Finite Group Databasebeta

Subgroups — #5592

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	C₂²×#67	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	#940	C₂	1	2	yes	maximal, complemented
6	#939	C₂	1	2	yes	maximal, complemented
7	#937	C₂	1	1	yes	maximal, characteristic, complemented
8	#885	C₂	1	2	yes	maximal, complemented
9	#884	C₂	1	2	yes	maximal, complemented
10	#882	C₂	1	1	yes	maximal, characteristic, complemented
11	#877	C₂	1	1	yes	maximal, characteristic, complemented
12	#535	—	3	1	no	maximal, Sylow
13	C₂⁴×C₃	C₂²	1	1	yes	characteristic
14	C₂³×D₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₄	C₂²	1	2	yes	complemented
17	C₂×#67	—	2	4	no	—
18	C₂×#67	C₂²	1	1	yes	characteristic
19	C₂×#67	C₂²	1	2	yes	—
20	C₂×#67	C₂²	1	2	yes	—
21	C₂×#67	C₂²	1	1	yes	characteristic, complemented
22	C₂×#67	C₂²	1	2	yes	complemented
23	C₂×#67	C₂²	1	2	yes	complemented
24	C₂²×Dic₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₃	C₂²	1	1	yes	characteristic, complemented
26	#237	—	2	4	no	—
27	C₂×D₁₂	C₂²	1	2	yes	complemented
28	C₂×C₄×D₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×Dic₆	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
32	#217	C₂²	1	1	yes	characteristic
33	#217	C₂²	1	2	yes	complemented
34	(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
35	(C₂×D₃)⋊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	C₂²×D₄	—	3	1	no	—
40	#127	—	3	2	no	—
41	#124	—	3	2	no	—
42	#121	—	3	2	no	—
43	#121	—	3	2	no	—
44	#120	—	3	1	no	—
45	#120	—	3	1	no	—
46	#120	—	3	1	no	—
47	#120	D₃	1	1	yes	characteristic, complemented
48	#117	—	3	1	no	—
49	C₂³×C₃	—	2	2	no	—
50	C₂³×C₃	—	4	2	no	—
51	C₂³×C₃	C₂³	1	2	yes	—
52	C₂³×C₃	C₂³	1	1	yes	characteristic
53	C₂³×C₃	C₂³	1	1	yes	characteristic
54	C₂²×D₃	—	2	2	no	—
55	C₂²×D₃	—	4	2	no	—
56	C₂²×D₃	C₂³	1	2	yes	—
57	C₂²×D₃	C₂³	1	1	yes	characteristic
58	C₃×D₄	—	2	2	no	—
59	C₃×D₄	—	4	2	no	—
60	C₂×C₃×C₄	C₂³	1	2	yes	—
61	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
62	#67	—	2	2	no	—
63	#67	—	2	2	no	—
64	#67	—	2	4	no	—
65	#67	—	2	4	no	—
66	#67	—	4	2	no	—
67	#67	—	4	2	no	—
68	C₂×Dic₃	—	2	2	no	—
69	C₂×Dic₃	D₄	1	4	yes	complemented
70	C₂×Dic₃	C₂³	1	2	yes	—
71	C₂×Dic₃	C₂³	1	1	yes	characteristic
72	C₂×Dic₃	C₂³	1	1	yes	characteristic
73	C₂×Dic₃	C₂³	1	1	yes	characteristic
74	D₁₂	—	4	2	no	—
75	C₄×D₃	—	2	2	no	—
76	Dic₆	—	2	2	no	—
77	C₂⁴	—	3	1	no	—
78	C₂⁴	C₂×D₃	1	1	yes	characteristic
79	#41	—	6	4	no	—
80	C₂×Q₈	—	3	1	no	—
81	C₂×D₄	—	3	1	no	—
82	C₂×D₄	—	3	2	no	—
83	C₂×D₄	—	3	2	no	—
84	C₂×D₄	—	3	1	no	—
85	C₂×D₄	—	3	2	no	—
86	C₂×D₄	—	3	2	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	—	3	1	no	—
94	C₄⋊C₄	—	3	2	no	—
95	C₂²⋊C₄	—	3	1	no	—
96	C₂²⋊C₄	—	3	2	no	—
97	C₂²⋊C₄	—	3	2	no	—
98	C₂²⋊C₄	—	3	2	no	—
99	C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
100	C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
101	C₄²	—	3	2	no	—
102	C₂²×C₃	—	2	2	no	—
103	C₂²×C₃	—	2	1	no	—
104	C₂²×C₃	—	2	2	no	—
105	C₂²×C₃	—	2	2	no	—
106	C₂²×C₃	—	2	2	no	—
107	C₂²×C₃	—	2	2	no	—
108	C₂²×C₃	—	4	4	no	—
109	C₂²×C₃	C₂×D₄	1	2	yes	—
110	C₂²×C₃	C₂⁴	1	1	yes	characteristic
111	C₂×D₃	—	2	1	no	—
112	C₂×D₃	—	2	2	no	—
113	C₂×D₃	—	2	2	no	—
114	C₂×D₃	—	2	2	no	—
115	C₂×D₃	—	2	2	no	—
116	C₂×D₃	—	4	4	no	—
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	2	2	no	—
119	Dic₃	—	2	1	no	—
120	Dic₃	—	2	2	no	—
121	Dic₃	C₂×D₄	1	2	yes	—
122	Dic₃	C₂×D₄	1	2	yes	—
123	C₂³	—	2	2	no	—
124	C₂³	—	3	2	no	—
125	C₂³	—	3	1	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	6	2	no	—
128	C₂³	—	12	2	no	—
129	C₂³	C₂²×D₃	1	2	yes	—
130	C₂³	C₂²×D₃	1	1	yes	characteristic
131	C₂³	C₂²×D₃	1	1	yes	characteristic
132	Q₈	—	6	2	no	—
133	D₄	—	2	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	6	4	no	—
136	D₄	—	6	4	no	—
137	D₄	—	6	2	no	—
138	D₄	—	6	2	no	—
139	D₄	—	12	2	no	—
140	D₄	—	12	2	no	—
141	D₄	—	12	2	no	—
142	C₂×C₄	—	3	4	no	—
143	C₂×C₄	—	3	1	no	—
144	C₂×C₄	—	3	1	no	—
145	C₂×C₄	—	3	2	no	—
146	C₂×C₄	—	3	1	no	—
147	C₂×C₄	—	6	2	no	—
148	C₂×C₄	—	6	2	no	—
149	C₂×C₄	C₂²×D₃	1	2	yes	—
150	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₃	—	2	2	no	—
152	C₂×C₃	—	4	1	no	—
153	C₂×C₃	—	4	2	no	—
154	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
156	D₃	—	4	1	no	—
157	D₃	—	4	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	4	4	no	—
165	C₂²	—	6	1	no	—
166	C₂²	—	6	2	no	—
167	C₂²	—	6	2	no	—
168	C₂²	—	6	2	no	—
169	C₂²	—	6	2	no	—
170	C₂²	—	12	4	no	—
171	C₂²	D₃×D₄	1	2	yes	—
172	C₂²	C₂³×D₃	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	2	no	—
175	C₄	—	3	2	no	—
176	C₄	—	3	2	no	—
177	C₄	—	6	1	no	—
178	C₄	—	6	2	no	—
179	C₃	#535	1	1	yes	characteristic, complemented, Sylow
180	C₂	—	2	2	no	—
181	C₂	—	4	2	no	—
182	C₂	—	4	1	no	—
183	C₂	—	12	2	no	—
184	C₂	—	12	1	no	—
185	C₂	C₂×D₃×D₄	1	1	yes	characteristic
186	C₂	#1004	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	#535
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×#237	C₂	semidirect
3. C₃×#120	C₂	semidirect
4. #953	C₂	semidirect
5. #940	C₂	semidirect
6. #939	C₂	semidirect
7. #937	C₂	semidirect
8. #885	C₂	semidirect
9. #884	C₂	semidirect
10. #882	C₂	semidirect
11. #877	C₂	semidirect
16. C₂×C₃×D₄	C₂²	semidirect
21. C₂×#67	C₂²	semidirect
22. C₂×#67	C₂²	semidirect
23. C₂×#67	C₂²	semidirect
25. C₂²×Dic₃	C₂²	semidirect
27. C₂×D₁₂	C₂²	semidirect
28. C₂×C₄×D₃	C₂²	semidirect
29. C₂×Dic₆	C₂²	semidirect
31. C₃×C₂²⋊C₄	C₂²	semidirect
33. #217	C₂²	semidirect
34. (C₂×D₃)⋊C₄	C₂²	semidirect
35. (C₂×D₃)⋊C₄	C₂²	semidirect
36. Dic₃⋊C₄	C₂²	semidirect
37. C₄×Dic₃	C₂²	semidirect
47. #120	D₃	semidirect
69. C₂×Dic₃	D₄	semidirect
90. C₂×D₄	C₂×D₃	semidirect
100. C₂²⋊C₄	C₂×D₃	semidirect
179. C₃	#535	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×#237	2
C₃×#120	2
#953	2
#940	2
#939	2
#937	2
#885	2
#884	2
#882	2
#877	2
#535	3