Subgroups — AGL(1,ℤ/8)×C₃⋊(C₃⋊D₃) #13179897

Order 1728.

4172 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723990	C₂	1	1	yes	maximal, characteristic, complemented
5	#11723973	C₂	1	1	yes	maximal, characteristic, complemented
6	#11723955	C₂	1	2	yes	maximal, complemented
7	SD₁₆×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
8	#11723952	C₂	1	2	yes	maximal, complemented
9	D₈×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
10	#11723949	C₂	1	1	yes	maximal, characteristic, complemented
11	M₁₆×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃⋊D₃×AGL(1,ℤ/8)	—	3	13	no	maximal
13	C₂³×C₃⋊(C₃⋊D₃)	—	2	1	no	—
14	C₃³×#41	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×#6312	—	2	1	no	—
17	#89080	C₂²	1	1	yes	characteristic, complemented
18	Q₈×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
19	#89078	—	2	1	no	—
20	#89078	C₂²	1	1	yes	characteristic, complemented
21	D₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
22	D₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
23	D₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
24	D₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
25	D₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
26	D₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
27	#89076	—	2	1	no	—
28	#89076	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
31	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
32	C₃³×SD₁₆	C₂²	1	2	yes	complemented
33	C₃³×D₈	C₂²	1	2	yes	complemented
34	C₃³×M₁₆	C₂²	1	1	yes	characteristic, complemented
35	#88862	C₂²	1	2	yes	complemented
36	#88861	C₂²	1	2	yes	complemented
37	#88860	C₂²	1	2	yes	complemented
38	#88860	C₂²	1	2	yes	complemented
39	#88855	C₂²	1	1	yes	characteristic, complemented
40	#88852	C₂²	1	2	yes	complemented
41	#88851	C₂²	1	2	yes	complemented
42	#88850	C₂²	1	2	yes	complemented
43	C₈×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
44	#41×C₃⋊D₃	—	3	13	no	—
45	C₂×D₄×C₃⋊D₃	—	3	13	no	—
46	C₃²×AGL(1,ℤ/8)	D₃	1	13	yes	complemented
47	#64158	—	3	13	no	—
48	#64141	—	3	13	no	—
49	#64123	—	3	26	no	—
50	SD₁₆×C₃⋊D₃	—	3	26	no	—
51	#64120	—	3	26	no	—
52	D₈×C₃⋊D₃	—	3	26	no	—
53	#64117	—	3	13	no	—
54	M₁₆×C₃⋊D₃	—	3	13	no	—
55	C₂³×C₃³	—	2	1	no	—
56	C₂²×C₃⋊(C₃⋊D₃)	—	2	1	no	—
57	C₂²×C₃⋊(C₃⋊D₃)	—	2	2	no	—
58	C₂²×C₃⋊(C₃⋊D₃)	—	2	2	no	—
59	C₂²×C₃⋊(C₃⋊D₃)	—	2	1	no	—
60	C₂²×C₃⋊(C₃⋊D₃)	—	4	4	no	—
61	C₂²×C₃⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
62	C₃³×Q₈	C₂³	1	1	yes	characteristic, complemented
63	C₃³×D₄	—	2	1	no	—
64	C₃³×D₄	—	2	1	no	—
65	C₃³×D₄	C₂³	1	2	yes	—
66	C₃³×D₄	C₂³	1	1	yes	characteristic, complemented
67	C₂×C₃³×C₄	—	2	1	no	—
68	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
69	#6312	—	2	1	no	—
70	#6312	—	2	1	no	—
71	#6312	—	2	1	no	—
72	#6312	—	2	2	no	—
73	#6312	—	2	2	no	—
74	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
75	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
76	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	1	no	—
77	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	1	no	—
78	C₄⋊(C₃⋊(C₃⋊D₃))	C₂³	1	2	yes	—
79	C₄⋊(C₃⋊(C₃⋊D₃))	C₂³	1	1	yes	characteristic, complemented
80	C₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
81	C₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
82	C₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
83	C₄×C₃⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
84	C₄×C₃⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
85	C₄×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
86	C₄×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
87	#6308	—	2	1	no	—
88	#6308	C₂³	1	1	yes	characteristic, complemented
89	C₃³×C₈	C₂³	1	2	yes	complemented
90	C₃⋊(C₃⋊(C₃⋊C₈))	C₂³	1	2	yes	complemented
91	D₃×AGL(1,ℤ/8)	—	9	13	no	—
92	C₂³×C₃⋊D₃	—	6	13	no	—
93	C₃²×#41	C₂×D₃	1	13	yes	complemented
94	C₂×C₃²×D₄	C₂×D₃	1	13	yes	complemented
95	C₂×#638	—	6	13	no	—
96	#3832	—	3	13	no	—
97	Q₈×C₃⋊D₃	—	3	13	no	—
98	#3830	—	3	13	no	—
99	#3830	—	6	13	no	—
100	D₄×C₃⋊D₃	—	3	26	no	—
101	D₄×C₃⋊D₃	—	3	26	no	—
102	D₄×C₃⋊D₃	—	3	13	no	—
103	D₄×C₃⋊D₃	—	6	13	no	—
104	D₄×C₃⋊D₃	—	6	13	no	—
105	D₄×C₃⋊D₃	—	6	13	no	—
106	#3828	—	3	13	no	—
107	#3828	—	6	13	no	—
108	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
109	C₂×C₄×C₃⋊D₃	—	3	13	no	—
110	C₂×C₄×C₃⋊D₃	—	6	13	no	—
111	C₃²×SD₁₆	C₂×D₃	1	26	yes	complemented
112	C₃²×D₈	C₂×D₃	1	26	yes	complemented
113	C₃²×M₁₆	C₂×D₃	1	13	yes	complemented
114	#3755	—	3	26	no	—
115	#3754	—	3	26	no	—
116	#3753	—	3	26	no	—
117	#3753	—	3	26	no	—
118	#3748	—	3	13	no	—
119	#3745	—	3	26	no	—
120	#3744	—	3	26	no	—
121	#3743	—	3	26	no	—
122	C₈×C₃⋊D₃	—	3	26	no	—
123	C₂²×C₃³	—	2	1	no	—
124	C₂²×C₃³	—	2	2	no	—
125	C₂²×C₃³	—	4	2	no	—
126	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
128	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
129	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
130	C₂×C₃⋊(C₃⋊D₃)	—	4	4	no	—
131	C₂×C₃⋊(C₃⋊D₃)	—	4	2	no	—
132	C₂×C₃⋊(C₃⋊D₃)	—	4	2	no	—
133	C₂×C₃⋊(C₃⋊D₃)	—	4	4	no	—
134	C₂×C₃⋊(C₃⋊D₃)	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃⋊(C₃⋊D₃)	C₂×D₄	1	1	yes	characteristic
136	C₃³×C₄	—	2	1	no	—
137	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
138	C₃³×C₄	C₂⁴	1	1	yes	characteristic
139	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
140	C₃⋊(C₃⋊Dic₃)	C₂×D₄	1	1	yes	characteristic
141	C₃⋊(C₃⋊Dic₃)	C₂×D₄	1	1	yes	characteristic
142	D₃×#41	—	9	13	no	—
143	C₂×D₃×D₄	—	9	13	no	—
144	C₃×AGL(1,ℤ/8)	C₃⋊D₃	1	13	yes	complemented
145	#949	—	9	13	no	—
146	#932	—	9	13	no	—
147	#914	—	9	26	no	—
148	D₃×SD₁₆	—	9	26	no	—
149	#911	—	9	26	no	—
150	D₃×D₈	—	9	26	no	—
151	#908	—	9	13	no	—
152	D₃×M₁₆	—	9	13	no	—
153	C₂³×C₃²	—	2	13	no	—
154	C₂²×C₃⋊D₃	—	3	13	no	—
155	C₂²×C₃⋊D₃	—	6	26	no	—
156	C₂²×C₃⋊D₃	—	6	26	no	—
157	C₂²×C₃⋊D₃	—	6	13	no	—
158	C₂²×C₃⋊D₃	—	6	13	no	—
159	C₂²×C₃⋊D₃	—	12	52	no	—
160	C₃²×Q₈	C₂²×D₃	1	13	yes	complemented
161	C₃²×D₄	—	2	13	no	—
162	C₃²×D₄	—	2	13	no	—
163	C₃²×D₄	C₂²×D₃	1	26	yes	—
164	C₃²×D₄	C₂²×D₃	1	13	yes	complemented
165	C₂×C₃²×C₄	—	2	13	no	—
166	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
167	#638	—	6	13	no	—
168	#638	—	6	26	no	—
169	#638	—	6	26	no	—
170	#638	—	6	13	no	—
171	#638	—	6	13	no	—
172	C₂×C₃⋊Dic₃	—	3	13	no	—
173	C₂×C₃⋊Dic₃	—	6	13	no	—
174	C₄⋊(C₃⋊D₃)	—	3	26	no	—
175	C₄⋊(C₃⋊D₃)	—	3	13	no	—
176	C₄⋊(C₃⋊D₃)	—	6	13	no	—
177	C₄⋊(C₃⋊D₃)	—	6	13	no	—
178	C₄×C₃⋊D₃	—	3	13	no	—
179	C₄×C₃⋊D₃	—	3	13	no	—
180	C₄×C₃⋊D₃	—	3	13	no	—
181	C₄×C₃⋊D₃	—	3	13	no	—
182	C₄×C₃⋊D₃	—	6	13	no	—
183	C₄×C₃⋊D₃	—	6	13	no	—
184	C₄×C₃⋊D₃	—	6	13	no	—
185	#634	—	3	13	no	—
186	#634	—	6	13	no	—
187	C₃²×C₈	C₂²×D₃	1	26	yes	complemented
188	C₃⋊(C₃⋊C₈)	—	3	26	no	—
189	C₂×AGL(1,ℤ/8)	—	27	1	no	Sylow
190	C₂×C₃³	—	2	1	no	—
191	C₂×C₃³	—	4	2	no	—
192	C₂×C₃³	—	4	1	no	—
193	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
194	C₃⋊(C₃⋊D₃)	—	2	1	no	—
195	C₃⋊(C₃⋊D₃)	—	4	1	no	—
196	C₃⋊(C₃⋊D₃)	—	4	2	no	—
197	C₃⋊(C₃⋊D₃)	AGL(1,ℤ/8)	1	2	yes	direct product factor
198	C₂³×D₃	—	18	13	no	—
199	C₃×#41	C₂×C₃⋊D₃	1	13	yes	complemented
200	C₂×C₃×D₄	C₂×C₃⋊D₃	1	13	yes	complemented
201	C₂×#67	—	18	13	no	—
202	#239	—	9	13	no	—
203	D₃×Q₈	—	9	13	no	—
204	#237	—	9	13	no	—
205	#237	—	18	13	no	—
206	D₃×D₄	—	9	26	no	—
207	D₃×D₄	—	9	26	no	—
208	D₃×D₄	—	9	13	no	—
209	D₃×D₄	—	18	13	no	—
210	D₃×D₄	—	18	13	no	—
211	D₃×D₄	—	18	13	no	—
212	#235	—	9	13	no	—
213	#235	—	18	13	no	—
214	C₂×D₁₂	—	9	13	no	—
215	C₂×C₄×D₃	—	9	13	no	—
216	C₂×C₄×D₃	—	18	13	no	—
217	C₃×SD₁₆	C₂×C₃⋊D₃	1	26	yes	complemented
218	C₃×D₈	C₂×C₃⋊D₃	1	26	yes	complemented
219	C₃×M₁₆	C₂×C₃⋊D₃	1	13	yes	complemented
220	#215	—	9	26	no	—
221	#214	—	9	26	no	—
222	#213	—	9	26	no	—
223	#213	—	9	26	no	—
224	#208	—	9	13	no	—
225	D₂₄	—	9	26	no	—
226	#204	—	9	26	no	—
227	ΓL(1,𝔽₂₅)	—	9	26	no	—
228	D₃×C₈	—	9	26	no	—
229	C₂²×C₃²	—	2	13	no	—
230	C₂²×C₃²	—	2	26	no	—
231	C₂²×C₃²	—	4	26	no	—
232	C₂²×C₃²	D₃×D₄	1	13	yes	—
233	C₂×C₃⋊D₃	—	3	13	no	—
234	C₂×C₃⋊D₃	—	3	13	no	—
235	C₂×C₃⋊D₃	—	6	26	no	—
236	C₂×C₃⋊D₃	—	6	13	no	—
237	C₂×C₃⋊D₃	—	6	26	no	—
238	C₂×C₃⋊D₃	—	12	26	no	—
239	C₂×C₃⋊D₃	—	12	52	no	—
240	C₂×C₃⋊D₃	—	12	52	no	—
241	C₂×C₃⋊D₃	—	12	26	no	—
242	C₃²×C₄	—	2	13	no	—
243	C₃²×C₄	D₃×D₄	1	13	yes	—
244	C₃²×C₄	C₂³×D₃	1	13	yes	—
245	C₃⋊Dic₃	—	3	13	no	—
246	C₃⋊Dic₃	—	3	13	no	—
247	C₃⋊Dic₃	—	6	13	no	—
248	C₂×#41	—	27	1	no	—
249	C₂²×D₄	—	27	1	no	—
250	AGL(1,ℤ/8)	—	27	1	no	—
251	AGL(1,ℤ/8)	—	27	1	no	—
252	AGL(1,ℤ/8)	—	27	2	no	—
253	AGL(1,ℤ/8)	—	27	2	no	—
254	AGL(1,ℤ/8)	—	27	1	no	—
255	AGL(1,ℤ/8)	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
256	C₂×SD₁₆	—	27	2	no	—
257	C₂×D₈	—	27	2	no	—
258	C₂×M₁₆	—	27	1	no	—
259	C₃³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
260	C₂³×C₃	—	2	13	no	—
261	C₂²×D₃	—	9	13	no	—
262	C₂²×D₃	—	18	13	no	—
263	C₂²×D₃	—	18	26	no	—
264	C₂²×D₃	—	18	26	no	—
265	C₂²×D₃	—	18	13	no	—
266	C₂²×D₃	—	36	52	no	—
267	C₃×Q₈	C₂²×C₃⋊D₃	1	13	yes	complemented
268	C₃×D₄	—	2	13	no	—
269	C₃×D₄	—	2	13	no	—
270	C₃×D₄	C₂²×C₃⋊D₃	1	26	yes	—
271	C₃×D₄	C₂²×C₃⋊D₃	1	13	yes	complemented
272	C₂×C₃×C₄	—	2	13	no	—
273	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
274	#67	—	18	13	no	—
275	#67	—	18	26	no	—
276	#67	—	18	26	no	—
277	#67	—	18	13	no	—
278	#67	—	18	13	no	—
279	C₂×Dic₃	—	9	13	no	—
280	C₂×Dic₃	—	18	13	no	—
281	D₁₂	—	9	26	no	—
282	D₁₂	—	9	13	no	—
283	D₁₂	—	18	13	no	—
284	D₁₂	—	18	13	no	—
285	C₄×D₃	—	9	13	no	—
286	C₄×D₃	—	9	13	no	—
287	C₄×D₃	—	9	13	no	—
288	C₄×D₃	—	9	13	no	—
289	C₄×D₃	—	18	13	no	—
290	C₄×D₃	—	18	13	no	—
291	C₄×D₃	—	18	13	no	—
292	Dic₆	—	9	13	no	—
293	Dic₆	—	18	13	no	—
294	C₃×C₈	C₂²×C₃⋊D₃	1	26	yes	complemented
295	C₃⋊C₈	—	9	26	no	—
296	C₂×C₃²	—	2	13	no	—
297	C₂×C₃²	—	4	26	no	—
298	C₂×C₃²	—	4	13	no	—
299	C₂×C₃²	C₂×D₃×D₄	1	13	yes	—
300	C₃⋊D₃	—	3	26	no	—
301	C₃⋊D₃	—	6	13	no	—
302	C₃⋊D₃	—	12	13	no	—
303	C₃⋊D₃	—	12	26	no	—
304	C₂⁴	—	54	1	no	—
305	#41	—	27	1	no	—
306	#41	—	27	1	no	—
307	#41	—	27	1	no	—
308	#41	—	54	1	no	—
309	#41	—	54	1	no	—
310	#41	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
311	C₂×Q₈	—	27	1	no	—
312	C₂×D₄	—	27	2	no	—
313	C₂×D₄	—	27	2	no	—
314	C₂×D₄	—	27	1	no	—
315	C₂×D₄	—	27	1	no	—
316	C₂×D₄	—	54	1	no	—
317	C₂×D₄	—	54	1	no	—
318	C₂×D₄	—	54	1	no	—
319	C₂×D₄	—	54	1	no	—
320	C₂×D₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
321	C₂²×C₄	—	27	1	no	—
322	C₂²×C₄	—	54	1	no	—
323	SD₁₆	—	27	2	no	—
324	SD₁₆	—	27	2	no	—
325	SD₁₆	—	27	2	no	—
326	SD₁₆	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
327	D₈	—	27	2	no	—
328	D₈	—	27	2	no	—
329	D₈	—	27	2	no	—
330	D₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
331	M₁₆	—	27	1	no	—
332	M₁₆	—	27	2	no	—
333	M₁₆	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
334	C₂×C₈	—	27	2	no	—
335	C₂²×C₃	—	2	13	no	—
336	C₂²×C₃	—	2	26	no	—
337	C₂²×C₃	—	4	26	no	—
338	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
339	C₂×D₃	—	9	13	no	—
340	C₂×D₃	—	9	13	no	—
341	C₂×D₃	—	18	26	no	—
342	C₂×D₃	—	18	13	no	—
343	C₂×D₃	—	18	26	no	—
344	C₂×D₃	—	36	52	no	—
345	C₂×D₃	—	36	26	no	—
346	C₂×D₃	—	36	52	no	—
347	C₂×D₃	—	36	26	no	—
348	C₃×C₄	—	2	13	no	—
349	C₃×C₄	D₄×C₃⋊D₃	1	13	yes	—
350	C₃×C₄	C₂³×C₃⋊D₃	1	13	yes	—
351	Dic₃	—	9	13	no	—
352	Dic₃	—	9	13	no	—
353	Dic₃	—	18	13	no	—
354	C₃²	D₃×AGL(1,ℤ/8)	1	13	yes	complemented
355	C₂³	—	2	1	no	—
356	C₂³	—	27	1	no	—
357	C₂³	—	54	1	no	—
358	C₂³	—	54	2	no	—
359	C₂³	—	54	2	no	—
360	C₂³	—	54	1	no	—
361	C₂³	—	108	4	no	—
362	Q₈	—	27	1	no	—
363	Q₈	—	54	1	no	—
364	Q₈	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
365	D₄	—	2	1	no	—
366	D₄	—	2	1	no	—
367	D₄	—	27	1	no	—
368	D₄	—	27	2	no	—
369	D₄	—	54	1	no	—
370	D₄	—	54	2	no	—
371	D₄	—	54	2	no	—
372	D₄	—	54	1	no	—
373	D₄	—	54	1	no	—
374	D₄	—	54	1	no	—
375	D₄	—	54	1	no	—
376	D₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
377	D₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	27	1	no	—
380	C₂×C₄	—	27	1	no	—
381	C₂×C₄	—	27	1	no	—
382	C₂×C₄	—	27	1	no	—
383	C₂×C₄	—	27	1	no	—
384	C₂×C₄	—	54	1	no	—
385	C₂×C₄	—	54	1	no	—
386	C₂×C₄	—	54	1	no	—
387	C₂×C₄	—	54	1	no	—
388	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
389	C₈	—	27	2	no	—
390	C₈	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
391	C₂×C₃	—	2	13	no	—
392	C₂×C₃	—	4	26	no	—
393	C₂×C₃	—	4	13	no	—
394	C₂×C₃	C₂×D₄×C₃⋊D₃	1	13	yes	—
395	D₃	—	9	26	no	—
396	D₃	—	18	13	no	—
397	D₃	—	36	26	no	—
398	D₃	—	36	13	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	27	1	no	—
403	C₂²	—	27	1	no	—
404	C₂²	—	54	2	no	—
405	C₂²	—	54	1	no	—
406	C₂²	—	54	2	no	—
407	C₂²	—	108	2	no	—
408	C₂²	—	108	4	no	—
409	C₂²	—	108	2	no	—
410	C₂²	—	108	4	no	—
411	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
412	C₄	—	2	1	no	—
413	C₄	—	27	1	no	—
414	C₄	—	27	1	no	—
415	C₄	—	54	1	no	—
416	C₄	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
417	C₄	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
418	C₃	C₃⋊D₃×AGL(1,ℤ/8)	1	13	yes	complemented
419	C₂	—	2	1	no	—
420	C₂	—	4	2	no	—
421	C₂	—	4	1	no	—
422	C₂	—	27	2	no	—
423	C₂	—	54	1	no	—
424	C₂	—	108	2	no	—
425	C₂	—	108	1	no	—
426	C₂	C₂×D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₃³×C₄	C₂⁴
Frattini	C₄	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₃³×AGL(1,ℤ/8)	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	AGL(1,ℤ/8)	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #41×C₃⋊(C₃⋊D₃)	C₂	semidirect
2. C₂×D₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
3. C₃³×AGL(1,ℤ/8)	C₂	semidirect
4. #11723990	C₂	semidirect
5. #11723973	C₂	semidirect
6. #11723955	C₂	semidirect
7. SD₁₆×C₃⋊(C₃⋊D₃)	C₂	semidirect
8. #11723952	C₂	semidirect
9. D₈×C₃⋊(C₃⋊D₃)	C₂	semidirect
10. #11723949	C₂	semidirect
11. M₁₆×C₃⋊(C₃⋊D₃)	C₂	semidirect
14. C₃³×#41	C₂²	semidirect
15. C₂×C₃³×D₄	C₂²	semidirect
17. #89080	C₂²	semidirect
18. Q₈×C₃⋊(C₃⋊D₃)	C₂²	semidirect
20. #89078	C₂²	semidirect
25. D₄×C₃⋊(C₃⋊D₃)	C₂²	semidirect
26. D₄×C₃⋊(C₃⋊D₃)	C₂²	semidirect
28. #89076	C₂²	semidirect
29. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	semidirect
32. C₃³×SD₁₆	C₂²	semidirect
33. C₃³×D₈	C₂²	semidirect
34. C₃³×M₁₆	C₂²	semidirect
35. #88862	C₂²	semidirect
36. #88861	C₂²	semidirect
37. #88860	C₂²	semidirect
38. #88860	C₂²	semidirect
39. #88855	C₂²	semidirect
40. #88852	C₂²	semidirect
41. #88851	C₂²	semidirect
42. #88850	C₂²	semidirect
43. C₈×C₃⋊(C₃⋊D₃)	C₂²	semidirect
46. C₃²×AGL(1,ℤ/8)	D₃	semidirect
62. C₃³×Q₈	C₂³	semidirect
66. C₃³×D₄	C₂³	semidirect
79. C₄⋊(C₃⋊(C₃⋊D₃))	C₂³	semidirect
88. #6308	C₂³	semidirect
89. C₃³×C₈	C₂³	semidirect
90. C₃⋊(C₃⋊(C₃⋊C₈))	C₂³	semidirect
93. C₃²×#41	C₂×D₃	semidirect
94. C₂×C₃²×D₄	C₂×D₃	semidirect
111. C₃²×SD₁₆	C₂×D₃	semidirect
112. C₃²×D₈	C₂×D₃	semidirect
113. C₃²×M₁₆	C₂×D₃	semidirect
144. C₃×AGL(1,ℤ/8)	C₃⋊D₃	semidirect
160. C₃²×Q₈	C₂²×D₃	semidirect
164. C₃²×D₄	C₂²×D₃	semidirect
187. C₃²×C₈	C₂²×D₃	semidirect
197. C₃⋊(C₃⋊D₃)	AGL(1,ℤ/8)	direct (N × H)
199. C₃×#41	C₂×C₃⋊D₃	semidirect
200. C₂×C₃×D₄	C₂×C₃⋊D₃	semidirect
217. C₃×SD₁₆	C₂×C₃⋊D₃	semidirect
218. C₃×D₈	C₂×C₃⋊D₃	semidirect
219. C₃×M₁₆	C₂×C₃⋊D₃	semidirect
255. AGL(1,ℤ/8)	C₃⋊(C₃⋊D₃)	direct (N × H)
259. C₃³	C₂×AGL(1,ℤ/8)	semidirect
267. C₃×Q₈	C₂²×C₃⋊D₃	semidirect
271. C₃×D₄	C₂²×C₃⋊D₃	semidirect
294. C₃×C₈	C₂²×C₃⋊D₃	semidirect
310. #41	C₂×C₃⋊(C₃⋊D₃)	semidirect
320. C₂×D₄	C₂×C₃⋊(C₃⋊D₃)	semidirect
326. SD₁₆	C₂×C₃⋊(C₃⋊D₃)	semidirect
330. D₈	C₂×C₃⋊(C₃⋊D₃)	semidirect
333. M₁₆	C₂×C₃⋊(C₃⋊D₃)	semidirect
354. C₃²	D₃×AGL(1,ℤ/8)	semidirect
364. Q₈	C₂²×C₃⋊(C₃⋊D₃)	semidirect
377. D₄	C₂²×C₃⋊(C₃⋊D₃)	semidirect
390. C₈	C₂²×C₃⋊(C₃⋊D₃)	semidirect
418. C₃	C₃⋊D₃×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
#41×C₃⋊(C₃⋊D₃)	2
C₂×D₄×C₃⋊(C₃⋊D₃)	2
C₃³×AGL(1,ℤ/8)	2
#11723990	2
#11723973	2
#11723955	2
SD₁₆×C₃⋊(C₃⋊D₃)	2
#11723952	2
D₈×C₃⋊(C₃⋊D₃)	2
#11723949	2
M₁₆×C₃⋊(C₃⋊D₃)	2
C₃⋊D₃×AGL(1,ℤ/8)	3

Subgroups — AGL(1,ℤ/8)×C3⋊(C3⋊D3) #13179897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,ℤ/8)×C₃⋊(C₃⋊D₃) #13179897