Subgroups — #73656

Order 384.

496 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	Dic₃×C₄⋊C₄	C₂	1	2	yes	maximal
4	#4972	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Dic₃	C₂	1	2	yes	maximal
7	#4928	C₂	1	2	yes	maximal
8	#4646	C₂	1	4	yes	maximal
9	#2369	—	3	1	no	maximal, Sylow
10	C₂²×Dic₆	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
20	C₂×C₄×Dic₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₃	C₂²	1	4	yes	—
22	#869	C₄	1	8	yes	complemented
23	C₃×#95	C₂²	1	2	yes	—
24	C₃⋊#95	C₂²	1	2	yes	—
25	C₃⋊#95	C₂²	1	4	yes	—
26	C₃⋊#95	C₂²	1	4	yes	—
27	C₂×C₄⋊Q₈	—	3	1	no	—
28	#391	—	3	4	no	—
29	#389	—	3	2	no	—
30	#389	—	3	2	no	—
31	C₄×C₄⋊C₄	—	3	2	no	—
32	C₄×C₄⋊C₄	—	3	2	no	—
33	C₄×C₄⋊C₄	—	3	1	no	—
34	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂³	1	2	yes	—
36	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂²×C₃×C₄	C₂³	1	2	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	C₂³	1	2	yes	—
41	C₂²×Dic₃	C₂³	1	4	yes	—
42	C₂²×Dic₃	C₂³	1	2	yes	—
43	C₂×Dic₆	—	4	2	no	—
44	C₂×Dic₆	—	4	2	no	—
45	C₂×Dic₆	C₂×C₄	1	8	yes	—
46	C₃×C₄⋊C₄	—	2	2	no	—
47	C₃×C₄⋊C₄	Q₈	1	4	yes	complemented
48	C₃×C₄²	—	2	4	no	—
49	C₃×C₄²	C₂×C₄	1	4	yes	—
50	C₄⋊Dic₃	—	2	2	no	—
51	C₄⋊Dic₃	C₂×C₄	1	8	yes	—
52	C₄⋊Dic₃	C₂×C₄	1	8	yes	—
53	C₄⋊Dic₃	Q₈	1	4	yes	complemented
54	Dic₃⋊C₄	—	4	4	no	—
55	C₄×Dic₃	—	2	4	no	—
56	C₄×Dic₃	—	4	4	no	—
57	C₂²×Q₈	—	3	2	no	—
58	C₄⋊Q₈	—	3	8	no	—
59	C₂×C₄⋊C₄	—	3	2	no	—
60	C₂×C₄⋊C₄	—	3	2	no	—
61	C₂×C₄⋊C₄	—	3	1	no	—
62	C₂×C₄⋊C₄	—	3	1	no	—
63	C₂×C₄⋊C₄	—	3	4	no	—
64	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄²	—	3	2	no	—
67	C₂×C₄²	—	3	4	no	—
68	C₂×C₄²	C₂×D₃	1	2	yes	—
69	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
70	#95	—	3	4	no	—
71	#95	—	3	4	no	—
72	#95	—	3	2	no	—
73	#95	C₂×D₃	1	2	yes	—
74	C₂³×C₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	2	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	1	no	—
78	C₂×C₃×C₄	—	2	1	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
84	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
86	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
87	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
88	C₂×C₃×C₄	#41	1	2	yes	—
89	C₂×C₃×C₄	#41	1	2	yes	—
90	C₂×Dic₃	—	2	2	no	—
91	C₂×Dic₃	—	2	2	no	—
92	C₂×Dic₃	—	2	4	no	—
93	C₂×Dic₃	—	2	4	no	—
94	C₂×Dic₃	—	2	2	no	—
95	C₂×Dic₃	—	2	2	no	—
96	C₂×Dic₃	—	2	2	no	—
97	C₂×Dic₃	C₂²×C₄	1	8	yes	—
98	C₂×Dic₃	C₂×Q₈	1	4	yes	—
99	Dic₆	—	4	8	no	—
100	C₂×Q₈	—	3	8	no	—
101	C₂×Q₈	—	12	2	no	—
102	C₂×Q₈	—	12	2	no	—
103	C₂²×C₄	—	3	2	no	—
104	C₂²×C₄	—	3	4	no	—
105	C₂²×C₄	—	3	2	no	—
106	C₂²×C₄	C₂²×D₃	1	2	yes	—
107	C₂²×C₄	C₂²×D₃	1	2	yes	—
108	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	3	8	no	—
113	C₄⋊C₄	—	3	8	no	—
114	C₄⋊C₄	—	3	4	no	—
115	C₄⋊C₄	—	6	2	no	—
116	C₄⋊C₄	—	12	4	no	—
117	C₄⋊C₄	Dic₆	1	4	yes	complemented
118	C₄²	—	2	4	no	—
119	C₄²	—	6	4	no	—
120	C₄²	—	12	4	no	—
121	C₄²	C₄×D₃	1	4	yes	—
122	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
123	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₃	C₂×#41	1	1	yes	characteristic
126	C₂²×C₃	C₂×#41	1	1	yes	characteristic
127	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₃×C₄	—	2	2	no	—
130	C₃×C₄	—	2	4	no	—
131	C₃×C₄	—	4	2	no	—
132	C₃×C₄	C₄×Q₈	1	4	yes	—
133	C₃×C₄	C₄×Q₈	1	4	yes	—
134	Dic₃	—	2	8	no	—
135	Dic₃	—	2	4	no	—
136	Dic₃	—	4	2	no	—
137	C₂³	C₂³×D₃	1	1	yes	characteristic
138	Q₈	—	12	8	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	3	8	no	—
147	C₂×C₄	—	3	4	no	—
148	C₂×C₄	—	6	2	no	—
149	C₂×C₄	—	6	2	no	—
150	C₂×C₄	—	6	4	no	—
151	C₂×C₄	—	6	4	no	—
152	C₂×C₄	—	6	2	no	—
153	C₂×C₄	—	6	2	no	—
154	C₂×C₄	—	6	2	no	—
155	C₂×C₄	C₂×Dic₆	1	4	yes	—
156	C₂×C₄	C₂×Dic₆	1	2	yes	—
157	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
160	C₂×C₄	#235	1	2	yes	—
161	C₂×C₄	#237	1	2	yes	—
162	C₂×C₄	D₃×Q₈	1	2	yes	—
163	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₂×C₃	#518	1	1	yes	characteristic
166	C₂×C₃	#554	1	1	yes	characteristic
167	C₂×C₃	#554	1	1	yes	characteristic
168	C₂×C₃	#555	1	1	yes	characteristic
169	C₂×C₃	Q₈²	1	1	yes	characteristic
170	C₂²	C₂²×Dic₆	1	1	yes	characteristic
171	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
172	C₂²	C₂×#235	1	1	yes	characteristic
173	C₂²	C₂×#237	1	1	yes	characteristic
174	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
175	C₂²	#1009	1	1	yes	characteristic
176	C₂²	#1009	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	4	no	—
179	C₄	—	4	2	no	—
180	C₄	—	6	8	no	—
181	C₄	—	6	4	no	—
182	C₄	—	12	2	no	—
183	C₄	C₄×Dic₆	1	4	yes	—
184	C₄	#887	1	4	yes	—
185	C₃	#2369	1	1	yes	characteristic, complemented, Sylow
186	C₂	C₂×C₄×Dic₆	1	1	yes	characteristic
187	C₂	C₂×#887	1	1	yes	characteristic
188	C₂	#5521	1	1	yes	characteristic
189	C₂	#5543	1	1	yes	characteristic
190	C₂	#5563	1	1	yes	characteristic
191	C₂	Q₈×Dic₆	1	1	yes	characteristic
192	C₂	#5579	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#2369
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
22. #869	C₄	semidirect
47. C₃×C₄⋊C₄	Q₈	semidirect
53. C₄⋊Dic₃	Q₈	semidirect
117. C₄⋊C₄	Dic₆	semidirect
185. C₃	#2369	semidirect

Maximal subgroups

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

subgroup	index
C₂×#869	2
C₃×C₄×C₄⋊C₄	2
Dic₃×C₄⋊C₄	2
#4972	2
C₄×C₄⋊Dic₃	2
#4928	2
#4646	2
#2369	3