Finite Group Databasebeta

Subgroups — #5608

Order 192.

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

normal subgroups only

This group has 294 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×#237	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	#939	C₂	1	2	yes	maximal, complemented
5	#938	C₂	1	2	yes	maximal, complemented
6	#938	C₂	1	1	yes	maximal, characteristic, complemented
7	#924	C₂	1	1	yes	maximal, characteristic, complemented
8	#897	C₂	1	1	yes	maximal, characteristic, complemented
9	D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#884	C₂	1	2	yes	maximal, complemented
11	#878	C₂	1	2	yes	maximal, complemented
12	#537	—	3	1	no	maximal, Sylow
13	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄	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
19	C₂×#67	C₂²	1	2	yes	—
20	C₂²×Dic₃	C₂²	1	2	yes	—
21	#237	—	2	4	no	—
22	#235	—	2	4	no	—
23	C₂×D₁₂	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₃	C₂²	1	2	yes	—
26	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
27	C₂×Dic₆	C₂²	1	1	yes	characteristic
28	C₂×Dic₆	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	#217	C₂²	1	2	yes	—
32	#217	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
34	C₄⋊Dic₃	C₂²	1	2	yes	—
35	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
36	Dic₃⋊C₄	C₂²	1	2	yes	—
37	Dic₃⋊C₄	C₂²	1	2	yes	—
38	C₂×#41	—	3	1	no	—
39	C₂×#41	—	3	1	no	—
40	C₂²⋊Q₈	—	3	1	no	—
41	C₂²⋊Q₈	—	3	1	no	—
42	C₂²⋊Q₈	—	3	2	no	—
43	#121	—	3	1	no	—
44	#121	—	3	2	no	—
45	#121	—	3	2	no	—
46	#121	—	3	2	no	—
47	#121	D₃	1	1	yes	characteristic, complemented
48	C₂×C₄⋊C₄	—	3	1	no	—
49	C₂³×C₃	C₂³	1	1	yes	characteristic
50	C₂³×C₃	C₂³	1	2	yes	—
51	C₂²×D₃	C₂³	1	1	yes	characteristic
52	C₂²×D₃	C₂³	1	1	yes	characteristic
53	C₃×D₄	—	2	2	no	—
54	C₃×D₄	—	4	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
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	#67	—	4	2	no	—
63	C₂×Dic₃	—	2	4	no	—
64	C₂×Dic₃	C₂³	1	2	yes	—
65	C₂×Dic₃	C₂³	1	2	yes	—
66	C₂×Dic₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₃	C₂³	1	1	yes	characteristic
68	D₁₂	—	2	2	no	—
69	C₄×D₃	—	2	2	no	—
70	C₄×D₃	—	4	2	no	—
71	C₄×D₃	D₄	1	4	yes	complemented
72	Dic₆	—	2	2	no	—
73	Dic₆	—	2	2	no	—
74	#41	—	6	4	no	—
75	#41	—	6	4	no	—
76	C₂×Q₈	—	3	1	no	—
77	C₂×Q₈	—	3	1	no	—
78	C₂×D₄	—	3	1	no	—
79	C₂×D₄	—	3	1	no	—
80	C₂×D₄	—	3	2	no	—
81	C₂×D₄	—	3	2	no	—
82	C₂×D₄	—	3	1	no	—
83	C₂×D₄	C₂×D₃	1	1	yes	characteristic
84	C₂×D₄	C₂×D₃	1	2	yes	—
85	C₂²×C₄	—	3	1	no	—
86	C₂²×C₄	—	3	2	no	—
87	C₂²×C₄	—	3	1	no	—
88	C₂²×C₄	—	3	2	no	—
89	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
90	C₄⋊C₄	—	3	2	no	—
91	C₄⋊C₄	—	3	1	no	—
92	C₄⋊C₄	—	3	2	no	—
93	C₄⋊C₄	—	3	2	no	—
94	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
95	C₂²⋊C₄	—	3	2	no	—
96	C₂²⋊C₄	—	3	2	no	—
97	C₂²⋊C₄	—	3	2	no	—
98	C₂²⋊C₄	C₂×D₃	1	2	yes	—
99	C₂²×C₃	—	2	2	no	—
100	C₂²×C₃	—	2	1	no	—
101	C₂²×C₃	—	2	2	no	—
102	C₂²×C₃	—	2	1	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	1	no	—
108	C₂×D₃	—	2	1	no	—
109	C₂×D₃	—	2	1	no	—
110	C₂×D₃	—	2	1	no	—
111	C₂×D₃	C₂×D₄	1	2	yes	—
112	C₃×C₄	—	2	1	no	—
113	C₃×C₄	—	2	2	no	—
114	C₃×C₄	C₂×D₄	1	2	yes	—
115	Dic₃	—	2	1	no	—
116	Dic₃	—	2	2	no	—
117	Dic₃	—	2	2	no	—
118	Dic₃	C₂×D₄	1	2	yes	—
119	C₂³	—	3	1	no	—
120	C₂³	—	3	1	no	—
121	C₂³	C₂²×D₃	1	1	yes	characteristic
122	C₂³	C₂²×D₃	1	2	yes	—
123	Q₈	—	6	2	no	—
124	Q₈	—	6	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	4	2	no	—
127	D₄	—	6	2	no	—
128	D₄	—	6	2	no	—
129	D₄	—	6	2	no	—
130	D₄	—	6	4	no	—
131	D₄	—	12	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	3	1	no	—
134	C₂×C₄	—	3	4	no	—
135	C₂×C₄	—	3	2	no	—
136	C₂×C₄	—	3	1	no	—
137	C₂×C₄	—	3	2	no	—
138	C₂×C₄	—	6	4	no	—
139	C₂×C₄	—	6	2	no	—
140	C₂×C₄	—	12	2	no	—
141	C₂×C₄	C₂²×D₃	1	2	yes	—
142	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₃	—	4	2	no	—
145	C₂×C₃	—	4	1	no	—
146	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
149	D₃	—	2	2	no	—
150	D₃	—	4	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	3	2	no	—
158	C₂²	—	6	1	no	—
159	C₂²	—	6	1	no	—
160	C₂²	—	6	1	no	—
161	C₂²	—	6	1	no	—
162	C₂²	—	6	1	no	—
163	C₂²	C₂³×D₃	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	2	2	no	—
166	C₄	—	3	2	no	—
167	C₄	—	6	2	no	—
168	C₄	—	6	1	no	—
169	C₄	—	6	2	no	—
170	C₄	D₃×D₄	1	2	yes	—
171	C₃	#537	1	1	yes	characteristic, complemented, Sylow
172	C₂	—	4	1	no	—
173	C₂	—	4	2	no	—
174	C₂	—	6	2	no	—
175	C₂	—	12	1	no	—
176	C₂	C₂×D₃×D₄	1	1	yes	characteristic
177	C₂	#1004	1	1	yes	characteristic
178	C₂	#1010	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	#537
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×#237	C₂	semidirect
2. C₂×#235	C₂	semidirect
3. C₃×#121	C₂	semidirect
4. #939	C₂	semidirect
5. #938	C₂	semidirect
6. #938	C₂	semidirect
7. #924	C₂	semidirect
8. #897	C₂	semidirect
9. D₃×C₄⋊C₄	C₂	semidirect
10. #884	C₂	semidirect
11. #878	C₂	semidirect
47. #121	D₃	semidirect
71. C₄×D₃	D₄	semidirect
171. C₃	#537	semidirect

Maximal subgroups

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

subgroup	index
C₂×#237	2
C₂×#235	2
C₃×#121	2
#939	2
#938	2
#924	2
#897	2
D₃×C₄⋊C₄	2
#884	2
#878	2
#537	3