Finite Group Databasebeta

Subgroups — #69012

Order 384.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×#825	C₂	1	1	yes	maximal, characteristic
2	#5132	C₂	1	1	yes	maximal, characteristic
3	#5124	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#5005	C₂	1	1	yes	maximal, characteristic
6	C₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	#4532	C₂	1	1	yes	maximal, characteristic, complemented
8	#1932	—	3	1	no	maximal, Sylow
9	C₂×#235	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	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	1	yes	characteristic
15	C₂×#208	C₂²	1	1	yes	characteristic
16	#825	—	2	1	no	—
17	#825	—	2	1	no	—
18	#824	C₄	1	2	yes	complemented
19	#823	C₄	1	2	yes	complemented
20	#822	—	2	2	no	—
21	#815	—	2	1	no	—
22	C₈×Dic₃	—	2	1	no	—
23	#805	—	2	2	no	—
24	#430	—	3	1	no	—
25	#421	—	3	1	no	—
26	C₂×#104	—	3	1	no	—
27	#413	—	3	1	no	—
28	C₄×M₁₆	—	3	1	no	—
29	#344	D₃	1	1	yes	characteristic, complemented
30	#339	—	3	1	no	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×#67	—	2	1	no	—
33	C₂²×Dic₃	—	2	1	no	—
34	#235	—	2	1	no	—
35	#235	—	2	1	no	—
36	#235	—	2	1	no	—
37	#235	—	2	1	no	—
38	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₃	—	2	1	no	—
40	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
41	C₃×M₁₆	—	2	1	no	—
42	C₃×M₁₆	—	2	1	no	—
43	C₃×M₁₆	—	2	1	no	—
44	C₃×M₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₈	—	2	1	no	—
46	C₂×C₃×C₈	—	2	1	no	—
47	C₃×C₄⋊C₄	—	2	1	no	—
48	C₃×C₂²⋊C₄	—	2	1	no	—
49	C₃×C₄²	—	2	1	no	—
50	C₄×Dic₃	—	2	1	no	—
51	C₄×Dic₃	D₄	1	1	yes	characteristic
52	C₄×Dic₃	D₄	1	1	yes	characteristic
53	#208	—	2	1	no	—
54	#208	—	2	1	no	—
55	#208	C₂×C₄	1	2	yes	—
56	#208	D₄	1	2	yes	—
57	C₂×C₃⋊C₈	—	2	1	no	—
58	C₂×C₃⋊C₈	—	2	1	no	—
59	C₂×#41	—	3	1	no	—
60	C₂×M₁₆	—	3	1	no	—
61	C₂×M₁₆	—	3	1	no	—
62	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
63	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
64	#117	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄²	—	3	1	no	—
66	#108	—	6	2	no	—
67	#104	—	6	2	no	—
68	#104	—	6	1	no	—
69	#104	—	6	1	no	—
70	#101	—	3	2	no	—
71	#100	—	3	2	no	—
72	#97	—	6	1	no	—
73	C₄×C₈	—	6	1	no	—
74	C₂³×C₃	#41	1	1	yes	characteristic
75	C₂²×D₃	—	2	1	no	—
76	C₂×C₃×C₄	—	4	1	no	—
77	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	#41	1	1	yes	characteristic
81	C₂×C₃×C₄	#41	1	1	yes	characteristic
82	C₂×C₃×C₄	#41	1	1	yes	characteristic
83	#67	—	4	1	no	—
84	#67	—	4	1	no	—
85	C₂×Dic₃	—	2	1	no	—
86	C₂×Dic₃	—	2	1	no	—
87	C₂×Dic₃	—	2	1	no	—
88	C₂×Dic₃	—	4	2	no	—
89	D₁₂	—	2	1	no	—
90	D₁₂	—	2	1	no	—
91	C₄×D₃	—	4	1	no	—
92	C₄×D₃	—	4	1	no	—
93	Dic₆	—	2	1	no	—
94	Dic₆	—	2	1	no	—
95	C₃×C₈	—	2	2	no	—
96	C₃×C₈	—	4	1	no	—
97	C₃⋊C₈	—	2	2	no	—
98	C₃⋊C₈	—	2	2	no	—
99	#41	—	6	1	no	—
100	#41	—	6	1	no	—
101	#41	—	6	1	no	—
102	#41	—	6	1	no	—
103	C₂×Q₈	—	3	1	no	—
104	C₂×D₄	—	3	1	no	—
105	C₂×D₄	—	6	1	no	—
106	C₂²×C₄	—	6	1	no	—
107	C₂²×C₄	—	6	1	no	—
108	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
109	M₁₆	—	2	1	no	—
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	1	no	—
112	M₁₆	—	3	2	no	—
113	M₁₆	—	3	2	no	—
114	M₁₆	—	6	1	no	—
115	M₁₆	—	6	1	no	—
116	M₁₆	C₄×D₃	1	2	yes	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	6	1	no	—
120	C₂×C₈	—	6	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	3	1	no	—
125	C₄²	—	3	1	no	—
126	C₄²	—	6	1	no	—
127	C₂²×C₃	—	4	1	no	—
128	C₂²×C₃	#117	1	1	yes	characteristic
129	C₂²×C₃	#123	1	1	yes	characteristic
130	C₂²×C₃	#124	1	1	yes	characteristic
131	C₂×D₃	—	4	1	no	—
132	C₂×D₃	—	8	1	no	—
133	C₃×C₄	—	8	1	no	—
134	C₃×C₄	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄	C₄×D₄	1	1	yes	characteristic
136	C₃×C₄	#121	1	1	yes	characteristic
137	C₃×C₄	#126	1	1	yes	characteristic
138	Dic₃	—	2	2	no	—
139	Dic₃	—	4	1	no	—
140	Dic₃	—	4	1	no	—
141	C₂³	—	6	1	no	—
142	C₂³	#235	1	1	yes	characteristic
143	Q₈	—	6	1	no	—
144	Q₈	—	6	1	no	—
145	D₄	—	6	1	no	—
146	D₄	—	6	1	no	—
147	D₄	—	12	1	no	—
148	D₄	—	12	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	6	1	no	—
151	C₂×C₄	—	6	1	no	—
152	C₂×C₄	—	6	1	no	—
153	C₂×C₄	—	12	2	no	—
154	C₂×C₄	—	12	1	no	—
155	C₂×C₄	—	12	1	no	—
156	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₄	#235	1	1	yes	characteristic
158	C₂×C₄	D₃×D₄	1	1	yes	characteristic
159	C₂×C₄	D₃×D₄	1	1	yes	characteristic
160	C₂×C₄	#237	1	1	yes	characteristic
161	C₂×C₄	#239	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	4	1	no	—
164	C₈	—	6	2	no	—
165	C₈	—	6	2	no	—
166	C₂×C₃	—	2	1	no	—
167	C₂×C₃	—	2	1	no	—
168	C₂×C₃	—	2	1	no	—
169	C₂×C₃	#388	1	1	yes	characteristic
170	D₃	—	8	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	12	1	no	—
173	C₂²	—	24	1	no	—
174	C₂²	#872	1	1	yes	characteristic
175	C₂²	#885	1	1	yes	characteristic
176	C₂²	#894	1	1	yes	characteristic
177	C₄	—	6	2	no	—
178	C₄	—	8	1	no	—
179	C₄	—	12	1	no	—
180	C₄	—	12	1	no	—
181	C₄	#67⋊C₄	1	1	yes	characteristic
182	C₄	#884	1	1	yes	characteristic
183	C₄	#893	1	1	yes	characteristic
184	C₄	#898	1	1	yes	characteristic
185	C₃	#1932	1	1	yes	characteristic, complemented, Sylow
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	—	24	1	no	—
190	C₂	#4669	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#884
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#344	C₂
socle	C₂×C₃	#388
prime radical	C₂×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#344	D₃
hypocenter	C₃	#1932
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
3. #5124	C₂	semidirect
4. Dic₃×M₁₆	C₂	semidirect
6. C₃×#344	C₂	semidirect
7. #4532	C₂	semidirect
18. #824	C₄	semidirect
19. #823	C₄	semidirect
29. #344	D₃	semidirect
185. C₃	#1932	semidirect

Maximal subgroups

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

subgroup	index
C₂×#825	2
#5132	2
#5124	2
Dic₃×M₁₆	2
#5005	2
C₃×#344	2
#4532	2
#1932	3