Finite Group Databasebeta

Subgroups — #5590

Order 192.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂²×#67	C₂	1	2	yes	maximal, complemented
2	C₂×D₃×D₄	C₂	1	2	yes	maximal, complemented
3	C₃×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	#940	C₂	1	1	yes	maximal, characteristic, complemented
5	#939	C₂	1	2	yes	maximal, complemented
6	#937	C₂	1	1	yes	maximal, characteristic, complemented
7	#884	C₂	1	2	yes	maximal, complemented
8	#882	C₂	1	2	yes	maximal, complemented
9	#67⋊C₄	C₂	1	2	yes	maximal, complemented
10	D₄²	—	3	1	no	maximal, Sylow
11	C₂⁴×C₃	C₂²	1	1	yes	characteristic, complemented
12	C₂³×D₃	C₂²	1	2	yes	complemented
13	C₂³×D₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×D₄	C₂²	1	2	yes	complemented
15	C₂×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×#67	—	2	8	no	—
17	C₂×#67	C₂²	1	2	yes	—
18	C₂×#67	C₂²	1	2	yes	complemented
19	C₂×#67	C₂²	1	2	yes	complemented
20	C₂×#67	C₂²	1	2	yes	complemented
21	C₂×#67	C₂²	1	2	yes	complemented
22	C₂²×Dic₃	C₂²	1	2	yes	complemented
23	D₃×D₄	—	2	8	no	—
24	C₂×D₁₂	C₂²	1	2	yes	complemented
25	C₂×D₁₂	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₄×D₃	C₂²	1	2	yes	complemented
27	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
28	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
29	#217	C₂²	1	1	yes	characteristic, complemented
30	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
31	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
32	Dic₃⋊C₄	C₂²	1	2	yes	complemented
33	C₄×Dic₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²×D₄	—	3	2	no	—
35	C₂²×D₄	—	3	2	no	—
36	#127	—	3	1	no	—
37	#121	—	3	2	no	—
38	#121	—	3	2	no	—
39	#120	—	3	1	no	—
40	#120	—	3	2	no	—
41	#120	D₃	1	1	yes	characteristic, complemented
42	C₄×D₄	—	3	2	no	—
43	C₂³×C₃	—	2	4	no	—
44	C₂³×C₃	—	4	1	no	—
45	C₂³×C₃	C₂³	1	1	yes	characteristic
46	C₂³×C₃	C₂³	1	1	yes	characteristic
47	C₂³×C₃	C₂³	1	2	yes	—
48	C₂²×D₃	—	2	4	no	—
49	C₂²×D₃	—	2	4	no	—
50	C₂²×D₃	—	2	4	no	—
51	C₂²×D₃	—	4	1	no	—
52	C₂²×D₃	—	4	2	no	—
53	C₂²×D₃	C₂³	1	2	yes	—
54	C₂²×D₃	C₂³	1	2	yes	—
55	C₃×D₄	—	2	4	no	—
56	C₃×D₄	—	4	1	no	—
57	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂³	1	2	yes	—
59	#67	—	2	4	no	—
60	#67	—	2	4	no	—
61	#67	—	2	4	no	—
62	#67	—	2	4	no	—
63	#67	D₄	1	8	yes	complemented
64	C₂×Dic₃	—	2	4	no	—
65	C₂×Dic₃	C₂³	1	2	yes	—
66	C₂×Dic₃	C₂³	1	2	yes	—
67	D₁₂	—	2	4	no	—
68	D₁₂	—	4	1	no	—
69	C₄×D₃	—	2	4	no	—
70	C₂⁴	—	3	1	no	—
71	C₂⁴	—	3	2	no	—
72	C₂⁴	C₂×D₃	1	1	yes	characteristic, complemented
73	C₂×D₄	—	3	2	no	—
74	C₂×D₄	—	3	2	no	—
75	C₂×D₄	—	3	2	no	—
76	C₂×D₄	—	3	2	no	—
77	C₂×D₄	—	3	2	no	—
78	C₂×D₄	—	3	2	no	—
79	C₂×D₄	—	3	1	no	—
80	C₂×D₄	—	6	8	no	—
81	C₂×D₄	—	6	8	no	—
82	C₂×D₄	C₂×D₃	1	2	yes	complemented
83	C₂×D₄	C₂×D₃	1	1	yes	characteristic, complemented
84	C₂²×C₄	—	3	2	no	—
85	C₂²×C₄	—	3	2	no	—
86	C₄⋊C₄	—	3	2	no	—
87	C₂²⋊C₄	—	3	1	no	—
88	C₂²⋊C₄	—	3	2	no	—
89	C₂²⋊C₄	—	3	2	no	—
90	C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
91	C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
92	C₄²	—	3	1	no	—
93	C₂²×C₃	—	2	2	no	—
94	C₂²×C₃	—	2	2	no	—
95	C₂²×C₃	—	2	2	no	—
96	C₂²×C₃	—	2	2	no	—
97	C₂²×C₃	—	2	1	no	—
98	C₂²×C₃	—	2	1	no	—
99	C₂²×C₃	—	4	4	no	—
100	C₂²×C₃	C₂×D₄	1	4	yes	complemented
101	C₂²×C₃	C₂⁴	1	1	yes	characteristic
102	C₂×D₃	—	2	2	no	—
103	C₂×D₃	—	2	2	no	—
104	C₂×D₃	—	2	2	no	—
105	C₂×D₃	—	2	2	no	—
106	C₂×D₃	—	2	2	no	—
107	C₂×D₃	—	4	4	no	—
108	C₂×D₃	—	4	8	no	—
109	C₂×D₃	C₂×D₄	1	4	yes	complemented
110	C₃×C₄	—	2	2	no	—
111	C₃×C₄	—	2	1	no	—
112	Dic₃	—	2	2	no	—
113	Dic₃	C₂×D₄	1	4	yes	complemented
114	C₂³	—	2	4	no	—
115	C₂³	—	3	2	no	—
116	C₂³	—	3	2	no	—
117	C₂³	—	4	1	no	—
118	C₂³	—	6	4	no	—
119	C₂³	—	6	4	no	—
120	C₂³	—	6	4	no	—
121	C₂³	—	12	2	no	—
122	C₂³	—	12	1	no	—
123	C₂³	C₂²×D₃	1	1	yes	characteristic
124	C₂³	C₂²×D₃	1	1	yes	characteristic
125	C₂³	C₂²×D₃	1	2	yes	—
126	D₄	—	2	4	no	—
127	D₄	—	3	8	no	—
128	D₄	—	4	1	no	—
129	D₄	—	6	4	no	—
130	D₄	—	6	4	no	—
131	D₄	—	6	4	no	—
132	D₄	—	6	4	no	—
133	D₄	—	6	4	no	—
134	D₄	—	12	1	no	—
135	C₂×C₄	—	3	2	no	—
136	C₂×C₄	—	3	2	no	—
137	C₂×C₄	—	6	4	no	—
138	C₂×C₄	—	6	4	no	—
139	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₄	C₂²×D₃	1	2	yes	—
141	C₂×C₃	—	2	4	no	—
142	C₂×C₃	—	4	1	no	—
143	C₂×C₃	—	4	1	no	—
144	C₂×C₃	C₂²×D₄	1	2	yes	—
145	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
146	D₃	—	2	4	no	—
147	D₃	—	4	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	3	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	6	2	no	—
157	C₂²	—	6	2	no	—
158	C₂²	—	6	2	no	—
159	C₂²	—	6	2	no	—
160	C₂²	—	6	2	no	—
161	C₂²	—	12	8	no	—
162	C₂²	—	12	4	no	—
163	C₂²	D₃×D₄	1	4	yes	complemented
164	C₂²	C₂³×D₃	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	1	no	—
167	C₄	—	3	4	no	—
168	C₄	—	6	2	no	—
169	C₃	D₄²	1	1	yes	characteristic, complemented, Sylow
170	C₂	—	2	4	no	—
171	C₂	—	4	1	no	—
172	C₂	—	4	1	no	—
173	C₂	—	6	4	no	—
174	C₂	—	12	2	no	—
175	C₂	C₂×D₃×D₄	1	2	yes	—
176	C₂	#1004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#120	C₂
hypercenter	#120	D₃
hypocenter	C₃	D₄²
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₂²×#67	C₂	semidirect
2. C₂×D₃×D₄	C₂	semidirect
3. C₃×#120	C₂	semidirect
4. #940	C₂	semidirect
5. #939	C₂	semidirect
6. #937	C₂	semidirect
7. #884	C₂	semidirect
8. #882	C₂	semidirect
9. #67⋊C₄	C₂	semidirect
11. C₂⁴×C₃	C₂²	semidirect
12. C₂³×D₃	C₂²	semidirect
13. C₂³×D₃	C₂²	semidirect
14. C₂×C₃×D₄	C₂²	semidirect
15. C₂×C₃×D₄	C₂²	semidirect
18. C₂×#67	C₂²	semidirect
19. C₂×#67	C₂²	semidirect
20. C₂×#67	C₂²	semidirect
21. C₂×#67	C₂²	semidirect
22. C₂²×Dic₃	C₂²	semidirect
24. C₂×D₁₂	C₂²	semidirect
25. C₂×D₁₂	C₂²	semidirect
26. C₂×C₄×D₃	C₂²	semidirect
27. C₃×C₂²⋊C₄	C₂²	semidirect
28. C₃×C₂²⋊C₄	C₂²	semidirect
29. #217	C₂²	semidirect
30. (C₂×D₃)⋊C₄	C₂²	semidirect
31. (C₂×D₃)⋊C₄	C₂²	semidirect
32. Dic₃⋊C₄	C₂²	semidirect
33. C₄×Dic₃	C₂²	semidirect
41. #120	D₃	semidirect
63. #67	D₄	semidirect
72. C₂⁴	C₂×D₃	semidirect
82. C₂×D₄	C₂×D₃	semidirect
83. C₂×D₄	C₂×D₃	semidirect
90. C₂²⋊C₄	C₂×D₃	semidirect
91. C₂²⋊C₄	C₂×D₃	semidirect
100. C₂²×C₃	C₂×D₄	semidirect
109. C₂×D₃	C₂×D₄	semidirect
113. Dic₃	C₂×D₄	semidirect
163. C₂²	D₃×D₄	semidirect
169. C₃	D₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67	2
C₂×D₃×D₄	2
C₃×#120	2
#940	2
#939	2
#937	2
#884	2
#882	2
#67⋊C₄	2
D₄²	3