Finite Group Databasebeta

Subgroups — #72013

Order 384.

256 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 105 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	#5236	C₂	1	2	yes	maximal
3	C₂×#819	C₂	1	1	yes	maximal, characteristic, complemented
4	#5106	C₂	1	1	yes	maximal, characteristic, complemented
5	#4559	C₂	1	2	yes	maximal, complemented
6	#1814	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	C₃×#131	—	2	1	no	—
9	C₃×#131	—	2	1	no	—
10	C₃×#131	C₄	1	2	yes	complemented
11	C₃×#131	C₄	1	2	yes	complemented
12	C₂×C₃×M₁₆	—	2	1	no	—
13	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	—	2	1	no	—
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	#926	C₂²	1	2	yes	—
17	C₂×#208	C₂²	1	2	yes	—
18	#837	—	2	4	no	—
19	#819	—	2	1	no	—
20	#819	—	2	1	no	—
21	#818	—	2	1	no	—
22	#817	—	2	1	no	—
23	C₂×#131	D₃	1	1	yes	characteristic
24	C₂×#108	—	3	1	no	—
25	#427	—	3	1	no	—
26	#421	—	3	2	no	—
27	#344	—	3	2	no	—
28	C₃×#41	—	2	1	no	—
29	C₃×#41	—	2	1	no	—
30	C₃×#41	C₂×C₄	1	2	yes	—
31	C₃×#41	D₄	1	2	yes	—
32	C₂×C₃×Q₈	Q₈	1	1	yes	characteristic
33	C₂×C₃×D₄	—	2	1	no	—
34	C₂×C₃×D₄	Q₈	1	1	yes	characteristic
35	C₂²×C₃×C₄	—	2	1	no	—
36	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₃×M₁₆	—	2	1	no	—
38	C₃×M₁₆	—	2	2	no	—
39	C₃×M₁₆	—	2	2	no	—
40	C₃×M₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₈	—	2	1	no	—
42	C₂×C₃×C₈	—	2	2	no	—
43	C₂×C₃×C₈	—	2	2	no	—
44	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃×C₈	D₄	1	1	yes	characteristic
48	C₂×C₃×C₈	D₄	1	1	yes	characteristic
49	C₂×C₃×C₈	D₄	1	1	yes	characteristic
50	#217	—	2	2	no	—
51	C₄⋊Dic₃	—	2	2	no	—
52	C₄×Dic₃	—	2	2	no	—
53	#208	—	2	2	no	—
54	#208	—	2	2	no	—
55	C₂×C₃⋊C₈	—	2	2	no	—
56	C₂×#41	C₂×D₃	1	1	yes	characteristic
57	#131	—	2	1	no	—
58	#131	—	2	1	no	—
59	#131	Dic₃	1	2	yes	complemented
60	#131	Dic₃	1	2	yes	complemented
61	C₂×M₁₆	—	2	1	no	—
62	C₂×M₁₆	—	3	2	no	—
63	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
66	#117	—	3	2	no	—
67	#108	—	6	1	no	—
68	#108	—	6	1	no	—
69	#107	—	6	1	no	—
70	#106	—	6	1	no	—
71	#104	—	6	4	no	—
72	C₂³×C₃	—	2	1	no	—
73	C₂³×C₃	#41	1	1	yes	characteristic
74	C₃×Q₈	—	2	1	no	—
75	C₃×Q₈	C₄⋊C₄	1	2	yes	—
76	C₃×D₄	—	2	2	no	—
77	C₃×D₄	—	2	2	no	—
78	C₃×D₄	—	2	1	no	—
79	C₃×D₄	C₄⋊C₄	1	2	yes	—
80	C₂×C₃×C₄	—	2	1	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₃×C₄	#41	1	1	yes	characteristic
89	C₂×Dic₃	—	4	2	no	—
90	C₃×C₈	—	2	2	no	—
91	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₃⋊C₈	—	4	2	no	—
96	#41	—	2	1	no	—
97	#41	—	2	1	no	—
98	#41	D₁₂	1	2	yes	—
99	#41	C₂×Dic₃	1	2	yes	—
100	C₂×Q₈	Dic₆	1	1	yes	characteristic
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	Dic₆	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
105	M₁₆	—	2	1	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	6	2	no	—
109	M₁₆	—	6	2	no	—
110	M₁₆	C₂×Dic₃	1	2	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	6	2	no	—
115	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₈	#67	1	1	yes	characteristic
118	C₂×C₈	#67	1	1	yes	characteristic
119	C₂×C₈	#67	1	1	yes	characteristic
120	C₂×C₈	#67	1	1	yes	characteristic
121	C₄⋊C₄	—	6	2	no	—
122	C₂²⋊C₄	—	6	2	no	—
123	C₄²	—	6	2	no	—
124	C₂²×C₃	—	2	2	no	—
125	C₂²×C₃	—	4	1	no	—
126	C₂²×C₃	—	4	2	no	—
127	C₂²×C₃	#117	1	1	yes	characteristic
128	C₂²×C₃	#121	1	1	yes	characteristic
129	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
130	C₃×C₄	—	2	2	no	—
131	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₃×C₄	#121	1	1	yes	characteristic
134	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
135	Dic₃	—	8	2	no	—
136	C₂³	—	2	1	no	—
137	C₂³	#235	1	1	yes	characteristic
138	Q₈	—	2	1	no	—
139	Q₈	C₄⋊Dic₃	1	2	yes	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	1	no	—
143	D₄	C₄⋊Dic₃	1	2	yes	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	12	2	no	—
148	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
149	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
150	C₂×C₄	#235	1	1	yes	characteristic
151	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
152	C₂×C₄	C₂×#67	1	1	yes	characteristic
153	C₂×C₄	C₂×#67	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	12	2	no	—
156	C₈	#217	1	1	yes	characteristic
157	C₈	#217	1	1	yes	characteristic
158	C₈	#217	1	1	yes	characteristic
159	C₈	#217	1	1	yes	characteristic
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	—	4	2	no	—
164	C₂×C₃	#380	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	2	no	—
168	C₂²	#924	1	1	yes	characteristic
169	C₂²	#926	1	1	yes	characteristic
170	C₂²	#930	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	24	2	no	—
173	C₄	#924	1	1	yes	characteristic
174	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
175	C₄	#930	1	1	yes	characteristic
176	C₄	C₂×#217	1	1	yes	characteristic
177	C₃	#1814	1	1	yes	characteristic, complemented, Sylow
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	#5212	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#930
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#131	C₂
socle	C₂×C₃	#380
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#131	D₃
hypocenter	C₃	#1814
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ 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. #5106	C₂	semidirect
5. #4559	C₂	semidirect
10. C₃×#131	C₄	semidirect
11. C₃×#131	C₄	semidirect
59. #131	Dic₃	semidirect
60. #131	Dic₃	semidirect
177. C₃	#1814	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
#5236	2
C₂×#819	2
#5106	2
#4559	2
#1814	3