Finite Group Databasebeta

Subgroups — #5607

Order 192.

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

normal subgroups only

This group has 334 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	1	yes	maximal, characteristic, complemented
2	C₂×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	#940	C₂	1	2	yes	maximal, complemented
5	#938	C₂	1	1	yes	maximal, characteristic, complemented
6	#937	C₂	1	2	yes	maximal, complemented
7	#930	C₂	1	1	yes	maximal, characteristic, complemented
8	#895	C₂	1	1	yes	maximal, characteristic, complemented
9	#892	C₂	1	1	yes	maximal, characteristic, complemented
10	#885	C₂	1	2	yes	maximal, complemented
11	#882	C₂	1	2	yes	maximal, complemented
12	#535	—	3	1	no	maximal, Sylow
13	C₂³×D₃	C₂²	1	2	yes	—
14	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₄	C₂²	1	2	yes	complemented
16	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×#67	C₂²	1	2	yes	—
18	C₂×#67	C₂²	1	1	yes	characteristic
19	C₂×#67	C₂²	1	1	yes	characteristic, complemented
20	C₂×#67	C₂²	1	2	yes	complemented
21	C₂×#67	C₂²	1	2	yes	complemented
22	D₃×D₄	—	2	4	no	—
23	#235	—	2	4	no	—
24	C₂×D₁₂	C₂²	1	1	yes	characteristic
25	C₂×D₁₂	C₂²	1	1	yes	characteristic
26	C₂×D₁₂	C₂²	1	2	yes	complemented
27	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×Dic₆	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
32	#217	C₂²	1	2	yes	complemented
33	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
35	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
36	C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
37	C₄×Dic₃	C₂²	1	2	yes	complemented
38	C₂×#41	—	3	1	no	—
39	C₂²×D₄	—	3	1	no	—
40	#127	—	3	2	no	—
41	#124	—	3	2	no	—
42	#121	—	3	1	no	—
43	#121	—	3	1	no	—
44	#121	—	3	1	no	—
45	#121	D₃	1	1	yes	characteristic, complemented
46	#120	—	3	2	no	—
47	#120	—	3	2	no	—
48	#117	—	3	1	no	—
49	C₂³×C₃	C₂³	1	2	yes	—
50	C₂³×C₃	C₂³	1	1	yes	characteristic
51	C₂²×D₃	—	2	4	no	—
52	C₂²×D₃	—	4	2	no	—
53	C₂²×D₃	—	4	2	no	—
54	C₂²×D₃	C₂³	1	2	yes	—
55	C₂²×D₃	C₂³	1	1	yes	characteristic
56	C₂²×D₃	C₂³	1	1	yes	characteristic
57	C₃×D₄	—	2	2	no	—
58	C₃×D₄	—	4	2	no	—
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×C₃×C₄	C₂³	1	2	yes	—
61	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
63	#67	—	2	2	no	—
64	#67	—	2	2	no	—
65	#67	—	2	4	no	—
66	#67	—	4	2	no	—
67	#67	—	4	2	no	—
68	C₂×Dic₃	C₂³	1	2	yes	—
69	C₂×Dic₃	C₂³	1	1	yes	characteristic
70	C₂×Dic₃	C₂³	1	1	yes	characteristic
71	D₁₂	—	2	2	no	—
72	D₁₂	—	2	2	no	—
73	D₁₂	—	4	2	no	—
74	C₄×D₃	—	2	2	no	—
75	C₄×D₃	D₄	1	4	yes	complemented
76	Dic₆	—	2	2	no	—
77	C₂⁴	—	3	2	no	—
78	#41	—	6	4	no	—
79	C₂×Q₈	—	3	1	no	—
80	C₂×D₄	—	3	1	no	—
81	C₂×D₄	—	3	2	no	—
82	C₂×D₄	—	3	1	no	—
83	C₂×D₄	—	3	2	no	—
84	C₂×D₄	—	3	2	no	—
85	C₂×D₄	—	3	1	no	—
86	C₂×D₄	—	3	1	no	—
87	C₂×D₄	—	3	2	no	—
88	C₂×D₄	—	6	4	no	—
89	C₂×D₄	C₂×D₃	1	1	yes	characteristic
90	C₂×D₄	C₂×D₃	1	2	yes	complemented
91	C₂²×C₄	—	3	1	no	—
92	C₂²×C₄	—	3	1	no	—
93	C₂²×C₄	C₂×D₃	1	1	yes	characteristic, complemented
94	C₄⋊C₄	—	3	1	no	—
95	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
96	C₂²⋊C₄	—	3	2	no	—
97	C₂²⋊C₄	—	3	2	no	—
98	C₂²⋊C₄	—	3	2	no	—
99	C₂²⋊C₄	—	3	2	no	—
100	C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
101	C₄²	—	3	2	no	—
102	C₂²×C₃	—	2	1	no	—
103	C₂²×C₃	—	2	2	no	—
104	C₂²×C₃	—	2	1	no	—
105	C₂²×C₃	—	2	2	no	—
106	C₂²×C₃	—	2	1	no	—
107	C₂²×C₃	—	2	2	no	—
108	C₂²×C₃	C₂⁴	1	1	yes	characteristic
109	C₂×D₃	—	2	1	no	—
110	C₂×D₃	—	2	1	no	—
111	C₂×D₃	—	2	1	no	—
112	C₂×D₃	—	2	2	no	—
113	C₂×D₃	—	2	1	no	—
114	C₂×D₃	—	2	2	no	—
115	C₂×D₃	—	2	1	no	—
116	C₂×D₃	—	2	2	no	—
117	C₂×D₃	—	4	8	no	—
118	C₂×D₃	C₂×D₄	1	2	yes	—
119	C₃×C₄	—	2	1	no	—
120	C₃×C₄	—	2	2	no	—
121	C₃×C₄	C₂×D₄	1	2	yes	—
122	Dic₃	—	2	1	no	—
123	Dic₃	—	2	2	no	—
124	Dic₃	C₂×D₄	1	2	yes	—
125	C₂³	—	3	1	no	—
126	C₂³	—	3	2	no	—
127	C₂³	—	3	1	no	—
128	C₂³	—	6	4	no	—
129	C₂³	—	12	2	no	—
130	C₂³	—	12	2	no	—
131	C₂³	C₂²×D₃	1	2	yes	—
132	C₂³	C₂²×D₃	1	1	yes	characteristic
133	Q₈	—	6	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	6	2	no	—
137	D₄	—	6	4	no	—
138	D₄	—	6	2	no	—
139	D₄	—	6	2	no	—
140	D₄	—	6	2	no	—
141	D₄	—	12	2	no	—
142	D₄	—	12	2	no	—
143	D₄	—	12	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	3	1	no	—
146	C₂×C₄	—	3	4	no	—
147	C₂×C₄	—	3	2	no	—
148	C₂×C₄	—	3	1	no	—
149	C₂×C₄	—	6	2	no	—
150	C₂×C₄	C₂²×D₃	1	2	yes	—
151	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
152	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
153	C₂×C₃	—	4	2	no	—
154	C₂×C₃	—	4	1	no	—
155	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₃	H(5,𝔽₂)	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	—	2	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	3	2	no	—
168	C₂²	—	6	1	no	—
169	C₂²	—	6	1	no	—
170	C₂²	—	6	1	no	—
171	C₂²	—	6	2	no	—
172	C₂²	—	6	1	no	—
173	C₂²	—	6	2	no	—
174	C₂²	—	6	1	no	—
175	C₂²	—	6	2	no	—
176	C₂²	—	12	8	no	—
177	C₂²	C₂³×D₃	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	2	no	—
180	C₄	—	3	2	no	—
181	C₄	—	6	1	no	—
182	C₄	—	6	2	no	—
183	C₄	D₃×D₄	1	2	yes	—
184	C₃	#535	1	1	yes	characteristic, complemented, Sylow
185	C₂	—	4	2	no	—
186	C₂	—	4	1	no	—
187	C₂	—	6	2	no	—
188	C₂	—	12	2	no	—
189	C₂	—	12	1	no	—
190	C₂	C₂×D₃×D₄	1	1	yes	characteristic
191	C₂	#1004	1	1	yes	characteristic
192	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	#535
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. C₂×#235	C₂	semidirect
3. C₃×#121	C₂	semidirect
4. #940	C₂	semidirect
5. #938	C₂	semidirect
6. #937	C₂	semidirect
7. #930	C₂	semidirect
8. #895	C₂	semidirect
9. #892	C₂	semidirect
10. #885	C₂	semidirect
11. #882	C₂	semidirect
15. C₂×C₃×D₄	C₂²	semidirect
16. C₂²×C₃×C₄	C₂²	semidirect
19. C₂×#67	C₂²	semidirect
20. C₂×#67	C₂²	semidirect
21. C₂×#67	C₂²	semidirect
26. C₂×D₁₂	C₂²	semidirect
28. C₂×C₄×D₃	C₂²	semidirect
29. C₂×Dic₆	C₂²	semidirect
30. C₃×C₄⋊C₄	C₂²	semidirect
31. C₃×C₂²⋊C₄	C₂²	semidirect
32. #217	C₂²	semidirect
34. (C₂×D₃)⋊C₄	C₂²	semidirect
35. (C₂×D₃)⋊C₄	C₂²	semidirect
36. C₄⋊Dic₃	C₂²	semidirect
37. C₄×Dic₃	C₂²	semidirect
45. #121	D₃	semidirect
75. C₄×D₃	D₄	semidirect
90. C₂×D₄	C₂×D₃	semidirect
93. C₂²×C₄	C₂×D₃	semidirect
95. C₄⋊C₄	C₂×D₃	semidirect
100. C₂²⋊C₄	C₂×D₃	semidirect
184. C₃	#535	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
C₂×#235	2
C₃×#121	2
#940	2
#938	2
#937	2
#930	2
#895	2
#892	2
#885	2
#882	2
#535	3