²×C₃×C₄	—	2	12	no	—
77	C₂²×C₃×C₄	—	2	24	no	—
78	C₂²×C₃×C₄	—	2	24	no	—
79	C₂²×C₃×C₄	—	2	48	no	—
80	C₂²×C₃×C₄	—	2	24	no	—
81	C₂²×C₃×C₄	—	2	24	no	—
82	C₂²×C₃×C₄	—	2	24	no	—
83	C₂²×C₃×C₄	—	4	16	no	—
84	C₂²×C₃×C₄	—	4	32	no	—
85	C₂²×C₃×C₄	—	4	32	no	—
86	C₂²×C₃×C₄	C₂×D₄	1	24	yes	—
87	C₂²×C₃×C₄	C₂×Q₈	1	24	yes	—
88	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
89	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
90	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
91	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
92	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
93	C₃×C₄⋊C₄	—	2	32	no	—
94	C₃×C₄⋊C₄	—	2	32	no	—
95	C₂⁵	C₂³×C₃	1	1	yes	characteristic
96	C₂³×C₄	—	2	4	no	—
97	C₂³×C₄	—	2	4	no	—
98	C₂³×C₄	—	2	8	no	—
99	C₂³×C₄	—	2	4	no	—
100	C₂³×C₄	—	2	8	no	—
101	C₂³×C₄	—	2	8	no	—
102	C₂³×C₄	—	2	16	no	—
103	C₂³×C₄	—	2	8	no	—
104	C₂³×C₄	—	2	8	no	—
105	C₂³×C₄	—	2	8	no	—
106	C₂³×C₄	C₃×D₄	1	8	yes	—
107	C₂³×C₄	C₃×Q₈	1	8	yes	—
108	C₂³×C₄	C₂³×C₃	1	6	yes	—
109	C₂³×C₄	C₂³×C₃	1	12	yes	—
110	C₂³×C₄	C₂³×C₃	1	12	yes	—
111	C₂³×C₄	C₂³×C₃	1	6	yes	—
112	C₂³×C₄	C₂³×C₃	1	6	yes	—
113	C₂×C₄⋊C₄	—	2	48	no	—
114	C₂×C₄⋊C₄	—	2	48	no	—
115	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	—
116	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	—
117	#95	C₂³×C₃	1	16	yes	—
118	#95	C₂³×C₃	1	32	yes	—
119	#95	C₂³×C₃	1	32	yes	—
120	C₂³×C₃	C₂²⋊Q₈	1	16	yes	—
121	C₂³×C₃	#123	1	16	yes	—
122	C₂³×C₃	#124	1	16	yes	—
123	C₂³×C₃	#125	1	32	yes	—
124	C₂³×C₃	#126	1	32	yes	—
125	C₂³×C₃	C₂²×D₄	1	6	yes	—
126	C₂³×C₃	C₂²×Q₈	1	6	yes	—
127	C₂³×C₃	C₂×#41	1	6	yes	—
128	C₂³×C₃	C₂×#41	1	12	yes	—
129	C₂³×C₃	C₂×#41	1	12	yes	—
130	C₂³×C₃	C₂⁵	1	1	yes	characteristic
131	C₂×C₃×C₄	—	2	48	no	—
132	C₂×C₃×C₄	—	2	48	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	8	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	8	no	—
138	C₂×C₃×C₄	—	2	8	no	—
139	C₂×C₃×C₄	—	2	16	no	—
140	C₂×C₃×C₄	—	2	8	no	—
141	C₂×C₃×C₄	—	2	8	no	—
142	C₂×C₃×C₄	—	2	8	no	—
143	C₂×C₃×C₄	—	4	24	no	—
144	C₂×C₃×C₄	—	4	48	no	—
145	C₂×C₃×C₄	—	4	48	no	—
146	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
147	C₂×C₃×C₄	C₂²×Q₈	1	8	yes	—
148	C₂⁴	C₂×C₃×D₄	1	4	yes	—
149	C₂⁴	C₂×C₃×Q₈	1	4	yes	—
150	C₂⁴	C₃×#41	1	4	yes	—
151	C₂⁴	C₃×#41	1	8	yes	—
152	C₂⁴	C₃×#41	1	8	yes	—
153	C₂⁴	C₂⁴×C₃	1	3	yes	—
154	C₂²×C₄	—	2	32	no	—
155	C₂²×C₄	—	2	32	no	—
156	C₂²×C₄	—	2	12	no	—
157	C₂²×C₄	—	2	12	no	—
158	C₂²×C₄	—	2	24	no	—
159	C₂²×C₄	—	2	12	no	—
160	C₂²×C₄	—	2	24	no	—
161	C₂²×C₄	—	2	24	no	—
162	C₂²×C₄	—	2	48	no	—
163	C₂²×C₄	—	2	24	no	—
164	C₂²×C₄	—	2	24	no	—
165	C₂²×C₄	—	2	24	no	—
166	C₂²×C₄	—	4	16	no	—
167	C₂²×C₄	—	4	32	no	—
168	C₂²×C₄	—	4	32	no	—
169	C₂²×C₄	C₂×C₃×D₄	1	24	yes	—
170	C₂²×C₄	C₂×C₃×Q₈	1	24	yes	—
171	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
172	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
173	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
174	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
175	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
176	C₄⋊C₄	—	2	32	no	—
177	C₄⋊C₄	—	2	32	no	—
178	C₂²×C₃	#400	1	64	yes	direct product factor
179	C₂²×C₃	C₂×C₂²⋊Q₈	1	12	yes	—
180	C₂²×C₃	C₂×#123	1	12	yes	—
181	C₂²×C₃	C₂×#124	1	12	yes	—
182	C₂²×C₃	C₂×#125	1	24	yes	—
183	C₂²×C₃	C₂×#126	1	24	yes	—
184	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
185	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
186	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
187	C₂²×C₃	C₂²×#41	1	2	yes	—
188	C₂²×C₃	C₂²×#41	1	2	yes	—
189	C₃×C₄	—	2	8	no	—
190	C₃×C₄	—	2	8	no	—
191	C₃×C₄	—	4	4	no	—
192	C₃×C₄	—	4	8	no	—
193	C₃×C₄	—	4	8	no	—
194	C₂³	C₃×C₂²⋊Q₈	1	16	yes	—
195	C₂³	C₃×#123	1	16	yes	—
196	C₂³	C₃×#124	1	16	yes	—
197	C₂³	C₃×#125	1	32	yes	—
198	C₂³	C₃×#126	1	32	yes	—
199	C₂³	C₂²×C₃×D₄	1	6	yes	—
200	C₂³	C₂²×C₃×Q₈	1	6	yes	—
201	C₂³	C₂×C₃×#41	1	6	yes	—
202	C₂³	C₂×C₃×#41	1	12	yes	—
203	C₂³	C₂×C₃×#41	1	12	yes	—
204	C₂³	C₂⁵×C₃	1	1	yes	characteristic
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	48	no	—
209	C₂×C₄	—	2	48	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	4	24	no	—
218	C₂×C₄	—	4	48	no	—
219	C₂×C₄	—	4	48	no	—
220	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
221	C₂×C₄	C₂²×C₃×Q₈	1	8	yes	—
222	C₂×C₃	C₂×#400	1	24	yes	direct product factor
223	C₂×C₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
224	C₂×C₃	C₂²×#123	1	1	yes	characteristic
225	C₂×C₃	C₂²×#124	1	1	yes	characteristic
226	C₂×C₃	C₂²×#125	1	2	yes	—
227	C₂×C₃	C₂²×#126	1	2	yes	—
228	C₂²	C₃×#400	1	64	yes	direct product factor
229	C₂²	C₂×C₃×C₂²⋊Q₈	1	12	yes	—
230	C₂²	C₂×C₃×#123	1	12	yes	—
231	C₂²	C₂×C₃×#124	1	12	yes	—
232	C₂²	C₂×C₃×#125	1	24	yes	—
233	C₂²	C₂×C₃×#126	1	24	yes	—
234	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
235	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
236	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
237	C₂²	C₂²×C₃×#41	1	2	yes	—
238	C₂²	C₂²×C₃×#41	1	2	yes	—
239	C₄	—	2	8	no	—
240	C₄	—	2	8	no	—
241	C₄	—	4	4	no	—
242	C₄	—	4	8	no	—
243	C₄	—	4	8	no	—
244	C₃	C₂²×#400	1	1	yes	characteristic, direct product factor, Sylow
245	C₂	C₂×C₃×#400	1	24	yes	direct product factor
246	C₂	C₂²×C₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂	C₂²×C₃×#123	1	1	yes	characteristic
248	C₂	C₂²×C₃×#124	1	1	yes	characteristic
249	C₂	C₂²×C₃×#125	1	2	yes	—
250	C₂	C₂²×C₃×#126	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#400	C₂	direct (N × H)
7. C₂²×#400	C₃	direct (N × H)
17. C₃×#400	C₂²	direct (N × H)
23. C₂×#400	C₂×C₃	direct (N × H)
61. #400	C₂²×C₃	direct (N × H)
178. C₂²×C₃	#400	direct (N × H)
222. C₂×C₃	C₂×#400	direct (N × H)
228. C₂²	C₃×#400	direct (N × H)
244. C₃	C₂²×#400	direct (N × H)
245. C₂	C₂×C₃×#400	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄⋊C₄	2
C₂×C₃×#400	2
C₂²×C₃×#95	2
C₂²×#400	3

Subgroups — C22×C3×#400 #10653148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#400 #10653148