Finite Group Databasebeta

Subgroups — #72122

Order 384.

220 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 53 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	C₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	#5193	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#826	C₂	1	1	yes	maximal, characteristic, complemented
4	#5142	C₂	1	1	yes	maximal, characteristic, complemented
5	#4567	C₂	1	1	yes	maximal, characteristic, complemented
6	#4563	C₂	1	1	yes	maximal, characteristic, complemented
7	#4561	C₂	1	1	yes	maximal, characteristic, complemented
8	#2192	—	3	1	no	maximal, Sylow
9	C₃×H(5,𝔽₂)	—	2	1	no	—
10	C₃×AGL(1,ℤ/8)	—	2	2	no	—
11	C₃×AGL(1,ℤ/8)	—	2	1	no	—
12	C₃×#135	—	2	1	no	—
13	C₃×#135	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₈	—	2	1	no	—
15	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
16	C₃×#131	C₂²	1	1	yes	characteristic
17	#949	—	2	1	no	—
18	C₂×D₂₄	C₂²	1	1	yes	characteristic, complemented
19	#828	—	2	1	no	—
20	#826	—	2	2	no	—
21	#826	—	2	1	no	—
22	#823	—	2	1	no	—
23	#819	C₂²	1	1	yes	characteristic, complemented
24	#812	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
26	#576	D₃	1	1	yes	characteristic, complemented
27	#509	—	3	1	no	—
28	C₂×D₁₆	—	3	1	no	—
29	#472	—	3	1	no	—
30	#361	—	3	1	no	—
31	#359	—	3	1	no	—
32	#350	—	3	1	no	—
33	C₃×#41	—	2	2	no	—
34	C₃×#41	—	2	1	no	—
35	C₃×#41	—	4	1	no	—
36	C₃×#41	D₄	1	1	yes	characteristic
37	C₂×C₃×D₄	—	2	2	no	—
38	C₂×C₃×D₄	—	2	1	no	—
39	C₂×C₃×D₄	—	4	2	no	—
40	C₂×C₃×D₄	—	4	1	no	—
41	C₂×D₁₂	—	2	1	no	—
42	C₃×Q₁₆	D₄	1	1	yes	characteristic
43	C₃×SD₁₆	—	2	2	no	—
44	C₃×SD₁₆	—	2	1	no	—
45	C₃×D₈	—	2	2	no	—
46	C₃×D₈	—	2	1	no	—
47	C₃×D₈	D₄	1	1	yes	characteristic
48	C₃×D₈	D₄	1	2	yes	—
49	C₃×M₁₆	—	2	1	no	—
50	C₃×M₁₆	D₄	1	1	yes	characteristic
51	C₂×C₃×C₈	—	2	1	no	—
52	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
53	#215	—	4	1	no	—
54	#213	—	4	1	no	—
55	#208	—	2	1	no	—
56	D₂₄	—	2	1	no	—
57	D₂₄	—	2	1	no	—
58	C₃⋊C₁₆	—	2	1	no	—
59	C₃⋊C₁₆	—	2	1	no	—
60	H(5,𝔽₂)	—	2	1	no	—
61	AGL(1,ℤ/8)	—	2	2	no	—
62	AGL(1,ℤ/8)	—	2	1	no	—
63	AGL(1,ℤ/8)	—	6	1	no	—
64	#135	—	2	1	no	—
65	#135	C₂×D₃	1	1	yes	characteristic
66	C₂×D₈	—	2	1	no	—
67	C₂×D₈	—	3	1	no	—
68	C₂×D₈	C₂×D₃	1	1	yes	characteristic
69	#131	C₂×D₃	1	1	yes	characteristic
70	SD₃₂	—	6	1	no	—
71	D₁₆	—	6	2	no	—
72	D₁₆	—	6	1	no	—
73	M₃₂	—	3	1	no	—
74	C₂×C₁₆	—	3	1	no	—
75	#108	—	3	1	no	—
76	#100	—	6	1	no	—
77	C₂³×C₃	—	4	2	no	—
78	C₂³×C₃	—	4	1	no	—
79	C₂²×D₃	—	4	1	no	—
80	C₃×Q₈	—	2	1	no	—
81	C₃×Q₈	D₈	1	1	yes	characteristic
82	C₃×D₄	—	2	1	no	—
83	C₃×D₄	—	2	1	no	—
84	C₃×D₄	—	2	2	no	—
85	C₃×D₄	—	4	2	no	—
86	C₃×D₄	—	4	2	no	—
87	C₃×D₄	—	4	1	no	—
88	C₃×D₄	—	4	1	no	—
89	C₃×D₄	D₈	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	4	2	no	—
92	C₂×C₃×C₄	—	4	1	no	—
93	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
94	D₁₂	—	4	1	no	—
95	D₁₂	—	4	1	no	—
96	C₃×C₈	—	2	1	no	—
97	C₃×C₈	C₂×D₄	1	1	yes	characteristic
98	C₃×C₈	C₂×D₄	1	1	yes	characteristic
99	C₃⋊C₈	—	4	1	no	—
100	#41	—	2	2	no	—
101	#41	—	2	1	no	—
102	#41	—	4	1	no	—
103	#41	#67	1	1	yes	characteristic
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	—	4	1	no	—
108	C₂×D₄	—	6	1	no	—
109	Q₁₆	#67	1	1	yes	characteristic
110	SD₁₆	—	2	2	no	—
111	SD₁₆	—	2	1	no	—
112	SD₁₆	—	12	1	no	—
113	D₈	—	2	2	no	—
114	D₈	—	2	1	no	—
115	D₈	—	6	1	no	—
116	D₈	—	6	1	no	—
117	D₈	—	12	1	no	—
118	D₈	#67	1	1	yes	characteristic
119	D₈	#67	1	2	yes	—
120	M₁₆	—	2	1	no	—
121	M₁₆	—	6	1	no	—
122	M₁₆	#67	1	1	yes	characteristic
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₁₆	—	6	1	no	—
126	C₁₆	—	6	1	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	—	4	2	no	—
129	C₂²×C₃	—	4	1	no	—
130	C₂²×C₃	—	16	1	no	—
131	C₂²×C₃	—	16	2	no	—
132	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×D₃	—	8	1	no	—
134	C₂×D₃	—	16	1	no	—
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	—	4	1	no	—
137	C₃×C₄	#120	1	1	yes	characteristic
138	C₃×C₄	C₂×D₈	1	1	yes	characteristic
139	C₂³	—	4	2	no	—
140	C₂³	—	4	1	no	—
141	C₂³	—	12	1	no	—
142	Q₈	—	2	1	no	—
143	Q₈	#213	1	1	yes	characteristic
144	D₄	—	2	1	no	—
145	D₄	—	2	1	no	—
146	D₄	—	2	2	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	12	1	no	—
152	D₄	—	12	1	no	—
153	D₄	#213	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₂×#67	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	12	1	no	—
160	C₈	C₂×#67	1	1	yes	characteristic
161	C₈	C₂×#67	1	1	yes	characteristic
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	—	4	1	no	—
164	C₂×C₃	—	8	2	no	—
165	C₂×C₃	—	8	1	no	—
166	C₂×C₃	#447	1	1	yes	characteristic
167	D₃	—	16	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	4	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	16	1	no	—
172	C₂²	—	16	2	no	—
173	C₂²	—	24	1	no	—
174	C₂²	—	48	1	no	—
175	C₂²	#932	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	4	1	no	—
178	C₄	C₂×#213	1	1	yes	characteristic
179	C₄	#953	1	1	yes	characteristic
180	C₃	#2192	1	1	yes	characteristic, complemented, Sylow
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	8	2	no	—
184	C₂	—	8	1	no	—
185	C₂	—	48	1	no	—
186	C₂	#5217	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#5217
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#576	C₂
socle	C₂×C₃	#447
hypercenter	#576	D₃
hypocenter	C₃	#2192
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #576

Complements

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

N	H ≅ G/N	type
1. C₃×#576	C₂	semidirect
2. #5193	C₂	semidirect
3. C₂×#826	C₂	semidirect
4. #5142	C₂	semidirect
5. #4567	C₂	semidirect
6. #4563	C₂	semidirect
7. #4561	C₂	semidirect
18. C₂×D₂₄	C₂²	semidirect
23. #819	C₂²	semidirect
24. #812	C₂²	semidirect
25. C₂×C₃⋊C₁₆	C₂²	semidirect
26. #576	D₃	semidirect
180. C₃	#2192	semidirect

Maximal subgroups

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

subgroup	index
C₃×#576	2
#5193	2
C₂×#826	2
#5142	2
#4567	2
#4563	2
#4561	2
#2192	3