Subgroups — #27187664

Order 12928.

548 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₂×#87938)⋊Q₈	C₂	1	2	yes	maximal, complemented
2	#20266126	C₂	1	2	yes	maximal, complemented
3	#20266126	C₂	1	2	yes	maximal, complemented
4	#20266124	C₂	1	2	yes	maximal
5	C₄⋊C₄×#87938	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₄×#87938	C₂	1	2	yes	maximal
7	C₄×C₂²⋊#87938	C₂	1	1	yes	maximal, characteristic
8	C₂×(C₂×#87938)⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
10	C₂³×#87938	C₂²	1	2	yes	—
11	C₂×C₄⋊#87938	C₂²	1	2	yes	—
12	C₂×C₄⋊#87938	C₂²	1	2	yes	—
13	C₂×C₂²⋊#87938	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊#87938	C₂²	1	1	yes	characteristic
15	C₂×C₄×#87938	C₂²	1	2	yes	—
16	C₂×C₄×#87938	C₂²	1	2	yes	—
17	C₂×C₄×#87938	C₂²	1	2	yes	—
18	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
19	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
20	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
21	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
22	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
23	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
24	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
25	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×D₁₀₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
28	#15692235	C₄	1	1	yes	characteristic, complemented
29	#15692234	C₄	1	1	yes	characteristic, complemented
30	#15692233	C₄	1	2	yes	complemented
31	#15692232	C₄	1	2	yes	complemented
32	#15692231	C₄	1	1	yes	characteristic, complemented
33	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	—
34	C₄⋊C₄×D₁₀₁	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₁₀₁	C₂²	1	2	yes	—
36	C₂²⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic, complemented
37	C₂²×#87938	—	2	4	no	—
38	C₂²×#87938	—	2	4	no	—
39	C₂²×#87938	—	4	2	no	—
40	C₂²×#87938	C₂³	1	2	yes	—
41	C₂²×#87938	C₂³	1	2	yes	—
42	C₂²×#87938	C₂³	1	2	yes	—
43	C₂²×#87938	C₂³	1	2	yes	—
44	C₄⋊#87938	—	2	4	no	—
45	C₄⋊#87938	—	4	2	no	—
46	C₂²⋊#87938	—	2	2	no	—
47	C₂²⋊#87938	Q₈	1	4	yes	complemented
48	C₄×#87938	—	2	4	no	—
49	C₄×#87938	—	4	2	no	—
50	C₄×#87938	—	4	2	no	—
51	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	complemented
53	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic, complemented
54	Q₈×D₁₀₁	—	4	2	no	—
55	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₀₁	—	4	2	no	—
57	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
62	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
63	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
64	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	complemented
65	#13108053	C₂×C₄	1	2	yes	complemented
66	#13108053	C₂×C₄	1	2	yes	complemented
67	#13108053	C₂×C₄	1	2	yes	complemented
68	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
69	#13108051	C₂×C₄	1	2	yes	—
70	#13108051	C₂×C₄	1	2	yes	—
71	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	complemented
74	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
75	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
76	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₀₁	—	2	2	no	—
78	C₂²×D₁₀₁	—	2	2	no	—
79	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
81	C₂²×D₁₀₁	#41	1	2	yes	—
82	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
83	C₂×#87938	—	2	2	no	—
84	C₂×#87938	—	2	2	no	—
85	C₂×#87938	—	2	2	no	—
86	C₂×#87938	—	2	2	no	—
87	C₂×#87938	—	2	2	no	—
88	C₂×#87938	—	2	2	no	—
89	C₂×#87938	—	2	2	no	—
90	C₂×#87938	—	2	2	no	—
91	C₂×#87938	—	2	2	no	—
92	C₂×#87938	—	2	2	no	—
93	C₂×#87938	—	4	8	no	—
94	C₂×#87938	C₂×Q₈	1	4	yes	—
95	C₂×#87938	#41	1	4	yes	—
96	Q₈×C₁₀₁	—	4	1	no	—
97	C₂×C₄×C₁₀₁	—	4	1	no	—
98	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₀₁	—	4	1	no	—
103	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
104	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
105	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₀₁	—	2	2	no	—
108	C₄×D₁₀₁	—	2	2	no	—
109	C₄×D₁₀₁	—	2	2	no	—
110	C₄×D₁₀₁	—	2	2	no	—
111	C₄×D₁₀₁	—	2	2	no	—
112	C₄×D₁₀₁	—	2	2	no	—
113	C₄×D₁₀₁	—	4	1	no	—
114	C₄×D₁₀₁	—	4	1	no	—
115	Dic₂₀₂	—	4	2	no	—
116	Dic₂₀₂	—	4	1	no	—
117	C₂²×C₁₀₁	—	2	1	no	—
118	C₂²×C₁₀₁	—	2	1	no	—
119	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
120	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₁₀₁	—	2	1	no	—
122	C₂×D₁₀₁	—	2	1	no	—
123	C₂×D₁₀₁	—	2	4	no	—
124	C₂×D₁₀₁	—	2	4	no	—
125	C₂×D₁₀₁	C₄×Q₈	1	2	yes	—
126	C₂×D₁₀₁	C₂²×Q₈	1	1	yes	characteristic
127	C₂×D₁₀₁	C₂×#41	1	1	yes	characteristic
128	C₂×D₁₀₁	C₂×#41	1	2	yes	—
129	C₂×D₁₀₁	H(5,𝔽₂)	1	2	yes	—
130	#87938	—	2	4	no	—
131	#87938	—	2	4	no	—
132	#87938	—	4	2	no	—
133	#87938	—	4	2	no	—
134	C₄×C₁₀₁	—	2	2	no	—
135	C₄×C₁₀₁	—	2	1	no	—
136	C₄×C₁₀₁	—	2	2	no	—
137	C₄×C₁₀₁	—	2	1	no	—
138	Dic₁₀₁	—	2	2	no	—
139	Dic₁₀₁	—	2	1	no	—
140	Dic₁₀₁	—	2	2	no	—
141	Dic₁₀₁	—	2	1	no	—
142	C₂×C₁₀₁	—	2	2	no	—
143	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
145	C₂×C₁₀₁	#518	1	1	yes	characteristic
146	D₁₀₁	—	2	2	no	—
147	D₁₀₁	#551	1	2	yes	—
148	D₁₀₁	#554	1	2	yes	—
149	#2368	—	101	1	no	maximal, Sylow
150	C₁₀₁	#2368	1	1	yes	characteristic, complemented, Sylow
151	C₂×C₂²⋊Q₈	—	101	1	no	—
152	#391	—	101	2	no	—
153	#387	—	101	2	no	—
154	#387	—	101	2	no	—
155	#385	—	101	2	no	—
156	C₄×C₄⋊C₄	—	101	2	no	—
157	C₄×C₂²⋊C₄	—	101	1	no	—
158	C₄×C₂²⋊C₄	—	101	2	no	—
159	C₂×#95	—	101	1	no	—
160	C₂²×Q₈	—	101	1	no	—
161	C₂³×C₄	—	101	1	no	—
162	C₂³×C₄	—	101	2	no	—
163	C₂²⋊Q₈	—	101	1	no	—
164	C₂²⋊Q₈	—	101	1	no	—
165	C₂²⋊Q₈	—	101	2	no	—
166	C₂²⋊Q₈	—	101	1	no	—
167	C₂²⋊Q₈	—	101	2	no	—
168	C₂²⋊Q₈	#87938	1	1	yes	characteristic, complemented
169	C₂×C₄⋊C₄	—	101	2	no	—
170	C₂×C₄⋊C₄	—	101	1	no	—
171	C₂×C₄⋊C₄	—	101	2	no	—
172	C₂×C₄⋊C₄	—	101	2	no	—
173	C₂×C₂²⋊C₄	—	101	1	no	—
174	C₂×C₂²⋊C₄	—	101	2	no	—
175	C₂×C₂²⋊C₄	—	101	1	no	—
176	C₂×C₄²	—	101	2	no	—
177	C₂×C₄²	—	101	2	no	—
178	C₂×C₄²	—	101	2	no	—
179	#95	—	101	1	no	—
180	#95	—	101	1	no	—
181	#95	—	101	2	no	—
182	#95	—	101	2	no	—
183	#95	—	101	2	no	—
184	#95	—	101	2	no	—
185	#95	—	101	2	no	—
186	#95	—	101	2	no	—
187	C₂⁴	—	101	1	no	—
188	C₂×Q₈	—	101	1	no	—
189	C₂×Q₈	—	101	2	no	—
190	C₂×Q₈	—	404	2	no	—
191	C₂×Q₈	C₂×#87938	1	1	yes	characteristic, complemented
192	C₂²×C₄	—	101	1	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	1	no	—
196	C₂²×C₄	—	101	1	no	—
197	C₂²×C₄	—	101	1	no	—
198	C₂²×C₄	—	101	1	no	—
199	C₂²×C₄	—	101	2	no	—
200	C₂²×C₄	—	101	2	no	—
201	C₂²×C₄	—	101	2	no	—
202	C₂²×C₄	—	101	2	no	—
203	C₂²×C₄	—	404	2	no	—
204	C₂²×C₄	—	404	2	no	—
205	C₂²×C₄	—	202	4	no	—
206	C₂²×C₄	—	202	4	no	—
207	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
208	C₄⋊C₄	—	101	2	no	—
209	C₄⋊C₄	—	101	2	no	—
210	C₄⋊C₄	—	101	2	no	—
211	C₄⋊C₄	—	101	1	no	—
212	C₄⋊C₄	—	101	2	no	—
213	C₄⋊C₄	—	404	2	no	—
214	C₄⋊C₄	—	202	4	no	—
215	C₄⋊C₄	C₂×#87938	1	1	yes	characteristic, complemented
216	C₄⋊C₄	C₂×#87938	1	2	yes	complemented
217	C₂²⋊C₄	—	101	4	no	—
218	C₂²⋊C₄	—	101	2	no	—
219	C₂²⋊C₄	—	101	2	no	—
220	C₂²⋊C₄	—	101	2	no	—
221	C₂²⋊C₄	—	202	2	no	—
222	C₂²⋊C₄	C₂×#87938	1	2	yes	—
223	C₄²	—	404	2	no	—
224	C₄²	—	404	2	no	—
225	C₄²	—	202	4	no	—
226	C₂³	—	101	1	no	—
227	C₂³	—	101	2	no	—
228	C₂³	—	101	2	no	—
229	C₂³	—	101	1	no	—
230	C₂³	—	202	2	no	—
231	C₂³	—	202	2	no	—
232	C₂³	C₂²×#87938	1	1	yes	characteristic
233	Q₈	—	4	1	no	—
234	Q₈	—	404	1	no	—
235	Q₈	—	404	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	101	4	no	—
238	C₂×C₄	—	101	4	no	—
239	C₂×C₄	—	101	2	no	—
240	C₂×C₄	—	101	2	no	—
241	C₂×C₄	—	101	1	no	—
242	C₂×C₄	—	101	1	no	—
243	C₂×C₄	—	404	1	no	—
244	C₂×C₄	—	404	1	no	—
245	C₂×C₄	—	404	8	no	—
246	C₂×C₄	—	404	1	no	—
247	C₂×C₄	—	202	2	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	2	no	—
251	C₂×C₄	—	202	2	no	—
252	C₂×C₄	—	202	2	no	—
253	C₂×C₄	—	202	2	no	—
254	C₂×C₄	—	202	2	no	—
255	C₂×C₄	—	202	2	no	—
256	C₂×C₄	—	202	2	no	—
257	C₂×C₄	—	202	2	no	—
258	C₂×C₄	—	202	2	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	—	202	2	no	—
262	C₂×C₄	—	202	2	no	—
263	C₂×C₄	C₂²×#87938	1	2	yes	—
264	C₂×C₄	C₂²×#87938	1	2	yes	—
265	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
266	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	101	1	no	—
270	C₂²	—	101	1	no	—
271	C₂²	—	101	2	no	—
272	C₂²	—	101	2	no	—
273	C₂²	—	101	2	no	—
274	C₂²	—	202	4	no	—
275	C₂²	—	202	4	no	—
276	C₂²	—	202	1	no	—
277	C₂²	—	202	1	no	—
278	C₂²	Q₈×#87938	1	2	yes	—
279	C₂²	C₂³×#87938	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	404	2	no	—
285	C₄	—	404	2	no	—
286	C₄	—	202	4	no	—
287	C₄	—	202	4	no	—
288	C₄	—	202	2	no	—
289	C₄	—	202	1	no	—
290	C₄	—	202	1	no	—
291	C₄	—	202	2	no	—
292	C₂	—	2	2	no	—
293	C₂	—	101	2	no	—
294	C₂	—	101	2	no	—
295	C₂	—	202	2	no	—
296	C₂	C₂×Q₈×#87938	1	1	yes	characteristic
297	C₂	#41×#87938	1	1	yes	characteristic
298	C₂	#20266244	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₀₁	C₄
prime radical	C₂³×D₁₀₁	C₂³
cosocle	C₂²×D₁₀₁	C₂⁴
hypercenter	C₂²⋊Q₈	#87938
hypocenter	C₁₀₁	#2368
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. (C₂×#87938)⋊Q₈	C₂	semidirect
2. #20266126	C₂	semidirect
3. #20266126	C₂	semidirect
5. C₄⋊C₄×#87938	C₂	semidirect
28. #15692235	C₄	semidirect
29. #15692234	C₄	semidirect
30. #15692233	C₄	semidirect
31. #15692232	C₄	semidirect
32. #15692231	C₄	semidirect
36. C₂²⋊Q₈×C₁₀₁	C₄	semidirect
47. C₂²⋊#87938	Q₈	semidirect
52. C₂×Dic₂₀₂	C₂×C₄	semidirect
53. C₂×Dic₂₀₂	C₂×C₄	semidirect
63. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
64. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
65. #13108053	C₂×C₄	semidirect
66. #13108053	C₂×C₄	semidirect
67. #13108053	C₂×C₄	semidirect
71. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
73. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
74. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
150. C₁₀₁	#2368	semidirect
168. C₂²⋊Q₈	#87938	semidirect
191. C₂×Q₈	C₂×#87938	semidirect
215. C₄⋊C₄	C₂×#87938	semidirect
216. C₄⋊C₄	C₂×#87938	semidirect

Maximal subgroups

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

subgroup	index
(C₂×#87938)⋊Q₈	2
#20266126	2
#20266124	2
C₄⋊C₄×#87938	2
C₂²⋊C₄×#87938	2
C₄×C₂²⋊#87938	2
C₂×(C₂×#87938)⋊C₄	2
C₂²⋊Q₈×D₁₀₁	2
#2368	101