Subgroups — #73715

Order 384.

556 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#888	C₂	1	2	yes	maximal
2	C₂×#869	C₂	1	2	yes	maximal
3	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#4984	C₂	1	2	yes	maximal
5	#4935	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
7	#4928	C₂	1	2	yes	maximal
8	#4656	C₂	1	4	yes	maximal
9	#2551	—	3	1	no	maximal, Sylow
10	C₂²×Dic₆	C₂²	1	4	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	4	yes	—
16	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄×Dic₃	C₂²	1	2	yes	—
23	#888	—	2	8	no	—
24	#869	—	2	8	no	—
25	C₃×#95	C₂²	1	2	yes	—
26	C₃⋊#95	C₂²	1	2	yes	—
27	C₃⋊#95	C₂²	1	4	yes	—
28	C₂×C₄⋊Q₈	—	3	2	no	—
29	C₂×C₄⋊Q₈	—	3	2	no	—
30	#395	—	3	4	no	—
31	#389	—	3	2	no	—
32	#389	—	3	2	no	—
33	#384	—	3	1	no	—
34	C₄×C₄⋊C₄	—	3	1	no	—
35	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂²×Dic₃	C₂³	1	4	yes	—
42	C₂²×Dic₃	C₂³	1	2	yes	—
43	C₂²×Dic₃	C₂³	1	2	yes	—
44	C₂×Dic₆	—	2	8	no	—
45	C₂×Dic₆	—	2	8	no	—
46	C₂×Dic₆	—	4	4	no	—
47	C₃×C₄⋊C₄	D₄	1	4	yes	—
48	C₃×C₄⋊C₄	Q₈	1	4	yes	complemented
49	C₃×C₄²	—	2	4	no	—
50	C₃×C₄²	Q₈	1	4	yes	complemented
51	C₄⋊Dic₃	—	2	4	no	—
52	C₄⋊Dic₃	—	2	4	no	—
53	C₄⋊Dic₃	—	2	8	no	—
54	C₄⋊Dic₃	—	4	2	no	—
55	C₄⋊Dic₃	D₄	1	4	yes	—
56	C₄⋊Dic₃	Q₈	1	4	yes	complemented
57	Dic₃⋊C₄	—	2	8	no	—
58	C₄×Dic₃	—	2	4	no	—
59	C₂²×Q₈	—	3	4	no	—
60	C₄⋊Q₈	—	6	8	no	—
61	C₄⋊Q₈	—	6	8	no	—
62	C₂×C₄⋊C₄	—	3	1	no	—
63	C₂×C₄⋊C₄	—	3	1	no	—
64	C₂×C₄⋊C₄	—	3	2	no	—
65	C₂×C₄⋊C₄	—	3	2	no	—
66	C₂×C₄⋊C₄	—	3	4	no	—
67	C₂×C₄⋊C₄	—	3	4	no	—
68	C₂×C₄⋊C₄	—	3	2	no	—
69	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₂×C₄²	—	3	2	no	—
72	C₂×C₄²	C₂×D₃	1	2	yes	—
73	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
74	#95	—	3	2	no	—
75	#95	—	3	4	no	—
76	#95	C₂×D₃	1	2	yes	—
77	C₂³×C₃	C₂⁴	1	1	yes	characteristic
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	1	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	2	1	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
85	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
87	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
88	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
89	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
90	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
91	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
92	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
93	C₂×Dic₃	—	2	2	no	—
94	C₂×Dic₃	—	2	2	no	—
95	C₂×Dic₃	—	2	2	no	—
96	C₂×Dic₃	—	2	2	no	—
97	C₂×Dic₃	—	2	4	no	—
98	C₂×Dic₃	—	2	4	no	—
99	C₂×Dic₃	—	2	4	no	—
100	C₂×Dic₃	C₂×D₄	1	4	yes	—
101	C₂×Dic₃	C₂×Q₈	1	4	yes	—
102	Dic₆	—	4	16	no	—
103	C₂×Q₈	—	6	8	no	—
104	C₂×Q₈	—	6	8	no	—
105	C₂×Q₈	—	12	4	no	—
106	C₂²×C₄	—	3	2	no	—
107	C₂²×C₄	—	3	2	no	—
108	C₂²×C₄	—	3	4	no	—
109	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₄⋊C₄	—	3	4	no	—
115	C₄⋊C₄	—	3	4	no	—
116	C₄⋊C₄	—	6	8	no	—
117	C₄⋊C₄	—	6	8	no	—
118	C₄⋊C₄	—	6	4	no	—
119	C₄⋊C₄	—	6	4	no	—
120	C₄⋊C₄	—	12	2	no	—
121	C₄⋊C₄	Dic₆	1	4	yes	complemented
122	C₄⋊C₄	D₁₂	1	4	yes	—
123	C₄²	—	2	4	no	—
124	C₄²	—	6	4	no	—
125	C₄²	Dic₆	1	4	yes	complemented
126	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₃×C₄	—	2	2	no	—
134	C₃×C₄	—	2	4	no	—
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	C₄⋊Q₈	1	4	yes	complemented
137	C₃×C₄	C₄⋊Q₈	1	4	yes	complemented
138	Dic₃	—	2	4	no	—
139	Dic₃	—	2	4	no	—
140	Dic₃	—	4	4	no	—
141	C₂³	C₂³×D₃	1	1	yes	characteristic
142	Q₈	—	12	16	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	3	4	no	—
150	C₂×C₄	—	3	4	no	—
151	C₂×C₄	—	6	2	no	—
152	C₂×C₄	—	6	2	no	—
153	C₂×C₄	—	6	2	no	—
154	C₂×C₄	—	6	2	no	—
155	C₂×C₄	—	6	4	no	—
156	C₂×C₄	—	6	4	no	—
157	C₂×C₄	—	6	4	no	—
158	C₂×C₄	C₂×Dic₆	1	2	yes	—
159	C₂×C₄	C₂×Dic₆	1	4	yes	—
160	C₂×C₄	C₂×Dic₆	1	2	yes	—
161	C₂×C₄	C₂×Dic₆	1	2	yes	—
162	C₂×C₄	C₂×Dic₆	1	2	yes	—
163	C₂×C₄	C₂×D₁₂	1	4	yes	—
164	C₂×C₄	C₂×D₁₂	1	2	yes	—
165	C₂×C₄	D₃×D₄	1	2	yes	—
166	C₂×C₄	D₃×Q₈	1	2	yes	—
167	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
169	C₂×C₃	#544	1	1	yes	characteristic
170	C₂×C₃	D₄²	1	1	yes	characteristic
171	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
172	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
173	C₂×C₃	Q₈²	1	1	yes	characteristic
174	C₂²	C₂²×Dic₆	1	1	yes	characteristic
175	C₂²	C₂²×Dic₆	1	1	yes	characteristic
176	C₂²	C₂²×D₁₂	1	1	yes	characteristic
177	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
178	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
179	C₂²	#1009	1	1	yes	characteristic
180	C₂²	#1010	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	4	no	—
184	C₄	—	6	4	no	—
185	C₄	—	6	4	no	—
186	C₄	—	12	4	no	—
187	C₄	#869	1	4	yes	complemented
188	C₄	#888	1	4	yes	complemented
189	C₃	#2551	1	1	yes	characteristic, complemented, Sylow
190	C₂	C₂×#869	1	1	yes	characteristic
191	C₂	C₂×#888	1	1	yes	characteristic
192	C₂	#5519	1	1	yes	characteristic
193	C₂	D₄×Dic₆	1	1	yes	characteristic
194	C₂	D₄×D₁₂	1	1	yes	characteristic
195	C₂	Q₈×Dic₆	1	1	yes	characteristic
196	C₂	Q₈×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×C₄×C₄⋊C₄	C₂
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#2551
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
48. C₃×C₄⋊C₄	Q₈	semidirect
50. C₃×C₄²	Q₈	semidirect
56. C₄⋊Dic₃	Q₈	semidirect
121. C₄⋊C₄	Dic₆	semidirect
125. C₄²	Dic₆	semidirect
136. C₃×C₄	C₄⋊Q₈	semidirect
137. C₃×C₄	C₄⋊Q₈	semidirect
187. C₄	#869	semidirect
188. C₄	#888	semidirect
189. C₃	#2551	semidirect

Maximal subgroups

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

subgroup	index
C₂×#888	2
C₂×#869	2
C₃×C₄×C₄⋊C₄	2
#4984	2
#4935	2
C₄×C₄⋊Dic₃	2
#4928	2
#4656	2
#2551	3