Subgroups — #82303

Order 384.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	#5207	C₂	1	2	yes	maximal, complemented
3	C₂×#830	C₂	1	2	yes	maximal, complemented
4	#5012	C₂	1	2	yes	maximal, complemented
5	#5011	C₂	1	4	yes	maximal, complemented
6	D₄×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
7	#4926	C₂	1	1	yes	maximal, characteristic, complemented
8	#4925	C₂	1	1	yes	maximal, characteristic, complemented
9	#2974	—	3	1	no	maximal, Sylow
10	C₂²×C₃×D₄	C₄	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₃×C₄×D₄	C₄	1	2	yes	—
13	C₃×C₄×D₄	C₄	1	1	yes	characteristic
14	C₃×C₄×D₄	C₄	1	1	yes	characteristic
15	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
16	C₃×C₄×D₄	C₂²	1	2	yes	—
17	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
19	C₂×C₃×C₂²⋊C₄	C₄	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₂×#208	C₂²	1	4	yes	complemented
22	C₂²×C₃⋊C₈	C₂²	1	4	yes	complemented
23	#830	C₂²	1	1	yes	characteristic
24	#830	C₂²	1	1	yes	characteristic
25	#830	C₂²	1	2	yes	—
26	#830	C₂²	1	4	yes	complemented
27	#830	C₂²	1	4	yes	complemented
28	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
29	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
30	#803	C₂²	1	2	yes	complemented
31	C₄×C₃⋊C₈	C₂²	1	2	yes	complemented
32	C₂×C₄×D₄	D₃	1	1	yes	characteristic
33	#436	—	3	2	no	—
34	#435	—	3	4	no	—
35	C₈×D₄	—	3	2	no	—
36	#432	—	3	1	no	—
37	#431	—	3	1	no	—
38	#407	—	3	2	no	—
39	C₂×C₂²⋊C₈	—	3	2	no	—
40	C₂⁴×C₃	C₂×C₄	1	2	yes	—
41	C₂×C₃×D₄	—	2	4	no	—
42	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
43	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₃×C₄	—	4	4	no	—
46	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
49	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
50	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
60	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
62	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
63	C₃×C₄²	C₂³	1	1	yes	characteristic
64	C₃×C₄²	C₂³	1	1	yes	characteristic
65	#208	—	4	4	no	—
66	C₂×C₃⋊C₈	—	4	4	no	—
67	C₂×C₃⋊C₈	C₂³	1	2	yes	—
68	C₂×C₃⋊C₈	C₂³	1	2	yes	—
69	C₂×C₃⋊C₈	C₂³	1	4	yes	—
70	C₂²×D₄	Dic₃	1	1	yes	characteristic
71	C₂³×C₄	C₂×D₃	1	2	yes	—
72	C₂×M₁₆	—	3	4	no	—
73	C₂²×C₈	—	3	4	no	—
74	C₄×D₄	Dic₃	1	2	yes	—
75	C₄×D₄	Dic₃	1	1	yes	characteristic
76	C₄×D₄	Dic₃	1	1	yes	characteristic
77	C₄×D₄	C₂×D₃	1	1	yes	characteristic
78	C₄×D₄	C₂×D₃	1	2	yes	—
79	C₄×D₄	C₂×D₃	1	1	yes	characteristic
80	C₂×C₄⋊C₄	Dic₃	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	Dic₃	1	2	yes	—
82	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
83	C₄⋊C₈	—	3	2	no	—
84	C₄⋊C₈	—	3	2	no	—
85	C₂²⋊C₈	—	3	1	no	—
86	C₂²⋊C₈	—	3	2	no	—
87	C₂²⋊C₈	—	3	1	no	—
88	C₂²⋊C₈	—	3	4	no	—
89	C₂²⋊C₈	—	3	4	no	—
90	#97	—	3	2	no	—
91	C₄×C₈	—	3	2	no	—
92	C₂³×C₃	—	2	4	no	—
93	C₂³×C₃	—	2	4	no	—
94	C₂³×C₃	—	4	2	no	—
95	C₂³×C₃	C₂²×C₄	1	2	yes	—
96	C₂³×C₃	C₂²×C₄	1	2	yes	—
97	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
98	C₃×D₄	—	2	4	no	—
99	C₃×D₄	—	2	2	no	—
100	C₃×D₄	M₁₆	1	4	yes	complemented
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	4	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	4	2	no	—
107	C₂×C₃×C₄	—	4	2	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
114	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
115	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
117	C₃⋊C₈	—	2	2	no	—
118	C₃⋊C₈	—	2	4	no	—
119	C₃⋊C₈	—	2	2	no	—
120	C₂⁴	C₂×Dic₃	1	2	yes	—
121	C₂×D₄	—	2	4	no	—
122	C₂×D₄	C₂×Dic₃	1	2	yes	—
123	C₂×D₄	C₂×Dic₃	1	1	yes	characteristic
124	C₂×D₄	C₂×Dic₃	1	1	yes	characteristic
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
127	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₃	1	2	yes	—
129	C₂²×C₄	C₂×Dic₃	1	2	yes	—
130	C₂²×C₄	C₂×Dic₃	1	2	yes	—
131	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₃	1	2	yes	—
133	C₂²×C₄	C₂²×D₃	1	2	yes	—
134	M₁₆	—	12	4	no	—
135	C₂×C₈	—	3	2	no	—
136	C₂×C₈	—	3	2	no	—
137	C₂×C₈	—	3	4	no	—
138	C₂×C₈	—	12	4	no	—
139	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
140	C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
141	C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
142	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
143	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
144	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
145	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
146	C₄²	C₂×Dic₃	1	1	yes	characteristic
147	C₄²	C₂×Dic₃	1	1	yes	characteristic
148	C₄²	C₂²×D₃	1	1	yes	characteristic
149	C₄²	C₂²×D₃	1	1	yes	characteristic
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	2	1	no	—
153	C₂²×C₃	—	2	2	no	—
154	C₂²×C₃	—	2	2	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	4	8	no	—
158	C₂²×C₃	C₂×M₁₆	1	4	yes	—
159	C₂²×C₃	#131	1	2	yes	—
160	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	C₂×M₁₆	1	2	yes	—
168	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂²×Dic₃	1	2	yes	—
173	C₂³	C₂²×Dic₃	1	2	yes	—
174	C₂³	C₂²×Dic₃	1	1	yes	characteristic
175	D₄	—	2	4	no	—
176	D₄	—	2	2	no	—
177	D₄	#208	1	4	yes	complemented
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₃	1	2	yes	—
191	C₂×C₄	C₂²×Dic₃	1	2	yes	—
192	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
193	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
194	C₈	—	6	2	no	—
195	C₈	—	6	4	no	—
196	C₈	—	6	2	no	—
197	C₂×C₃	—	2	2	no	—
198	C₂×C₃	—	2	4	no	—
199	C₂×C₃	—	4	2	no	—
200	C₂×C₃	#518	1	1	yes	characteristic
201	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
202	C₂×C₃	C₂×#131	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	8	no	—
211	C₂²	C₂×#208	1	4	yes	—
212	C₂²	#948	1	2	yes	—
213	C₂²	C₂³×Dic₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	C₂×#208	1	2	yes	—
221	C₄	#1004	1	2	yes	—
222	C₃	#2974	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×#208	1	1	yes	characteristic
227	C₂	#5798	1	1	yes	characteristic
228	C₂	C₂×#948	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄×D₄	C₂
prime radical	C₂²×C₃×D₄	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄×D₄	D₃
hypocenter	C₃	#2974
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #5207	C₂	semidirect
3. C₂×#830	C₂	semidirect
4. #5012	C₂	semidirect
5. #5011	C₂	semidirect
6. D₄×C₃⋊C₈	C₂	semidirect
7. #4926	C₂	semidirect
8. #4925	C₂	semidirect
21. C₂×#208	C₂²	semidirect
22. C₂²×C₃⋊C₈	C₂²	semidirect
26. #830	C₂²	semidirect
27. #830	C₂²	semidirect
28. C₄⋊(C₃⋊C₈)	C₂²	semidirect
29. C₄⋊(C₃⋊C₈)	C₂²	semidirect
30. #803	C₂²	semidirect
31. C₄×C₃⋊C₈	C₂²	semidirect
100. C₃×D₄	M₁₆	semidirect
177. D₄	#208	semidirect
222. C₃	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×D₄	2
#5207	2
C₂×#830	2
#5012	2
#5011	2
D₄×C₃⋊C₈	2
#4926	2
#4925	2
#2974	3