Finite Group Databasebeta

Subgroups — #71523

Order 384.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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	#5769	C₂	1	1	yes	maximal, characteristic
2	C₃×#510	C₂	1	1	yes	maximal, characteristic, complemented
3	#5192	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#828	C₂	1	1	yes	maximal, characteristic, complemented
5	#4567	C₂	1	1	yes	maximal, characteristic, complemented
6	#4516	C₂	1	1	yes	maximal, characteristic, complemented
7	#4515	C₂	1	1	yes	maximal, characteristic, complemented
8	#2194	—	3	1	no	maximal, Sylow
9	#1009	—	2	1	no	—
10	C₃×#135	C₂²	1	1	yes	characteristic
11	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
12	#949	—	2	1	no	—
13	#942	—	2	1	no	—
14	#919	—	2	1	no	—
15	#914	—	2	2	no	—
16	D₃×D₈	—	2	1	no	—
17	C₂×D₂₄	C₂²	1	1	yes	characteristic
18	#901	C₂²	1	1	yes	characteristic
19	C₃×Q₃₂	—	2	1	no	—
20	C₃×SD₃₂	—	2	1	no	—
21	C₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
22	#836	—	2	1	no	—
23	#828	—	2	2	no	—
24	#819	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
26	#576	—	3	1	no	—
27	#510	D₃	1	1	yes	characteristic, complemented
28	C₂×SD₃₂	—	3	1	no	—
29	#473	—	3	1	no	—
30	#361	—	3	1	no	—
31	#359	—	3	1	no	—
32	#350	—	3	1	no	—
33	C₃×#41	—	2	1	no	—
34	C₂×C₃×Q₈	—	2	1	no	—
35	#239	—	2	2	no	—
36	D₃×D₄	—	2	2	no	—
37	D₃×D₄	—	4	2	no	—
38	#235	—	4	1	no	—
39	#235	D₄	1	1	yes	characteristic
40	C₂×D₁₂	—	2	1	no	—
41	C₂×D₁₂	—	4	1	no	—
42	C₃×Q₁₆	—	2	1	no	—
43	C₃×Q₁₆	—	2	1	no	—
44	C₃×Q₁₆	D₄	1	1	yes	characteristic
45	C₃×SD₁₆	—	2	1	no	—
46	C₃×D₈	D₄	1	1	yes	characteristic
47	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
48	#216	—	4	1	no	—
49	#215	—	2	2	no	—
50	#215	—	4	1	no	—
51	#213	—	2	2	no	—
52	#208	—	2	1	no	—
53	#208	D₄	1	1	yes	characteristic
54	D₂₄	—	2	1	no	—
55	D₂₄	D₄	1	2	yes	—
56	ΓL(1,𝔽₂₅)	—	2	1	no	—
57	D₃×C₈	—	2	1	no	—
58	C₃×C₁₆	—	2	1	no	—
59	C₃⋊C₁₆	—	2	1	no	—
60	H(5,𝔽₂)	—	6	1	no	—
61	#137	—	6	1	no	—
62	AGL(1,ℤ/8)	—	6	1	no	—
63	AGL(1,ℤ/8)	—	6	2	no	—
64	#135	—	6	1	no	—
65	#135	C₂×D₃	1	1	yes	characteristic
66	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
67	C₂×D₈	—	3	1	no	—
68	C₂×D₈	—	6	1	no	—
69	#131	—	3	1	no	—
70	Q₃₂	—	2	1	no	—
71	SD₃₂	—	2	1	no	—
72	SD₃₂	—	6	2	no	—
73	M₃₂	C₂×D₃	1	1	yes	characteristic, complemented
74	C₂×C₁₆	—	3	1	no	—
75	#108	—	3	1	no	—
76	#101	—	6	1	no	—
77	C₂²×D₃	—	4	2	no	—
78	C₂²×D₃	—	4	1	no	—
79	C₃×Q₈	—	2	1	no	—
80	C₃×Q₈	—	4	1	no	—
81	C₃×Q₈	—	4	1	no	—
82	C₃×D₄	—	2	1	no	—
83	C₃×D₄	—	4	1	no	—
84	C₂×C₃×C₄	—	4	1	no	—
85	C₂×C₃×C₄	—	4	1	no	—
86	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	#67	—	2	1	no	—
88	#67	—	4	2	no	—
89	D₁₂	—	2	2	no	—
90	D₁₂	—	4	2	no	—
91	D₁₂	—	4	1	no	—
92	D₁₂	D₈	1	1	yes	characteristic
93	C₄×D₃	—	2	1	no	—
94	C₄×D₃	—	4	2	no	—
95	Dic₆	D₈	1	1	yes	characteristic
96	C₃×C₈	C₂×D₄	1	1	yes	characteristic
97	C₃×C₈	C₂×D₄	1	1	yes	characteristic
98	C₃⋊C₈	—	2	1	no	—
99	C₃⋊C₈	—	4	1	no	—
100	#41	—	2	1	no	—
101	#41	—	3	1	no	—
102	#41	—	6	2	no	—
103	#41	—	12	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×D₄	—	6	2	no	—
106	C₂×D₄	—	6	1	no	—
107	C₂×D₄	—	12	1	no	—
108	C₂×D₄	—	12	2	no	—
109	Q₁₆	—	2	1	no	—
110	Q₁₆	—	2	1	no	—
111	Q₁₆	—	12	1	no	—
112	Q₁₆	#67	1	1	yes	characteristic
113	SD₁₆	—	2	1	no	—
114	SD₁₆	—	6	2	no	—
115	SD₁₆	—	12	1	no	—
116	D₈	—	3	2	no	—
117	D₈	—	6	2	no	—
118	D₈	—	6	1	no	—
119	D₈	#67	1	1	yes	characteristic
120	M₁₆	—	3	1	no	—
121	M₁₆	—	6	1	no	—
122	M₁₆	—	6	1	no	—
123	C₂×C₈	—	6	1	no	—
124	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	6	1	no	—
127	C₂²×C₃	—	4	1	no	—
128	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×D₃	—	2	1	no	—
130	C₂×D₃	—	4	2	no	—
131	C₂×D₃	—	16	2	no	—
132	C₂×D₃	—	16	1	no	—
133	C₃×C₄	—	4	1	no	—
134	C₃×C₄	—	8	1	no	—
135	C₃×C₄	#120	1	1	yes	characteristic
136	C₃×C₄	C₂×D₈	1	1	yes	characteristic
137	Dic₃	—	2	1	no	—
138	C₂³	—	12	2	no	—
139	C₂³	—	12	1	no	—
140	Q₈	—	2	1	no	—
141	Q₈	—	3	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	4	1	no	—
144	D₄	—	2	1	no	—
145	D₄	—	3	1	no	—
146	D₄	—	4	1	no	—
147	D₄	—	6	1	no	—
148	D₄	—	6	2	no	—
149	D₄	—	12	2	no	—
150	D₄	—	12	2	no	—
151	D₄	—	12	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	6	1	no	—
155	C₂×C₄	—	12	2	no	—
156	C₂×C₄	D₃×D₄	1	1	yes	characteristic
157	C₈	—	6	1	no	—
158	C₈	—	12	1	no	—
159	C₈	D₃×D₄	1	1	yes	characteristic
160	C₈	C₂×#67	1	1	yes	characteristic
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	8	1	no	—
163	C₂×C₃	#447	1	1	yes	characteristic
164	D₃	—	4	1	no	—
165	D₃	—	8	2	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	6	1	no	—
168	C₂²	—	12	2	no	—
169	C₂²	—	48	1	no	—
170	C₂²	—	48	2	no	—
171	C₂²	#911	1	1	yes	characteristic
172	C₄	—	4	1	no	—
173	C₄	—	6	1	no	—
174	C₄	—	8	1	no	—
175	C₄	D₃×D₈	1	1	yes	characteristic
176	C₄	#937	1	1	yes	characteristic
177	C₃	#2194	1	1	yes	characteristic, complemented, Sylow
178	C₂	—	2	1	no	—
179	C₂	—	8	1	no	—
180	C₂	—	12	1	no	—
181	C₂	—	24	2	no	—
182	C₂	#5156	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#5156
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#510	C₂
socle	C₂×C₃	#447
prime radical	#5769	C₂
hypercenter	#510	D₃
hypocenter	C₃	#2194
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

Complements

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

N	H ≅ G/N	type
2. C₃×#510	C₂	semidirect
3. #5192	C₂	semidirect
4. C₂×#828	C₂	semidirect
5. #4567	C₂	semidirect
6. #4516	C₂	semidirect
7. #4515	C₂	semidirect
11. C₂×C₃×Q₁₆	C₂²	semidirect
21. C₃×M₃₂	C₂²	semidirect
24. #819	C₂²	semidirect
25. C₂×C₃⋊C₁₆	C₂²	semidirect
27. #510	D₃	semidirect
66. C₂×Q₁₆	C₂×D₃	semidirect
73. M₃₂	C₂×D₃	semidirect
177. C₃	#2194	semidirect

Maximal subgroups

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

subgroup	index
#5769	2
C₃×#510	2
#5192	2
C₂×#828	2
#4567	2
#4516	2
#4515	2
#2194	3