Finite Group Databasebeta

Subgroups — #72084

Order 384.

300 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#837	C₂	1	2	yes	maximal, complemented
3	#5234	C₂	1	2	yes	maximal, complemented
4	C₂×#819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#1974	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	2	yes	—
8	C₃×#135	—	2	2	no	—
9	C₃×#135	—	2	2	no	—
10	C₂×C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃×SD₁₆	C₄	1	2	yes	complemented
12	C₂×C₃×D₈	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×#208	C₂²	1	2	yes	—
16	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	#837	—	2	4	no	—
18	#830	—	2	4	no	—
19	#819	—	2	2	no	—
20	C₈×Dic₃	—	2	2	no	—
21	C₂×#135	D₃	1	1	yes	characteristic
22	C₂×#108	—	3	1	no	—
23	C₂×#104	—	3	2	no	—
24	#408	—	3	2	no	—
25	C₂×C₄×C₈	—	3	1	no	—
26	C₃×#41	—	2	2	no	—
27	C₃×#41	—	2	2	no	—
28	C₃×#41	—	2	2	no	—
29	C₃×#41	—	2	2	no	—
30	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×D₄	—	2	2	no	—
32	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄	—	2	2	no	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	—	2	1	no	—
36	C₃×Q₁₆	—	2	2	no	—
37	C₃×SD₁₆	—	2	4	no	—
38	C₃×D₈	—	2	2	no	—
39	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₈	D₄	1	2	yes	—
42	C₂×C₃×C₈	D₄	1	2	yes	—
43	C₄×Dic₃	—	2	1	no	—
44	C₄×Dic₃	D₄	1	2	yes	—
45	#208	—	2	2	no	—
46	#208	—	2	2	no	—
47	C₂×C₃⋊C₈	—	2	1	no	—
48	C₂×C₃⋊C₈	—	2	1	no	—
49	C₂×C₃⋊C₈	—	2	2	no	—
50	C₂×C₃⋊C₈	D₄	1	2	yes	—
51	C₂×#41	C₂×D₃	1	2	yes	—
52	#135	—	2	2	no	—
53	#135	—	2	2	no	—
54	C₂×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
55	C₂×SD₁₆	Dic₃	1	2	yes	complemented
56	C₂×D₈	Dic₃	1	1	yes	characteristic, complemented
57	C₂×M₁₆	—	3	2	no	—
58	C₂²×C₈	—	3	1	no	—
59	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄²	—	3	1	no	—
61	#108	—	6	2	no	—
62	#104	—	6	4	no	—
63	C₂²⋊C₈	—	6	4	no	—
64	C₄×C₈	—	6	2	no	—
65	C₂³×C₃	—	2	2	no	—
66	C₂³×C₃	#41	1	1	yes	characteristic
67	C₃×Q₈	—	2	2	no	—
68	C₃×Q₈	—	2	2	no	—
69	C₃×D₄	—	2	2	no	—
70	C₃×D₄	—	2	2	no	—
71	C₃×D₄	—	4	2	no	—
72	C₃×D₄	—	4	2	no	—
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	4	2	no	—
75	C₂×C₃×C₄	—	4	2	no	—
76	C₂×C₃×C₄	C₂²×C₄	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×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄	#41	1	1	yes	characteristic
82	C₂×Dic₃	—	2	4	no	—
83	C₂×Dic₃	—	2	2	no	—
84	C₃×C₈	C₂²⋊C₄	1	2	yes	—
85	C₃×C₈	C₂²⋊C₄	1	2	yes	—
86	C₃⋊C₈	—	2	2	no	—
87	C₃⋊C₈	—	4	2	no	—
88	#41	—	2	2	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	#41	—	2	2	no	—
92	C₂×Q₈	C₂×Dic₃	1	2	yes	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₃	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	6	1	no	—
97	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
98	Q₁₆	—	2	2	no	—
99	SD₁₆	—	2	4	no	—
100	D₈	—	2	2	no	—
101	M₁₆	—	6	2	no	—
102	M₁₆	—	6	2	no	—
103	C₂×C₈	—	3	2	no	—
104	C₂×C₈	—	6	1	no	—
105	C₂×C₈	—	6	1	no	—
106	C₂×C₈	—	6	2	no	—
107	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
108	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₈	#67	1	2	yes	—
110	C₂×C₈	#67	1	2	yes	—
111	C₄²	—	3	2	no	—
112	C₄²	—	6	1	no	—
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	—	4	2	no	—
115	C₂²×C₃	—	4	4	no	—
116	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×C₃	#121	1	1	yes	characteristic
118	C₂²×C₃	#124	1	1	yes	characteristic
119	C₃×C₄	—	4	2	no	—
120	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₃×C₄	C₄×D₄	1	1	yes	characteristic
122	C₃×C₄	#121	1	1	yes	characteristic
123	C₃×C₄	#127	1	1	yes	characteristic
124	Dic₃	—	2	4	no	—
125	C₂³	—	2	2	no	—
126	C₂³	#237	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	2	no	—
129	D₄	—	2	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	6	4	no	—
137	C₂×C₄	—	6	2	no	—
138	C₂×C₄	D₃×D₄	1	1	yes	characteristic
139	C₂×C₄	D₃×D₄	1	1	yes	characteristic
140	C₂×C₄	#237	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	—	6	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	—	8	2	no	—
151	C₂×C₃	#390	1	1	yes	characteristic
152	C₂×C₃	#443	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	2	no	—
156	C₂²	D₄×Dic₃	1	1	yes	characteristic
157	C₂²	#936	1	1	yes	characteristic
158	C₂²	#938	1	1	yes	characteristic
159	C₄	—	4	2	no	—
160	C₄	—	6	4	no	—
161	C₄	D₄×Dic₃	1	1	yes	characteristic
162	C₄	#938	1	1	yes	characteristic
163	C₄	#940	1	1	yes	characteristic
164	C₄	C₂×#217	1	1	yes	characteristic
165	C₃	#1974	1	1	yes	characteristic, complemented, Sylow
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	8	2	no	—
169	C₂	#5196	1	2	yes	—
170	C₂	#5221	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting, prime radical	C₂×C₃×#135	C₂
socle	C₂²×C₃	C₄×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#135	D₃
hypocenter	C₃	#1974
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#837	C₂	semidirect
3. #5234	C₂	semidirect
4. C₂×#819	C₂	semidirect
5. C₂×C₈×Dic₃	C₂	semidirect
10. C₂×C₃×Q₁₆	C₄	semidirect
11. C₂×C₃×SD₁₆	C₄	semidirect
12. C₂×C₃×D₈	C₄	semidirect
54. C₂×Q₁₆	Dic₃	semidirect
55. C₂×SD₁₆	Dic₃	semidirect
56. C₂×D₈	Dic₃	semidirect
165. C₃	#1974	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
C₂×#837	2
#5234	2
C₂×#819	2
C₂×C₈×Dic₃	2
#1974	3