Subgroups — #13179752

Order 1728.

3024 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11723988	C₂	1	1	yes	maximal, characteristic, complemented
3	#11723985	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723970	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6312	C₂	1	1	yes	maximal, characteristic, complemented
6	#11723939	C₂	1	1	yes	maximal, characteristic, complemented
7	#11723937	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723937	C₂	1	1	yes	maximal, characteristic, complemented
9	#11723935	C₂	1	1	yes	maximal, characteristic, complemented
10	#11723933	C₂	1	1	yes	maximal, characteristic, complemented
11	#11723928	C₂	1	1	yes	maximal, characteristic, complemented
12	#11723926	C₂	1	1	yes	maximal, characteristic, complemented
13	#11723926	C₂	1	1	yes	maximal, characteristic, complemented
14	#11723925	C₂	1	1	yes	maximal, characteristic
15	#11723920	C₂	1	1	yes	maximal, characteristic, complemented
16	#10596111	—	3	13	no	maximal
17	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
18	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
19	C₂×#6312	C₂²	1	1	yes	characteristic
20	C₂×#6312	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
24	C₂×#6308	C₂²	1	1	yes	characteristic
25	C₂×#6308	C₂²	1	1	yes	characteristic
26	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₃³×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₃³×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	#88864	C₂²	1	1	yes	characteristic
32	#88864	C₂²	1	1	yes	characteristic
33	#88859	C₂²	1	1	yes	characteristic
34	#88859	C₂²	1	1	yes	characteristic
35	#88859	C₂²	1	1	yes	characteristic
36	#88859	C₂²	1	1	yes	characteristic
37	#88859	C₂²	1	1	yes	characteristic
38	#88859	C₂²	1	1	yes	characteristic
39	#88859	C₂²	1	1	yes	characteristic
40	#88859	C₂²	1	1	yes	characteristic
41	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
42	#88857	C₂²	1	1	yes	characteristic
43	#88857	C₂²	1	1	yes	characteristic
44	#88857	C₂²	1	1	yes	characteristic
45	#88857	C₂²	1	1	yes	characteristic
46	#88857	C₂²	1	1	yes	characteristic
47	#88857	C₂²	1	1	yes	characteristic
48	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
49	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
50	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
51	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
52	C₃²×C₂²⋊Q₈	D₃	1	13	yes	complemented
53	#64156	—	3	13	no	—
54	#64153	—	3	13	no	—
55	#64138	—	3	13	no	—
56	C₄×#638	—	3	13	no	—
57	#64107	—	3	13	no	—
58	#64105	—	3	13	no	—
59	#64105	—	3	13	no	—
60	#64103	—	3	13	no	—
61	#64101	—	3	13	no	—
62	#64096	—	3	13	no	—
63	#64094	—	3	13	no	—
64	#64094	—	3	13	no	—
65	#64093	—	3	13	no	—
66	#64088	—	3	13	no	—
67	C₂³×C₃³	C₂³	1	1	yes	characteristic
68	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
69	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
70	C₃³×Q₈	—	4	1	no	—
71	C₂×C₃³×C₄	—	4	1	no	—
72	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
73	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
74	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
75	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
76	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
77	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
78	#6312	—	2	2	no	—
79	#6312	—	4	1	no	—
80	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
83	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
84	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
85	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
86	C₄⋊(C₃⋊(C₃⋊D₃))	—	4	1	no	—
87	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
88	C₄×C₃⋊(C₃⋊D₃)	—	4	1	no	—
89	#6308	—	2	2	no	—
90	#6308	—	4	1	no	—
91	#5640	—	9	13	no	—
92	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
93	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
94	C₂×#638	—	3	13	no	—
95	C₂×#638	—	3	13	no	—
96	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
97	C₂×C₄×C₃⋊D₃	—	3	13	no	—
98	C₂×C₄×C₃⋊D₃	—	3	13	no	—
99	C₂×#634	—	3	13	no	—
100	C₂×#634	—	3	13	no	—
101	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
102	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
103	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
104	C₃²×C₂²⋊C₄	C₂×D₃	1	13	yes	—
105	C₃²×C₂²⋊C₄	C₂×D₃	1	13	yes	—
106	#3757	—	3	13	no	—
107	#3757	—	3	13	no	—
108	#3752	—	3	13	no	—
109	#3752	—	3	13	no	—
110	#3752	—	3	13	no	—
111	#3752	—	3	13	no	—
112	#3752	—	3	13	no	—
113	#3752	—	3	13	no	—
114	#3752	—	3	13	no	—
115	#3752	—	3	13	no	—
116	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
117	#3750	—	3	13	no	—
118	#3750	—	3	13	no	—
119	#3750	—	3	13	no	—
120	#3750	—	3	13	no	—
121	#3750	—	3	13	no	—
122	#3750	—	3	13	no	—
123	C₄×C₃⋊Dic₃	—	3	13	no	—
124	C₄×C₃⋊Dic₃	—	3	13	no	—
125	C₄×C₃⋊Dic₃	—	3	13	no	—
126	C₄×C₃⋊Dic₃	—	3	13	no	—
127	C₂²×C₃³	—	2	1	no	—
128	C₂²×C₃³	—	2	1	no	—
129	C₂²×C₃³	—	2	1	no	—
130	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
131	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
132	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
133	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
134	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
135	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
136	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
137	C₃³×C₄	—	2	1	no	—
138	C₃³×C₄	—	2	1	no	—
139	C₃³×C₄	—	2	1	no	—
140	C₃³×C₄	—	2	1	no	—
141	C₃³×C₄	—	2	1	no	—
142	C₃³×C₄	—	2	1	no	—
143	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
144	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
145	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
146	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
147	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
148	C₃⋊(C₃⋊Dic₃)	#41	1	2	yes	—
149	C₃×C₂²⋊Q₈	C₃⋊D₃	1	13	yes	complemented
150	#947	—	9	13	no	—
151	#944	—	9	13	no	—
152	#929	—	9	13	no	—
153	C₄×#67	—	9	13	no	—
154	#898	—	9	13	no	—
155	#896	—	9	13	no	—
156	#896	—	9	13	no	—
157	#894	—	9	13	no	—
158	#892	—	9	13	no	—
159	#887	—	9	13	no	—
160	#885	—	9	13	no	—
161	#885	—	9	13	no	—
162	#884	—	9	13	no	—
163	#879	—	9	13	no	—
164	C₂³×C₃²	C₂²×D₃	1	13	yes	—
165	C₂²×C₃⋊D₃	—	3	13	no	—
166	C₂²×C₃⋊D₃	—	3	13	no	—
167	C₃²×Q₈	—	4	13	no	—
168	C₂×C₃²×C₄	—	4	13	no	—
169	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
170	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
171	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
172	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
173	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
174	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
175	#638	—	6	26	no	—
176	#638	—	12	13	no	—
177	C₂×C₃⋊Dic₃	—	3	13	no	—
178	C₂×C₃⋊Dic₃	—	3	13	no	—
179	C₂×C₃⋊Dic₃	—	3	13	no	—
180	C₂×C₃⋊Dic₃	—	3	13	no	—
181	C₂×C₃⋊Dic₃	—	3	13	no	—
182	C₂×C₃⋊Dic₃	—	3	13	no	—
183	C₄⋊(C₃⋊D₃)	—	12	13	no	—
184	C₄×C₃⋊D₃	—	6	26	no	—
185	C₄×C₃⋊D₃	—	12	13	no	—
186	#634	—	6	26	no	—
187	#634	—	12	13	no	—
188	#542	—	27	1	no	Sylow
189	C₂×C₃³	—	4	1	no	—
190	C₂×C₃³	C₂×#41	1	1	yes	characteristic
191	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
193	C₃⋊(C₃⋊D₃)	—	4	1	no	—
194	C₃⋊(C₃⋊D₃)	—	4	1	no	—
195	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
196	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
197	C₂×#67	—	9	13	no	—
198	C₂×#67	—	9	13	no	—
199	C₂×D₁₂	—	9	13	no	—
200	C₂×C₄×D₃	—	9	13	no	—
201	C₂×C₄×D₃	—	9	13	no	—
202	C₂×Dic₆	—	9	13	no	—
203	C₂×Dic₆	—	9	13	no	—
204	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
205	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
206	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
207	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
208	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
209	#217	—	9	13	no	—
210	#217	—	9	13	no	—
211	(C₂×D₃)⋊C₄	—	9	13	no	—
212	(C₂×D₃)⋊C₄	—	9	13	no	—
213	(C₂×D₃)⋊C₄	—	9	13	no	—
214	(C₂×D₃)⋊C₄	—	9	13	no	—
215	(C₂×D₃)⋊C₄	—	9	13	no	—
216	(C₂×D₃)⋊C₄	—	9	13	no	—
217	(C₂×D₃)⋊C₄	—	9	13	no	—
218	(C₂×D₃)⋊C₄	—	9	13	no	—
219	C₄⋊Dic₃	—	9	13	no	—
220	Dic₃⋊C₄	—	9	13	no	—
221	Dic₃⋊C₄	—	9	13	no	—
222	Dic₃⋊C₄	—	9	13	no	—
223	Dic₃⋊C₄	—	9	13	no	—
224	Dic₃⋊C₄	—	9	13	no	—
225	Dic₃⋊C₄	—	9	13	no	—
226	C₄×Dic₃	—	9	13	no	—
227	C₄×Dic₃	—	9	13	no	—
228	C₄×Dic₃	—	9	13	no	—
229	C₄×Dic₃	—	9	13	no	—
230	C₂²×C₃²	—	2	13	no	—
231	C₂²×C₃²	—	2	13	no	—
232	C₂²×C₃²	—	2	13	no	—
233	C₂²×C₃²	C₂³×D₃	1	13	yes	—
234	C₂×C₃⋊D₃	—	6	13	no	—
235	C₂×C₃⋊D₃	—	6	13	no	—
236	C₂×C₃⋊D₃	—	6	13	no	—
237	C₂×C₃⋊D₃	—	6	13	no	—
238	C₂×C₃⋊D₃	—	6	13	no	—
239	C₂×C₃⋊D₃	—	6	13	no	—
240	C₃²×C₄	—	2	13	no	—
241	C₃²×C₄	—	2	13	no	—
242	C₃²×C₄	—	2	13	no	—
243	C₃²×C₄	—	2	13	no	—
244	C₃²×C₄	—	2	13	no	—
245	C₃²×C₄	—	2	13	no	—
246	C₃⋊Dic₃	—	3	26	no	—
247	C₃⋊Dic₃	—	6	13	no	—
248	C₃⋊Dic₃	—	6	13	no	—
249	C₃⋊Dic₃	—	6	13	no	—
250	C₃⋊Dic₃	—	6	13	no	—
251	C₃⋊Dic₃	—	6	13	no	—
252	C₄⋊Q₈	—	27	1	no	—
253	#126	—	27	1	no	—
254	#126	—	27	1	no	—
255	#124	—	27	1	no	—
256	#124	—	27	1	no	—
257	#124	—	27	1	no	—
258	#123	—	27	1	no	—
259	#123	—	27	1	no	—
260	C₂²⋊Q₈	—	27	1	no	—
261	C₂²⋊Q₈	—	27	1	no	—
262	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
263	#121	—	27	1	no	—
264	C₄×Q₈	—	27	1	no	—
265	C₄×D₄	—	27	1	no	—
266	#117	—	27	1	no	—
267	C₃³	#542	1	1	yes	characteristic, complemented, Sylow
268	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
269	C₂²×D₃	—	9	13	no	—
270	C₂²×D₃	—	9	13	no	—
271	C₃×Q₈	—	4	13	no	—
272	C₂×C₃×C₄	—	4	13	no	—
273	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
274	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
275	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
276	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
277	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
278	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
279	#67	—	18	26	no	—
280	#67	—	36	13	no	—
281	C₂×Dic₃	—	9	13	no	—
282	C₂×Dic₃	—	9	13	no	—
283	C₂×Dic₃	—	9	13	no	—
284	C₂×Dic₃	—	9	13	no	—
285	C₂×Dic₃	—	9	13	no	—
286	C₂×Dic₃	—	9	13	no	—
287	D₁₂	—	36	13	no	—
288	C₄×D₃	—	18	26	no	—
289	C₄×D₃	—	36	13	no	—
290	Dic₆	—	18	26	no	—
291	Dic₆	—	36	13	no	—
292	C₂×C₃²	—	4	13	no	—
293	C₂×C₃²	#1004	1	13	yes	—
294	C₂×C₃²	#1007	1	13	yes	—
295	C₂×C₃²	D₃×#41	1	13	yes	—
296	C₃⋊D₃	—	12	13	no	—
297	C₃⋊D₃	—	12	13	no	—
298	C₂×Q₈	—	27	1	no	—
299	C₂×Q₈	—	27	1	no	—
300	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
301	C₂×D₄	—	27	1	no	—
302	C₂×D₄	—	27	1	no	—
303	C₂×D₄	—	27	1	no	—
304	C₂²×C₄	—	27	1	no	—
305	C₂²×C₄	—	27	1	no	—
306	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
307	C₄⋊C₄	—	27	1	no	—
308	C₄⋊C₄	—	27	1	no	—
309	C₄⋊C₄	—	27	1	no	—
310	C₄⋊C₄	—	27	1	no	—
311	C₄⋊C₄	—	27	1	no	—
312	C₄⋊C₄	—	27	1	no	—
313	C₄⋊C₄	—	27	1	no	—
314	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
315	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
316	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
317	C₂²⋊C₄	—	27	1	no	—
318	C₂²⋊C₄	—	27	1	no	—
319	C₂²⋊C₄	—	27	1	no	—
320	C₂²⋊C₄	—	27	1	no	—
321	C₂²⋊C₄	—	27	1	no	—
322	C₂²⋊C₄	—	27	1	no	—
323	C₂²⋊C₄	—	27	1	no	—
324	C₂²⋊C₄	—	27	1	no	—
325	C₂²⋊C₄	—	27	1	no	—
326	C₂²⋊C₄	—	27	1	no	—
327	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
328	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
329	C₄²	—	27	1	no	—
330	C₄²	—	27	1	no	—
331	C₄²	—	27	1	no	—
332	C₄²	—	27	1	no	—
333	C₂²×C₃	—	2	13	no	—
334	C₂²×C₃	—	2	13	no	—
335	C₂²×C₃	—	2	13	no	—
336	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
337	C₂×D₃	—	18	13	no	—
338	C₂×D₃	—	18	13	no	—
339	C₂×D₃	—	18	13	no	—
340	C₂×D₃	—	18	13	no	—
341	C₂×D₃	—	18	13	no	—
342	C₂×D₃	—	18	13	no	—
343	C₃×C₄	—	2	13	no	—
344	C₃×C₄	—	2	13	no	—
345	C₃×C₄	—	2	13	no	—
346	C₃×C₄	—	2	13	no	—
347	C₃×C₄	—	2	13	no	—
348	C₃×C₄	—	2	13	no	—
349	Dic₃	—	9	26	no	—
350	Dic₃	—	18	13	no	—
351	Dic₃	—	18	13	no	—
352	Dic₃	—	18	13	no	—
353	Dic₃	—	18	13	no	—
354	Dic₃	—	18	13	no	—
355	C₃²	#5640	1	13	yes	complemented
356	C₂³	—	27	1	no	—
357	C₂³	—	27	1	no	—
358	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
359	Q₈	—	4	1	no	—
360	Q₈	—	54	2	no	—
361	Q₈	—	108	1	no	—
362	D₄	—	54	2	no	—
363	D₄	—	108	1	no	—
364	D₄	—	108	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	27	1	no	—
367	C₂×C₄	—	27	1	no	—
368	C₂×C₄	—	27	1	no	—
369	C₂×C₄	—	27	1	no	—
370	C₂×C₄	—	27	1	no	—
371	C₂×C₄	—	27	1	no	—
372	C₂×C₄	—	54	2	no	—
373	C₂×C₄	—	108	1	no	—
374	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
375	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
376	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
377	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
378	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
379	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
380	C₂×C₃	—	4	13	no	—
381	C₂×C₃	#64361	1	13	yes	—
382	C₂×C₃	#64364	1	13	yes	—
383	C₂×C₃	#41×C₃⋊D₃	1	13	yes	—
384	D₃	—	36	13	no	—
385	D₃	—	36	13	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	54	1	no	—
390	C₂²	—	54	1	no	—
391	C₂²	—	54	1	no	—
392	C₂²	—	54	1	no	—
393	C₂²	—	54	1	no	—
394	C₂²	—	54	1	no	—
395	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	27	2	no	—
403	C₄	—	54	1	no	—
404	C₄	—	54	1	no	—
405	C₄	—	54	1	no	—
406	C₄	—	54	1	no	—
407	C₄	—	54	1	no	—
408	C₃	#10596111	1	13	yes	complemented
409	C₂	—	4	1	no	—
410	C₂	—	108	1	no	—
411	C₂	—	108	1	no	—
412	C₂	#11724679	1	1	yes	characteristic
413	C₂	#11724682	1	1	yes	characteristic
414	C₂	#41×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₃³×C₂²⋊Q₈	C₂
prime radical	#11723925	C₂
hypercenter	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#542
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₂²⋊Q₈	C₂	semidirect
2. #11723988	C₂	semidirect
3. #11723985	C₂	semidirect
4. #11723970	C₂	semidirect
5. C₄×#6312	C₂	semidirect
6. #11723939	C₂	semidirect
7. #11723937	C₂	semidirect
8. #11723937	C₂	semidirect
9. #11723935	C₂	semidirect
10. #11723933	C₂	semidirect
11. #11723928	C₂	semidirect
12. #11723926	C₂	semidirect
13. #11723926	C₂	semidirect
15. #11723920	C₂	semidirect
52. C₃²×C₂²⋊Q₈	D₃	semidirect
149. C₃×C₂²⋊Q₈	C₃⋊D₃	semidirect
262. C₂²⋊Q₈	C₃⋊(C₃⋊D₃)	semidirect
267. C₃³	#542	semidirect
355. C₃²	#5640	semidirect
408. C₃	#10596111	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₂²⋊Q₈	2
#11723988	2
#11723985	2
#11723970	2
C₄×#6312	2
#11723939	2
#11723937	2
#11723935	2
#11723933	2
#11723928	2
#11723926	2
#11723925	2
#11723920	2
#10596111	3