Finite Group Databasebeta

Subgroups — #78546

Order 384.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#5519	C₂	1	2	yes	maximal
2	C₃×#405	C₂	1	1	yes	maximal, characteristic
3	#5121	C₂	1	2	yes	maximal
4	#5106	C₂	1	1	yes	maximal, characteristic
5	C₂×#818	C₂	1	1	yes	maximal, characteristic
6	#4702	C₂	1	4	yes	maximal, complemented
7	#4683	C₂	1	2	yes	maximal, complemented
8	#4682	C₂	1	2	yes	maximal, complemented
9	#3164	—	3	1	no	maximal, Sylow
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×#117	C₂²	1	1	yes	characteristic
13	#926	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
15	#890	—	2	2	no	—
16	#888	—	2	2	no	—
17	#878	—	2	4	no	—
18	#870	C₂²	1	4	yes	—
19	#869	C₂²	1	4	yes	—
20	C₃×#97	C₂²	1	2	yes	—
21	C₃×C₄×C₈	C₂²	1	2	yes	—
22	#818	C₂²	1	4	yes	—
23	#818	C₂²	1	1	yes	characteristic
24	#818	C₂²	1	2	yes	—
25	#818	C₂²	1	1	yes	characteristic
26	#818	C₂²	1	2	yes	—
27	#817	C₂²	1	4	yes	—
28	#817	C₂²	1	2	yes	—
29	#544	—	3	2	no	—
30	#501	—	3	4	no	—
31	#500	—	3	2	no	—
32	#499	—	3	2	no	—
33	#428	—	3	2	no	—
34	#427	—	3	1	no	—
35	C₂×#107	—	3	1	no	—
36	#405	D₃	1	1	yes	characteristic
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×Dic₃	—	2	2	no	—
39	C₂×Dic₆	—	2	4	no	—
40	C₃×M₁₆	Q₈	1	4	yes	complemented
41	C₂×C₃×C₈	Q₈	1	4	yes	complemented
42	C₂×C₃×C₈	C₂³	1	2	yes	—
43	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₃×C₂²⋊C₄	D₄	1	2	yes	—
48	C₃×C₄²	C₂³	1	2	yes	—
49	#217	—	2	2	no	—
50	C₄⋊Dic₃	—	2	4	no	—
51	C₄⋊Dic₃	—	2	2	no	—
52	C₄⋊Dic₃	C₂³	1	2	yes	—
53	C₄⋊Dic₃	C₂³	1	2	yes	—
54	C₄⋊Dic₃	C₂³	1	4	yes	—
55	Dic₃⋊C₄	—	2	4	no	—
56	Dic₃⋊C₄	—	2	4	no	—
57	C₄×Dic₃	—	2	2	no	—
58	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
60	C₄⋊Q₈	—	3	4	no	—
61	C₄⋊Q₈	—	6	2	no	—
62	#125	—	3	4	no	—
63	#125	—	6	2	no	—
64	C₂²⋊Q₈	—	6	4	no	—
65	#117	—	3	2	no	—
66	#117	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	3	2	no	—
68	#107	—	3	2	no	—
69	#107	—	3	2	no	—
70	#107	—	3	1	no	—
71	#107	—	3	1	no	—
72	#107	—	3	4	no	—
73	#106	—	3	2	no	—
74	#106	—	3	4	no	—
75	#97	C₂×D₃	1	2	yes	—
76	C₄×C₈	C₂×D₃	1	2	yes	—
77	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
82	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
83	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
84	C₂×Dic₃	—	2	2	no	—
85	C₂×Dic₃	—	2	2	no	—
86	C₂×Dic₃	—	2	4	no	—
87	C₂×Dic₃	—	8	2	no	—
88	Dic₆	—	8	4	no	—
89	C₃×C₈	C₂×Q₈	1	2	yes	—
90	C₃×C₈	C₂×Q₈	1	2	yes	—
91	C₃×C₈	C₂×Q₈	1	4	yes	—
92	C₂×Q₈	—	6	4	no	—
93	C₂²×C₄	—	6	2	no	—
94	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
95	M₁₆	Dic₆	1	4	yes	complemented
96	C₂×C₈	Dic₆	1	4	yes	complemented
97	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₈	C₂²×D₃	1	2	yes	—
99	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
100	C₄⋊C₄	—	3	2	no	—
101	C₄⋊C₄	—	3	4	no	—
102	C₄⋊C₄	—	3	2	no	—
103	C₄⋊C₄	—	6	2	no	—
104	C₄⋊C₄	—	6	4	no	—
105	C₄⋊C₄	—	6	4	no	—
106	C₄⋊C₄	—	6	4	no	—
107	C₄⋊C₄	D₁₂	1	1	yes	characteristic
108	C₄⋊C₄	D₁₂	1	1	yes	characteristic
109	C₂²⋊C₄	—	6	2	no	—
110	C₂²⋊C₄	D₁₂	1	2	yes	—
111	C₄²	—	6	2	no	—
112	C₄²	C₂²×D₃	1	2	yes	—
113	C₂²×C₃	—	2	1	no	—
114	C₂²×C₃	—	2	1	no	—
115	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
116	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
117	C₃×C₄	—	2	2	no	—
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	C₄⋊Q₈	1	2	yes	—
120	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
121	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
122	Dic₃	—	4	2	no	—
123	Dic₃	—	4	4	no	—
124	Dic₃	—	4	2	no	—
125	C₂³	C₂×D₁₂	1	1	yes	characteristic
126	Q₈	—	24	4	no	—
127	C₂×C₄	—	6	2	no	—
128	C₂×C₄	—	6	2	no	—
129	C₂×C₄	—	6	4	no	—
130	C₂×C₄	—	24	2	no	—
131	C₂×C₄	C₂×Dic₆	1	2	yes	—
132	C₂×C₄	C₂×Dic₆	1	2	yes	—
133	C₂×C₄	C₂×D₁₂	1	2	yes	—
134	C₂×C₄	C₂×D₁₂	1	2	yes	—
135	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
137	C₈	C₂×Dic₆	1	2	yes	—
138	C₈	C₂×Dic₆	1	2	yes	—
139	C₈	C₂×Dic₆	1	4	yes	—
140	C₂×C₃	—	2	2	no	—
141	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₃	#576	1	1	yes	characteristic
143	C₂×C₃	#578	1	1	yes	characteristic
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#869	1	2	yes	—
147	C₂²	C₂²×D₁₂	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	12	4	no	—
151	C₄	—	12	2	no	—
152	C₄	—	12	2	no	—
153	C₄	#869	1	2	yes	—
154	C₄	C₂²×Dic₆	1	1	yes	characteristic
155	C₄	C₂²×Dic₆	1	1	yes	characteristic
156	C₃	#3164	1	1	yes	characteristic, complemented, Sylow
157	C₂	—	2	2	no	—
158	C₂	C₂×#869	1	1	yes	characteristic
159	C₂	#5752	1	1	yes	characteristic
160	C₂	#5753	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#405	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	#405	D₃
hypocenter	C₃	#3164
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
6. #4702	C₂	semidirect
7. #4683	C₂	semidirect
8. #4682	C₂	semidirect
40. C₃×M₁₆	Q₈	semidirect
41. C₂×C₃×C₈	Q₈	semidirect
95. M₁₆	Dic₆	semidirect
96. C₂×C₈	Dic₆	semidirect
156. C₃	#3164	semidirect

Maximal subgroups

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

subgroup	index
#5519	2
C₃×#405	2
#5121	2
#5106	2
C₂×#818	2
#4702	2
#4683	2
#4682	2
#3164	3