Finite Group Databasebeta

← C2×D3×#126⋊C3 #11923232

Subgroups — C2×D3×#126⋊C3 #11923232

Order 1152.

615 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126⋊C3C211yesmaximal, characteristic, complemented
2D3×#126⋊C3C214yesmaximal, direct product factor
3C2×D3×C42⋊C3C211yesmaximal, characteristic, complemented
4C2×C3⋊(#126⋊C3)C211yesmaximal, characteristic, complemented
5C2×D3×#126C311yesmaximal, characteristic, complemented
6C22×#126⋊C3—31nomaximal
7C22×D3×A4—41nomaximal
8C3×#126⋊C3C2212yescomplemented
9C2×C3×C42⋊C3C2211yescharacteristic, complemented
10D3×C42⋊C3C2212yescomplemented
11C3⋊(#126⋊C3)C2212yescomplemented
12C2×C3×#126C2×C311yescharacteristic, complemented
13D3×#126—34no—
14D3×#126C2×C314yescomplemented
15C2×#898—31no—
16C2×D3×C4⋊C4—31no—
17C2×D3×C22⋊C4—31no—
18C2×#879—31no—
19C2×#876C2×C311yescharacteristic, complemented
20C2×C42×D3C2×C311yescharacteristic, complemented
21C2×#126⋊C3—21no—
22C2×#126⋊C3—34no—
23C2×#126⋊C3—31no—
24C2×#126⋊C3D311yescharacteristic, direct product factor
25C22×C42⋊C3—31no—
26C22×C3×A4—41no—
27C2×D3×A4—41no—
28C2×D3×A4—41no—
29C2×D3×A4—44no—
30C3×C42⋊C3C2311yescharacteristic, complemented
31C22×#126—31noSylow
32C24×D3A411yescharacteristic
33C23×A4—121no—
34C22×C4×D3—31no—
35C22×C4×D3—31no—
36C3×#126—32no—
37C3×#126C22×C312yescomplemented
38C2×C3×C4⋊C4—31no—
39C2×C3×C22⋊C4—31no—
40C2×C3×C42C22×C311yescharacteristic, complemented
41C2×#217—31no—
42C2×(C2×D3)⋊C4—31no—
43C2×(C2×D3)⋊C4—31no—
44C2×C4⋊Dic3—31no—
45C2×Dic3⋊C4—31no—
46C2×Dic3⋊C4—31no—
47C2×C4×Dic3—31no—
48#898—34no—
49#898—34no—
50D3×C4⋊C4—34no—
51D3×C4⋊C4—34no—
52D3×C22⋊C4—34no—
53D3×C22⋊C4—34no—
54#879—34no—
55#879—34no—
56#876—32no—
57#876C22×C312yescomplemented
58C42×D3—32no—
59C42×D3C22×C312yescomplemented
60#126⋊C3—22no—
61#126⋊C3—32no—
62#126⋊C3C2×D312yesdirect product factor
63C2×C42⋊C3—21no—
64C2×C42⋊C3—32no—
65C2×C42⋊C3C2×D311yescharacteristic, complemented
66C22×C3×D3—161no—
67C2×C3×A4—41no—
68C2×C3×A4—42no—
69D3×A4—42no—
70D3×A4—42no—
71C2×#126—34no—
72C2×#126—31no—
73C2×#126—91no—
74C2×#126—91no—
75C2×#126—94no—
76C2×#126C3×D311yescharacteristic, complemented
77C22×C4⋊C4—91no—
78C22×C22⋊C4—91no—
79C22×C42—31no—
80C24×C3C2×A411yescharacteristic
81C23×D3—61no—
82C23×D3—62no—
83C23×D3—61no—
84C23×D3C2×A411yescharacteristic
85C23×D3C2×A414yes—
86C23×D3C2×A411yescharacteristic
87C22×A4—41no—
88C22×A4—81no—
89C22×A4—121no—
90C22×A4—121no—
91C22×A4—124no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×Dic3—31no—
95C22×Dic3—31no—
96C2×C4×D3—32no—
97C2×C4×D3—32no—
98C2×C4×D3—34no—
99C2×C4×D3—61no—
100C2×C4×D3—61no—
101C2×C4×D3—62no—
102C2×C4×D3—61no—
103C2×C4×D3—62no—
104C2×C4×D3—61no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—32no—
107C3×C22⋊C4—32no—
108C3×C22⋊C4—32no—
109C3×C42—31no—
110C3×C42C23×C311yescharacteristic, complemented
111#217—34no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—32no—
114(C2×D3)⋊C4—34no—
115C4⋊Dic3—34no—
116Dic3⋊C4—32no—
117Dic3⋊C4—34no—
118Dic3⋊C4—32no—
119C4×Dic3—32no—
120C4×Dic3—32no—
121C42⋊C3—21no—
122C42⋊C3C22×D311yescharacteristic, complemented
123C22×C32—161no—
124C2×C3×D3—161no—
125C2×C3×D3—164no—
126C2×C3×D3—161no—
127C3×A4—41no—
128C25—31no—
129C23×C4—91no—
130C23×C4—91no—
131#126—32no—
132#126—32no—
133#126—92no—
134#126—94no—
135#126—94no—
136#126—94no—
137#126—94no—
138#126C2×C3×D312yescomplemented
139C2×C4⋊C4—31no—
140C2×C4⋊C4—94no—
141C2×C4⋊C4—94no—
142C2×C4⋊C4—91no—
143C2×C4⋊C4—91no—
144C2×C4⋊C4—91no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—94no—
147C2×C22⋊C4—94no—
148C2×C22⋊C4—91no—
149C2×C22⋊C4—91no—
150C2×C22⋊C4—91no—
151C2×C42—32no—
152C2×C42—92no—
153C2×C42—91no—
154C2×C42C2×C3×D311yescharacteristic, complemented
155C23×C3—61no—
156C23×C3—61no—
157C23×C3—481no—
158C23×C3C22×A411yescharacteristic
159C23×C3C22×A412yes—
160C22×D3—31no—
161C22×D3—31no—
162C22×D3—32no—
163C22×D3—41no—
164C22×D3—64no—
165C22×D3—64no—
166C22×D3—61no—
167C22×D3—61no—
168C22×D3—122no—
169C22×D3—121no—
170C22×D3—122no—
171C22×D3C22×A412yes—
172C22×D3C22×A412yes—
173C2×A4—41no—
174C2×A4—42no—
175C2×A4—81no—
176C2×A4—82no—
177C2×A4—122no—
178C2×A4—122no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—32no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×Dic3—32no—
187C2×Dic3—32no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C4×D3—62no—
193C4×D3—62no—
194C4×D3—64no—
195C4×D3—62no—
196C4×D3—62no—
197C4×D3—64no—
198C2×C32—162no—
199C2×C32—161no—
200C3×D3—162no—
201C3×D3—162no—
202C24—31no—
203C24—34no—
204C24—31no—
205C24—181no—
206C24—181no—
207C24—182no—
208C24D3×A411yescharacteristic
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—92no—
212C22×C4—92no—
213C22×C4—94no—
214C22×C4—91no—
215C22×C4—91no—
216C22×C4—181no—
217C22×C4—181no—
218C22×C4—182no—
219C22×C4—181no—
220C22×C4—182no—
221C22×C4—181no—
222C4⋊C4—32no—
223C4⋊C4—32no—
224C4⋊C4—92no—
225C4⋊C4—92no—
226C4⋊C4—94no—
227C4⋊C4—94no—
228C22⋊C4—32no—
229C22⋊C4—32no—
230C22⋊C4—92no—
231C22⋊C4—92no—
232C22⋊C4—94no—
233C22⋊C4—94no—
234C42—31no—
235C42—92no—
236C42—92no—
237C42C22×C3×D311yescharacteristic, complemented
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—41no—
241C22×C3—62no—
242C22×C3—121no—
243C22×C3—161no—
244C22×C3—321no—
245C22×C3—481no—
246C22×C3—484no—
247C22×C3—481no—
248C22×C3C23×A411yescharacteristic
249C2×D3—32no—
250C2×D3—31no—
251C2×D3—32no—
252C2×D3—32no—
253C2×D3—32no—
254C2×D3—41no—
255C2×D3—44no—
256C2×D3—62no—
257C2×D3—124no—
258C2×D3—121no—
259C2×D3#126⋊C311yescharacteristic, direct product factor
260A4—41no—
261A4—81no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—62no—
265Dic3—62no—
266Dic3—62no—
267C32—161noSylow
268C23—32no—
269C23—32no—
270C23—61no—
271C23—61no—
272C23—91no—
273C23—92no—
274C23—91no—
275C23—121no—
276C23—181no—
277C23—181no—
278C23—184no—
279C23—184no—
280C23—361no—
281C23—362no—
282C23—362no—
283C23C2×D3×A411yescharacteristic
284C23C2×D3×A412yes—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—32no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—92no—
293C2×C4—92no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—184no—
297C2×C4—182no—
298C2×C4—182no—
299C2×C4—184no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—42no—
307C2×C3—162no—
308C2×C3—161no—
309C2×C3—322no—
310C2×C3—321no—
311C2×C3—482no—
312C2×C3—482no—
313C2×C3C2×#126⋊C311yescharacteristic, complemented
314D3—32no—
315D3—42no—
316D3C2×#126⋊C312yesdirect product factor
317C22—31no—
318C22—31no—
319C22—31no—
320C22—41no—
321C22—62no—
322C22—92no—
323C22—92no—
324C22—92no—
325C22—92no—
326C22—91no—
327C22—121no—
328C22—124no—
329C22—121no—
330C22—182no—
331C22—361no—
332C22—364no—
333C22C22×D3×A411yescharacteristic
334C4—61no—
335C4—61no—
336C4—62no—
337C4—182no—
338C4—182no—
339C3—161no—
340C3—321no—
341C3C22×#126⋊C311yescharacteristic, complemented
342C2—31no—
343C2—31no—
344C2—32no—
345C2—42no—
346C2—92no—
347C2—122no—
348C2D3×#126⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2D3×#126⋊C3
commutator, cosocle, hypocenterC3×C42C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×#126C2×C3
socleC23×C3C22×A4
LCSG ▷ C3×C42
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126⋊C3C2semidirect
2. D3×#126⋊C3C2direct (N × H)
3. C2×D3×C42⋊C3C2semidirect
4. C2×C3⋊(#126⋊C3)C2semidirect
5. C2×D3×#126C3semidirect
8. C3×#126⋊C3C22semidirect
9. C2×C3×C42⋊C3C22semidirect
10. D3×C42⋊C3C22semidirect
11. C3⋊(#126⋊C3)C22semidirect
12. C2×C3×#126C2×C3semidirect
14. D3×#126C2×C3semidirect
19. C2×#876C2×C3semidirect
20. C2×C42×D3C2×C3semidirect
24. C2×#126⋊C3D3direct (N × H)
30. C3×C42⋊C3C23semidirect
37. C3×#126C22×C3semidirect
40. C2×C3×C42C22×C3semidirect
57. #876C22×C3semidirect
59. C42×D3C22×C3semidirect
62. #126⋊C3C2×D3direct (N × H)
65. C2×C42⋊C3C2×D3semidirect
76. C2×#126C3×D3semidirect
110. C3×C42C23×C3semidirect
122. C42⋊C3C22×D3semidirect
138. #126C2×C3×D3semidirect
154. C2×C42C2×C3×D3semidirect
237. C42C22×C3×D3semidirect
259. C2×D3#126⋊C3direct (N × H)
313. C2×C3C2×#126⋊C3semidirect
316. D3C2×#126⋊C3direct (N × H)
341. C3C22×#126⋊C3semidirect
348. C2D3×#126⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#126⋊C32
D3×#126⋊C32
C2×D3×C42⋊C32
C2×C3⋊(#126⋊C3)2
C2×D3×#1263
C22×#126⋊C33
C22×D3×A44