Finite Group Databasebeta

← C2×C3⋊#5450 #11923231

Subgroups — C2×C3⋊#5450 #11923231

Order 1152.

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

normal subgroups only

This group has 655 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×#120⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊(#120⋊C3)C212yesmaximal, complemented
3C3⋊#5450C214yesmaximal, direct product factor
4C2×#5594C311yesmaximal, characteristic, complemented
5C2×#5450—31nomaximal
6C2×C3×C22⋊A4C2211yescharacteristic, complemented
7C22×D3×A4—41nomaximal
8C3×#120⋊C3C2212yescomplemented
9C3⋊(#120⋊C3)C2214yescomplemented
10C2×C3×#120C2×C311yescharacteristic, complemented
11C2×#953C2×C312yescomplemented
12C2×#937—31no—
13C2×#936—31no—
14#5594—34no—
15#5594C2×C314yescomplemented
16C2×#885—32no—
17#5450—34no—
18C2×#120⋊C3—21no—
19C2×#120⋊C3—32no—
20C2×#120⋊C3D311yescharacteristic, complemented
21C3×C22⋊A4C2311yescharacteristic, complemented
22C22×C3×A4—41no—
23C2×D3×A4—42no—
24C2×D3×A4—44no—
25C2×H(3,𝔽4)—31noSylow
26C25×C3C22×C311yescharacteristic, complemented
27C24×D3A411yescharacteristic, complemented
28C2×C22⋊A4—21no—
29C2×C22⋊A4C2×D311yescharacteristic, complemented
30C23×A4—121no—
31C22×C3×D4—31no—
32C22×#67—32no—
33C22×Dic6—31no—
34C3×#120—32no—
35C3×#120C22×C312yescomplemented
36C2×C3×C22⋊C4—31no—
37#953—34no—
38#953C22×C314yescomplemented
39C2×#217—31no—
40C2×#217—32no—
41#937—34no—
42#937—34no—
43#936—34no—
44#936—34no—
45C2×(C2×D3)⋊C4—32no—
46C2×C4×Dic3—31no—
47#885—38no—
48#885—38no—
49#120⋊C3—22no—
50#120⋊C3—34no—
51#120⋊C3C2×D312yescomplemented
52C22×C3×D3—161no—
53C2×C3×A4—41no—
54C2×C3×A4—42no—
55C2×C3×A4—161no—
56D3×A4—44no—
57H(3,𝔽4)—34no—
58H(3,𝔽4)—94no—
59C2×#124—92no—
60C2×#124—91no—
61C2×#120—32no—
62C2×#120—91no—
63C2×#120C3×D311yescharacteristic, complemented
64C24×C3—31no—
65C24×C3—31no—
66C24×C3—121no—
67C24×C3—121no—
68C24×C3C2×A411yescharacteristic
69C24×C3C23×C311yescharacteristic, complemented
70C23×D3—121no—
71C23×D3—121no—
72C23×D3C2×A412yes—
73C23×D3C2×A414yescomplemented
74C22⋊A4—21no—
75C22⋊A4C22×D311yescharacteristic, complemented
76C22×A4—41no—
77C22×A4—81no—
78C22×A4—122no—
79C22×A4—124no—
80C2×C3×D4—32no—
81C2×C3×D4—32no—
82C2×C3×D4—121no—
83C2×C3×D4—121no—
84C22×C3×C4—31no—
85C2×#67—34no—
86C2×#67—34no—
87C2×#67—122no—
88C2×#67—122no—
89C22×Dic3—32no—
90C2×Dic6—34no—
91C2×Dic6—121no—
92C2×Dic6—121no—
93C3×C22⋊C4—32no—
94C3×C22⋊C4—32no—
95#217—34no—
96#217—34no—
97#217—34no—
98(C2×D3)⋊C4—38no—
99C4×Dic3—34no—
100C22×C32—161no—
101C2×C3×D3—164no—
102C2×C3×D3—162no—
103C3×A4—41no—
104C3×A4—161no—
105C25—31no—
106C25C2×C3×D311yescharacteristic, complemented
107C22×Q8—91no—
108C22×D4—31no—
109C22×D4—92no—
110#124—98no—
111#124—98no—
112#124—94no—
113#124—94no—
114#120—32no—
115#120—34no—
116#120—94no—
117#120—94no—
118#120—94no—
119#120C2×C3×D312yescomplemented
120C2×C22⋊C4—31no—
121C2×C22⋊C4—92no—
122C2×C22⋊C4—91no—
123C2×C22⋊C4—92no—
124C2×C42—91no—
125C23×C3—31no—
126C23×C3—31no—
127C23×C3—41no—
128C23×C3—61no—
129C23×C3—61no—
130C23×C3—62no—
131C23×C3—62no—
132C23×C3—61no—
133C23×C3—61no—
134C23×C3—121no—
135C23×C3—121no—
136C23×C3—122no—
137C23×C3—121no—
138C23×C3—121no—
139C23×C3—123no—
140C23×C3—481no—
141C23×C3C22×A411yescharacteristic
142C23×C3C22×A412yes—
143C22×D3—41no—
144C22×D3—62no—
145C22×D3—62no—
146C22×D3—121no—
147C22×D3—124no—
148C22×D3—124no—
149C22×D3C22×A414yes—
150C2×A4—42no—
151C2×A4—41no—
152C2×A4—82no—
153C2×A4—81no—
154C2×A4—124no—
155C2×A4—161no—
156C2×A4—162no—
157C3×D4—122no—
158C3×D4—122no—
159C2×C3×C4—32no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162#67—124no—
163#67—124no—
164C2×Dic3—34no—
165C2×Dic3—62no—
166C2×Dic3—62no—
167Dic6—124no—
168C2×C32—162no—
169C2×C32—161no—
170C3×D3—164no—
171C24—31no—
172C24—31no—
173C24—32no—
174C24—34no—
175C24—121no—
176C24—121no—
177C24—361no—
178C24—361no—
179C24D3×A411yescharacteristic
180C24C22×C3×D311yescharacteristic, complemented
181C2×Q8—94no—
182C2×Q8—361no—
183C2×Q8—361no—
184C2×D4—32no—
185C2×D4—32no—
186C2×D4—94no—
187C2×D4—94no—
188C2×D4—121no—
189C2×D4—121no—
190C2×D4—362no—
191C2×D4—362no—
192C22×C4—31no—
193C22×C4—92no—
194C22⋊C4—32no—
195C22⋊C4—32no—
196C22⋊C4—98no—
197C22⋊C4—94no—
198C22⋊C4—94no—
199C22⋊C4—94no—
200C42—94no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—62no—
206C22×C3—62no—
207C22×C3—62no—
208C22×C3—61no—
209C22×C3—61no—
210C22×C3—121no—
211C22×C3—121no—
212C22×C3—122no—
213C22×C3—121no—
214C22×C3—121no—
215C22×C3—123no—
216C22×C3—121no—
217C22×C3—161no—
218C22×C3—321no—
219C22×C3—484no—
220C22×C3—482no—
221C22×C3C23×A411yescharacteristic
222C2×D3—44no—
223C2×D3—42no—
224C2×D3—64no—
225C2×D3—122no—
226C2×D3—124no—
227A4—41no—
228A4—81no—
229A4—161no—
230A4—162no—
231C3×C4—62no—
232Dic3—64no—
233C32—161noSylow
234C23—31no—
235C23—31no—
236C23—34no—
237C23—41no—
238C23—61no—
239C23—61no—
240C23—61no—
241C23—61no—
242C23—62no—
243C23—62no—
244C23—121no—
245C23—121no—
246C23—121no—
247C23—121no—
248C23—122no—
249C23—121no—
250C23—123no—
251C23—182no—
252C23—182no—
253C23—361no—
254C23—364no—
255C23—364no—
256C23C2×D3×A411yescharacteristic
257C23C2×D3×A412yes—
258Q8—364no—
259D4—122no—
260D4—122no—
261D4—364no—
262D4—364no—
263C2×C4—32no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—94no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—42no—
272C2×C3—121no—
273C2×C3—121no—
274C2×C3—162no—
275C2×C3—161no—
276C2×C3—322no—
277C2×C3—321no—
278C2×C3—484no—
279C2×C3#545011yescharacteristic, complemented
280D3—44no—
281C22—31no—
282C22—31no—
283C22—41no—
284C22—41no—
285C22—62no—
286C22—61no—
287C22—61no—
288C22—62no—
289C22—62no—
290C22—121no—
291C22—122no—
292C22—121no—
293C22—124no—
294C22—121no—
295C22—121no—
296C22—123no—
297C22—121no—
298C22—122no—
299C22—184no—
300C22—362no—
301C22—364no—
302C22C22×D3×A411yescharacteristic
303C4—62no—
304C4—184no—
305C3—161no—
306C3—321no—
307C3C2×#545011yescharacteristic, complemented
308C2—31no—
309C2—31no—
310C2—42no—
311C2—121no—
312C2—121no—
313C2—124no—
314C2C3⋊#545011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊#5450
commutator, cosocle, hypocenterC24×C3C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×#120C2×C3
socleC23×C3C22×A4
LCSG ▷ C24×C3
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×#120⋊C3C2semidirect
2. C2×C3⋊(#120⋊C3)C2semidirect
3. C3⋊#5450C2direct (N × H)
4. C2×#5594C3semidirect
6. C2×C3×C22⋊A4C22semidirect
8. C3×#120⋊C3C22semidirect
9. C3⋊(#120⋊C3)C22semidirect
10. C2×C3×#120C2×C3semidirect
11. C2×#953C2×C3semidirect
15. #5594C2×C3semidirect
20. C2×#120⋊C3D3semidirect
21. C3×C22⋊A4C23semidirect
26. C25×C3C22×C3semidirect
27. C24×D3A4semidirect
29. C2×C22⋊A4C2×D3semidirect
35. C3×#120C22×C3semidirect
38. #953C22×C3semidirect
51. #120⋊C3C2×D3semidirect
63. C2×#120C3×D3semidirect
69. C24×C3C23×C3semidirect
73. C23×D3C2×A4semidirect
75. C22⋊A4C22×D3semidirect
106. C25C2×C3×D3semidirect
119. #120C2×C3×D3semidirect
180. C24C22×C3×D3semidirect
279. C2×C3#5450semidirect
307. C3C2×#5450semidirect
314. C2C3⋊#5450direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#120⋊C32
C2×C3⋊(#120⋊C3)2
C3⋊#54502
C2×#55943
C2×#54503
C22×D3×A44