Subgroups — #27187656

Order 12928.

620 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#87938	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊#87938	C₂	1	1	yes	maximal, characteristic
3	(C₂×#87938)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#20266128	C₂	1	2	yes	maximal, complemented
5	#20266128	C₂	1	1	yes	maximal, characteristic, complemented
6	#20266126	C₂	1	2	yes	maximal, complemented
7	#20266124	C₂	1	2	yes	maximal
8	C₄⋊C₄×#87938	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊C₄×#87938	C₂	1	2	yes	maximal
10	C₄×C₂²⋊#87938	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×#87938	C₂²	1	2	yes	—
13	Q₈×#87938	—	2	4	no	—
14	C₂×C₄⋊#87938	—	2	4	no	—
15	C₂×C₄⋊#87938	C₂²	1	2	yes	—
16	C₂×C₄⋊#87938	C₂²	1	2	yes	—
17	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#87938	C₂²	1	2	yes	—
19	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊#87938	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊#87938	C₂²	1	1	yes	characteristic
22	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
23	C₂×C₄×#87938	C₂²	1	2	yes	—
24	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
25	C₂×C₄×#87938	C₂²	1	2	yes	—
26	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
27	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
28	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
29	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
30	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
31	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
32	C₂×Q₈×D₁₀₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
34	#15692235	C₄	1	1	yes	characteristic, complemented
35	#15692234	C₄	1	1	yes	characteristic, complemented
36	#15692233	C₄	1	2	yes	complemented
37	#15692232	C₄	1	2	yes	complemented
38	#15692231	C₄	1	1	yes	characteristic, complemented
39	C₄⋊C₄×D₁₀₁	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	—
41	C₂²⋊C₄×D₁₀₁	C₂²	1	2	yes	—
42	C₂²⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic, complemented
43	C₂²×#87938	—	2	4	no	—
44	C₂²×#87938	—	2	4	no	—
45	C₂²×#87938	—	4	2	no	—
46	C₂²×#87938	C₂³	1	1	yes	characteristic
47	C₂²×#87938	C₂³	1	2	yes	—
48	C₂²×#87938	C₂³	1	1	yes	characteristic
49	C₂²×#87938	C₂³	1	2	yes	—
50	C₂²×#87938	C₂³	1	1	yes	characteristic
51	C₂²×#87938	C₂³	1	1	yes	characteristic
52	C₄⋊#87938	—	2	4	no	—
53	C₄⋊#87938	—	2	4	no	—
54	C₄⋊#87938	—	2	2	no	—
55	C₄⋊#87938	—	4	2	no	—
56	C₄⋊#87938	D₄	1	4	yes	—
57	C₂²⋊#87938	—	2	2	no	—
58	C₂²⋊#87938	Q₈	1	4	yes	complemented
59	C₄×#87938	—	2	4	no	—
60	C₄×#87938	—	2	2	no	—
61	C₄×#87938	—	2	2	no	—
62	C₄×#87938	—	4	2	no	—
63	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	complemented
65	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic, complemented
66	Q₈×D₁₀₁	—	2	2	no	—
67	Q₈×D₁₀₁	—	2	2	no	—
68	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₁	—	2	2	no	—
70	C₂×C₄×D₁₀₁	—	2	2	no	—
71	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
75	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
76	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
78	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	complemented
79	#13108053	C₂×C₄	1	2	yes	complemented
80	#13108053	C₂×C₄	1	2	yes	complemented
81	#13108053	C₂×C₄	1	2	yes	complemented
82	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
83	#13108051	C₂×C₄	1	2	yes	—
84	#13108051	C₂×C₄	1	2	yes	—
85	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
86	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	complemented
88	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
89	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
90	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₁₀₁	—	2	1	no	—
92	C₂²×D₁₀₁	—	2	1	no	—
93	C₂²×D₁₀₁	—	2	1	no	—
94	C₂²×D₁₀₁	—	2	1	no	—
95	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
97	C₂²×D₁₀₁	#41	1	2	yes	—
98	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
99	C₂×#87938	—	2	1	no	—
100	C₂×#87938	—	2	1	no	—
101	C₂×#87938	—	2	2	no	—
102	C₂×#87938	—	2	1	no	—
103	C₂×#87938	—	2	2	no	—
104	C₂×#87938	—	2	1	no	—
105	C₂×#87938	—	2	1	no	—
106	C₂×#87938	—	2	2	no	—
107	C₂×#87938	—	2	1	no	—
108	C₂×#87938	—	2	1	no	—
109	C₂×#87938	—	2	1	no	—
110	C₂×#87938	—	2	2	no	—
111	C₂×#87938	—	2	1	no	—
112	C₂×#87938	—	2	2	no	—
113	C₂×#87938	—	2	1	no	—
114	C₂×#87938	—	4	8	no	—
115	C₂×#87938	C₂×D₄	1	4	yes	—
116	C₂×#87938	C₂×Q₈	1	2	yes	—
117	C₂×#87938	C₂×Q₈	1	2	yes	—
118	Q₈×C₁₀₁	—	2	2	no	—
119	C₂×C₄×C₁₀₁	—	2	2	no	—
120	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
122	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₁₀₁	—	2	2	no	—
125	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
126	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
127	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
129	C₄×D₁₀₁	—	2	1	no	—
130	C₄×D₁₀₁	—	2	2	no	—
131	C₄×D₁₀₁	—	2	1	no	—
132	C₄×D₁₀₁	—	2	2	no	—
133	C₄×D₁₀₁	—	2	2	no	—
134	C₄×D₁₀₁	—	2	2	no	—
135	C₄×D₁₀₁	—	2	2	no	—
136	C₄×D₁₀₁	—	2	2	no	—
137	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
138	C₄×D₁₀₁	#41	1	2	yes	—
139	Dic₂₀₂	—	2	2	no	—
140	Dic₂₀₂	—	2	4	no	—
141	C₂²×C₁₀₁	—	2	1	no	—
142	C₂²×C₁₀₁	—	2	1	no	—
143	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	complemented
144	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
145	C₂×D₁₀₁	—	2	1	no	—
146	C₂×D₁₀₁	—	2	1	no	—
147	C₂×D₁₀₁	—	2	2	no	—
148	C₂×D₁₀₁	—	2	2	no	—
149	C₂×D₁₀₁	—	2	2	no	—
150	C₂×D₁₀₁	—	2	2	no	—
151	C₂×D₁₀₁	C₄×Q₈	1	2	yes	complemented
152	C₂×D₁₀₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₀₁	C₂²×Q₈	1	1	yes	characteristic
154	C₂×D₁₀₁	C₂×#41	1	1	yes	characteristic
155	C₂×D₁₀₁	C₂×#41	1	1	yes	characteristic
156	C₂×D₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×D₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
158	#87938	—	2	4	no	—
159	#87938	—	2	2	no	—
160	#87938	—	2	2	no	—
161	#87938	—	4	1	no	—
162	#87938	—	4	2	no	—
163	#87938	—	4	1	no	—
164	C₄×C₁₀₁	—	2	2	no	—
165	C₄×C₁₀₁	—	2	1	no	—
166	C₄×C₁₀₁	—	2	2	no	—
167	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
168	Dic₁₀₁	—	2	2	no	—
169	Dic₁₀₁	—	2	1	no	—
170	Dic₁₀₁	—	2	2	no	—
171	Dic₁₀₁	C₄×D₄	1	2	yes	—
172	C₂×C₁₀₁	—	2	2	no	—
173	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₂×C₁₀₁	#520	1	1	yes	characteristic
176	D₁₀₁	—	2	2	no	—
177	D₁₀₁	#546	1	1	yes	characteristic
178	D₁₀₁	D₄×Q₈	1	1	yes	characteristic
179	D₁₀₁	#552	1	1	yes	characteristic
180	D₁₀₁	#554	1	1	yes	characteristic
181	#2360	—	101	1	no	maximal, Sylow
182	C₁₀₁	#2360	1	1	yes	characteristic, complemented, Sylow
183	C₂×C₂²⋊Q₈	—	101	1	no	—
184	C₂×C₄×Q₈	—	101	1	no	—
185	C₂²×C₄⋊C₄	—	101	1	no	—
186	#391	—	101	1	no	—
187	#389	—	101	1	no	—
188	#389	—	101	2	no	—
189	#387	—	101	2	no	—
190	#385	—	101	2	no	—
191	C₄×C₄⋊C₄	—	101	1	no	—
192	C₄×C₂²⋊C₄	—	101	1	no	—
193	C₄×C₂²⋊C₄	—	101	2	no	—
194	C₂²×Q₈	—	101	1	no	—
195	C₂³×C₄	—	101	1	no	—
196	C₂³×C₄	—	101	2	no	—
197	C₂²⋊Q₈	—	101	1	no	—
198	C₂²⋊Q₈	—	101	1	no	—
199	C₂²⋊Q₈	—	101	2	no	—
200	C₂²⋊Q₈	—	101	1	no	—
201	C₂²⋊Q₈	—	101	2	no	—
202	C₂²⋊Q₈	#87938	1	1	yes	characteristic, complemented
203	C₄×Q₈	—	202	4	no	—
204	C₂×C₄⋊C₄	—	101	1	no	—
205	C₂×C₄⋊C₄	—	101	1	no	—
206	C₂×C₄⋊C₄	—	101	2	no	—
207	C₂×C₄⋊C₄	—	101	2	no	—
208	C₂×C₄⋊C₄	—	101	2	no	—
209	C₂×C₄⋊C₄	—	101	1	no	—
210	C₂×C₄⋊C₄	—	101	2	no	—
211	C₂×C₄⋊C₄	—	202	4	no	—
212	C₂×C₂²⋊C₄	—	101	1	no	—
213	C₂×C₂²⋊C₄	—	101	2	no	—
214	C₂×C₂²⋊C₄	—	101	1	no	—
215	C₂×C₄²	—	101	2	no	—
216	C₂×C₄²	—	101	2	no	—
217	C₂×C₄²	—	101	1	no	—
218	C₂×C₄²	—	101	1	no	—
219	#95	—	101	1	no	—
220	#95	—	101	2	no	—
221	#95	—	101	1	no	—
222	#95	—	101	2	no	—
223	#95	—	101	2	no	—
224	#95	—	101	2	no	—
225	C₂⁴	—	101	1	no	—
226	C₂×Q₈	—	101	2	no	—
227	C₂×Q₈	—	101	1	no	—
228	C₂×Q₈	—	202	2	no	—
229	C₂×Q₈	—	202	2	no	—
230	C₂×Q₈	C₂×#87938	1	1	yes	characteristic, complemented
231	C₂²×C₄	—	101	1	no	—
232	C₂²×C₄	—	101	1	no	—
233	C₂²×C₄	—	101	1	no	—
234	C₂²×C₄	—	101	2	no	—
235	C₂²×C₄	—	101	1	no	—
236	C₂²×C₄	—	101	2	no	—
237	C₂²×C₄	—	101	2	no	—
238	C₂²×C₄	—	101	1	no	—
239	C₂²×C₄	—	101	1	no	—
240	C₂²×C₄	—	101	1	no	—
241	C₂²×C₄	—	101	1	no	—
242	C₂²×C₄	—	101	2	no	—
243	C₂²×C₄	—	101	1	no	—
244	C₂²×C₄	—	404	2	no	—
245	C₂²×C₄	—	202	2	no	—
246	C₂²×C₄	—	202	2	no	—
247	C₂²×C₄	—	202	4	no	—
248	C₂²×C₄	—	202	4	no	—
249	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
250	C₄⋊C₄	—	101	4	no	—
251	C₄⋊C₄	—	101	2	no	—
252	C₄⋊C₄	—	101	2	no	—
253	C₄⋊C₄	—	101	2	no	—
254	C₄⋊C₄	—	101	1	no	—
255	C₄⋊C₄	—	101	2	no	—
256	C₄⋊C₄	—	404	2	no	—
257	C₄⋊C₄	—	202	2	no	—
258	C₄⋊C₄	—	202	4	no	—
259	C₄⋊C₄	—	202	4	no	—
260	C₄⋊C₄	C₂×#87938	1	1	yes	characteristic, complemented
261	C₄⋊C₄	C₂×#87938	1	2	yes	complemented
262	C₂²⋊C₄	—	101	4	no	—
263	C₂²⋊C₄	—	101	2	no	—
264	C₂²⋊C₄	—	101	2	no	—
265	C₂²⋊C₄	—	101	2	no	—
266	C₂²⋊C₄	—	202	2	no	—
267	C₂²⋊C₄	C₂×#87938	1	2	yes	—
268	C₄²	—	404	2	no	—
269	C₄²	—	202	2	no	—
270	C₄²	—	202	2	no	—
271	C₄²	—	202	4	no	—
272	C₂³	—	101	1	no	—
273	C₂³	—	101	2	no	—
274	C₂³	—	101	2	no	—
275	C₂³	—	101	1	no	—
276	C₂³	—	202	1	no	—
277	C₂³	—	202	1	no	—
278	C₂³	—	202	1	no	—
279	C₂³	—	202	1	no	—
280	C₂³	C₂²×#87938	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	202	4	no	—
283	Q₈	—	202	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	101	1	no	—
286	C₂×C₄	—	101	2	no	—
287	C₂×C₄	—	101	2	no	—
288	C₂×C₄	—	101	4	no	—
289	C₂×C₄	—	101	2	no	—
290	C₂×C₄	—	101	2	no	—
291	C₂×C₄	—	101	2	no	—
292	C₂×C₄	—	101	2	no	—
293	C₂×C₄	—	101	1	no	—
294	C₂×C₄	—	404	8	no	—
295	C₂×C₄	—	202	2	no	—
296	C₂×C₄	—	202	2	no	—
297	C₂×C₄	—	202	2	no	—
298	C₂×C₄	—	202	1	no	—
299	C₂×C₄	—	202	1	no	—
300	C₂×C₄	—	202	2	no	—
301	C₂×C₄	—	202	2	no	—
302	C₂×C₄	—	202	2	no	—
303	C₂×C₄	—	202	1	no	—
304	C₂×C₄	—	202	1	no	—
305	C₂×C₄	—	202	1	no	—
306	C₂×C₄	—	202	1	no	—
307	C₂×C₄	—	202	1	no	—
308	C₂×C₄	—	202	2	no	—
309	C₂×C₄	—	202	2	no	—
310	C₂×C₄	—	202	2	no	—
311	C₂×C₄	—	202	1	no	—
312	C₂×C₄	—	202	1	no	—
313	C₂×C₄	—	202	2	no	—
314	C₂×C₄	—	202	1	no	—
315	C₂×C₄	—	202	1	no	—
316	C₂×C₄	—	202	2	no	—
317	C₂×C₄	—	202	1	no	—
318	C₂×C₄	—	202	2	no	—
319	C₂×C₄	C₂²×#87938	1	2	yes	—
320	C₂×C₄	C₂²×#87938	1	2	yes	—
321	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
322	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	101	1	no	—
326	C₂²	—	101	1	no	—
327	C₂²	—	101	1	no	—
328	C₂²	—	101	1	no	—
329	C₂²	—	101	1	no	—
330	C₂²	—	101	1	no	—
331	C₂²	—	101	2	no	—
332	C₂²	—	202	2	no	—
333	C₂²	—	202	2	no	—
334	C₂²	—	202	2	no	—
335	C₂²	—	202	2	no	—
336	C₂²	—	202	1	no	—
337	C₂²	—	202	1	no	—
338	C₂²	Q₈×#87938	1	2	yes	complemented
339	C₂²	C₂³×#87938	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	101	2	no	—
344	C₄	—	404	2	no	—
345	C₄	—	404	1	no	—
346	C₄	—	404	1	no	—
347	C₄	—	202	4	no	—
348	C₄	—	202	2	no	—
349	C₄	—	202	2	no	—
350	C₄	—	202	1	no	—
351	C₄	—	202	2	no	—
352	C₄	—	202	2	no	—
353	C₄	D₄×#87938	1	2	yes	—
354	C₂	—	2	2	no	—
355	C₂	—	101	1	no	—
356	C₂	—	101	1	no	—
357	C₂	—	101	1	no	—
358	C₂	—	101	1	no	—
359	C₂	—	202	2	no	—
360	C₂	C₂×D₄×#87938	1	1	yes	characteristic
361	C₂	C₂×Q₈×#87938	1	1	yes	characteristic
362	C₂	#20266246	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	#2360
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×Q₈×#87938	C₂	semidirect
3. (C₂×#87938)⋊Q₈	C₂	semidirect
4. #20266128	C₂	semidirect
5. #20266128	C₂	semidirect
6. #20266126	C₂	semidirect
8. C₄⋊C₄×#87938	C₂	semidirect
34. #15692235	C₄	semidirect
35. #15692234	C₄	semidirect
36. #15692233	C₄	semidirect
37. #15692232	C₄	semidirect
38. #15692231	C₄	semidirect
42. C₂²⋊Q₈×C₁₀₁	C₄	semidirect
58. C₂²⋊#87938	Q₈	semidirect
64. C₂×Dic₂₀₂	C₂×C₄	semidirect
65. C₂×Dic₂₀₂	C₂×C₄	semidirect
77. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
78. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
79. #13108053	C₂×C₄	semidirect
80. #13108053	C₂×C₄	semidirect
81. #13108053	C₂×C₄	semidirect
85. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
87. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
88. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
143. C₂²×C₁₀₁	C₄×Q₈	semidirect
151. C₂×D₁₀₁	C₄×Q₈	semidirect
182. C₁₀₁	#2360	semidirect
202. C₂²⋊Q₈	#87938	semidirect
230. C₂×Q₈	C₂×#87938	semidirect
260. C₄⋊C₄	C₂×#87938	semidirect
261. C₄⋊C₄	C₂×#87938	semidirect
338. C₂²	Q₈×#87938	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#87938	2
C₂²×C₄⋊#87938	2
(C₂×#87938)⋊Q₈	2
#20266128	2
#20266126	2
#20266124	2
C₄⋊C₄×#87938	2
C₂²⋊C₄×#87938	2
C₄×C₂²⋊#87938	2
C₂²⋊Q₈×D₁₀₁	2
#2360	101