Finite Group Databasebeta

Subgroups — #5605

Order 192.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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×D₃×D₄	C₂	1	2	yes	maximal, complemented
2	C₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	#939	C₂	1	2	yes	maximal, complemented
4	#937	C₂	1	2	yes	maximal, complemented
5	#930	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#894	C₂	1	2	yes	maximal, complemented
8	#883	C₂	1	2	yes	maximal, complemented
9	#882	C₂	1	2	yes	maximal, complemented
10	#534	—	3	1	no	maximal, Sylow
11	C₂³×D₃	C₂²	1	1	yes	characteristic
12	C₂³×D₃	C₂²	1	2	yes	complemented
13	C₂×C₃×D₄	C₂²	1	2	yes	complemented
14	C₂×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×#67	C₂²	1	2	yes	—
17	C₂×#67	C₂²	1	2	yes	complemented
18	C₂×#67	C₂²	1	2	yes	complemented
19	C₂²×Dic₃	C₂²	1	1	yes	characteristic, complemented
20	D₃×D₄	—	2	8	no	—
21	C₂×D₁₂	C₂²	1	2	yes	—
22	C₂×D₁₂	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄×D₃	C₂²	1	2	yes	complemented
24	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
25	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
26	#217	C₂²	1	2	yes	complemented
27	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
28	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
29	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
30	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
31	C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
32	Dic₃⋊C₄	C₂²	1	2	yes	complemented
33	C₂²×D₄	—	3	2	no	—
34	#123	—	3	2	no	—
35	#123	—	3	2	no	—
36	#121	—	3	2	no	—
37	#121	—	3	1	no	—
38	#121	D₃	1	1	yes	characteristic, complemented
39	#120	—	3	2	no	—
40	#120	—	3	2	no	—
41	C₂×C₂²⋊C₄	—	3	1	no	—
42	C₂³×C₃	C₂³	1	2	yes	—
43	C₂³×C₃	C₂³	1	1	yes	characteristic
44	C₂²×D₃	—	2	4	no	—
45	C₂²×D₃	—	4	1	no	—
46	C₂²×D₃	—	4	1	no	—
47	C₂²×D₃	—	4	2	no	—
48	C₂²×D₃	—	4	2	no	—
49	C₂²×D₃	D₄	1	4	yes	complemented
50	C₂²×D₃	C₂³	1	2	yes	—
51	C₂²×D₃	C₂³	1	2	yes	—
52	C₃×D₄	—	2	4	no	—
53	C₃×D₄	—	4	1	no	—
54	C₂×C₃×C₄	—	4	1	no	—
55	C₂×C₃×C₄	C₂³	1	2	yes	—
56	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
58	#67	—	2	4	no	—
59	#67	—	2	4	no	—
60	#67	—	4	2	no	—
61	C₂×Dic₃	—	4	1	no	—
62	C₂×Dic₃	C₂³	1	2	yes	—
63	C₂×Dic₃	C₂³	1	2	yes	—
64	D₁₂	—	2	4	no	—
65	D₁₂	—	4	1	no	—
66	C₄×D₃	—	2	4	no	—
67	C₂⁴	—	3	1	no	—
68	C₂⁴	—	3	2	no	—
69	C₂×D₄	—	3	2	no	—
70	C₂×D₄	—	3	2	no	—
71	C₂×D₄	—	3	2	no	—
72	C₂×D₄	—	3	2	no	—
73	C₂×D₄	—	3	1	no	—
74	C₂×D₄	—	6	8	no	—
75	C₂×D₄	C₂×D₃	1	2	yes	complemented
76	C₂×D₄	C₂×D₃	1	1	yes	characteristic, complemented
77	C₂²×C₄	—	3	1	no	—
78	C₂²×C₄	—	3	2	no	—
79	C₂²×C₄	C₂×D₃	1	1	yes	characteristic, complemented
80	C₄⋊C₄	—	3	2	no	—
81	C₄⋊C₄	—	3	1	no	—
82	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
83	C₂²⋊C₄	—	3	2	no	—
84	C₂²⋊C₄	—	3	2	no	—
85	C₂²⋊C₄	—	3	2	no	—
86	C₂²⋊C₄	—	3	2	no	—
87	C₂²⋊C₄	—	3	2	no	—
88	C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
89	C₂²×C₃	—	2	1	no	—
90	C₂²×C₃	—	2	1	no	—
91	C₂²×C₃	—	2	2	no	—
92	C₂²×C₃	—	2	2	no	—
93	C₂²×C₃	—	2	2	no	—
94	C₂²×C₃	C₂×D₄	1	2	yes	—
95	C₂²×C₃	C₂⁴	1	1	yes	characteristic
96	C₂×D₃	—	2	2	no	—
97	C₂×D₃	—	2	2	no	—
98	C₂×D₃	—	2	2	no	—
99	C₂×D₃	—	2	2	no	—
100	C₂×D₃	—	2	2	no	—
101	C₂×D₃	—	4	4	no	—
102	C₂×D₃	—	4	8	no	—
103	C₂×D₃	C₂×D₄	1	4	yes	—
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	—	2	1	no	—
106	C₃×C₄	—	2	1	no	—
107	Dic₃	—	2	2	no	—
108	Dic₃	—	2	2	no	—
109	C₂³	—	3	2	no	—
110	C₂³	—	3	4	no	—
111	C₂³	—	3	2	no	—
112	C₂³	—	6	4	no	—
113	C₂³	—	12	1	no	—
114	C₂³	—	12	1	no	—
115	C₂³	—	12	2	no	—
116	C₂³	—	12	2	no	—
117	C₂³	C₂²×D₃	1	2	yes	—
118	C₂³	C₂²×D₃	1	1	yes	characteristic
119	D₄	—	2	4	no	—
120	D₄	—	4	1	no	—
121	D₄	—	6	4	no	—
122	D₄	—	6	4	no	—
123	D₄	—	6	4	no	—
124	D₄	—	12	1	no	—
125	D₄	—	12	2	no	—
126	C₂×C₄	—	3	2	no	—
127	C₂×C₄	—	3	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	6	4	no	—
130	C₂×C₄	—	12	1	no	—
131	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₃	1	2	yes	—
134	C₂×C₃	—	2	2	no	—
135	C₂×C₃	—	4	2	no	—
136	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
139	D₃	—	2	4	no	—
140	D₃	—	4	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	3	4	no	—
147	C₂²	—	6	2	no	—
148	C₂²	—	6	2	no	—
149	C₂²	—	6	2	no	—
150	C₂²	—	6	2	no	—
151	C₂²	—	6	2	no	—
152	C₂²	—	12	4	no	—
153	C₂²	—	12	8	no	—
154	C₂²	D₃×D₄	1	2	yes	—
155	C₂²	C₂³×D₃	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	6	2	no	—
160	C₄	—	6	2	no	—
161	C₃	#534	1	1	yes	characteristic, complemented, Sylow
162	C₂	—	2	2	no	—
163	C₂	—	4	2	no	—
164	C₂	—	6	4	no	—
165	C₂	—	12	2	no	—
166	C₂	C₂×D₃×D₄	1	1	yes	characteristic
167	C₂	#1004	1	1	yes	characteristic
168	C₂	#1009	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#121	C₂
hypercenter	#121	D₃
hypocenter	C₃	#534
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×D₄	C₂	semidirect
2. C₃×#121	C₂	semidirect
3. #939	C₂	semidirect
4. #937	C₂	semidirect
5. #930	C₂	semidirect
6. C₂×(C₂×D₃)⋊C₄	C₂	semidirect
7. #894	C₂	semidirect
8. #883	C₂	semidirect
9. #882	C₂	semidirect
12. C₂³×D₃	C₂²	semidirect
13. C₂×C₃×D₄	C₂²	semidirect
14. C₂×C₃×D₄	C₂²	semidirect
15. C₂²×C₃×C₄	C₂²	semidirect
17. C₂×#67	C₂²	semidirect
18. C₂×#67	C₂²	semidirect
19. C₂²×Dic₃	C₂²	semidirect
22. C₂×D₁₂	C₂²	semidirect
23. C₂×C₄×D₃	C₂²	semidirect
24. C₃×C₄⋊C₄	C₂²	semidirect
25. C₃×C₂²⋊C₄	C₂²	semidirect
26. #217	C₂²	semidirect
27. (C₂×D₃)⋊C₄	C₂²	semidirect
28. (C₂×D₃)⋊C₄	C₂²	semidirect
29. (C₂×D₃)⋊C₄	C₂²	semidirect
30. (C₂×D₃)⋊C₄	C₂²	semidirect
31. C₄⋊Dic₃	C₂²	semidirect
32. Dic₃⋊C₄	C₂²	semidirect
38. #121	D₃	semidirect
49. C₂²×D₃	D₄	semidirect
75. C₂×D₄	C₂×D₃	semidirect
76. C₂×D₄	C₂×D₃	semidirect
79. C₂²×C₄	C₂×D₃	semidirect
82. C₄⋊C₄	C₂×D₃	semidirect
88. C₂²⋊C₄	C₂×D₃	semidirect
161. C₃	#534	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₄	2
C₃×#121	2
#939	2
#937	2
#930	2
C₂×(C₂×D₃)⋊C₄	2
#894	2
#883	2
#882	2
#534	3