Subgroups — #13136471

Order 1728.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11720196	C₂	1	2	yes	maximal, complemented
3	#11720195	C₂	1	2	yes	maximal, complemented
4	#11720147	C₂	1	2	yes	maximal, complemented
5	#11720146	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720144	C₂	1	1	yes	maximal, characteristic
7	#11720143	C₂	1	2	yes	maximal, complemented
8	#11720142	C₂	1	1	yes	maximal, characteristic, complemented
9	#11720136	C₂	1	1	yes	maximal, characteristic, complemented
10	#11720124	C₂	1	1	yes	maximal, characteristic, complemented
11	#11720121	C₂	1	1	yes	maximal, characteristic, complemented
12	#10589835	—	3	1	no	maximal
13	C₂×Q₈×C₂₇	C₂²	1	2	yes	—
14	C₂×D₁₀₈	C₂²	1	1	yes	characteristic
15	C₂×D₁₀₈	C₂²	1	2	yes	—
16	C₂×C₄×D₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₂₇	C₂²	1	2	yes	—
18	C₂×Dic₅₄	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₂₇	C₂²	1	2	yes	—
22	C₄²×C₂₇	C₂²	1	1	yes	characteristic
23	(C₂×D₂₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₂₇)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₂₇)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₂₇)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₂₇)⋊C₄	C₂²	1	2	yes	—
28	(C₂×D₂₇)⋊C₄	C₂²	1	1	yes	characteristic
29	(C₂×D₂₇)⋊C₄	C₂²	1	2	yes	—
30	C₄⋊Dic₂₇	C₂²	1	2	yes	—
31	Dic₂₇⋊C₄	C₂²	1	1	yes	characteristic
32	Dic₂₇⋊C₄	C₂²	1	2	yes	—
33	Dic₂₇⋊C₄	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₇	C₂²	1	2	yes	—
35	C₄×Dic₂₇	C₂²	1	1	yes	characteristic
36	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic
37	#63509	—	3	2	no	—
38	#63508	—	3	2	no	—
39	#63460	—	3	2	no	—
40	#63459	—	3	1	no	—
41	#63457	—	3	1	no	—
42	#63456	—	3	2	no	—
43	#63455	—	3	1	no	—
44	#63449	—	3	1	no	—
45	#63437	—	3	1	no	—
46	#63434	—	3	1	no	—
47	C₂²×D₂₇	C₂³	1	1	yes	characteristic
48	C₂²×D₂₇	C₂³	1	2	yes	—
49	Q₈×C₂₇	—	4	2	no	—
50	C₂×C₄×C₂₇	C₂³	1	2	yes	—
51	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₂₇	C₂³	1	2	yes	—
54	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₇	C₂³	1	1	yes	characteristic
56	C₂×Dic₂₇	C₂³	1	1	yes	characteristic
57	C₂×Dic₂₇	C₂³	1	2	yes	—
58	C₂×Dic₂₇	C₂³	1	1	yes	characteristic
59	D₁₀₈	—	2	2	no	—
60	D₁₀₈	—	4	2	no	—
61	C₄×D₂₇	—	2	2	no	—
62	C₄×D₂₇	—	4	2	no	—
63	Dic₅₄	—	2	2	no	—
64	#5733	—	9	1	no	—
65	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
66	C₂×D₃₆	—	3	1	no	—
67	C₂×D₃₆	—	3	2	no	—
68	C₂×C₄×D₉	—	3	1	no	—
69	C₂×C₄×D₉	—	3	2	no	—
70	C₂×Dic₁₈	—	3	1	no	—
71	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
72	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
74	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
75	(C₂×D₉)⋊C₄	—	3	2	no	—
76	(C₂×D₉)⋊C₄	—	3	2	no	—
77	(C₂×D₉)⋊C₄	—	3	1	no	—
78	(C₂×D₉)⋊C₄	—	3	2	no	—
79	(C₂×D₉)⋊C₄	—	3	2	no	—
80	(C₂×D₉)⋊C₄	—	3	1	no	—
81	(C₂×D₉)⋊C₄	—	3	2	no	—
82	C₄⋊Dic₉	—	3	2	no	—
83	Dic₉⋊C₄	—	3	1	no	—
84	Dic₉⋊C₄	—	3	2	no	—
85	Dic₉⋊C₄	—	3	1	no	—
86	C₄×Dic₉	—	3	2	no	—
87	C₄×Dic₉	—	3	1	no	—
88	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
89	C₂×D₂₇	—	2	1	no	—
90	C₂×D₂₇	—	2	2	no	—
91	C₂×D₂₇	—	2	2	no	—
92	C₂×D₂₇	—	2	1	no	—
93	C₂×D₂₇	—	2	1	no	—
94	C₂×D₂₇	—	2	2	no	—
95	C₄×C₂₇	—	2	2	no	—
96	C₄×C₂₇	—	2	2	no	—
97	C₄×C₂₇	—	2	1	no	—
98	C₄×C₂₇	—	2	1	no	—
99	C₄×C₂₇	#41	1	2	yes	—
100	Dic₂₇	—	2	2	no	—
101	Dic₂₇	—	2	1	no	—
102	Dic₂₇	—	2	1	no	—
103	Dic₂₇	—	2	1	no	—
104	C₃×C₄⋊Q₈	D₉	1	1	yes	characteristic
105	#947	—	9	2	no	—
106	#946	—	9	2	no	—
107	#898	—	9	2	no	—
108	#897	—	9	1	no	—
109	#895	—	9	1	no	—
110	#894	—	9	2	no	—
111	#893	—	9	1	no	—
112	#887	—	9	1	no	—
113	#875	—	9	1	no	—
114	#872	—	9	1	no	—
115	C₂²×D₉	—	3	2	no	—
116	C₂²×D₉	—	3	1	no	—
117	Q₈×C₉	—	4	2	no	—
118	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
119	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
122	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
123	C₂×Dic₉	—	3	1	no	—
124	C₂×Dic₉	—	3	1	no	—
125	C₂×Dic₉	—	3	2	no	—
126	C₂×Dic₉	—	3	1	no	—
127	D₃₆	—	6	2	no	—
128	D₃₆	—	12	2	no	—
129	C₄×D₉	—	6	2	no	—
130	C₄×D₉	—	12	2	no	—
131	Dic₁₈	—	6	2	no	—
132	#542	—	27	1	no	Sylow
133	C₂×C₂₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₂₇	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₂₇	2₋¹⁺⁴	1	1	yes	characteristic
136	D₂₇	—	4	1	no	—
137	D₂₇	—	4	2	no	—
138	C₂×C₃×Q₈	C₂×D₉	1	2	yes	—
139	C₂×D₁₂	—	9	1	no	—
140	C₂×D₁₂	—	9	2	no	—
141	C₂×C₄×D₃	—	9	1	no	—
142	C₂×C₄×D₃	—	9	2	no	—
143	C₂×Dic₆	—	9	1	no	—
144	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
145	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
146	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
147	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
148	(C₂×D₃)⋊C₄	—	9	1	no	—
149	(C₂×D₃)⋊C₄	—	9	1	no	—
150	(C₂×D₃)⋊C₄	—	9	2	no	—
151	(C₂×D₃)⋊C₄	—	9	2	no	—
152	(C₂×D₃)⋊C₄	—	9	2	no	—
153	(C₂×D₃)⋊C₄	—	9	2	no	—
154	(C₂×D₃)⋊C₄	—	9	2	no	—
155	C₄⋊Dic₃	—	9	2	no	—
156	Dic₃⋊C₄	—	9	1	no	—
157	Dic₃⋊C₄	—	9	1	no	—
158	Dic₃⋊C₄	—	9	2	no	—
159	C₄×Dic₃	—	9	1	no	—
160	C₄×Dic₃	—	9	2	no	—
161	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
162	C₂×D₉	—	6	2	no	—
163	C₂×D₉	—	6	1	no	—
164	C₂×D₉	—	6	1	no	—
165	C₂×D₉	—	6	2	no	—
166	C₂×D₉	—	6	2	no	—
167	C₂×D₉	—	6	1	no	—
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	—	2	2	no	—
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	#239	1	2	yes	—
173	Dic₉	—	6	1	no	—
174	Dic₉	—	6	2	no	—
175	Dic₉	—	6	1	no	—
176	Dic₉	—	6	1	no	—
177	C₄⋊Q₈	D₂₇	1	1	yes	characteristic, complemented
178	#126	—	27	2	no	—
179	#124	—	27	2	no	—
180	#124	—	27	1	no	—
181	#123	—	27	2	no	—
182	C₂²⋊Q₈	—	27	2	no	—
183	C₂²⋊Q₈	—	27	1	no	—
184	#121	—	27	1	no	—
185	C₄×Q₈	—	27	1	no	—
186	C₄×D₄	—	27	1	no	—
187	#117	—	27	1	no	—
188	C₂₇	#542	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₃	—	9	2	no	—
190	C₂²×D₃	—	9	1	no	—
191	C₃×Q₈	—	4	2	no	—
192	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
193	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
196	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
197	C₂×Dic₃	—	9	1	no	—
198	C₂×Dic₃	—	9	1	no	—
199	C₂×Dic₃	—	9	2	no	—
200	C₂×Dic₃	—	9	1	no	—
201	D₁₂	—	18	2	no	—
202	D₁₂	—	36	2	no	—
203	C₄×D₃	—	18	2	no	—
204	C₄×D₃	—	36	2	no	—
205	Dic₆	—	18	2	no	—
206	C₂×C₉	#1004	1	1	yes	characteristic
207	C₂×C₉	C₂×#239	1	1	yes	characteristic
208	C₂×C₉	#1007	1	1	yes	characteristic
209	D₉	—	12	2	no	—
210	D₉	—	12	1	no	—
211	C₂×Q₈	—	27	1	no	—
212	C₂×Q₈	C₂×D₂₇	1	2	yes	—
213	C₂×D₄	—	27	1	no	—
214	C₂×D₄	—	27	2	no	—
215	C₂²×C₄	—	27	1	no	—
216	C₂²×C₄	—	27	2	no	—
217	C₄⋊C₄	—	27	1	no	—
218	C₄⋊C₄	—	27	1	no	—
219	C₄⋊C₄	—	27	2	no	—
220	C₄⋊C₄	—	27	2	no	—
221	C₄⋊C₄	C₂×D₂₇	1	1	yes	characteristic
222	C₄⋊C₄	C₂×D₂₇	1	1	yes	characteristic
223	C₄⋊C₄	C₂×D₂₇	1	2	yes	—
224	C₂²⋊C₄	—	27	1	no	—
225	C₂²⋊C₄	—	27	1	no	—
226	C₂²⋊C₄	—	27	2	no	—
227	C₂²⋊C₄	—	27	2	no	—
228	C₂²⋊C₄	—	27	2	no	—
229	C₂²⋊C₄	—	27	2	no	—
230	C₂²⋊C₄	—	27	2	no	—
231	C₄²	—	27	1	no	—
232	C₄²	—	27	2	no	—
233	C₄²	C₂×D₂₇	1	1	yes	characteristic
234	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
235	C₂×D₃	—	18	2	no	—
236	C₂×D₃	—	18	1	no	—
237	C₂×D₃	—	18	1	no	—
238	C₂×D₃	—	18	2	no	—
239	C₂×D₃	—	18	2	no	—
240	C₂×D₃	—	18	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	#3701	1	2	yes	—
246	Dic₃	—	18	1	no	—
247	Dic₃	—	18	2	no	—
248	Dic₃	—	18	1	no	—
249	Dic₃	—	18	1	no	—
250	C₉	#5733	1	1	yes	characteristic
251	C₂³	—	27	2	no	—
252	C₂³	—	27	1	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	54	2	no	—
255	D₄	—	54	2	no	—
256	D₄	—	108	2	no	—
257	C₂×C₄	—	27	1	no	—
258	C₂×C₄	—	27	2	no	—
259	C₂×C₄	—	27	1	no	—
260	C₂×C₄	—	27	1	no	—
261	C₂×C₄	—	54	2	no	—
262	C₂×C₄	—	108	2	no	—
263	C₂×C₄	C₂²×D₂₇	1	2	yes	—
264	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₇	1	2	yes	—
267	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
268	C₂×C₃	#63710	1	1	yes	characteristic
269	C₂×C₃	C₂×#3701	1	1	yes	characteristic
270	C₂×C₃	#63713	1	1	yes	characteristic
271	D₃	—	36	2	no	—
272	D₃	—	36	1	no	—
273	C₂²	—	54	2	no	—
274	C₂²	—	54	1	no	—
275	C₂²	—	54	1	no	—
276	C₂²	—	54	2	no	—
277	C₂²	—	54	2	no	—
278	C₂²	—	54	1	no	—
279	C₂²	C₂³×D₂₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	54	1	no	—
285	C₄	—	54	1	no	—
286	C₄	—	54	2	no	—
287	C₄	—	54	1	no	—
288	C₄	#88403	1	2	yes	—
289	C₃	#10589835	1	1	yes	characteristic
290	C₂	—	108	2	no	—
291	C₂	—	108	1	no	—
292	C₂	#11720670	1	1	yes	characteristic
293	C₂	C₂×#88403	1	1	yes	characteristic
294	C₂	#11720673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₇
commutator, cosocle	C₂²×C₂₇	C₂⁴
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂₇×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂³×D₉
prime radical	#11720144	C₂
hypercenter	C₄⋊Q₈	D₂₇
hypocenter	C₂₇	#542
LCS	G ▷ C₂²×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂₇×C₄⋊Q₈	C₂	semidirect
2. #11720196	C₂	semidirect
3. #11720195	C₂	semidirect
4. #11720147	C₂	semidirect
5. #11720146	C₂	semidirect
7. #11720143	C₂	semidirect
8. #11720142	C₂	semidirect
9. #11720136	C₂	semidirect
10. #11720124	C₂	semidirect
11. #11720121	C₂	semidirect
177. C₄⋊Q₈	D₂₇	semidirect
188. C₂₇	#542	semidirect

Maximal subgroups

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

subgroup	index
C₂₇×C₄⋊Q₈	2
#11720196	2
#11720195	2
#11720147	2
#11720146	2
#11720144	2
#11720143	2
#11720142	2
#11720136	2
#11720124	2
#11720121	2
#10589835	3