Finite Group Databasebeta

← C2×(#126⋊C3)⋊D3 #11923233

Subgroups — C2×(#126⋊C3)⋊D3 #11923233

Order 1152.

623 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 67 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
2(#126⋊C3)⋊D3C214yesmaximal, direct product factor
3C2×C3⋊(#126⋊C3)C211yesmaximal, characteristic, complemented
4C2×C3⋊(#127⋊C3)C211yesmaximal, characteristic, complemented
5C2×#5711C311yesmaximal, characteristic, complemented
6C2×#560⋊C3—31nomaximal
7C22×D3×A4—41nomaximal
8C3×#126⋊C3C2212yescomplemented
9C2×C3×C42⋊C3C2211yescharacteristic, complemented
10C3⋊(#126⋊C3)C2212yescomplemented
11C3⋊(#127⋊C3)C2212yescomplemented
12C2×C3×#126C2×C311yescharacteristic, complemented
13#5711—34no—
14#5711C2×C314yescomplemented
15C2×#895—31no—
16C2×#894—31no—
17C2×#884—31no—
18C2×#882—31no—
19C2×#876C2×C311yescharacteristic, complemented
20C2×#874C2×C311yescharacteristic, complemented
21#560⋊C3—34no—
22C2×#127⋊C3—31no—
23C2×#126⋊C3—21no—
24C2×#126⋊C3—31no—
25C2×#126⋊C3D311yescharacteristic, complemented
26C22×C3×A4—41no—
27C2×D3×A4—41no—
28C2×D3×A4—41no—
29C2×D3×A4—44no—
30C3×C42⋊C3C2311yescharacteristic, complemented
31C2×#560—31noSylow
32C24×D3A411yescharacteristic
33C23×A4—121no—
34C22×#67—31no—
35C22×D12—31no—
36C22×D12—31no—
37C22×D12—31no—
38C22×C4×D3—31no—
39C3×#126—32no—
40C3×#126C22×C312yescomplemented
41C2×C3×C4⋊C4—31no—
42C2×C3×C22⋊C4—31no—
43C2×C3×C42C22×C311yescharacteristic, complemented
44C2×(C2×D3)⋊C4—31no—
45C2×(C2×D3)⋊C4—31no—
46C2×(C2×D3)⋊C4—31no—
47C2×Dic3⋊C4—31no—
48#895—34no—
49#895—34no—
50#894—34no—
51#894—34no—
52#884—34no—
53#884—34no—
54#882—34no—
55#882—34no—
56#876—32no—
57#876C22×C312yescomplemented
58#874—32no—
59#874C22×C312yescomplemented
60#127⋊C3—32no—
61#126⋊C3—22no—
62#126⋊C3—32no—
63#126⋊C3C2×D312yescomplemented
64C2×C42⋊C3—21no—
65C2×C42⋊C3C2×D311yescharacteristic, complemented
66C22×C3×D3—161no—
67C2×C3×A4—41no—
68C2×C3×A4—42no—
69D3×A4—42no—
70D3×A4—42no—
71#560—34no—
72#560—94no—
73C2×#127—31no—
74C2×#126—31no—
75C2×#126C3×D311yescharacteristic, complemented
76C2×#123—91no—
77C2×#121—91no—
78C2×#121—91no—
79C2×#120—91no—
80C24×C3C2×A411yescharacteristic
81C23×D3—31no—
82C23×D3—121no—
83C23×D3—121no—
84C23×D3C2×A411yescharacteristic
85C23×D3C2×A411yescharacteristic
86C23×D3C2×A414yes—
87C22×A4—41no—
88C22×A4—81no—
89C22×A4—121no—
90C22×A4—121no—
91C22×A4—124no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C2×#67—34no—
95C2×#67—121no—
96C2×#67—121no—
97C22×Dic3—31no—
98C2×D12—32no—
99C2×D12—32no—
100C2×D12—32no—
101C2×D12—32no—
102C2×D12—34no—
103C2×D12—121no—
104C2×D12—121no—
105C2×D12—121no—
106C2×D12—121no—
107C2×D12—121no—
108C2×D12—121no—
109C2×C4×D3—34no—
110C2×C4×D3—121no—
111C2×C4×D3—121no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—32no—
114C3×C22⋊C4—32no—
115C3×C22⋊C4—32no—
116C3×C42—31no—
117C3×C42C23×C311yescharacteristic, complemented
118(C2×D3)⋊C4—34no—
119(C2×D3)⋊C4—32no—
120(C2×D3)⋊C4—32no—
121(C2×D3)⋊C4—34no—
122Dic3⋊C4—32no—
123Dic3⋊C4—32no—
124C42⋊C3—21no—
125C42⋊C3C22×D311yescharacteristic, complemented
126C22×C32—161no—
127C2×C3×D3—164no—
128C2×C3×D3—161no—
129C2×C3×D3—161no—
130C3×A4—41no—
131C25—31no—
132C22×D4—91no—
133C22×D4—91no—
134C22×D4—91no—
135C22×D4—91no—
136C23×C4—91no—
137#127—32no—
138#127—92no—
139#126—32no—
140#126—32no—
141#126—92no—
142#126C2×C3×D312yescomplemented
143#123—94no—
144#123—94no—
145#121—94no—
146#121—94no—
147#121—94no—
148#121—94no—
149#120—94no—
150#120—94no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—91no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—91no—
155C2×C22⋊C4—91no—
156C2×C22⋊C4—91no—
157C2×C42C2×C3×D311yescharacteristic, complemented
158C23×C3—61no—
159C23×C3—61no—
160C23×C3—481no—
161C23×C3C22×A411yescharacteristic
162C23×C3C22×A412yes—
163C22×D3—32no—
164C22×D3—41no—
165C22×D3—61no—
166C22×D3—61no—
167C22×D3—61no—
168C22×D3—61no—
169C22×D3—61no—
170C22×D3—61no—
171C22×D3—61no—
172C22×D3—61no—
173C22×D3—61no—
174C22×D3—61no—
175C22×D3—121no—
176C22×D3—124no—
177C22×D3—124no—
178C22×D3C22×A412yes—
179C22×D3C22×A412yes—
180C2×A4—41no—
181C2×A4—42no—
182C2×A4—81no—
183C2×A4—82no—
184C2×A4—122no—
185C2×A4—122no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—32no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193#67—124no—
194C2×Dic3—32no—
195C2×Dic3—61no—
196C2×Dic3—61no—
197D12—124no—
198D12—122no—
199D12—122no—
200D12—122no—
201D12—122no—
202C4×D3—124no—
203C2×C32—162no—
204C2×C32—161no—
205C3×D3—162no—
206C3×D3—162no—
207C24—31no—
208C24—31no—
209C24—34no—
210C24—91no—
211C24—361no—
212C24—361no—
213C24D3×A411yescharacteristic
214C2×D4—94no—
215C2×D4—92no—
216C2×D4—92no—
217C2×D4—92no—
218C2×D4—92no—
219C2×D4—94no—
220C2×D4—361no—
221C2×D4—361no—
222C2×D4—361no—
223C2×D4—361no—
224C2×D4—361no—
225C2×D4—361no—
226C2×D4—361no—
227C2×D4—361no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—94no—
231C22×C4—91no—
232C22×C4—361no—
233C22×C4—361no—
234C4⋊C4—32no—
235C4⋊C4—32no—
236C4⋊C4—92no—
237C4⋊C4—92no—
238C22⋊C4—32no—
239C22⋊C4—32no—
240C22⋊C4—92no—
241C22⋊C4—92no—
242C22⋊C4—94no—
243C22⋊C4—94no—
244C42—31no—
245C42C22×C3×D311yescharacteristic, complemented
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—41no—
249C22×C3—62no—
250C22×C3—121no—
251C22×C3—161no—
252C22×C3—321no—
253C22×C3—484no—
254C22×C3—481no—
255C22×C3—481no—
256C22×C3C23×A411yescharacteristic
257C2×D3—44no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—62no—
264C2×D3—62no—
265C2×D3—121no—
266C2×D3—121no—
267C2×D3—124no—
268C2×D3—121no—
269C2×D3—121no—
270C2×D3—121no—
271C2×D3—121no—
272A4—41no—
273A4—81no—
274C3×C4—62no—
275C3×C4—61no—
276C3×C4—61no—
277Dic3—62no—
278C32—161noSylow
279C23—32no—
280C23—32no—
281C23—61no—
282C23—61no—
283C23—92no—
284C23—121no—
285C23—181no—
286C23—181no—
287C23—181no—
288C23—181no—
289C23—181no—
290C23—181no—
291C23—181no—
292C23—181no—
293C23—181no—
294C23—181no—
295C23—361no—
296C23—364no—
297C23—364no—
298C23C2×D3×A411yescharacteristic
299C23C2×D3×A412yes—
300D4—364no—
301D4—362no—
302D4—362no—
303D4—362no—
304D4—362no—
305D4—364no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—92no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—364no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—42no—
320C2×C3—162no—
321C2×C3—161no—
322C2×C3—322no—
323C2×C3—321no—
324C2×C3—482no—
325C2×C3—482no—
326C2×C3#560⋊C311yescharacteristic, complemented
327D3—42no—
328D3—42no—
329D3—122no—
330C22—31no—
331C22—31no—
332C22—41no—
333C22—62no—
334C22—121no—
335C22—124no—
336C22—121no—
337C22—121no—
338C22—182no—
339C22—182no—
340C22—182no—
341C22—182no—
342C22—182no—
343C22—361no—
344C22—361no—
345C22—361no—
346C22—361no—
347C22—361no—
348C22—364no—
349C22—361no—
350C22C22×D3×A411yescharacteristic
351C4—62no—
352C4—61no—
353C4—61no—
354C4—182no—
355C3—161no—
356C3—321no—
357C3C2×#560⋊C311yescharacteristic, complemented
358C2—31no—
359C2—31no—
360C2—42no—
361C2—122no—
362C2—122no—
363C2—362no—
364C2(#126⋊C3)⋊D311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2(#126⋊C3)⋊D3
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. (#126⋊C3)⋊D3C2direct (N × H)
3. C2×C3⋊(#126⋊C3)C2semidirect
4. C2×C3⋊(#127⋊C3)C2semidirect
5. C2×#5711C3semidirect
8. C3×#126⋊C3C22semidirect
9. C2×C3×C42⋊C3C22semidirect
10. C3⋊(#126⋊C3)C22semidirect
11. C3⋊(#127⋊C3)C22semidirect
12. C2×C3×#126C2×C3semidirect
14. #5711C2×C3semidirect
19. C2×#876C2×C3semidirect
20. C2×#874C2×C3semidirect
25. C2×#126⋊C3D3semidirect
30. C3×C42⋊C3C23semidirect
40. C3×#126C22×C3semidirect
43. C2×C3×C42C22×C3semidirect
57. #876C22×C3semidirect
59. #874C22×C3semidirect
63. #126⋊C3C2×D3semidirect
65. C2×C42⋊C3C2×D3semidirect
75. C2×#126C3×D3semidirect
117. C3×C42C23×C3semidirect
125. C42⋊C3C22×D3semidirect
142. #126C2×C3×D3semidirect
157. C2×C42C2×C3×D3semidirect
245. C42C22×C3×D3semidirect
326. C2×C3#560⋊C3semidirect
357. C3C2×#560⋊C3semidirect
364. C2(#126⋊C3)⋊D3direct (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
(#126⋊C3)⋊D32
C2×C3⋊(#126⋊C3)2
C2×C3⋊(#127⋊C3)2
C2×#57113
C2×#560⋊C33
C22×D3×A44