Finite Group Databasebeta

Subgroups — #13146697

Order 1728.

319 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6188	C₂	1	1	yes	maximal, characteristic
2	C₂×#88589	C₂	1	1	yes	maximal, characteristic, complemented
3	#11721309	C₂	1	2	yes	maximal
4	#11720704	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720702	C₂	1	2	yes	maximal, complemented
6	C₂²×#1084	C₂²	1	1	yes	characteristic
7	C₂×#6196	C₂²	1	2	yes	—
8	C₂×#6188	—	2	2	no	—
9	C₂×#6188	C₂²	1	1	yes	characteristic
10	C₂×#6188	C₂²	1	1	yes	characteristic
11	#88589	—	2	2	no	—
12	#3757⋊C₃	—	2	2	no	—
13	#88449	C₂²	1	2	yes	—
14	#88448	—	2	2	no	—
15	#63964	—	6	2	no	—
16	C₂³×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	C₂×#1084	—	2	1	no	—
18	C₂×#1084	—	2	1	no	—
19	C₂×#1084	D₄	1	1	yes	characteristic
20	C₂×#1084	C₂³	1	1	yes	characteristic
21	C₂×(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
22	C₂×(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
23	#6196	—	2	2	no	—
24	#6196	—	2	2	no	—
25	#6188	—	2	1	no	—
26	#6188	D₄	1	2	yes	—
27	#5116	—	9	1	no	maximal
28	C₂²×C₃×Dic₃	—	6	2	no	—
29	C₂×#627	—	6	2	no	—
30	#3757	—	2	2	no	—
31	C₃×#217	—	6	2	no	—
32	#3724	—	6	2	no	—
33	Dic₃⋊Dic₃	—	6	2	no	—
34	Dic₃⋊Dic₃	—	6	2	no	—
35	C₂²×H(3,𝔽₃)	—	2	1	no	—
36	C₂²×H(3,𝔽₃)	—	2	1	no	—
37	C₂²×H(3,𝔽₃)	Q₁₆	1	2	yes	complemented
38	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
39	#1084	—	2	1	no	—
40	#1084	C₂×D₄	1	1	yes	characteristic
41	#1084	#41	1	1	yes	characteristic
42	(C₃⋊Dic₃)⋊C₃	—	4	2	no	—
43	(C₃⋊Dic₃)⋊C₃	—	4	2	no	—
44	C₂²×C₃×C₈	—	9	1	no	—
45	#924	—	9	2	no	—
46	C₂×Dic₁₂	—	9	1	no	—
47	#878	—	18	2	no	—
48	#818	—	9	1	no	—
49	#816	—	9	2	no	—
50	C₂³×C₃²	—	2	2	no	—
51	C₂×C₃⋊Dic₃	—	2	2	no	—
52	C₂×C₃⋊Dic₃	—	2	2	no	—
53	C₂×C₃×Dic₃	—	6	2	no	—
54	C₂×C₃×Dic₃	—	6	2	no	—
55	C₂×C₃×Dic₃	—	6	2	no	—
56	C₂×C₃×Dic₃	—	6	2	no	—
57	C₂×C₃×Dic₃	—	12	2	no	—
58	C₂×C₃×Dic₃	—	12	2	no	—
59	#627	—	12	2	no	—
60	#627	—	12	2	no	—
61	#467	—	27	1	no	Sylow
62	C₂×H(3,𝔽₃)	—	2	2	no	—
63	C₂×H(3,𝔽₃)	#121	1	1	yes	characteristic
64	C₂×H(3,𝔽₃)	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×H(3,𝔽₃)	#135	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	9	1	no	—
67	C₂²×Dic₃	—	6	2	no	—
68	C₂×Dic₆	—	9	2	no	—
69	C₂×Dic₆	—	18	2	no	—
70	C₂×C₃×C₈	—	9	1	no	—
71	C₂×C₃×C₈	—	9	1	no	—
72	C₂×C₃×C₈	—	18	2	no	—
73	C₃×C₂²⋊C₄	—	18	2	no	—
74	#217	—	6	2	no	—
75	#217	—	18	2	no	—
76	C₄⋊Dic₃	—	9	2	no	—
77	C₄⋊Dic₃	—	18	2	no	—
78	Dic₃⋊C₄	—	18	2	no	—
79	Dic₃⋊C₄	—	18	2	no	—
80	Dic₃⋊C₄	—	18	2	no	—
81	Dic₁₂	—	18	2	no	—
82	C₂²×C₃²	—	2	2	no	—
83	C₂²×C₃²	—	2	4	no	—
84	C₂²×C₃²	—	4	2	no	—
85	C₂²×C₃²	—	4	2	no	—
86	C₃⋊Dic₃	—	4	2	no	—
87	C₃⋊Dic₃	—	4	2	no	—
88	C₃×Dic₃	—	6	2	no	—
89	C₃×Dic₃	—	6	2	no	—
90	C₃×Dic₃	—	12	2	no	—
91	C₃×Dic₃	—	12	2	no	—
92	C₃×Dic₃	—	12	2	no	—
93	C₂×Q₁₆	—	27	1	no	—
94	C₂²×C₈	—	9	1	no	—
95	C₂²⋊Q₈	—	27	2	no	—
96	#107	—	27	1	no	—
97	#103	—	27	2	no	—
98	H(3,𝔽₃)	#467	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₃	—	6	2	no	—
100	C₂³×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
101	C₂×C₃×C₄	—	9	1	no	—
102	C₂×C₃×C₄	—	9	1	no	—
103	C₂×C₃×C₄	—	18	1	no	—
104	C₂×C₃×C₄	—	18	1	no	—
105	C₂×C₃×C₄	—	18	2	no	—
106	C₂×C₃×C₄	—	18	2	no	—
107	C₂×Dic₃	—	6	2	no	—
108	C₂×Dic₃	—	6	2	no	—
109	C₂×Dic₃	—	6	2	no	—
110	C₂×Dic₃	—	6	2	no	—
111	C₂×Dic₃	—	12	2	no	—
112	C₂×Dic₃	—	12	2	no	—
113	C₂×Dic₃	—	18	2	no	—
114	C₂×Dic₃	—	18	2	no	—
115	Dic₆	—	18	2	no	—
116	Dic₆	—	18	2	no	—
117	Dic₆	—	36	2	no	—
118	Dic₆	—	36	2	no	—
119	C₃×C₈	—	9	2	no	—
120	C₃×C₈	—	18	1	no	—
121	C₂×C₃²	—	2	2	no	—
122	C₂×C₃²	—	2	2	no	—
123	C₂×C₃²	—	2	2	no	—
124	C₂×C₃²	—	4	4	no	—
125	C₂×Q₈	—	27	2	no	—
126	C₂²×C₄	—	9	1	no	—
127	Q₁₆	—	54	2	no	—
128	C₂×C₈	—	9	1	no	—
129	C₂×C₈	—	9	1	no	—
130	C₂×C₈	—	18	2	no	—
131	C₄⋊C₄	—	27	2	no	—
132	C₄⋊C₄	—	54	2	no	—
133	C₂²⋊C₄	—	18	2	no	—
134	C₂²×C₃	—	2	1	no	—
135	C₂²×C₃	—	2	1	no	—
136	C₂²×C₃	—	6	4	no	—
137	C₂²×C₃	—	6	2	no	—
138	C₂²×C₃	—	12	2	no	—
139	C₂²×C₃	—	12	2	no	—
140	C₂²×C₃	#3776	1	2	yes	—
141	C₂²×C₃	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
142	C₃×C₄	—	9	1	no	—
143	C₃×C₄	—	9	1	no	—
144	C₃×C₄	—	18	1	no	—
145	C₃×C₄	—	36	2	no	—
146	C₃×C₄	—	36	2	no	—
147	Dic₃	—	6	2	no	—
148	Dic₃	—	6	2	no	—
149	Dic₃	—	12	2	no	—
150	Dic₃	—	12	2	no	—
151	Dic₃	—	12	2	no	—
152	Dic₃	—	36	2	no	—
153	Dic₃	—	36	2	no	—
154	C₃²	—	2	2	no	—
155	C₂³	#1088⋊C₂	1	1	yes	characteristic
156	Q₈	—	54	2	no	—
157	Q₈	—	54	2	no	—
158	C₂×C₄	—	9	1	no	—
159	C₂×C₄	—	9	1	no	—
160	C₂×C₄	—	18	1	no	—
161	C₂×C₄	—	18	1	no	—
162	C₂×C₄	—	18	2	no	—
163	C₂×C₄	—	18	2	no	—
164	C₈	—	9	2	no	—
165	C₈	—	18	1	no	—
166	C₂×C₃	—	2	2	no	—
167	C₂×C₃	—	6	2	no	—
168	C₂×C₃	—	6	2	no	—
169	C₂×C₃	—	6	2	no	—
170	C₂×C₃	—	12	4	no	—
171	C₂×C₃	#64237	1	1	yes	characteristic
172	C₂×C₃	C₂×#3776	1	1	yes	characteristic
173	C₂×C₃	#64242	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#88589	1	2	yes	complemented
177	C₂²	C₂×#1088⋊C₂	1	1	yes	characteristic
178	C₄	—	9	1	no	—
179	C₄	—	9	1	no	—
180	C₄	—	18	1	no	—
181	C₄	—	36	2	no	—
182	C₄	—	36	2	no	—
183	C₃	—	6	2	no	—
184	C₃	#10593527	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#11722310	1	1	yes	characteristic
187	C₂	C₂×#88589	1	1	yes	characteristic
188	C₂	#11722316	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#1088⋊C₂
commutator, cosocle	C₂×#1084	C₂³
Frattini, socle	C₂²×C₃	C₂×GO⁺(4,𝔽₂)
Fitting, prime radical	C₂³×H(3,𝔽₃)	D₄
hypercenter	C₂³	#1088⋊C₂
hypocenter	H(3,𝔽₃)	#467
LCS	G ▷ C₂×#1084 ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×#88589	C₂	semidirect
4. #11720704	C₂	semidirect
5. #11720702	C₂	semidirect
37. C₂²×H(3,𝔽₃)	Q₁₆	semidirect
98. H(3,𝔽₃)	#467	semidirect
176. C₂²	#88589	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#6188	2
C₂×#88589	2
#11721309	2
#11720704	2
#11720702	2
#5116	9