Subgroups — D₄×C₄⋊#87604 #27158248

Order 12416.

776 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×#87604	C₂	1	2	yes	maximal
2	C₂²×C₄⋊#87604	C₂	1	2	yes	maximal, complemented
3	#20261686	C₂	1	2	yes	maximal, complemented
4	#20261682	C₂	1	4	yes	maximal, complemented
5	#20261681	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261677	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊#87604	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄×D₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×#87604	C₂²	1	4	yes	—
10	D₄×#87604	—	2	8	no	—
11	C₂×C₄⋊#87604	—	2	8	no	—
12	C₂×C₄⋊#87604	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#87604	C₂²	1	2	yes	—
14	C₂×C₄⋊#87604	C₂²	1	2	yes	—
15	C₂×C₄⋊#87604	C₂²	1	4	yes	—
16	C₂×C₄⋊#87604	C₂²	1	2	yes	—
17	C₂×C₄⋊#87604	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊#87604	C₂²	1	4	yes	—
19	C₂×C₄×#87604	C₂²	1	2	yes	—
20	(C₂×#87604)⋊C₄	C₂²	1	2	yes	—
21	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
22	C₂×D₄×D₉₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
24	C₄×D₃₈₈	C₄	1	1	yes	characteristic, complemented
25	D₄×Dic₉₇	C₄	1	1	yes	characteristic, complemented
26	#15690772	C₄	1	1	yes	characteristic, complemented
27	C₄×C₂²⋊D₉₇	C₄	1	2	yes	complemented
28	#15690770	C₄	1	2	yes	complemented
29	C₄⋊C₄×D₉₇	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×D₉₇	C₂²	1	2	yes	—
31	C₄²×D₉₇	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₉₇	C₄	1	1	yes	characteristic, complemented
33	C₂²×#87604	—	2	8	no	—
34	C₂²×#87604	—	2	8	no	—
35	C₂²×#87604	—	4	4	no	—
36	C₂²×#87604	C₂³	1	2	yes	—
37	C₂²×#87604	C₂³	1	4	yes	—
38	C₂²×#87604	C₂³	1	2	yes	—
39	C₄⋊#87604	—	2	8	no	—
40	C₄⋊#87604	—	2	4	no	—
41	C₄⋊#87604	—	2	4	no	—
42	C₄⋊#87604	—	2	2	no	—
43	C₄⋊#87604	—	4	2	no	—
44	C₄⋊#87604	D₄	1	4	yes	direct product factor
45	C₂²⋊#87604	—	2	8	no	—
46	C₄×#87604	—	2	4	no	—
47	C₂³×D₉₇	C₂³	1	2	yes	—
48	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
50	D₄×D₉₇	D₄	1	4	yes	—
51	D₄×D₉₇	Q₈	1	4	yes	—
52	C₂²×Dic₉₇	C₂×C₄	1	2	yes	complemented
53	C₂×C₄×D₉₇	—	2	4	no	—
54	C₂×C₄×D₉₇	—	2	4	no	—
55	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	complemented
56	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	complemented
57	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₉₇	C₂³	1	2	yes	—
59	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
62	#13095490	C₂×C₄	1	2	yes	complemented
63	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	complemented
64	#13095488	C₂×C₄	1	2	yes	complemented
65	#13095488	C₂×C₄	1	2	yes	complemented
66	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
67	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
68	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic, complemented
69	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	complemented
71	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	complemented
73	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
75	C₂²×D₉₇	—	2	2	no	—
76	C₂²×D₉₇	—	2	2	no	—
77	C₂²×D₉₇	—	2	2	no	—
78	C₂²×D₉₇	—	2	2	no	—
79	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
80	C₂²×D₉₇	C₂×D₄	1	4	yes	—
81	C₂²×D₉₇	C₂×Q₈	1	4	yes	—
82	C₂²×D₉₇	C₂⁴	1	1	yes	characteristic
83	C₂×#87604	—	2	2	no	—
84	C₂×#87604	—	2	2	no	—
85	C₂×#87604	—	2	2	no	—
86	C₂×#87604	—	2	4	no	—
87	C₂×#87604	—	2	4	no	—
88	C₂×#87604	—	2	2	no	—
89	C₂×#87604	—	2	4	no	—
90	C₂×#87604	—	2	2	no	—
91	C₂×#87604	—	4	16	no	—
92	C₂×#87604	C₂×D₄	1	4	yes	—
93	D₄×C₉₇	C₄⋊C₄	1	4	yes	complemented
94	C₂×C₄×C₉₇	—	2	4	no	—
95	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²⋊D₉₇	C₄⋊C₄	1	8	yes	complemented
100	C₂×Dic₉₇	—	2	4	no	—
101	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
103	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
105	D₃₈₈	C₄⋊C₄	1	4	yes	complemented
106	C₄×D₉₇	—	2	2	no	—
107	C₄×D₉₇	—	2	2	no	—
108	C₄×D₉₇	—	2	1	no	—
109	C₄×D₉₇	—	2	1	no	—
110	C₄×D₉₇	—	2	4	no	—
111	C₄×D₉₇	—	2	4	no	—
112	C₄×D₉₇	C₂×D₄	1	2	yes	—
113	C₄×D₉₇	C₂×D₄	1	2	yes	—
114	C₄×D₉₇	C₂×Q₈	1	2	yes	—
115	C₄×D₉₇	#41	1	2	yes	—
116	C₂²×C₉₇	—	2	2	no	—
117	C₂²×C₉₇	—	2	2	no	—
118	C₂²×C₉₇	C₂×C₄⋊C₄	1	4	yes	complemented
119	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×D₉₇	—	2	2	no	—
121	C₂×D₉₇	—	2	2	no	—
122	C₂×D₉₇	—	2	4	no	—
123	C₂×D₉₇	—	2	4	no	—
124	C₂×D₉₇	—	2	4	no	—
125	C₂×D₉₇	—	2	4	no	—
126	C₂×D₉₇	C₂×C₄⋊C₄	1	4	yes	complemented
127	C₂×D₉₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₉₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×D₉₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×D₉₇	C₂×#41	1	1	yes	characteristic
131	C₂×D₉₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₉₇	2₋¹⁺⁴	1	1	yes	characteristic
133	#87604	—	2	4	no	—
134	#87604	—	4	4	no	—
135	#87604	—	4	2	no	—
136	C₄×C₉₇	—	2	2	no	—
137	C₄×C₉₇	—	2	1	no	—
138	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	complemented
139	C₄×C₉₇	C₄×D₄	1	2	yes	complemented
140	Dic₉₇	—	2	2	no	—
141	Dic₉₇	—	2	1	no	—
142	Dic₉₇	C₂×C₄⋊C₄	1	2	yes	complemented
143	Dic₉₇	C₄×D₄	1	2	yes	complemented
144	C₂×C₉₇	—	2	4	no	—
145	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₉₇	#520	1	1	yes	characteristic
148	D₉₇	—	2	4	no	—
149	D₉₇	D₄²	1	1	yes	characteristic
150	D₉₇	#547	1	1	yes	characteristic
151	D₉₇	D₄×Q₈	1	1	yes	characteristic
152	D₉₇	#554	1	1	yes	characteristic
153	D₄×C₄⋊C₄	—	97	1	no	maximal, Sylow
154	C₉₇	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄×D₄	—	97	1	no	—
156	C₂×C₄×D₄	—	97	2	no	—
157	C₂²×C₄⋊C₄	—	97	2	no	—
158	#389	—	97	2	no	—
159	#385	—	97	4	no	—
160	#384	—	97	1	no	—
161	#380	—	97	2	no	—
162	C₄×C₄⋊C₄	—	97	1	no	—
163	C₂²×D₄	—	97	1	no	—
164	C₂³×C₄	—	97	2	no	—
165	C₂³×C₄	—	97	4	no	—
166	C₄×D₄	—	97	1	no	—
167	C₄×D₄	—	97	2	no	—
168	C₄×D₄	—	97	1	no	—
169	C₄×D₄	—	97	2	no	—
170	C₄×D₄	—	97	1	no	—
171	C₄×D₄	—	194	8	no	—
172	C₄×D₄	#87604	1	1	yes	characteristic, complemented
173	C₂×C₄⋊C₄	—	97	1	no	—
174	C₂×C₄⋊C₄	—	97	1	no	—
175	C₂×C₄⋊C₄	—	97	2	no	—
176	C₂×C₄⋊C₄	—	97	2	no	—
177	C₂×C₄⋊C₄	—	97	4	no	—
178	C₂×C₄⋊C₄	—	97	2	no	—
179	C₂×C₄⋊C₄	—	97	1	no	—
180	C₂×C₄⋊C₄	—	194	8	no	—
181	C₂×C₂²⋊C₄	—	97	2	no	—
182	C₂×C₂²⋊C₄	—	97	4	no	—
183	C₂×C₄²	—	97	1	no	—
184	C₂×C₄²	—	97	2	no	—
185	#95	—	97	2	no	—
186	#95	—	97	4	no	—
187	C₂⁴	—	97	2	no	—
188	C₂×D₄	—	97	2	no	—
189	C₂×D₄	—	97	4	no	—
190	C₂×D₄	—	97	4	no	—
191	C₂×D₄	—	97	1	no	—
192	C₂×D₄	C₂×#87604	1	1	yes	characteristic
193	C₂²×C₄	—	97	1	no	—
194	C₂²×C₄	—	97	1	no	—
195	C₂²×C₄	—	97	2	no	—
196	C₂²×C₄	—	97	2	no	—
197	C₂²×C₄	—	97	2	no	—
198	C₂²×C₄	—	97	2	no	—
199	C₂²×C₄	—	97	1	no	—
200	C₂²×C₄	—	97	2	no	—
201	C₂²×C₄	—	97	2	no	—
202	C₂²×C₄	—	97	4	no	—
203	C₂²×C₄	—	388	4	no	—
204	C₂²×C₄	—	194	4	no	—
205	C₂²×C₄	—	194	4	no	—
206	C₂²×C₄	—	194	8	no	—
207	C₂²×C₄	—	194	8	no	—
208	C₂²×C₄	C₂×#87604	1	2	yes	complemented
209	C₄⋊C₄	—	97	4	no	—
210	C₄⋊C₄	—	97	2	no	—
211	C₄⋊C₄	—	97	1	no	—
212	C₄⋊C₄	—	388	2	no	—
213	C₄⋊C₄	—	194	2	no	—
214	C₄⋊C₄	—	194	4	no	—
215	C₄⋊C₄	—	194	8	no	—
216	C₄⋊C₄	—	194	4	no	—
217	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic, complemented
218	C₂²⋊C₄	—	97	2	no	—
219	C₂²⋊C₄	—	97	2	no	—
220	C₂²⋊C₄	—	97	2	no	—
221	C₂²⋊C₄	—	194	8	no	—
222	C₂²⋊C₄	C₂×#87604	1	2	yes	complemented
223	C₄²	—	97	1	no	—
224	C₄²	—	97	1	no	—
225	C₄²	—	97	1	no	—
226	C₄²	—	194	4	no	—
227	C₄²	C₂×#87604	1	1	yes	characteristic, complemented
228	C₂³	—	97	1	no	—
229	C₂³	—	97	4	no	—
230	C₂³	—	97	4	no	—
231	C₂³	—	97	2	no	—
232	C₂³	—	194	2	no	—
233	C₂³	—	194	2	no	—
234	C₂³	—	194	2	no	—
235	C₂³	—	194	2	no	—
236	C₂³	C₂²×#87604	1	2	yes	—
237	D₄	—	97	8	no	—
238	D₄	—	97	4	no	—
239	D₄	C₄⋊#87604	1	4	yes	direct product factor
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	97	1	no	—
242	C₂×C₄	—	97	1	no	—
243	C₂×C₄	—	97	2	no	—
244	C₂×C₄	—	97	2	no	—
245	C₂×C₄	—	97	2	no	—
246	C₂×C₄	—	97	2	no	—
247	C₂×C₄	—	97	4	no	—
248	C₂×C₄	—	97	1	no	—
249	C₂×C₄	—	97	2	no	—
250	C₂×C₄	—	388	16	no	—
251	C₂×C₄	—	194	4	no	—
252	C₂×C₄	—	194	1	no	—
253	C₂×C₄	—	194	2	no	—
254	C₂×C₄	—	194	2	no	—
255	C₂×C₄	—	194	1	no	—
256	C₂×C₄	—	194	4	no	—
257	C₂×C₄	—	194	2	no	—
258	C₂×C₄	—	194	4	no	—
259	C₂×C₄	—	194	4	no	—
260	C₂×C₄	—	194	4	no	—
261	C₂×C₄	—	194	2	no	—
262	C₂×C₄	—	194	2	no	—
263	C₂×C₄	—	194	2	no	—
264	C₂×C₄	—	194	2	no	—
265	C₂×C₄	—	194	4	no	—
266	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
267	C₂×C₄	C₂²×#87604	1	2	yes	—
268	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
269	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	97	1	no	—
273	C₂²	—	97	1	no	—
274	C₂²	—	97	1	no	—
275	C₂²	—	97	1	no	—
276	C₂²	—	97	1	no	—
277	C₂²	—	97	1	no	—
278	C₂²	—	97	4	no	—
279	C₂²	—	194	4	no	—
280	C₂²	—	194	4	no	—
281	C₂²	—	194	4	no	—
282	C₂²	—	194	4	no	—
283	C₂²	—	194	2	no	—
284	C₂²	—	194	2	no	—
285	C₂²	C₂×C₄⋊#87604	1	4	yes	complemented
286	C₂²	C₂³×#87604	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	97	2	no	—
290	C₄	—	97	2	no	—
291	C₄	—	388	4	no	—
292	C₄	—	388	2	no	—
293	C₄	—	194	4	no	—
294	C₄	—	194	2	no	—
295	C₄	—	194	1	no	—
296	C₄	C₂×C₄⋊#87604	1	2	yes	complemented
297	C₄	D₄×#87604	1	2	yes	complemented
298	C₂	—	2	4	no	—
299	C₂	—	97	1	no	—
300	C₂	—	97	1	no	—
301	C₂	—	97	1	no	—
302	C₂	—	97	1	no	—
303	C₂	—	194	4	no	—
304	C₂	C₂²×C₄⋊#87604	1	1	yes	characteristic
305	C₂	C₂×D₄×#87604	1	1	yes	characteristic
306	C₂	#20261804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87604
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₄×D₄×C₉₇	C₄
prime radical	C₂×D₄×D₉₇	C₂²
cosocle	C₂²×D₉₇	C₂⁴
hypercenter	C₄×D₄	#87604
hypocenter	C₉₇	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊#87604	C₂	semidirect
3. #20261686	C₂	semidirect
4. #20261682	C₂	semidirect
5. #20261681	C₂	semidirect
6. #20261677	C₂	semidirect
7. C₄×C₄⋊#87604	C₂	semidirect
24. C₄×D₃₈₈	C₄	semidirect
25. D₄×Dic₉₇	C₄	semidirect
26. #15690772	C₄	semidirect
27. C₄×C₂²⋊D₉₇	C₄	semidirect
28. #15690770	C₄	semidirect
32. C₄×D₄×C₉₇	C₄	semidirect
44. C₄⋊#87604	D₄	direct (N × H)
52. C₂²×Dic₉₇	C₂×C₄	semidirect
55. C₂×C₄×D₉₇	C₂×C₄	semidirect
56. C₂×C₄×D₉₇	C₂×C₄	semidirect
61. C₄⋊Dic₉₇	C₂×C₄	semidirect
62. #13095490	C₂×C₄	semidirect
63. C₂²⋊Dic₉₇	C₂×C₄	semidirect
64. #13095488	C₂×C₄	semidirect
65. #13095488	C₂×C₄	semidirect
66. C₄×Dic₉₇	C₂×C₄	semidirect
67. C₄×Dic₉₇	C₂×C₄	semidirect
68. C₄×Dic₉₇	C₂×C₄	semidirect
70. C₂²×C₄×C₉₇	C₂×C₄	semidirect
71. C₄⋊C₄×C₉₇	C₂×C₄	semidirect
72. C₂²⋊C₄×C₉₇	C₂×C₄	semidirect
73. C₄²×C₉₇	C₂×C₄	semidirect
93. D₄×C₉₇	C₄⋊C₄	semidirect
99. C₂²⋊D₉₇	C₄⋊C₄	semidirect
105. D₃₈₈	C₄⋊C₄	semidirect
118. C₂²×C₉₇	C₂×C₄⋊C₄	semidirect
126. C₂×D₉₇	C₂×C₄⋊C₄	semidirect
138. C₄×C₉₇	C₂×C₄⋊C₄	semidirect
139. C₄×C₉₇	C₄×D₄	semidirect
142. Dic₉₇	C₂×C₄⋊C₄	semidirect
143. Dic₉₇	C₄×D₄	semidirect
154. C₉₇	D₄×C₄⋊C₄	semidirect
172. C₄×D₄	#87604	semidirect
208. C₂²×C₄	C₂×#87604	semidirect
217. C₄⋊C₄	C₂×#87604	semidirect
222. C₂²⋊C₄	C₂×#87604	semidirect
227. C₄²	C₂×#87604	semidirect
239. D₄	C₄⋊#87604	direct (N × H)
285. C₂²	C₂×C₄⋊#87604	semidirect
296. C₄	C₂×C₄⋊#87604	semidirect
297. C₄	D₄×#87604	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×#87604	2
C₂²×C₄⋊#87604	2
#20261686	2
#20261682	2
#20261681	2
#20261677	2
C₄×C₄⋊#87604	2
C₄×D₄×D₉₇	2
D₄×C₄⋊C₄	97

Subgroups — D4×C4⋊#87604 #27158248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₄⋊#87604 #27158248