Subgroups — C₂³×#878 #11701828

Order 768.

14078 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	C₂⁵×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂³×#217	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#878	C₂	1	56	yes	maximal, direct product factor
9	C₂³×C₂²⋊Q₈	—	3	1	no	maximal, Sylow
10	C₂⁶×C₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₃	—	2	16	no	—
14	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
15	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
16	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
17	C₂⁴×Dic₃	C₂²	1	28	yes	—
18	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
19	C₂³×Dic₆	—	2	16	no	—
20	C₂³×Dic₆	C₂²	1	28	yes	complemented
21	C₂²×C₃×C₂²⋊C₄	C₂²	1	28	yes	—
22	C₂²×#217	C₂²	1	28	yes	—
23	C₂²×C₄⋊Dic₃	C₂²	1	28	yes	complemented
24	C₂²×Dic₃⋊C₄	C₂²	1	28	yes	complemented
25	C₂²×Dic₃⋊C₄	C₂²	1	28	yes	complemented
26	C₂×#878	C₂²	1	448	yes	direct product factor
27	C₂⁴×Q₈	—	3	1	no	—
28	C₂⁵×C₄	—	3	1	no	—
29	C₂²×C₂²⋊Q₈	—	3	56	no	—
30	C₂³×C₄⋊C₄	—	3	1	no	—
31	C₂³×C₄⋊C₄	—	3	1	no	—
32	C₂³×C₄⋊C₄	—	3	1	no	—
33	C₂³×C₂²⋊C₄	—	3	1	no	—
34	C₂³×C₂²⋊C₄	D₃	1	1	yes	characteristic
35	C₂⁵×C₃	—	2	8	no	—
36	C₂⁵×C₃	—	2	8	no	—
37	C₂⁵×C₃	Q₈	1	16	yes	complemented
38	C₂⁵×C₃	C₂³	1	14	yes	—
39	C₂⁵×C₃	C₂³	1	1	yes	characteristic
40	C₂³×C₃×C₄	—	2	8	no	—
41	C₂³×C₃×C₄	—	2	8	no	—
42	C₂³×C₃×C₄	C₂³	1	14	yes	—
43	C₂³×C₃×C₄	C₂³	1	14	yes	—
44	C₂³×Dic₃	—	2	8	no	—
45	C₂³×Dic₃	—	2	8	no	—
46	C₂³×Dic₃	—	2	8	no	—
47	C₂³×Dic₃	—	2	224	no	—
48	C₂³×Dic₃	D₄	1	16	yes	—
49	C₂³×Dic₃	C₂³	1	14	yes	—
50	C₂³×Dic₃	C₂³	1	112	yes	—
51	C₂³×Dic₃	C₂³	1	14	yes	—
52	C₂³×Dic₃	C₂³	1	14	yes	—
53	C₂³×Dic₃	C₂³	1	14	yes	—
54	C₂²×Dic₆	—	2	224	no	—
55	C₂²×Dic₆	C₂³	1	112	yes	complemented
56	C₂×C₃×C₂²⋊C₄	C₂³	1	112	yes	—
57	C₂×#217	C₂³	1	112	yes	—
58	C₂×C₄⋊Dic₃	C₂³	1	112	yes	complemented
59	C₂×Dic₃⋊C₄	C₂³	1	112	yes	complemented
60	C₂×Dic₃⋊C₄	C₂³	1	112	yes	complemented
61	#878	C₂³	1	512	yes	direct product factor
62	C₂⁶	C₂×D₃	1	1	yes	characteristic
63	C₂³×Q₈	—	3	28	no	—
64	C₂³×Q₈	—	6	16	no	—
65	C₂⁴×C₄	—	3	1	no	—
66	C₂⁴×C₄	—	3	28	no	—
67	C₂⁴×C₄	—	3	1	no	—
68	C₂⁴×C₄	—	3	1	no	—
69	C₂⁴×C₄	—	3	1	no	—
70	C₂⁴×C₄	—	6	16	no	—
71	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
72	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
73	C₂×C₂²⋊Q₈	—	3	448	no	—
74	C₂²×C₄⋊C₄	—	3	28	no	—
75	C₂²×C₄⋊C₄	—	3	28	no	—
76	C₂²×C₄⋊C₄	—	3	28	no	—
77	C₂²×C₂²⋊C₄	—	3	28	no	—
78	C₂²×C₂²⋊C₄	C₂×D₃	1	28	yes	—
79	C₂⁴×C₃	—	2	128	no	—
80	C₂⁴×C₃	—	2	56	no	—
81	C₂⁴×C₃	—	2	56	no	—
82	C₂⁴×C₃	C₂×D₄	1	8	yes	—
83	C₂⁴×C₃	C₂×Q₈	1	8	yes	—
84	C₂⁴×C₃	C₂×Q₈	1	112	yes	complemented
85	C₂⁴×C₃	#41	1	8	yes	—
86	C₂⁴×C₃	C₂⁴	1	28	yes	—
87	C₂⁴×C₃	C₂⁴	1	7	yes	—
88	C₂²×C₃×C₄	—	2	56	no	—
89	C₂²×C₃×C₄	—	2	56	no	—
90	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
91	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
92	C₂²×Dic₃	—	2	56	no	—
93	C₂²×Dic₃	—	2	56	no	—
94	C₂²×Dic₃	—	2	56	no	—
95	C₂²×Dic₃	—	2	448	no	—
96	C₂²×Dic₃	C₂×D₄	1	112	yes	—
97	C₂²×Dic₃	C₂⁴	1	64	yes	—
98	C₂²×Dic₃	C₂⁴	1	28	yes	—
99	C₂²×Dic₃	C₂⁴	1	28	yes	—
100	C₂²×Dic₃	C₂⁴	1	28	yes	—
101	C₂²×Dic₃	C₂⁴	1	28	yes	—
102	C₂×Dic₆	—	2	448	no	—
103	C₂×Dic₆	C₂⁴	1	64	yes	complemented
104	C₃×C₂²⋊C₄	C₂⁴	1	64	yes	—
105	#217	C₂⁴	1	64	yes	—
106	C₄⋊Dic₃	C₂⁴	1	64	yes	complemented
107	Dic₃⋊C₄	C₂⁴	1	64	yes	complemented
108	Dic₃⋊C₄	C₂⁴	1	64	yes	complemented
109	C₂⁵	—	2	8	no	—
110	C₂⁵	—	2	8	no	—
111	C₂⁵	Dic₆	1	16	yes	complemented
112	C₂⁵	C₂²×D₃	1	14	yes	—
113	C₂⁵	C₂²×D₃	1	1	yes	characteristic
114	C₂²×Q₈	—	3	112	no	—
115	C₂²×Q₈	—	6	224	no	—
116	C₂³×C₄	—	2	8	no	—
117	C₂³×C₄	—	2	8	no	—
118	C₂³×C₄	—	3	16	no	—
119	C₂³×C₄	—	3	14	no	—
120	C₂³×C₄	—	3	14	no	—
121	C₂³×C₄	—	3	14	no	—
122	C₂³×C₄	—	3	14	no	—
123	C₂³×C₄	—	3	112	no	—
124	C₂³×C₄	—	6	224	no	—
125	C₂³×C₄	—	6	8	no	—
126	C₂³×C₄	—	6	8	no	—
127	C₂³×C₄	—	6	8	no	—
128	C₂³×C₄	C₂²×D₃	1	14	yes	—
129	C₂³×C₄	C₂²×D₃	1	14	yes	—
130	C₂²⋊Q₈	—	3	512	no	—
131	C₂×C₄⋊C₄	—	3	112	no	—
132	C₂×C₄⋊C₄	—	3	112	no	—
133	C₂×C₄⋊C₄	—	3	112	no	—
134	C₂×C₂²⋊C₄	—	3	112	no	—
135	C₂×C₂²⋊C₄	C₂²×D₃	1	112	yes	—
136	C₂³×C₃	—	2	448	no	—
137	C₂³×C₃	—	2	56	no	—
138	C₂³×C₃	—	2	56	no	—
139	C₂³×C₃	C₂²⋊Q₈	1	64	yes	complemented
140	C₂³×C₃	C₂²×D₄	1	28	yes	—
141	C₂³×C₃	C₂²×Q₈	1	28	yes	—
142	C₂³×C₃	C₂²×Q₈	1	112	yes	complemented
143	C₂³×C₃	C₂×#41	1	28	yes	—
144	C₂³×C₃	C₂⁵	1	8	yes	—
145	C₂³×C₃	C₂⁵	1	7	yes	—
146	C₂×C₃×C₄	—	2	56	no	—
147	C₂×C₃×C₄	—	2	56	no	—
148	C₂×C₃×C₄	C₂⁵	1	8	yes	—
149	C₂×C₃×C₄	C₂⁵	1	8	yes	—
150	C₂×Dic₃	—	2	56	no	—
151	C₂×Dic₃	—	2	56	no	—
152	C₂×Dic₃	—	2	56	no	—
153	C₂×Dic₃	—	2	128	no	—
154	C₂×Dic₃	C₂²×D₄	1	112	yes	—
155	C₂×Dic₃	C₂⁵	1	8	yes	—
156	C₂×Dic₃	C₂⁵	1	8	yes	—
157	C₂×Dic₃	C₂⁵	1	8	yes	—
158	C₂×Dic₃	C₂⁵	1	8	yes	—
159	Dic₆	—	2	128	no	—
160	C₂⁴	—	2	128	no	—
161	C₂⁴	—	2	56	no	—
162	C₂⁴	—	2	56	no	—
163	C₂⁴	C₂×Dic₆	1	8	yes	—
164	C₂⁴	C₂×Dic₆	1	112	yes	complemented
165	C₂⁴	D₃×D₄	1	8	yes	—
166	C₂⁴	#237	1	8	yes	—
167	C₂⁴	C₂³×D₃	1	28	yes	—
168	C₂⁴	C₂³×D₃	1	7	yes	—
169	C₂×Q₈	—	3	64	no	—
170	C₂×Q₈	—	6	448	no	—
171	C₂²×C₄	—	2	56	no	—
172	C₂²×C₄	—	2	56	no	—
173	C₂²×C₄	—	3	112	no	—
174	C₂²×C₄	—	3	28	no	—
175	C₂²×C₄	—	3	28	no	—
176	C₂²×C₄	—	3	28	no	—
177	C₂²×C₄	—	3	28	no	—
178	C₂²×C₄	—	3	64	no	—
179	C₂²×C₄	—	6	448	no	—
180	C₂²×C₄	—	6	56	no	—
181	C₂²×C₄	—	6	56	no	—
182	C₂²×C₄	—	6	56	no	—
183	C₂²×C₄	C₂³×D₃	1	28	yes	—
184	C₂²×C₄	C₂³×D₃	1	28	yes	—
185	C₄⋊C₄	—	3	64	no	—
186	C₄⋊C₄	—	3	64	no	—
187	C₄⋊C₄	—	3	64	no	—
188	C₂²⋊C₄	—	3	64	no	—
189	C₂²⋊C₄	C₂³×D₃	1	64	yes	—
190	C₂²×C₃	—	2	8	no	—
191	C₂²×C₃	—	2	8	no	—
192	C₂²×C₃	—	2	224	no	—
193	C₂²×C₃	C₂×C₂²⋊Q₈	1	112	yes	complemented
194	C₂²×C₃	C₂³×D₄	1	14	yes	—
195	C₂²×C₃	C₂³×Q₈	1	14	yes	—
196	C₂²×C₃	C₂³×Q₈	1	16	yes	complemented
197	C₂²×C₃	C₂²×#41	1	14	yes	—
198	C₂²×C₃	C₂⁶	1	1	yes	characteristic
199	C₃×C₄	—	2	8	no	—
200	C₃×C₄	—	2	8	no	—
201	Dic₃	—	2	8	no	—
202	Dic₃	—	2	8	no	—
203	Dic₃	—	2	8	no	—
204	Dic₃	C₂³×D₄	1	16	yes	—
205	C₂³	—	2	56	no	—
206	C₂³	—	2	56	no	—
207	C₂³	—	2	448	no	—
208	C₂³	#878	1	64	yes	direct product factor
209	C₂³	C₂²×Dic₆	1	28	yes	—
210	C₂³	C₂²×Dic₆	1	112	yes	complemented
211	C₂³	C₂×D₃×D₄	1	28	yes	—
212	C₂³	C₂×#237	1	28	yes	—
213	C₂³	C₂⁴×D₃	1	8	yes	—
214	C₂³	C₂⁴×D₃	1	7	yes	—
215	Q₈	—	6	128	no	—
216	C₂×C₄	—	2	56	no	—
217	C₂×C₄	—	2	56	no	—
218	C₂×C₄	—	3	112	no	—
219	C₂×C₄	—	3	8	no	—
220	C₂×C₄	—	3	8	no	—
221	C₂×C₄	—	3	8	no	—
222	C₂×C₄	—	3	8	no	—
223	C₂×C₄	—	6	128	no	—
224	C₂×C₄	—	6	56	no	—
225	C₂×C₄	—	6	56	no	—
226	C₂×C₄	—	6	56	no	—
227	C₂×C₄	C₂⁴×D₃	1	8	yes	—
228	C₂×C₄	C₂⁴×D₃	1	8	yes	—
229	C₂×C₃	—	2	16	no	—
230	C₂×C₃	C₂²×C₂²⋊Q₈	1	28	yes	complemented
231	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
232	C₂×C₃	C₂⁴×Q₈	1	1	yes	characteristic
233	C₂×C₃	C₂³×#41	1	1	yes	characteristic
234	C₂²	—	2	8	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	224	no	—
237	C₂²	C₂×#878	1	112	yes	direct product factor
238	C₂²	C₂³×Dic₆	1	14	yes	—
239	C₂²	C₂³×Dic₆	1	16	yes	complemented
240	C₂²	C₂²×D₃×D₄	1	14	yes	—
241	C₂²	C₂²×#237	1	14	yes	—
242	C₂²	C₂⁵×D₃	1	1	yes	characteristic
243	C₄	—	2	8	no	—
244	C₄	—	2	8	no	—
245	C₄	—	3	16	no	—
246	C₄	—	6	8	no	—
247	C₄	—	6	8	no	—
248	C₄	—	6	8	no	—
249	C₃	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
250	C₂	—	2	16	no	—
251	C₂	C₂²×#878	1	28	yes	direct product factor
252	C₂	C₂⁴×Dic₆	1	1	yes	characteristic
253	C₂	C₂³×#237	1	1	yes	characteristic
254	C₂	C₂³×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁶
Frattini	C₂²	C₂⁵×D₃
Fitting	C₂³×C₃×C₂²⋊C₄	C₂
socle	C₂⁵×C₃	C₂³
prime radical	C₂⁶×C₃	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₃
hypocenter	C₃	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₆	C₂	semidirect
5. C₂³×Dic₃⋊C₄	C₂	semidirect
6. C₂³×Dic₃⋊C₄	C₂	semidirect
7. C₂³×C₄⋊Dic₃	C₂	semidirect
8. C₂²×#878	C₂	direct (N × H)
20. C₂³×Dic₆	C₂²	semidirect
23. C₂²×C₄⋊Dic₃	C₂²	semidirect
24. C₂²×Dic₃⋊C₄	C₂²	semidirect
25. C₂²×Dic₃⋊C₄	C₂²	semidirect
26. C₂×#878	C₂²	direct (N × H)
37. C₂⁵×C₃	Q₈	semidirect
55. C₂²×Dic₆	C₂³	semidirect
58. C₂×C₄⋊Dic₃	C₂³	semidirect
59. C₂×Dic₃⋊C₄	C₂³	semidirect
60. C₂×Dic₃⋊C₄	C₂³	semidirect
61. #878	C₂³	direct (N × H)
84. C₂⁴×C₃	C₂×Q₈	semidirect
103. C₂×Dic₆	C₂⁴	semidirect
106. C₄⋊Dic₃	C₂⁴	semidirect
107. Dic₃⋊C₄	C₂⁴	semidirect
108. Dic₃⋊C₄	C₂⁴	semidirect
111. C₂⁵	Dic₆	semidirect
139. C₂³×C₃	C₂²⋊Q₈	semidirect
142. C₂³×C₃	C₂²×Q₈	semidirect
164. C₂⁴	C₂×Dic₆	semidirect
193. C₂²×C₃	C₂×C₂²⋊Q₈	semidirect
196. C₂²×C₃	C₂³×Q₈	semidirect
208. C₂³	#878	direct (N × H)
210. C₂³	C₂²×Dic₆	semidirect
230. C₂×C₃	C₂²×C₂²⋊Q₈	semidirect
237. C₂²	C₂×#878	direct (N × H)
239. C₂²	C₂³×Dic₆	semidirect
249. C₃	C₂³×C₂²⋊Q₈	semidirect
251. C₂	C₂²×#878	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Dic₃	2
C₂⁴×Dic₆	2
C₂³×C₃×C₂²⋊C₄	2
C₂³×#217	2
C₂³×Dic₃⋊C₄	2
C₂³×C₄⋊Dic₃	2
C₂²×#878	2
C₂³×C₂²⋊Q₈	3

Subgroups — C23×#878 #11701828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#878 #11701828