Finite Group Databasebeta

← C2×C3×#578⋊C3 #11919802

Subgroups — C2×C3×#578⋊C3 #11919802

Order 1152.

816 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 158 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×C3×2−1+4⋊C3C212yesmaximal, complemented
2C3×#578⋊C3C214yesmaximal, direct product factor
3C2×C3×#131⋊C3C211yesmaximal, characteristic, complemented
4C2×C3×#578C311yesmaximal, characteristic, complemented
5C2×#578⋊C3C313yesmaximal, direct product factor
6C3×2−1+4⋊C3—22no—
7C3×2−1+4⋊C3C2214yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)—22no—
10C2×C32×D8—41nomaximal
11C3×#131⋊C3C2212yescomplemented
12C2×C3×2−1+4C2×C312yescomplemented
13C2×2−1+4⋊C3C2×C316yescomplemented
14C3×#578—34no—
15C3×#578C2×C314yescomplemented
16C2×C3×#137—32no—
17C2×C3×#135—31no—
18C22×C3×Q16—31no—
19C2×C3×#131C2×C311yescharacteristic, complemented
20#578⋊C3C2×C3112yesdirect product factor
21C2×#131⋊C3C2×C313yescomplemented
22C2×C32×D4—42no—
23C3×#41⋊C3D411yescharacteristic
24C3×#41⋊C3C2311yescharacteristic
25C2×C3×SL(2,𝔽3)—24no—
26C2×C3×SL(2,𝔽3)D411yescharacteristic
27C32×D8—44no—
28C2×C32×C8—41no—
29C2×#578C3211yescharacteristic, complemented, Sylow
30C3×2−1+4—22no—
31C3×2−1+4—34no—
32C3×2−1+4—62no—
33C3×2−1+4C22×C314yescomplemented
34C2×C3×#41—32no—
35C2×C3×#41—62no—
36C2×C3×#41C22×C311yescharacteristic
37C22×C3×Q8—22no—
38C22×C3×Q8—32no—
392−1+4⋊C3—26no—
402−1+4⋊C3C22×C3112yescomplemented
41C2×#41⋊C3C22×C313yes—
42C22×SL(2,𝔽3)—26no—
43C3×#137—38no—
44C3×#137—38no—
45C3×#135—34no—
46C3×#135—34no—
47C2×C3×Q16—32no—
48C2×C3×Q16—34no—
49C2×C3×Q16—34no—
50C2×C3×Q16—31no—
51C2×C3×SD16—32no—
52C2×C3×D8—43no—
53C2×C3×D8A411yescharacteristic
54C3×#131—32no—
55C3×#131C22×C312yescomplemented
56C2×C3×M16—31no—
57C22×C3×C8—31no—
58#131⋊C3C22×C316yescomplemented
59C23×C32—82no—
60C32×D4—44no—
61C32×D4—82no—
62C2×C32×C4—41no—
63C3×SL(2,𝔽3)C2×D411yescharacteristic
64C32×C8—42no—
65C2×2−1+4C2×C3212yescomplemented
66#578—34no—
67#578C2×C3214yescomplemented
68C2×#137—32no—
69C2×#135—31no—
70C22×Q16—31no—
71C2×#131C2×C3211yescharacteristic, complemented
72C3×#41—31no—
73C3×#41—31no—
74C3×#41—34no—
75C3×#41—34no—
76C3×#41—62no—
77C3×#41—62no—
78C3×#41—68no—
79C3×#41—64no—
80C3×#41—64no—
81C3×#41C3×D411yescharacteristic
82C3×#41C23×C311yescharacteristic
83C2×C3×Q8—24no—
84C2×C3×Q8—34no—
85C2×C3×Q8—34no—
86C2×C3×Q8—32no—
87C2×C3×Q8—62no—
88C2×C3×Q8—62no—
89C2×C3×Q8—62no—
90C2×C3×Q8—64no—
91C2×C3×Q8C3×D411yescharacteristic
92C2×C3×D4—31no—
93C2×C3×D4—46no—
94C2×C3×D4—62no—
95C2×C3×D4C2×A412yes—
96C22×C3×C4—31no—
97C22×C3×C4—62no—
98C22×C3×C4—62no—
99#41⋊C3C3×D413yes—
100#41⋊C3C23×C313yes—
101C2×SL(2,𝔽3)—212no—
102C2×SL(2,𝔽3)C3×D413yes—
103C3×Q16—38no—
104C3×Q16—34no—
105C3×SD16—38no—
106C3×D8—412no—
107C3×D8C2×A414yes—
108C3×M16—32no—
109C3×M16—32no—
110C2×C3×C8—31no—
111C2×C3×C8—32no—
112C2×C3×C8—32no—
113C2×C3×C8—43no—
114C2×C3×C8C2×A411yescharacteristic
115C22×C32—41no—
116C22×C32—84no—
117C22×C32—164no—
118C32×C4—41no—
119C32×C4—41no—
1202−1+4—22no—
1212−1+4—34no—
1222−1+4—62no—
1232−1+4C22×C3214yescomplemented
124C2×#41—32no—
125C2×#41—62no—
126C2×#41C22×C3211yescharacteristic
127C22×Q8—22no—
128C22×Q8—32no—
129#137—38no—
130#137—38no—
131#135—34no—
132#135—34no—
133C2×Q16—32no—
134C2×Q16—34no—
135C2×Q16—34no—
136C2×Q16—31no—
137C2×SD16—32no—
138C2×D8C3×A411yescharacteristic
139#131—32no—
140#131C22×C3212yescomplemented
141C2×M16—31no—
142C22×C8—31no—
143C23×C3—22no—
144C23×C3—31no—
145C23×C3—86no—
146C3×Q8—31no—
147C3×Q8—34no—
148C3×Q8—64no—
149C3×Q8—64no—
150C3×Q8—62no—
151C3×Q8C2×C3×D411yescharacteristic
152C3×D4—22no—
153C3×D4—31no—
154C3×D4—31no—
155C3×D4—32no—
156C3×D4—412no—
157C3×D4—64no—
158C3×D4—64no—
159C3×D4—86no—
160C3×D4C22×A414yes—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—43no—
167C2×C3×C4—64no—
168C2×C3×C4—64no—
169C2×C3×C4—64no—
170C2×C3×C4—62no—
171C2×C3×C4—64no—
172C2×C3×C4C22×A411yescharacteristic
173SL(2,𝔽3)C2×C3×D413yes—
174C3×C8—32no—
175C3×C8—46no—
176C3×C8C22×A412yes—
177C2×C32—41no—
178C2×C32—42no—
179C2×C32—164no—
180#41—31no—
181#41—31no—
182#41—34no—
183#41—34no—
184#41—62no—
185#41—62no—
186#41—68no—
187#41—64no—
188#41—64no—
189#41C32×D411yescharacteristic
190#41C23×C3211yescharacteristic
191C2×Q8—24no—
192C2×Q8—34no—
193C2×Q8—34no—
194C2×Q8—32no—
195C2×Q8—62no—
196C2×Q8—62no—
197C2×Q8—62no—
198C2×Q8—64no—
199C2×Q8C32×D411yescharacteristic
200C2×D4—31no—
201C2×D4—62no—
202C2×D4C2×C3×A412yes—
203C22×C4—31no—
204C22×C4—62no—
205C22×C4—62no—
206Q16—38no—
207Q16—34no—
208SD16—38no—
209D8C2×C3×A414yes—
210M16—32no—
211M16—32no—
212C2×C8—31no—
213C2×C8—32no—
214C2×C8—32no—
215C2×C8C2×C3×A411yescharacteristic
216C22×C3—24no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—44no—
220C22×C3—43no—
221C22×C3—62no—
222C22×C3—812no—
223C22×C3—1612no—
224C22×C3D4×A411yescharacteristic
225C3×C4—31no—
226C3×C4—31no—
227C3×C4—43no—
228C3×C4—43no—
229C3×C4—64no—
230C3×C4D4×A411yescharacteristic
231C3×C4C23×A411yescharacteristic
232C32—41noSylow
233C23—22no—
234C23—31no—
235Q8—31no—
236Q8—34no—
237Q8—64no—
238Q8—64no—
239Q8—62no—
240Q8C2×C32×D411yescharacteristic
241D4—22no—
242D4—31no—
243D4—31no—
244D4—32no—
245D4—64no—
246D4—64no—
247D4C22×C3×A414yes—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—64no—
254C2×C4—64no—
255C2×C4—64no—
256C2×C4—62no—
257C2×C4—64no—
258C2×C4C22×C3×A411yescharacteristic
259C8—32no—
260C8C22×C3×A412yes—
261C2×C3—44no—
262C2×C3—43no—
263C2×C3—46no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—1612no—
267C2×C3#578⋊C312yesdirect product factor
268C2×C3C2×D4×A411yescharacteristic
269C22—24no—
270C22—31no—
271C22—31no—
272C22—44no—
273C22—62no—
274C22C3×D4×A411yescharacteristic
275C4—31no—
276C4—31no—
277C4—64no—
278C4C3×D4×A411yescharacteristic
279C4C23×C3×A411yescharacteristic
280C3—43no—
281C3C2×#578⋊C311yescharacteristic, direct product factor
282C2—44no—
283C2—61no—
284C2—61no—
285C2C3×#578⋊C312yesdirect product factor
286C2C2×C3×D4×A411yescharacteristic
subgroupG/H
center, socleC22×C3D4×A4
commutator, cosocle#41C23×C32
FrattiniC4C23×C3×A4
FittingC2×C3×#578C3
hypercenterC2×C3×D8A4
hypocenterQ8C2×C32×D4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D8

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C3×#578⋊C3C2direct (N × H)
3. C2×C3×#131⋊C3C2semidirect
4. C2×C3×#578C3semidirect
5. C2×#578⋊C3C3direct (N × H)
7. C3×2−1+4⋊C3C22semidirect
11. C3×#131⋊C3C22semidirect
12. C2×C3×2−1+4C2×C3semidirect
13. C2×2−1+4⋊C3C2×C3semidirect
15. C3×#578C2×C3semidirect
19. C2×C3×#131C2×C3semidirect
20. #578⋊C3C2×C3direct (N × H)
21. C2×#131⋊C3C2×C3semidirect
29. C2×#578C32semidirect
33. C3×2−1+4C22×C3semidirect
40. 2−1+4⋊C3C22×C3semidirect
55. C3×#131C22×C3semidirect
58. #131⋊C3C22×C3semidirect
65. C2×2−1+4C2×C32semidirect
67. #578C2×C32semidirect
71. C2×#131C2×C32semidirect
123. 2−1+4C22×C32semidirect
140. #131C22×C32semidirect
267. C2×C3#578⋊C3direct (N × H)
281. C3C2×#578⋊C3direct (N × H)
285. C2C3×#578⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C3×#578⋊C32
C2×C3×#131⋊C32
C2×C3×#5783
C2×#578⋊C33
C2×C32×D84