Finite Group Databasebeta

Subgroups — #5586

Order 192.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂³×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	#953	C₂	1	1	yes	maximal, characteristic, complemented
5	#939	C₂	1	2	yes	maximal, complemented
6	#935	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Dic₃	C₂	1	2	yes	maximal, complemented
8	#886	C₂	1	1	yes	maximal, characteristic, complemented
9	#67⋊C₄	C₂	1	2	yes	maximal, complemented
10	#878	C₂	1	2	yes	maximal, complemented
11	#877	C₂	1	1	yes	maximal, characteristic, complemented
12	#525	—	3	1	no	maximal, Sylow
13	C₂⁴×C₃	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₄	C₂²	1	2	yes	—
15	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂×#67	C₂²	1	2	yes	—
17	C₂×#67	C₂²	1	1	yes	characteristic
18	C₂×#67	C₂²	1	1	yes	characteristic, complemented
19	C₂²×Dic₃	—	2	4	no	—
20	C₂²×Dic₃	C₂²	1	2	yes	—
21	C₂²×Dic₃	C₂²	1	2	yes	—
22	C₂²×Dic₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₃	C₂²	1	1	yes	characteristic
24	C₂²×Dic₃	C₂²	1	1	yes	characteristic, complemented
25	#237	—	2	4	no	—
26	C₂×C₄×D₃	C₂²	1	1	yes	characteristic, complemented
27	C₂×Dic₆	C₂²	1	1	yes	characteristic, complemented
28	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	#217	C₂²	1	1	yes	characteristic
31	#217	C₂²	1	2	yes	—
32	#217	C₂²	1	2	yes	complemented
33	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
34	C₄⋊Dic₃	C₂²	1	2	yes	complemented
35	Dic₃⋊C₄	C₂²	1	2	yes	complemented
36	Dic₃⋊C₄	C₂²	1	2	yes	complemented
37	C₄×Dic₃	C₂²	1	2	yes	complemented
38	C₂×#41	—	3	1	no	—
39	C₂³×C₄	—	3	1	no	—
40	#123	—	3	1	no	—
41	#123	—	3	1	no	—
42	C₂²⋊Q₈	—	3	2	no	—
43	#121	—	3	2	no	—
44	#120	—	3	1	no	—
45	#120	D₃	1	1	yes	characteristic, complemented
46	C₄×D₄	—	3	2	no	—
47	C₄×D₄	—	3	2	no	—
48	#117	—	3	1	no	—
49	C₂³×C₃	—	2	2	no	—
50	C₂³×C₃	—	2	4	no	—
51	C₂³×C₃	C₂³	1	1	yes	characteristic
52	C₂³×C₃	C₂³	1	1	yes	characteristic
53	C₂³×C₃	C₂³	1	2	yes	—
54	C₂²×D₃	C₂³	1	1	yes	characteristic
55	C₃×D₄	—	2	2	no	—
56	C₃×D₄	—	2	4	no	—
57	C₂×C₃×C₄	C₂³	1	2	yes	—
58	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
59	#67	—	2	2	no	—
60	#67	—	2	4	no	—
61	#67	—	2	2	no	—
62	C₂×Dic₃	—	2	2	no	—
63	C₂×Dic₃	—	2	2	no	—
64	C₂×Dic₃	—	2	4	no	—
65	C₂×Dic₃	—	2	4	no	—
66	C₂×Dic₃	D₄	1	4	yes	complemented
67	C₂×Dic₃	C₂³	1	2	yes	—
68	C₂×Dic₃	C₂³	1	2	yes	—
69	C₂×Dic₃	C₂³	1	1	yes	characteristic
70	C₂×Dic₃	C₂³	1	1	yes	characteristic
71	C₂×Dic₃	C₂³	1	1	yes	characteristic
72	C₄×D₃	—	2	2	no	—
73	Dic₆	—	2	2	no	—
74	C₂⁴	C₂×D₃	1	1	yes	characteristic
75	#41	—	6	4	no	—
76	C₂×Q₈	—	3	1	no	—
77	C₂×D₄	—	3	2	no	—
78	C₂×D₄	—	3	1	no	—
79	C₂×D₄	—	3	1	no	—
80	C₂×D₄	C₂×D₃	1	2	yes	—
81	C₂×D₄	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₄	—	3	1	no	—
83	C₂²×C₄	—	3	2	no	—
84	C₂²×C₄	—	3	1	no	—
85	C₂²×C₄	—	3	2	no	—
86	C₂²×C₄	—	3	1	no	—
87	C₂²×C₄	—	3	1	no	—
88	C₂²×C₄	—	6	4	no	—
89	C₄⋊C₄	—	3	2	no	—
90	C₄⋊C₄	—	3	2	no	—
91	C₄⋊C₄	—	3	2	no	—
92	C₂²⋊C₄	—	3	1	no	—
93	C₂²⋊C₄	—	3	2	no	—
94	C₂²⋊C₄	—	3	2	no	—
95	C₂²⋊C₄	—	3	2	no	—
96	C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
97	C₂²⋊C₄	C₂×D₃	1	2	yes	—
98	C₄²	—	3	2	no	—
99	C₂²×C₃	—	2	8	no	—
100	C₂²×C₃	—	2	2	no	—
101	C₂²×C₃	—	2	2	no	—
102	C₂²×C₃	—	2	2	no	—
103	C₂²×C₃	—	2	1	no	—
104	C₂²×C₃	—	2	2	no	—
105	C₂²×C₃	C₂×D₄	1	2	yes	—
106	C₂²×C₃	#41	1	4	yes	complemented
107	C₂²×C₃	C₂⁴	1	1	yes	characteristic
108	C₂×D₃	—	2	1	no	—
109	C₂×D₃	—	2	2	no	—
110	C₃×C₄	—	2	1	no	—
111	C₃×C₄	—	2	2	no	—
112	Dic₃	—	2	1	no	—
113	Dic₃	—	2	2	no	—
114	Dic₃	—	2	2	no	—
115	Dic₃	C₂×D₄	1	2	yes	—
116	Dic₃	C₂×D₄	1	2	yes	—
117	C₂³	—	2	2	no	—
118	C₂³	—	2	4	no	—
119	C₂³	—	3	1	no	—
120	C₂³	C₂²×D₃	1	1	yes	characteristic
121	C₂³	C₂²×D₃	1	1	yes	characteristic
122	C₂³	C₂²×D₃	1	2	yes	—
123	Q₈	—	6	2	no	—
124	D₄	—	2	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	6	2	no	—
127	D₄	—	6	2	no	—
128	D₄	—	6	4	no	—
129	C₂×C₄	—	3	4	no	—
130	C₂×C₄	—	3	1	no	—
131	C₂×C₄	—	3	1	no	—
132	C₂×C₄	—	3	2	no	—
133	C₂×C₄	—	3	2	no	—
134	C₂×C₄	—	3	1	no	—
135	C₂×C₄	—	6	2	no	—
136	C₂×C₄	—	6	4	no	—
137	C₂×C₄	—	6	4	no	—
138	C₂×C₄	—	6	2	no	—
139	C₂×C₄	—	6	2	no	—
140	C₂×C₄	C₂²×D₃	1	2	yes	—
141	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₃	—	2	2	no	—
143	C₂×C₃	—	2	4	no	—
144	C₂×C₃	—	4	1	no	—
145	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₃	C₂×#41	1	2	yes	—
147	D₃	—	4	1	no	—
148	C₂²	—	2	8	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	6	1	no	—
155	C₂²	—	6	2	no	—
156	C₂²	D₃×D₄	1	2	yes	—
157	C₂²	#237	1	4	yes	complemented
158	C₂²	C₂³×D₃	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	3	2	no	—
162	C₄	—	3	2	no	—
163	C₄	—	6	2	no	—
164	C₄	—	6	1	no	—
165	C₄	—	6	2	no	—
166	C₃	#525	1	1	yes	characteristic, complemented, Sylow
167	C₂	—	2	2	no	—
168	C₂	—	2	4	no	—
169	C₂	—	4	1	no	—
170	C₂	—	12	1	no	—
171	C₂	C₂×D₃×D₄	1	1	yes	characteristic
172	C₂	C₂×#237	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#120	C₂
hypercenter	#120	D₃
hypocenter	C₃	#525
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. C₂³×Dic₃	C₂	semidirect
2. C₂×#237	C₂	semidirect
3. C₃×#120	C₂	semidirect
4. #953	C₂	semidirect
5. #939	C₂	semidirect
6. #935	C₂	semidirect
7. D₄×Dic₃	C₂	semidirect
8. #886	C₂	semidirect
9. #67⋊C₄	C₂	semidirect
10. #878	C₂	semidirect
11. #877	C₂	semidirect
18. C₂×#67	C₂²	semidirect
24. C₂²×Dic₃	C₂²	semidirect
26. C₂×C₄×D₃	C₂²	semidirect
27. C₂×Dic₆	C₂²	semidirect
32. #217	C₂²	semidirect
33. (C₂×D₃)⋊C₄	C₂²	semidirect
34. C₄⋊Dic₃	C₂²	semidirect
35. Dic₃⋊C₄	C₂²	semidirect
36. Dic₃⋊C₄	C₂²	semidirect
37. C₄×Dic₃	C₂²	semidirect
45. #120	D₃	semidirect
66. C₂×Dic₃	D₄	semidirect
106. C₂²×C₃	#41	semidirect
157. C₂²	#237	semidirect
166. C₃	#525	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Dic₃	2
C₂×#237	2
C₃×#120	2
#953	2
#939	2
#935	2
D₄×Dic₃	2
#886	2
#67⋊C₄	2
#878	2
#877	2
#525	3