Finite Group Databasebeta

Subgroups — #27157326

Order 12416.

176 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 64 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	Q₈×#11715897	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261763	C₂	1	1	yes	maximal, characteristic
3	#20261599	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#11715897	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261583	C₂	1	1	yes	maximal, characteristic
6	#20261576	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260820	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095519	—	2	1	no	—
9	C₄⋊#11715897	—	2	1	no	—
10	C₄⋊#11715897	C₂²	1	1	yes	characteristic
11	C₄⋊#11715897	C₂²	1	1	yes	characteristic
12	#15690867	—	2	1	no	—
13	C₄×#11715897	—	2	1	no	—
14	C₄×#11715897	C₂²	1	1	yes	characteristic
15	C₄×#11715897	C₂²	1	1	yes	characteristic
16	C₂×#13095498	C₄	1	1	yes	characteristic
17	Q₈×Dic₉₇	C₂²	1	1	yes	characteristic
18	#15690772	C₂²	1	1	yes	characteristic
19	#15690741	C₄	1	1	yes	characteristic
20	#15690732	C₄	1	1	yes	characteristic
21	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
22	#103×C₉₇	C₄	1	1	yes	characteristic
23	#13095519	—	4	2	no	—
24	C₂×#11715897	—	2	1	no	—
25	C₂×#11715897	—	2	1	no	—
26	C₂×#11715897	—	2	1	no	—
27	C₂×#11715897	D₄	1	1	yes	characteristic
28	C₂×#11715897	D₄	1	1	yes	characteristic
29	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₉₇	—	2	1	no	—
31	#13095498	C₈	1	4	yes	complemented
32	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₉₇	—	2	1	no	—
34	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
35	#13095488	—	2	1	no	—
36	C₄×Dic₉₇	—	2	1	no	—
37	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₉₇	—	2	1	no	—
42	Q₈×C₉₇	—	2	1	no	—
43	Q₈×C₉₇	C₂×C₈	1	2	yes	complemented
44	C₂×C₄×C₉₇	—	2	1	no	—
45	C₂×C₄×C₉₇	—	2	1	no	—
46	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₉₇	—	2	1	no	—
48	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₉₇	#41	1	1	yes	characteristic
50	D₃₈₈	—	2	1	no	—
51	D₃₈₈	C₂×C₈	1	2	yes	—
52	C₄×D₉₇	—	4	1	no	—
53	C₄×D₉₇	—	4	1	no	—
54	#11715897	—	2	1	no	—
55	#11715897	—	2	1	no	—
56	#11715897	—	4	1	no	—
57	#11715897	—	4	1	no	—
58	#11715897	—	4	1	no	—
59	C₈×C₉₇	—	2	1	no	—
60	#11715895	C₂×C₈	1	2	yes	complemented
61	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
62	C₂×D₉₇	—	2	2	no	—
63	C₂×D₉₇	—	4	1	no	—
64	C₂×D₉₇	—	4	1	no	—
65	C₄×C₉₇	—	2	2	no	—
66	C₄×C₉₇	—	4	1	no	—
67	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
68	C₄×C₉₇	#131	1	1	yes	characteristic
69	Dic₉₇	—	2	1	no	—
70	Dic₉₇	—	4	1	no	—
71	Dic₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₉₇	#137	1	1	yes	characteristic
73	C₂×C₉₇	C₈×D₄	1	1	yes	characteristic
74	C₂×C₉₇	#440	1	1	yes	characteristic
75	C₂×C₉₇	#444	1	1	yes	characteristic
76	D₉₇	—	4	2	no	—
77	SD₁₆⋊C₈	—	97	1	no	maximal, Sylow
78	C₉₇	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Sylow
79	C₈×Q₈	—	97	1	no	—
80	C₄×SD₁₆	—	97	1	no	—
81	C₈×D₄	—	97	1	no	—
82	#335	—	97	1	no	—
83	Q₈⋊C₈	—	97	1	no	—
84	D₄⋊C₈	—	97	1	no	—
85	#322	—	97	1	no	—
86	C₂×SD₁₆	—	97	1	no	—
87	C₂²×C₈	—	194	1	no	—
88	C₄×Q₈	—	97	1	no	—
89	C₄×D₄	—	97	1	no	—
90	#106	—	97	1	no	—
91	C₄⋊C₈	—	97	1	no	—
92	C₄⋊C₈	—	97	1	no	—
93	C₄⋊C₈	—	194	1	no	—
94	#103	#87604	1	1	yes	characteristic
95	#102	—	97	1	no	—
96	C₂²⋊C₈	—	194	1	no	—
97	C₄×C₈	—	97	1	no	—
98	C₄×C₈	—	97	1	no	—
99	C₄×C₈	—	97	1	no	—
100	C₄×C₈	—	194	1	no	—
101	C₂×Q₈	C₂×#87604	1	1	yes	characteristic
102	C₂×D₄	—	97	1	no	—
103	C₂²×C₄	—	194	1	no	—
104	SD₁₆	—	97	4	no	—
105	C₂×C₈	—	97	1	no	—
106	C₂×C₈	—	97	1	no	—
107	C₂×C₈	—	97	1	no	—
108	C₂×C₈	—	388	2	no	—
109	C₂×C₈	—	194	1	no	—
110	C₂×C₈	—	194	1	no	—
111	C₂×C₈	—	194	1	no	—
112	C₂×C₈	C₂×#87604	1	1	yes	characteristic
113	C₄⋊C₄	—	97	1	no	—
114	C₄⋊C₄	—	194	1	no	—
115	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic
116	C₂²⋊C₄	—	194	1	no	—
117	C₄²	—	97	1	no	—
118	C₄²	—	194	1	no	—
119	C₂³	—	194	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#13095519	1	2	yes	complemented
122	D₄	—	97	2	no	—
123	D₄	—	194	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	97	1	no	—
127	C₂×C₄	—	97	1	no	—
128	C₂×C₄	—	388	1	no	—
129	C₂×C₄	—	388	1	no	—
130	C₂×C₄	—	194	1	no	—
131	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	97	2	no	—
134	C₈	—	388	1	no	—
135	C₈	—	388	1	no	—
136	C₈	—	388	1	no	—
137	C₈	—	194	1	no	—
138	C₈	—	194	1	no	—
139	C₂²	—	388	1	no	—
140	C₂²	—	388	1	no	—
141	C₂²	—	194	2	no	—
142	C₂²	D₄×#87604	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	1	no	—
145	C₄	—	97	1	no	—
146	C₄	—	97	1	no	—
147	C₄	—	388	1	no	—
148	C₄	—	194	1	no	—
149	C₄	C₂×#13095519	1	1	yes	characteristic
150	C₄	#15690897	1	1	yes	characteristic
151	C₂	—	388	2	no	—
152	C₂	#20261762	1	1	yes	characteristic
153	C₂	#20261774	1	1	yes	characteristic
154	C₂	#20261781	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#87604
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting	#103×C₉₇	C₄
socle	C₂²×C₉₇	C₄×D₄
prime radical	C₂×D₃₈₈	C₂×C₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	#103	#87604
hypocenter	C₉₇	SD₁₆⋊C₈
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. Q₈×#11715897	C₂	semidirect
3. #20261599	C₂	semidirect
4. Q₈⋊#11715897	C₂	semidirect
6. #20261576	C₂	semidirect
31. #13095498	C₈	semidirect
43. Q₈×C₉₇	C₂×C₈	semidirect
60. #11715895	C₂×C₈	semidirect
78. C₉₇	SD₁₆⋊C₈	semidirect
121. Q₈	#13095519	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11715897	2
#20261763	2
#20261599	2
Q₈⋊#11715897	2
#20261583	2
#20261576	2
#20260820	2
SD₁₆⋊C₈	97