Finite Group Databasebeta

Subgroups — #5612

Order 192.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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	#938	C₂	1	2	yes	maximal, complemented
5	#937	C₂	1	2	yes	maximal, complemented
6	#936	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#67	C₂	1	1	yes	maximal, characteristic, complemented
8	#897	C₂	1	1	yes	maximal, characteristic, complemented
9	#893	C₂	1	1	yes	maximal, characteristic, complemented
10	#883	C₂	1	2	yes	maximal, complemented
11	D₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	#546	—	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	2	yes	complemented
20	D₃×D₄	—	2	4	no	—
21	#235	—	2	4	no	—
22	C₂×D₁₂	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₃	C₂²	1	2	yes	complemented
25	C₂×C₄×D₃	C₂²	1	2	yes	complemented
26	C₂×Dic₆	C₂²	1	1	yes	characteristic, complemented
27	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
29	#217	C₂²	1	2	yes	complemented
30	#217	C₂²	1	2	yes	complemented
31	#217	C₂²	1	1	yes	characteristic, complemented
32	(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
33	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
35	C₄⋊Dic₃	C₂²	1	2	yes	complemented
36	Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
37	C₄×Dic₃	C₂²	1	1	yes	characteristic, complemented
38	C₂×#41	—	3	1	no	—
39	C₂²×D₄	—	3	1	no	—
40	#124	—	3	1	no	—
41	#123	—	3	2	no	—
42	C₂²⋊Q₈	—	3	1	no	—
43	#121	—	3	2	no	—
44	#121	D₃	1	1	yes	characteristic, complemented
45	#120	—	3	2	no	—
46	C₄×D₄	—	3	1	no	—
47	C₄×D₄	—	3	1	no	—
48	C₂×C₂²⋊C₄	—	3	2	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	—	2	4	no	—
53	C₂²×D₃	—	4	2	no	—
54	C₂²×D₃	C₂³	1	2	yes	—
55	C₂²×D₃	C₂³	1	1	yes	characteristic
56	C₃×D₄	—	2	2	no	—
57	C₃×D₄	—	4	2	no	—
58	C₂×C₃×C₄	—	2	2	no	—
59	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂³	1	2	yes	—
61	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
62	#67	—	2	4	no	—
63	#67	—	2	4	no	—
64	#67	—	2	2	no	—
65	C₂×Dic₃	C₂³	1	1	yes	characteristic
66	C₂×Dic₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₃	C₂³	1	2	yes	—
69	D₁₂	D₄	1	4	yes	complemented
70	C₄×D₃	—	2	4	no	—
71	C₄×D₃	—	2	2	no	—
72	C₄×D₃	—	4	2	no	—
73	Dic₆	—	2	2	no	—
74	C₂⁴	—	3	2	no	—
75	#41	—	6	4	no	—
76	C₂×Q₈	—	3	1	no	—
77	C₂×D₄	—	3	1	no	—
78	C₂×D₄	—	3	2	no	—
79	C₂×D₄	—	3	2	no	—
80	C₂×D₄	—	3	1	no	—
81	C₂×D₄	—	6	4	no	—
82	C₂×D₄	C₂×D₃	1	1	yes	characteristic
83	C₂×D₄	C₂×D₃	1	2	yes	complemented
84	C₂²×C₄	—	3	2	no	—
85	C₂²×C₄	—	3	1	no	—
86	C₂²×C₄	—	3	2	no	—
87	C₂²×C₄	C₂×D₃	1	1	yes	characteristic, complemented
88	C₄⋊C₄	—	3	1	no	—
89	C₄⋊C₄	—	3	2	no	—
90	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
91	C₂²⋊C₄	—	3	2	no	—
92	C₂²⋊C₄	—	3	1	no	—
93	C₂²⋊C₄	—	3	2	no	—
94	C₂²⋊C₄	—	3	1	no	—
95	C₂²⋊C₄	—	3	2	no	—
96	C₂²⋊C₄	—	3	2	no	—
97	C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
98	C₄²	—	3	1	no	—
99	C₂²×C₃	—	2	1	no	—
100	C₂²×C₃	—	2	2	no	—
101	C₂²×C₃	—	2	1	no	—
102	C₂²×C₃	—	2	2	no	—
103	C₂²×C₃	—	2	1	no	—
104	C₂²×C₃	—	2	2	no	—
105	C₂²×C₃	C₂⁴	1	1	yes	characteristic
106	C₂×D₃	—	2	1	no	—
107	C₂×D₃	—	2	2	no	—
108	C₂×D₃	—	2	2	no	—
109	C₂×D₃	—	2	1	no	—
110	C₂×D₃	—	4	8	no	—
111	C₂×D₃	C₂×D₄	1	4	yes	—
112	C₂×D₃	#41	1	2	yes	complemented
113	C₃×C₄	—	2	1	no	—
114	C₃×C₄	—	2	2	no	—
115	C₃×C₄	C₂×D₄	1	2	yes	—
116	Dic₃	—	2	2	no	—
117	Dic₃	—	2	1	no	—
118	Dic₃	—	2	1	no	—
119	Dic₃	—	2	1	no	—
120	C₂³	—	3	1	no	—
121	C₂³	—	3	2	no	—
122	C₂³	—	6	4	no	—
123	C₂³	—	6	4	no	—
124	C₂³	—	12	2	no	—
125	C₂³	C₂²×D₃	1	2	yes	—
126	C₂³	C₂²×D₃	1	1	yes	characteristic
127	Q₈	—	6	2	no	—
128	D₄	—	2	2	no	—
129	D₄	—	3	4	no	—
130	D₄	—	4	2	no	—
131	D₄	—	6	4	no	—
132	D₄	—	6	4	no	—
133	D₄	—	6	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	3	1	no	—
136	C₂×C₄	—	3	1	no	—
137	C₂×C₄	—	3	2	no	—
138	C₂×C₄	—	3	1	no	—
139	C₂×C₄	—	6	4	no	—
140	C₂×C₄	—	6	2	no	—
141	C₂×C₄	—	12	2	no	—
142	C₂×C₄	C₂²×D₃	1	2	yes	—
143	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₃	—	4	2	no	—
146	C₂×C₃	—	4	1	no	—
147	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
150	D₃	—	2	2	no	—
151	D₃	—	2	4	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	3	4	no	—
159	C₂²	—	3	2	no	—
160	C₂²	—	6	1	no	—
161	C₂²	—	6	2	no	—
162	C₂²	—	6	1	no	—
163	C₂²	—	6	2	no	—
164	C₂²	—	12	8	no	—
165	C₂²	C₂³×D₃	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	6	1	no	—
169	C₄	—	6	2	no	—
170	C₄	—	6	1	no	—
171	C₄	—	6	1	no	—
172	C₄	D₃×D₄	1	2	yes	—
173	C₃	#546	1	1	yes	characteristic, complemented, Sylow
174	C₂	—	4	2	no	—
175	C₂	—	4	1	no	—
176	C₂	—	6	2	no	—
177	C₂	—	6	4	no	—
178	C₂	C₂×D₃×D₄	1	1	yes	characteristic
179	C₂	#1004	1	1	yes	characteristic
180	C₂	D₃×#41	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	#546
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. #938	C₂	semidirect
5. #937	C₂	semidirect
6. #936	C₂	semidirect
7. C₄×#67	C₂	semidirect
8. #897	C₂	semidirect
9. #893	C₂	semidirect
10. #883	C₂	semidirect
11. D₃×C₂²⋊C₄	C₂	semidirect
15. C₂×C₃×D₄	C₂²	semidirect
16. C₂²×C₃×C₄	C₂²	semidirect
19. C₂×#67	C₂²	semidirect
24. C₂×C₄×D₃	C₂²	semidirect
25. C₂×C₄×D₃	C₂²	semidirect
26. C₂×Dic₆	C₂²	semidirect
27. C₃×C₄⋊C₄	C₂²	semidirect
28. C₃×C₂²⋊C₄	C₂²	semidirect
29. #217	C₂²	semidirect
30. #217	C₂²	semidirect
31. #217	C₂²	semidirect
34. (C₂×D₃)⋊C₄	C₂²	semidirect
35. C₄⋊Dic₃	C₂²	semidirect
36. Dic₃⋊C₄	C₂²	semidirect
37. C₄×Dic₃	C₂²	semidirect
44. #121	D₃	semidirect
69. D₁₂	D₄	semidirect
83. C₂×D₄	C₂×D₃	semidirect
87. C₂²×C₄	C₂×D₃	semidirect
90. C₄⋊C₄	C₂×D₃	semidirect
97. C₂²⋊C₄	C₂×D₃	semidirect
112. C₂×D₃	#41	semidirect
173. C₃	#546	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
#938	2
#937	2
#936	2
C₄×#67	2
#897	2
#893	2
#883	2
D₃×C₂²⋊C₄	2
#546	3