Finite Group Databasebeta

Subgroups — #72076

Order 384.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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	#5106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	#4566	C₂	1	4	yes	maximal, complemented
5	#2145	—	3	1	no	maximal, Sylow
6	C₂×C₃×#41	—	2	1	no	—
7	C₃×#135	—	2	1	no	—
8	C₃×#135	—	2	1	no	—
9	C₃×#135	C₄	1	2	yes	complemented
10	C₃×#135	C₂²	1	2	yes	—
11	C₂×C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃×SD₁₆	—	2	1	no	—
13	C₂×C₃×D₈	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	#926	—	2	1	no	—
16	#818	—	2	1	no	—
17	#817	C₂²	1	2	yes	complemented
18	#812	—	2	1	no	—
19	#812	C₂²	1	2	yes	complemented
20	C₂×C₃⋊C₁₆	—	2	1	no	—
21	C₂×#135	D₃	1	1	yes	characteristic
22	C₂×M₃₂	—	3	1	no	—
23	#427	—	3	1	no	—
24	#360	—	3	4	no	—
25	C₃×#41	—	2	2	no	—
26	C₃×#41	—	2	2	no	—
27	C₃×#41	—	4	1	no	—
28	C₃×#41	—	4	1	no	—
29	C₂×C₃×Q₈	—	2	1	no	—
30	C₂×C₃×D₄	—	2	1	no	—
31	C₂×C₃×D₄	—	4	1	no	—
32	C₂²×C₃×C₄	—	4	1	no	—
33	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
34	C₃×Q₁₆	—	2	1	no	—
35	C₃×Q₁₆	C₂×C₄	1	2	yes	—
36	C₃×SD₁₆	—	2	2	no	—
37	C₃×SD₁₆	—	2	2	no	—
38	C₃×D₈	—	2	1	no	—
39	C₃×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₈	D₄	1	1	yes	characteristic
43	C₂×C₃×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃×C₈	D₄	1	1	yes	characteristic
45	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
46	#217	—	4	1	no	—
47	C₄⋊Dic₃	—	2	2	no	—
48	C₄×Dic₃	—	4	1	no	—
49	C₃⋊C₁₆	—	2	2	no	—
50	C₂×#41	—	2	1	no	—
51	#135	—	2	1	no	—
52	#135	—	2	1	no	—
53	#135	Dic₃	1	2	yes	complemented
54	#135	C₂×D₃	1	2	yes	—
55	C₂×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
56	C₂×SD₁₆	—	2	1	no	—
57	C₂×D₈	Dic₃	1	1	yes	characteristic, complemented
58	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
59	#117	—	6	1	no	—
60	M₃₂	—	3	2	no	—
61	M₃₂	—	6	1	no	—
62	C₂×C₁₆	—	6	1	no	—
63	#107	—	6	1	no	—
64	#106	—	3	2	no	—
65	C₂³×C₃	—	4	1	no	—
66	C₂³×C₃	D₈	1	1	yes	characteristic
67	C₃×Q₈	—	2	2	no	—
68	C₃×Q₈	—	4	1	no	—
69	C₃×D₄	—	2	2	no	—
70	C₃×D₄	—	4	2	no	—
71	C₃×D₄	—	4	2	no	—
72	C₃×D₄	—	4	1	no	—
73	C₂×C₃×C₄	—	4	1	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	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	D₈	1	1	yes	characteristic
79	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
80	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₃	—	4	2	no	—
83	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₃×C₈	C₂×D₄	1	1	yes	characteristic
87	#41	—	2	2	no	—
88	#41	—	2	2	no	—
89	#41	—	4	1	no	—
90	#41	—	4	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	#67	1	1	yes	characteristic
96	Q₁₆	—	2	1	no	—
97	Q₁₆	C₂×Dic₃	1	2	yes	—
98	SD₁₆	—	2	2	no	—
99	SD₁₆	—	2	2	no	—
100	D₈	—	2	1	no	—
101	D₈	C₂×Dic₃	1	2	yes	—
102	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₈	#67	1	1	yes	characteristic
105	C₂×C₈	#67	1	1	yes	characteristic
106	C₂×C₈	#67	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
108	C₄⋊C₄	—	6	2	no	—
109	C₂²⋊C₄	—	12	1	no	—
110	C₄²	—	12	1	no	—
111	C₁₆	—	6	2	no	—
112	C₂²×C₃	—	4	1	no	—
113	C₂²×C₃	—	4	2	no	—
114	C₂²×C₃	—	8	2	no	—
115	C₂²×C₃	#102	1	1	yes	characteristic
116	C₂²×C₃	#102	1	1	yes	characteristic
117	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
118	C₃×C₄	—	4	2	no	—
119	C₃×C₄	#102	1	1	yes	characteristic
120	C₃×C₄	#102	1	1	yes	characteristic
121	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
123	Dic₃	—	8	2	no	—
124	C₂³	—	4	1	no	—
125	C₂³	#213	1	1	yes	characteristic
126	Q₈	—	2	2	no	—
127	Q₈	—	4	1	no	—
128	D₄	—	2	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	12	2	no	—
136	C₂×C₄	#213	1	1	yes	characteristic
137	C₂×C₄	#214	1	1	yes	characteristic
138	C₂×C₄	#214	1	1	yes	characteristic
139	C₂×C₄	#217	1	1	yes	characteristic
140	C₂×C₄	#217	1	1	yes	characteristic
141	C₂×C₄	C₂×#67	1	1	yes	characteristic
142	C₈	#217	1	1	yes	characteristic
143	C₈	#217	1	1	yes	characteristic
144	C₈	C₂²×Dic₃	1	1	yes	characteristic
145	C₈	C₂×#67	1	1	yes	characteristic
146	C₂×C₃	—	2	1	no	—
147	C₂×C₃	—	2	1	no	—
148	C₂×C₃	—	2	1	no	—
149	C₂×C₃	—	8	2	no	—
150	C₂×C₃	C₂×#102	1	1	yes	characteristic
151	C₂²	—	4	1	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	#832	1	1	yes	characteristic
155	C₂²	#832	1	1	yes	characteristic
156	C₂²	C₂×#213	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	24	2	no	—
159	C₄	#832	1	1	yes	characteristic
160	C₄	#832	1	1	yes	characteristic
161	C₄	C₂×#214	1	1	yes	characteristic
162	C₄	C₂×#217	1	1	yes	characteristic
163	C₃	#2145	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	8	2	no	—
168	C₂	C₂×#832	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#832
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×#135	C₂
socle	C₂×C₃	C₂×#102
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×#135	D₃
hypocenter	C₃	#2145
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. #5106	C₂	semidirect
3. C₂×#812	C₂	semidirect
4. #4566	C₂	semidirect
9. C₃×#135	C₄	semidirect
11. C₂×C₃×Q₁₆	C₄	semidirect
13. C₂×C₃×D₈	C₄	semidirect
17. #817	C₂²	semidirect
19. #812	C₂²	semidirect
53. #135	Dic₃	semidirect
55. C₂×Q₁₆	Dic₃	semidirect
57. C₂×D₈	Dic₃	semidirect
163. C₃	#2145	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
#5106	2
C₂×#812	2
#4566	2
#2145	3