Finite Group Databasebeta

Subgroups — #5778

Order 192.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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	#1009	C₂	1	1	yes	maximal, characteristic, complemented
2	#1007	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#215	C₂	1	1	yes	maximal, characteristic, complemented
6	#919	C₂	1	2	yes	maximal, complemented
7	#918	C₂	1	2	yes	maximal, complemented
8	#914	C₂	1	2	yes	maximal, complemented
9	D₃×SD₁₆	C₂	1	2	yes	maximal, complemented
10	#908	C₂	1	1	yes	maximal, characteristic, complemented
11	#907	C₂	1	1	yes	maximal, characteristic, complemented
12	#577	—	3	1	no	maximal, Sylow
13	C₃×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
15	#239	—	2	1	no	—
16	#239	C₂²	1	2	yes	—
17	#239	C₂²	1	2	yes	complemented
18	D₃×Q₈	—	2	1	no	—
19	D₃×Q₈	C₂²	1	2	yes	complemented
20	D₃×D₄	—	2	2	no	—
21	D₃×D₄	C₂²	1	2	yes	—
22	#235	—	2	2	no	—
23	#235	—	2	1	no	—
24	#235	C₂²	1	1	yes	characteristic
25	#235	C₂²	1	1	yes	characteristic, complemented
26	C₂×D₁₂	—	2	1	no	—
27	C₂×D₁₂	C₂²	1	1	yes	characteristic
28	C₃×Q₁₆	C₂²	1	2	yes	complemented
29	C₃×SD₁₆	C₂²	1	2	yes	complemented
30	C₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
31	#216	C₂²	1	1	yes	characteristic, complemented
32	#215	C₂²	1	2	yes	complemented
33	#215	C₂²	1	2	yes	complemented
34	#215	C₂²	1	1	yes	characteristic, complemented
35	#214	C₂²	1	1	yes	characteristic, complemented
36	#213	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	D₂₄	C₂²	1	2	yes	complemented
39	#204	C₂²	1	2	yes	complemented
40	ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
41	D₃×C₈	C₂²	1	2	yes	complemented
42	2₋¹⁺⁴	—	3	1	no	—
43	H(5,𝔽₂)	—	3	1	no	—
44	#137	—	3	2	no	—
45	#137	D₃	1	1	yes	characteristic, complemented
46	AGL(1,ℤ/8)	—	3	2	no	—
47	AGL(1,ℤ/8)	—	3	1	no	—
48	#135	—	3	1	no	—
49	#135	—	3	2	no	—
50	C₂×SD₁₆	—	3	1	no	—
51	C₂×SD₁₆	—	3	2	no	—
52	#131	—	3	1	no	—
53	C₂²×D₃	—	2	1	no	—
54	C₂²×D₃	—	2	2	no	—
55	C₃×Q₈	—	2	1	no	—
56	C₃×Q₈	C₂³	1	2	yes	—
57	C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₃×D₄	—	2	1	no	—
59	C₃×D₄	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄	—	2	1	no	—
61	C₂×C₃×C₄	—	2	1	no	—
62	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
63	#67	—	2	1	no	—
64	#67	—	2	2	no	—
65	#67	D₄	1	2	yes	—
66	D₁₂	—	2	2	no	—
67	D₁₂	—	2	2	no	—
68	D₁₂	—	2	1	no	—
69	D₁₂	D₄	1	1	yes	characteristic
70	D₁₂	C₂³	1	2	yes	—
71	D₁₂	C₂³	1	1	yes	characteristic
72	C₄×D₃	—	2	2	no	—
73	C₄×D₃	—	2	1	no	—
74	C₄×D₃	—	2	2	no	—
75	C₄×D₃	—	2	2	no	—
76	C₄×D₃	C₂³	1	2	yes	—
77	Dic₆	—	2	2	no	—
78	Dic₆	D₄	1	1	yes	characteristic
79	Dic₆	C₂³	1	1	yes	characteristic
80	C₃×C₈	C₂³	1	2	yes	—
81	C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₃⋊C₈	C₂³	1	1	yes	characteristic
83	#41	—	3	1	no	—
84	#41	—	3	2	no	—
85	#41	—	3	2	no	—
86	#41	—	3	1	no	—
87	#41	—	6	2	no	—
88	#41	—	6	1	no	—
89	#41	—	6	1	no	—
90	#41	C₂×D₃	1	1	yes	characteristic
91	C₂×Q₈	—	3	2	no	—
92	C₂×Q₈	—	6	1	no	—
93	C₂×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
94	C₂×D₄	—	3	2	no	—
95	C₂×D₄	—	3	1	no	—
96	C₂×D₄	—	6	1	no	—
97	C₂×D₄	—	6	2	no	—
98	Q₁₆	—	3	1	no	—
99	Q₁₆	C₂×D₃	1	2	yes	complemented
100	SD₁₆	—	3	2	no	—
101	SD₁₆	—	3	2	no	—
102	SD₁₆	—	3	1	no	—
103	SD₁₆	—	3	1	no	—
104	SD₁₆	—	3	2	no	—
105	SD₁₆	C₂×D₃	1	2	yes	complemented
106	D₈	—	3	1	no	—
107	D₈	—	3	2	no	—
108	M₁₆	—	3	2	no	—
109	M₁₆	C₂×D₃	1	1	yes	characteristic, complemented
110	C₂×C₈	—	3	1	no	—
111	C₂×C₈	—	3	2	no	—
112	C₂²×C₃	—	2	1	no	—
113	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
114	C₂×D₃	—	2	1	no	—
115	C₂×D₃	—	2	2	no	—
116	C₂×D₃	—	8	2	no	—
117	C₂×D₃	—	8	1	no	—
118	C₂×D₃	C₂×D₄	1	2	yes	—
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	—	2	1	no	—
121	C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₃×C₄	C₂⁴	1	1	yes	characteristic
123	Dic₃	—	2	1	no	—
124	Dic₃	C₂×D₄	1	2	yes	—
125	C₂³	—	6	2	no	—
126	C₂³	—	6	1	no	—
127	Q₈	—	2	1	no	—
128	Q₈	—	3	1	no	—
129	Q₈	—	3	1	no	—
130	Q₈	—	6	2	no	—
131	Q₈	C₂²×D₃	1	2	yes	—
132	Q₈	C₂²×D₃	1	1	yes	characteristic
133	D₄	—	2	1	no	—
134	D₄	—	3	2	no	—
135	D₄	—	3	1	no	—
136	D₄	—	3	1	no	—
137	D₄	—	3	2	no	—
138	D₄	—	6	2	no	—
139	D₄	—	6	2	no	—
140	D₄	—	6	2	no	—
141	D₄	—	6	1	no	—
142	D₄	—	6	1	no	—
143	D₄	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	3	2	no	—
147	C₂×C₄	—	6	2	no	—
148	C₂×C₄	—	6	2	no	—
149	C₂×C₄	—	6	2	no	—
150	C₂×C₄	—	6	1	no	—
151	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
152	C₈	—	3	1	no	—
153	C₈	—	3	1	no	—
154	C₈	C₂²×D₃	1	2	yes	—
155	C₂×C₃	—	2	1	no	—
156	C₂×C₃	—	4	1	no	—
157	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
158	D₃	—	2	2	no	—
159	D₃	—	4	1	no	—
160	D₃	—	4	2	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	3	2	no	—
163	C₂²	—	6	1	no	—
164	C₂²	—	6	2	no	—
165	C₂²	—	24	1	no	—
166	C₂²	—	24	2	no	—
167	C₂²	D₃×D₄	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	2	no	—
170	C₄	—	3	2	no	—
171	C₄	—	6	1	no	—
172	C₄	D₃×D₄	1	1	yes	characteristic
173	C₄	C₂³×D₃	1	1	yes	characteristic
174	C₃	#577	1	1	yes	characteristic, complemented, Sylow
175	C₂	—	2	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	6	2	no	—
178	C₂	—	12	2	no	—
179	C₂	—	12	1	no	—
180	C₂	C₂×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₃×#137	C₂
socle	C₂×C₃	C₂²×D₄
hypercenter	#137	D₃
hypocenter	C₃	#577
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #1009	C₂	semidirect
2. #1007	C₂	semidirect
3. C₃×#137	C₂	semidirect
4. #950	C₂	semidirect
5. C₂×#215	C₂	semidirect
6. #919	C₂	semidirect
7. #918	C₂	semidirect
8. #914	C₂	semidirect
9. D₃×SD₁₆	C₂	semidirect
10. #908	C₂	semidirect
11. #907	C₂	semidirect
14. C₂×C₃×Q₈	C₂²	semidirect
17. #239	C₂²	semidirect
19. D₃×Q₈	C₂²	semidirect
25. #235	C₂²	semidirect
28. C₃×Q₁₆	C₂²	semidirect
29. C₃×SD₁₆	C₂²	semidirect
30. C₃×M₁₆	C₂²	semidirect
31. #216	C₂²	semidirect
32. #215	C₂²	semidirect
33. #215	C₂²	semidirect
34. #215	C₂²	semidirect
35. #214	C₂²	semidirect
36. #213	C₂²	semidirect
37. C₂×C₃⋊C₈	C₂²	semidirect
38. D₂₄	C₂²	semidirect
39. #204	C₂²	semidirect
40. ΓL(1,𝔽₂₅)	C₂²	semidirect
41. D₃×C₈	C₂²	semidirect
45. #137	D₃	semidirect
93. C₂×Q₈	C₂×D₃	semidirect
99. Q₁₆	C₂×D₃	semidirect
105. SD₁₆	C₂×D₃	semidirect
109. M₁₆	C₂×D₃	semidirect
174. C₃	#577	semidirect

Maximal subgroups

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

subgroup	index
#1009	2
#1007	2
C₃×#137	2
#950	2
C₂×#215	2
#919	2
#918	2
#914	2
D₃×SD₁₆	2
#908	2
#907	2
#577	3