Subgroups — #27158265

Order 12416.

552 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#87604	C₂	1	2	yes	maximal
2	(C₂×#87604)⋊Q₈	C₂	1	2	yes	maximal
3	#20261686	C₂	1	2	yes	maximal
4	#20261684	C₂	1	4	yes	maximal
5	C₄⋊C₄×#87604	C₂	1	2	yes	maximal
6	C₄×C₄⋊#87604	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊#87604	C₂	1	1	yes	maximal, characteristic
8	C₄⋊Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic
9	Q₈×#87604	—	2	8	no	—
10	C₂×C₄⋊#87604	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#87604	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#87604	C₂²	1	4	yes	—
13	C₂×C₄⋊#87604	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#87604	C₂²	1	1	yes	characteristic
15	C₂×C₄×#87604	C₂²	1	2	yes	—
16	C₂×C₄×#87604	C₂²	1	2	yes	—
17	C₂×C₄×#87604	C₂²	1	4	yes	—
18	(C₂×#87604)⋊C₄	C₂²	1	2	yes	—
19	(C₂×#87604)⋊C₄	C₂²	1	2	yes	—
20	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
21	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
22	C₂×Q₈×D₉₇	C₂²	1	2	yes	—
23	#15690810	C₄	1	1	yes	characteristic, complemented
24	#15690809	C₄	1	2	yes	complemented
25	#15690809	C₄	1	2	yes	complemented
26	#15690808	C₄	1	2	yes	complemented
27	C₄⋊C₄×D₉₇	C₂²	1	2	yes	—
28	C₄⋊C₄×D₉₇	C₂²	1	2	yes	—
29	C₄²×D₉₇	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic, complemented
31	C₂²×#87604	C₂³	1	2	yes	—
32	C₂²×#87604	C₂³	1	2	yes	—
33	C₂²×#87604	C₂³	1	2	yes	—
34	C₂²×#87604	C₂³	1	2	yes	—
35	C₄⋊#87604	—	2	8	no	—
36	C₄⋊#87604	—	2	2	no	—
37	C₄⋊#87604	—	2	2	no	—
38	C₄⋊#87604	D₄	1	4	yes	—
39	C₄⋊#87604	Q₈	1	4	yes	complemented
40	C₄×#87604	—	2	8	no	—
41	C₄×#87604	—	2	4	no	—
42	C₄×#87604	—	2	4	no	—
43	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
44	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
45	Q₈×D₉₇	—	2	4	no	—
46	Q₈×D₉₇	—	2	4	no	—
47	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₉₇	C₂³	1	4	yes	—
49	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
53	#13095490	C₂×C₄	1	4	yes	—
54	#13095490	C₂×C₄	1	4	yes	—
55	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
58	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
61	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×D₉₇	C₂⁴	1	1	yes	characteristic
63	C₂×#87604	—	2	2	no	—
64	C₂×#87604	—	2	2	no	—
65	C₂×#87604	—	2	2	no	—
66	C₂×#87604	—	2	2	no	—
67	C₂×#87604	—	2	2	no	—
68	C₂×#87604	—	2	2	no	—
69	C₂×#87604	—	2	2	no	—
70	C₂×#87604	—	2	2	no	—
71	C₂×#87604	—	2	2	no	—
72	C₂×#87604	—	2	2	no	—
73	C₂×#87604	C₂×D₄	1	4	yes	—
74	C₂×#87604	C₂×Q₈	1	4	yes	—
75	Q₈×C₉₇	—	2	4	no	—
76	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
82	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₉₇	—	2	4	no	—
85	C₄×D₉₇	—	2	4	no	—
86	C₄×D₉₇	—	2	1	no	—
87	C₄×D₉₇	—	2	1	no	—
88	C₄×D₉₇	C₂×D₄	1	2	yes	—
89	C₄×D₉₇	C₂×Q₈	1	2	yes	—
90	C₄×D₉₇	#41	1	2	yes	—
91	C₄×D₉₇	#41	1	2	yes	—
92	Dic₁₉₄	—	2	4	no	—
93	Dic₁₉₄	—	2	8	no	—
94	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₉₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×D₉₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×D₉₇	C₂×#41	1	1	yes	characteristic
98	C₂×D₉₇	C₂×#41	1	1	yes	characteristic
99	C₂×D₉₇	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂×D₉₇	2₋¹⁺⁴	1	1	yes	characteristic
101	#87604	—	2	4	no	—
102	#87604	—	2	4	no	—
103	#87604	—	4	2	no	—
104	#87604	—	4	2	no	—
105	C₄×C₉₇	—	2	4	no	—
106	C₄×C₉₇	—	2	1	no	—
107	C₄×C₉₇	C₄×D₄	1	2	yes	—
108	C₄×C₉₇	C₄×Q₈	1	2	yes	complemented
109	Dic₉₇	—	2	4	no	—
110	Dic₉₇	—	2	1	no	—
111	Dic₉₇	C₄×D₄	1	2	yes	—
112	Dic₉₇	C₄×Q₈	1	2	yes	complemented
113	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₂×C₉₇	#519	1	1	yes	characteristic
116	D₉₇	#547	1	1	yes	characteristic
117	D₉₇	D₄×Q₈	1	1	yes	characteristic
118	D₉₇	#556	1	1	yes	characteristic
119	D₉₇	#557	1	1	yes	characteristic
120	#2365	—	97	1	no	maximal, Sylow
121	C₉₇	#2365	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄⋊Q₈	—	97	1	no	—
123	C₂×C₄×Q₈	—	97	2	no	—
124	#391	—	97	2	no	—
125	#389	—	97	2	no	—
126	#387	—	97	4	no	—
127	C₄×C₄⋊C₄	—	97	1	no	—
128	C₄×C₄⋊C₄	—	97	1	no	—
129	C₄×C₄⋊C₄	—	97	2	no	—
130	C₂²×Q₈	—	97	2	no	—
131	C₄⋊Q₈	—	97	2	no	—
132	C₄⋊Q₈	—	97	2	no	—
133	C₄⋊Q₈	—	97	2	no	—
134	C₄⋊Q₈	—	97	1	no	—
135	C₄⋊Q₈	#87604	1	1	yes	characteristic, complemented
136	C₄×Q₈	—	194	8	no	—
137	C₂×C₄⋊C₄	—	97	1	no	—
138	C₂×C₄⋊C₄	—	97	1	no	—
139	C₂×C₄⋊C₄	—	97	2	no	—
140	C₂×C₄⋊C₄	—	97	2	no	—
141	C₂×C₄⋊C₄	—	97	4	no	—
142	C₂×C₄⋊C₄	—	97	1	no	—
143	C₂×C₄⋊C₄	—	97	1	no	—
144	C₂×C₄²	—	97	1	no	—
145	C₂×C₄²	—	97	4	no	—
146	C₂×C₄²	—	97	2	no	—
147	C₂×C₄²	—	97	2	no	—
148	#95	—	97	2	no	—
149	#95	—	97	2	no	—
150	#95	—	97	4	no	—
151	#95	—	97	4	no	—
152	C₂×Q₈	—	97	4	no	—
153	C₂×Q₈	—	97	2	no	—
154	C₂×Q₈	—	194	4	no	—
155	C₂×Q₈	—	194	4	no	—
156	C₂×Q₈	C₂×#87604	1	2	yes	—
157	C₂²×C₄	—	97	1	no	—
158	C₂²×C₄	—	97	1	no	—
159	C₂²×C₄	—	97	2	no	—
160	C₂²×C₄	—	97	2	no	—
161	C₂²×C₄	—	97	1	no	—
162	C₂²×C₄	—	97	4	no	—
163	C₂²×C₄	—	97	2	no	—
164	C₂²×C₄	—	97	2	no	—
165	C₄⋊C₄	—	97	4	no	—
166	C₄⋊C₄	—	97	4	no	—
167	C₄⋊C₄	—	97	4	no	—
168	C₄⋊C₄	—	97	4	no	—
169	C₄⋊C₄	—	97	2	no	—
170	C₄⋊C₄	—	97	2	no	—
171	C₄⋊C₄	—	194	2	no	—
172	C₄⋊C₄	—	194	2	no	—
173	C₄⋊C₄	—	194	8	no	—
174	C₄⋊C₄	C₂×#87604	1	2	yes	—
175	C₄⋊C₄	C₂×#87604	1	2	yes	—
176	C₄²	—	97	1	no	—
177	C₄²	—	97	1	no	—
178	C₄²	—	97	1	no	—
179	C₄²	—	194	4	no	—
180	C₄²	—	194	4	no	—
181	C₄²	—	194	8	no	—
182	C₄²	C₂×#87604	1	1	yes	characteristic
183	C₂³	—	97	1	no	—
184	Q₈	—	2	4	no	—
185	Q₈	—	194	8	no	—
186	Q₈	—	194	4	no	—
187	C₂×C₄	—	97	1	no	—
188	C₂×C₄	—	97	1	no	—
189	C₂×C₄	—	97	2	no	—
190	C₂×C₄	—	97	2	no	—
191	C₂×C₄	—	97	2	no	—
192	C₂×C₄	—	97	2	no	—
193	C₂×C₄	—	97	4	no	—
194	C₂×C₄	—	97	4	no	—
195	C₂×C₄	—	97	1	no	—
196	C₂×C₄	—	97	4	no	—
197	C₂×C₄	—	194	4	no	—
198	C₂×C₄	—	194	4	no	—
199	C₂×C₄	—	194	1	no	—
200	C₂×C₄	—	194	1	no	—
201	C₂×C₄	—	194	2	no	—
202	C₂×C₄	—	194	2	no	—
203	C₂×C₄	—	194	2	no	—
204	C₂×C₄	—	194	2	no	—
205	C₂×C₄	—	194	2	no	—
206	C₂×C₄	—	194	2	no	—
207	C₂×C₄	—	194	2	no	—
208	C₂×C₄	—	194	2	no	—
209	C₂×C₄	—	194	2	no	—
210	C₂×C₄	—	194	2	no	—
211	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
212	C₂×C₄	C₂²×#87604	1	4	yes	—
213	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
214	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
215	C₂²	—	97	1	no	—
216	C₂²	—	97	1	no	—
217	C₂²	—	97	1	no	—
218	C₂²	—	97	1	no	—
219	C₂²	—	97	1	no	—
220	C₂²	—	97	1	no	—
221	C₂²	C₂³×#87604	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	2	1	no	—
224	C₄	—	97	2	no	—
225	C₄	—	97	2	no	—
226	C₄	—	388	2	no	—
227	C₄	—	388	2	no	—
228	C₄	—	194	4	no	—
229	C₄	—	194	4	no	—
230	C₄	—	194	4	no	—
231	C₄	—	194	1	no	—
232	C₄	D₄×#87604	1	2	yes	—
233	C₄	Q₈×#87604	1	2	yes	complemented
234	C₂	—	97	1	no	—
235	C₂	—	97	1	no	—
236	C₂	—	97	1	no	—
237	C₂	—	97	1	no	—
238	C₂	C₂×D₄×#87604	1	1	yes	characteristic
239	C₂	C₂×Q₈×#87604	1	1	yes	characteristic
240	C₂	#20261803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87604
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₄⋊Q₈×C₉₇	C₄
prime radical, cosocle	C₂²×D₉₇	C₂⁴
hypercenter	C₄⋊Q₈	#87604
hypocenter	C₉₇	#2365
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. #15690810	C₄	semidirect
24. #15690809	C₄	semidirect
25. #15690809	C₄	semidirect
26. #15690808	C₄	semidirect
30. C₄⋊Q₈×C₉₇	C₄	semidirect
39. C₄⋊#87604	Q₈	semidirect
108. C₄×C₉₇	C₄×Q₈	semidirect
112. Dic₉₇	C₄×Q₈	semidirect
121. C₉₇	#2365	semidirect
135. C₄⋊Q₈	#87604	semidirect
233. C₄	Q₈×#87604	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#87604	2
(C₂×#87604)⋊Q₈	2
#20261686	2
#20261684	2
C₄⋊C₄×#87604	2
C₄×C₄⋊#87604	2
C₄⋊Q₈×D₉₇	2
#2365	97