Finite Group Databasebeta

Subgroups — #72016

Order 384.

264 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	#5232	C₂	1	2	yes	maximal
3	C₂×#819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#818	C₂	1	1	yes	maximal, characteristic, complemented
5	#4554	C₂	1	2	yes	maximal, complemented
6	#1816	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	C₃×#131	—	2	2	no	—
9	C₃×#131	C₄	1	4	yes	complemented
10	C₂×C₃×M₁₆	—	2	1	no	—
11	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₈	—	2	1	no	—
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×#208	C₂²	1	2	yes	—
16	#835	—	2	2	no	—
17	#832	—	2	2	no	—
18	#819	—	2	2	no	—
19	#818	—	2	1	no	—
20	#818	—	2	1	no	—
21	C₂×#131	D₃	1	1	yes	characteristic
22	C₂×#108	—	3	1	no	—
23	C₂×#107	—	3	1	no	—
24	#417	—	3	2	no	—
25	#343	—	3	2	no	—
26	C₃×#41	—	2	1	no	—
27	C₃×#41	—	2	1	no	—
28	C₃×#41	C₂×C₄	1	2	yes	—
29	C₃×#41	Q₈	1	2	yes	—
30	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×D₄	—	2	1	no	—
32	C₂×C₃×D₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	—	2	1	no	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	—	2	2	no	—
36	C₃×M₁₆	—	2	1	no	—
37	C₃×M₁₆	—	2	4	no	—
38	C₃×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₈	—	2	1	no	—
40	C₂×C₃×C₈	—	2	4	no	—
41	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₈	D₄	1	1	yes	characteristic
43	C₂×C₃×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃×C₈	D₄	1	2	yes	—
45	C₄⋊Dic₃	—	2	2	no	—
46	C₄⋊Dic₃	—	2	2	no	—
47	#208	—	2	2	no	—
48	#208	—	2	2	no	—
49	C₂×C₃⋊C₈	—	2	2	no	—
50	C₂×#41	C₂×D₃	1	1	yes	characteristic
51	#131	—	2	2	no	—
52	#131	Dic₃	1	4	yes	complemented
53	C₂×M₁₆	—	2	1	no	—
54	C₂×M₁₆	—	3	2	no	—
55	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₄⋊C₄	—	3	2	no	—
59	#108	—	6	2	no	—
60	#107	—	6	1	no	—
61	#107	—	6	1	no	—
62	#103	—	6	2	no	—
63	#102	—	6	2	no	—
64	C₂³×C₃	—	2	1	no	—
65	C₂³×C₃	#41	1	1	yes	characteristic
66	C₃×Q₈	—	2	1	no	—
67	C₃×Q₈	C₄⋊C₄	1	2	yes	—
68	C₃×D₄	—	2	2	no	—
69	C₃×D₄	—	2	2	no	—
70	C₃×D₄	—	2	1	no	—
71	C₃×D₄	C₄⋊C₄	1	2	yes	—
72	C₂×C₃×C₄	—	2	1	no	—
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₃×C₄	#41	1	1	yes	characteristic
81	C₂×Dic₃	—	4	2	no	—
82	C₂×Dic₃	—	8	2	no	—
83	C₃×C₈	—	2	2	no	—
84	C₃×C₈	C₂²⋊C₄	1	2	yes	—
85	C₃×C₈	C₂²⋊C₄	1	2	yes	—
86	C₃⋊C₈	—	4	2	no	—
87	#41	—	2	1	no	—
88	#41	—	2	1	no	—
89	#41	Dic₆	1	2	yes	—
90	#41	C₂×Dic₃	1	2	yes	—
91	C₂×Q₈	D₁₂	1	1	yes	characteristic
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	D₁₂	1	1	yes	characteristic
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	6	2	no	—
96	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	4	no	—
99	M₁₆	—	6	2	no	—
100	M₁₆	—	6	2	no	—
101	M₁₆	C₂×Dic₃	1	2	yes	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	6	2	no	—
105	C₂×C₈	C₂×Dic₃	1	2	yes	—
106	C₂×C₈	#67	1	1	yes	characteristic
107	C₂×C₈	#67	1	1	yes	characteristic
108	C₂×C₈	#67	1	2	yes	—
109	C₄⋊C₄	—	6	2	no	—
110	C₄⋊C₄	—	6	2	no	—
111	C₂²×C₃	—	2	1	no	—
112	C₂²×C₃	—	2	1	no	—
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	—	4	1	no	—
115	C₂²×C₃	—	4	1	no	—
116	C₂²×C₃	#117	1	1	yes	characteristic
117	C₂²×C₃	#121	1	1	yes	characteristic
118	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₃×C₄	#121	1	1	yes	characteristic
123	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
124	Dic₃	—	8	2	no	—
125	C₂³	—	2	1	no	—
126	C₂³	#235	1	1	yes	characteristic
127	Q₈	—	2	1	no	—
128	Q₈	C₄⋊Dic₃	1	2	yes	—
129	D₄	—	2	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	2	1	no	—
132	D₄	C₄⋊Dic₃	1	2	yes	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	12	2	no	—
137	C₂×C₄	—	24	2	no	—
138	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
139	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
140	C₂×C₄	#235	1	1	yes	characteristic
141	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
142	C₂×C₄	C₂×#67	1	1	yes	characteristic
143	C₂×C₄	C₂×#67	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	12	2	no	—
146	C₈	#217	1	2	yes	—
147	C₈	#217	1	2	yes	—
148	C₂×C₃	—	2	1	no	—
149	C₂×C₃	—	2	1	no	—
150	C₂×C₃	—	4	2	no	—
151	C₂×C₃	#380	1	1	yes	characteristic
152	C₂×C₃	#472	1	1	yes	characteristic
153	C₂×C₃	#473	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	#924	1	1	yes	characteristic
160	C₂²	#926	1	1	yes	characteristic
161	C₂²	#930	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	24	2	no	—
164	C₄	#924	1	1	yes	characteristic
165	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
166	C₄	#930	1	1	yes	characteristic
167	C₄	C₂×#217	1	1	yes	characteristic
168	C₃	#1816	1	1	yes	characteristic, complemented, Sylow
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	4	2	no	—
172	C₂	#5142	1	1	yes	characteristic
173	C₂	#5143	1	1	yes	characteristic
174	C₂	#5212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#930
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#131	C₂
socle	C₂²×C₃	#121
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#131	D₃
hypocenter	C₃	#1816
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
3. C₂×#819	C₂	semidirect
4. C₂×#818	C₂	semidirect
5. #4554	C₂	semidirect
9. C₃×#131	C₄	semidirect
52. #131	Dic₃	semidirect
168. C₃	#1816	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#131	2
#5232	2
C₂×#819	2
C₂×#818	2
#4554	2
#1816	3