Finite Group Databasebeta

Subgroups — #72086

Order 384.

284 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 93 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×#135	C₂	1	1	yes	maximal, characteristic
2	#5240	C₂	1	2	yes	maximal, complemented
3	#5236	C₂	1	2	yes	maximal, complemented
4	#5106	C₂	1	1	yes	maximal, characteristic, complemented
5	#5101	C₂	1	1	yes	maximal, characteristic, complemented
6	#1976	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	2	yes	—
8	C₃×#135	—	2	1	no	—
9	C₃×#135	—	2	1	no	—
10	C₃×#135	—	2	1	no	—
11	C₃×#135	—	2	1	no	—
12	C₂×C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
13	C₂×C₃×SD₁₆	C₄	1	2	yes	complemented
14	C₂×C₃×D₈	C₄	1	1	yes	characteristic, complemented
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	#926	C₂²	1	2	yes	—
17	#926	C₂²	1	1	yes	characteristic
18	C₂×#208	C₂²	1	1	yes	characteristic
19	#837	—	2	4	no	—
20	#834	—	2	4	no	—
21	#818	—	2	1	no	—
22	#817	—	2	1	no	—
23	#815	—	2	1	no	—
24	C₈×Dic₃	—	2	1	no	—
25	C₂×#135	D₃	1	1	yes	characteristic
26	#427	—	3	1	no	—
27	#421	—	3	2	no	—
28	#410	—	3	2	no	—
29	#405	—	3	1	no	—
30	C₃×#41	—	2	2	no	—
31	C₃×#41	—	2	2	no	—
32	C₃×#41	—	2	2	no	—
33	C₃×#41	—	2	2	no	—
34	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×D₄	—	2	2	no	—
36	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
37	C₂²×C₃×C₄	—	2	2	no	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₃×Q₁₆	—	2	1	no	—
40	C₃×Q₁₆	—	2	1	no	—
41	C₃×SD₁₆	—	2	2	no	—
42	C₃×SD₁₆	—	2	2	no	—
43	C₃×D₈	—	2	1	no	—
44	C₃×D₈	—	2	1	no	—
45	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₈	C₂×C₄	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	C₂×C₃×C₈	D₄	1	1	yes	characteristic
51	#217	—	2	2	no	—
52	#217	D₄	1	2	yes	—
53	C₄⋊Dic₃	—	2	1	no	—
54	C₄⋊Dic₃	—	2	2	no	—
55	C₄×Dic₃	—	2	1	no	—
56	C₄×Dic₃	—	2	2	no	—
57	#208	—	2	1	no	—
58	#208	D₄	1	2	yes	—
59	C₂×C₃⋊C₈	—	2	1	no	—
60	C₂×#41	C₂×D₃	1	2	yes	—
61	#135	—	2	1	no	—
62	#135	—	2	1	no	—
63	#135	—	2	1	no	—
64	#135	—	2	1	no	—
65	C₂×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
66	C₂×SD₁₆	Dic₃	1	2	yes	complemented
67	C₂×D₈	Dic₃	1	1	yes	characteristic, complemented
68	C₂×M₁₆	—	3	1	no	—
69	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
70	#117	—	3	1	no	—
71	#117	—	3	2	no	—
72	#107	—	6	1	no	—
73	#106	—	6	1	no	—
74	#104	—	6	4	no	—
75	#99	—	6	4	no	—
76	#97	—	6	1	no	—
77	C₄×C₈	—	6	1	no	—
78	C₂³×C₃	—	2	2	no	—
79	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
80	C₃×Q₈	—	2	2	no	—
81	C₃×Q₈	—	2	2	no	—
82	C₃×D₄	—	2	2	no	—
83	C₃×D₄	—	2	2	no	—
84	C₃×D₄	—	4	2	no	—
85	C₃×D₄	—	4	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	4	2	no	—
88	C₂×C₃×C₄	—	4	2	no	—
89	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄	#41	1	1	yes	characteristic
94	C₂×C₃×C₄	#41	1	1	yes	characteristic
95	C₂×Dic₃	—	2	2	no	—
96	C₂×Dic₃	—	4	2	no	—
97	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₃⋊C₈	—	2	2	no	—
102	#41	—	2	2	no	—
103	#41	—	2	2	no	—
104	#41	—	2	2	no	—
105	#41	—	2	2	no	—
106	C₂×Q₈	C₂×Dic₃	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	C₂×Dic₃	1	2	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
111	Q₁₆	—	2	1	no	—
112	Q₁₆	—	2	1	no	—
113	SD₁₆	—	2	2	no	—
114	SD₁₆	—	2	2	no	—
115	D₈	—	2	1	no	—
116	D₈	—	2	1	no	—
117	M₁₆	—	3	2	no	—
118	M₁₆	—	6	1	no	—
119	C₂×C₈	—	6	1	no	—
120	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
122	C₂×C₈	#67	1	1	yes	characteristic
123	C₂×C₈	#67	1	1	yes	characteristic
124	C₂×C₈	#67	1	1	yes	characteristic
125	C₂×C₈	#67	1	1	yes	characteristic
126	C₄⋊C₄	—	6	1	no	—
127	C₄⋊C₄	—	6	2	no	—
128	C₂²⋊C₄	—	3	2	no	—
129	C₂²⋊C₄	—	6	2	no	—
130	C₄²	—	6	1	no	—
131	C₄²	—	6	2	no	—
132	C₂²×C₃	—	4	1	no	—
133	C₂²×C₃	—	4	2	no	—
134	C₂²×C₃	—	8	2	no	—
135	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
136	C₂²×C₃	#121	1	1	yes	characteristic
137	C₂²×C₃	#127	1	1	yes	characteristic
138	C₃×C₄	—	4	2	no	—
139	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₃×C₄	C₄×D₄	1	1	yes	characteristic
141	C₃×C₄	#121	1	1	yes	characteristic
142	C₃×C₄	#124	1	1	yes	characteristic
143	Dic₃	—	4	2	no	—
144	Dic₃	—	8	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	D₃×D₄	1	1	yes	characteristic
147	Q₈	—	2	2	no	—
148	Q₈	—	2	2	no	—
149	D₄	—	2	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	6	2	no	—
157	C₂×C₄	—	12	2	no	—
158	C₂×C₄	D₃×D₄	1	1	yes	characteristic
159	C₂×C₄	#237	1	1	yes	characteristic
160	C₂×C₄	#237	1	1	yes	characteristic
161	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
162	C₂×C₄	C₂×#67	1	1	yes	characteristic
163	C₂×C₄	C₂×#67	1	1	yes	characteristic
164	C₈	—	6	2	no	—
165	C₈	#217	1	1	yes	characteristic
166	C₈	#217	1	1	yes	characteristic
167	C₈	#217	1	1	yes	characteristic
168	C₈	#217	1	1	yes	characteristic
169	C₂×C₃	—	2	1	no	—
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	—	2	1	no	—
172	C₂×C₃	—	8	2	no	—
173	C₂×C₃	#390	1	1	yes	characteristic
174	C₂²	—	4	1	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	8	2	no	—
177	C₂²	D₄×Dic₃	1	1	yes	characteristic
178	C₂²	#938	1	1	yes	characteristic
179	C₂²	#940	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	12	2	no	—
182	C₄	—	24	2	no	—
183	C₄	D₄×Dic₃	1	1	yes	characteristic
184	C₄	#936	1	1	yes	characteristic
185	C₄	#938	1	1	yes	characteristic
186	C₄	C₂×#217	1	1	yes	characteristic
187	C₃	#1976	1	1	yes	characteristic, complemented, Sylow
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	—	8	2	no	—
192	C₂	#5221	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#938
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting, prime radical	C₂×C₃×#135	C₂
socle	C₂×C₃	#390
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#135	D₃
hypocenter	C₃	#1976
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. #5240	C₂	semidirect
3. #5236	C₂	semidirect
4. #5106	C₂	semidirect
5. #5101	C₂	semidirect
12. C₂×C₃×Q₁₆	C₄	semidirect
13. C₂×C₃×SD₁₆	C₄	semidirect
14. C₂×C₃×D₈	C₄	semidirect
65. C₂×Q₁₆	Dic₃	semidirect
66. C₂×SD₁₆	Dic₃	semidirect
67. C₂×D₈	Dic₃	semidirect
187. C₃	#1976	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#135	2
#5240	2
#5236	2
#5106	2
#5101	2
#1976	3