Finite Group Databasebeta

Subgroups — #69026

Order 384.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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	#5521	C₂	1	1	yes	maximal, characteristic
2	C₂×#823	C₂	1	1	yes	maximal, characteristic
3	C₂×#823	C₂	1	1	yes	maximal, characteristic
4	C₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
5	#4559	C₂	1	1	yes	maximal, characteristic, complemented
6	#4556	C₂	1	1	yes	maximal, characteristic, complemented
7	#4533	C₂	1	1	yes	maximal, characteristic, complemented
8	#1829	—	3	1	no	maximal, Sylow
9	C₂²×D₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₃×#117	C₂²	1	1	yes	characteristic
13	#926	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	1	yes	characteristic
15	C₂×#208	C₂²	1	1	yes	characteristic
16	#893	—	2	2	no	—
17	D₃×C₂²⋊C₄	C₄	1	4	yes	—
18	C₄×D₁₂	—	2	2	no	—
19	#823	—	2	2	no	—
20	#823	—	2	2	no	—
21	#518	—	3	1	no	—
22	C₂×#100	—	3	1	no	—
23	C₂×#100	—	3	1	no	—
24	#344	—	3	1	no	—
25	#344	—	3	1	no	—
26	#344	—	3	1	no	—
27	#344	D₃	1	1	yes	characteristic, complemented
28	C₂³×D₃	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×D₁₂	—	2	1	no	—
31	C₂×D₁₂	—	2	1	no	—
32	C₂×D₁₂	—	2	1	no	—
33	C₂×D₁₂	—	2	1	no	—
34	C₂×D₁₂	D₄	1	2	yes	—
35	C₂×D₁₂	D₄	1	2	yes	—
36	C₂×C₄×D₃	—	2	4	no	—
37	C₃×M₁₆	—	2	1	no	—
38	C₃×M₁₆	—	2	1	no	—
39	C₃×M₁₆	—	2	1	no	—
40	C₃×M₁₆	—	2	1	no	—
41	C₂×C₃×C₈	—	2	1	no	—
42	C₂×C₃×C₈	—	2	1	no	—
43	C₃×C₄⋊C₄	—	2	1	no	—
44	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
45	C₃×C₄²	—	2	1	no	—
46	#217	C₂×C₄	1	2	yes	—
47	(C₂×D₃)⋊C₄	—	2	4	no	—
48	C₄⋊Dic₃	—	2	1	no	—
49	C₄×Dic₃	—	2	1	no	—
50	#208	—	2	1	no	—
51	#208	—	2	1	no	—
52	#208	—	2	1	no	—
53	#208	—	2	1	no	—
54	C₂×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃⋊C₈	—	2	1	no	—
56	C₂²×D₄	—	3	1	no	—
57	C₂×M₁₆	—	3	1	no	—
58	C₂×M₁₆	—	3	1	no	—
59	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
60	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
61	C₄×D₄	—	6	2	no	—
62	C₄×D₄	—	6	2	no	—
63	#117	—	3	1	no	—
64	#117	C₂×D₃	1	1	yes	characteristic
65	C₂×C₂²⋊C₄	—	3	4	no	—
66	#100	—	6	2	no	—
67	#100	—	6	2	no	—
68	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₃	—	2	2	no	—
70	C₂²×D₃	—	2	2	no	—
71	C₂²×D₃	—	8	2	no	—
72	C₂²×D₃	C₂²⋊C₄	1	4	yes	—
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄	#41	1	1	yes	characteristic
77	C₂×C₃×C₄	#41	1	1	yes	characteristic
78	C₂×C₃×C₄	#41	1	1	yes	characteristic
79	C₂×C₃×C₄	#41	1	1	yes	characteristic
80	C₂×Dic₃	—	2	2	no	—
81	D₁₂	—	2	2	no	—
82	D₁₂	—	2	2	no	—
83	D₁₂	—	2	2	no	—
84	D₁₂	—	2	2	no	—
85	C₄×D₃	—	8	4	no	—
86	C₃×C₈	—	4	1	no	—
87	C₃×C₈	—	4	1	no	—
88	C₃⋊C₈	—	4	1	no	—
89	C₃⋊C₈	—	4	1	no	—
90	C₂⁴	—	3	2	no	—
91	C₂×D₄	—	3	2	no	—
92	C₂×D₄	—	3	2	no	—
93	C₂×D₄	—	6	1	no	—
94	C₂×D₄	—	6	1	no	—
95	C₂×D₄	—	6	1	no	—
96	C₂×D₄	—	6	1	no	—
97	C₂²×C₄	—	6	4	no	—
98	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	6	1	no	—
104	M₁₆	—	6	1	no	—
105	M₁₆	—	6	1	no	—
106	M₁₆	—	6	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	6	1	no	—
110	C₂×C₈	—	6	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	6	1	no	—
113	C₂²⋊C₄	—	3	2	no	—
114	C₂²⋊C₄	—	6	4	no	—
115	C₂²⋊C₄	C₄×D₃	1	2	yes	—
116	C₄²	—	2	1	no	—
117	C₄²	—	6	1	no	—
118	C₂²×C₃	—	4	1	no	—
119	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₃	#117	1	1	yes	characteristic
121	C₂²×C₃	#117	1	1	yes	characteristic
122	C₂×D₃	—	2	4	no	—
123	C₂×D₃	—	4	4	no	—
124	C₂×D₃	—	8	2	no	—
125	C₂×D₃	—	8	2	no	—
126	C₃×C₄	—	4	2	no	—
127	C₃×C₄	#123	1	1	yes	characteristic
128	C₃×C₄	#123	1	1	yes	characteristic
129	C₃×C₄	#123	1	1	yes	characteristic
130	C₃×C₄	#123	1	1	yes	characteristic
131	Dic₃	—	4	2	no	—
132	C₂³	—	3	4	no	—
133	C₂³	—	6	2	no	—
134	C₂³	—	6	2	no	—
135	C₂³	—	24	2	no	—
136	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
137	D₄	—	6	2	no	—
138	D₄	—	6	2	no	—
139	D₄	—	6	2	no	—
140	D₄	—	6	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	6	2	no	—
143	C₂×C₄	—	24	4	no	—
144	C₂×C₄	#235	1	1	yes	characteristic
145	C₂×C₄	#235	1	1	yes	characteristic
146	C₂×C₄	D₃×D₄	1	1	yes	characteristic
147	C₂×C₄	D₃×D₄	1	1	yes	characteristic
148	C₂×C₄	#237	1	1	yes	characteristic
149	C₂×C₄	#239	1	1	yes	characteristic
150	C₈	—	4	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	12	1	no	—
153	C₈	—	12	1	no	—
154	C₂×C₃	—	2	1	no	—
155	C₂×C₃	—	2	1	no	—
156	C₂×C₃	—	2	1	no	—
157	C₂×C₃	#381	1	1	yes	characteristic
158	D₃	—	4	4	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	6	4	no	—
161	C₂²	—	12	4	no	—
162	C₂²	—	24	2	no	—
163	C₂²	—	24	2	no	—
164	C₂²	#872	1	1	yes	characteristic
165	C₂²	D₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²	#892	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	12	2	no	—
169	C₄	#883	1	1	yes	characteristic
170	C₄	#883	1	1	yes	characteristic
171	C₄	#894	1	1	yes	characteristic
172	C₄	#894	1	1	yes	characteristic
173	C₃	#1829	1	1	yes	characteristic, complemented, Sylow
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	12	4	no	—
178	C₂	#4664	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4664
commutator	C₂³×C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#344	C₂
socle	C₂×C₃	#381
prime radical	C₂²×D₁₂	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#344	D₃
hypocenter	C₃	#1829
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
4. C₃×#344	C₂	semidirect
5. #4559	C₂	semidirect
6. #4556	C₂	semidirect
7. #4533	C₂	semidirect
27. #344	D₃	semidirect
173. C₃	#1829	semidirect

Maximal subgroups

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

subgroup	index
#5521	2
C₂×#823	2
C₃×#344	2
#4559	2
#4556	2
#4533	2
#1829	3