Finite Group Databasebeta

← C3×#3499⋊C3 #11919748

Subgroups — C3×#3499⋊C3 #11919748

Order 1152.

924 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 166 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×D4×SL(2,𝔽3)C214yesmaximal, complemented
3C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
4C3×#3499C311yesmaximal, characteristic, complemented
5#3499⋊C3C313yesmaximal, direct product factor
6C3×2−1+4⋊C3—24no—
7C2×C3×#41⋊C3C2211yescharacteristic
8C22×C3×SL(2,𝔽3)C2214yes—
9C32×#127—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C2×C3×2−1+4C2×C312yescomplemented
12C2×2−1+4⋊C3C2×C316yescomplemented
13C3×D4×Q8C2×C314yescomplemented
14C3×#547—34no—
15C3×#536—32no—
16C3×#532—31no—
17C2×C3×C4⋊Q8—31no—
18C3×C4×#41C2×C311yescharacteristic, complemented
19D4×SL(2,𝔽3)C2×C3112yescomplemented
20C4×#41⋊C3C2×C313yescomplemented
21C2×C32×D4—42no—
22C2×C32×D4—44no—
23C3×#41⋊C3D412yescomplemented
24C2×C3×SL(2,𝔽3)—24no—
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26C32×C42—41no—
27#3499C3211yescharacteristic, complemented, Sylow
28C3×2−1+4—24no—
29C3×2−1+4—64no—
30C2×C3×#41—32no—
31C2×C3×#41—34no—
32C2×C3×#41C22×C311yescharacteristic
33C22×C3×Q8—32no—
34C22×C3×Q8C22×C314yes—
352−1+4⋊C3—212no—
36C2×#41⋊C3C22×C313yes—
37C22×SL(2,𝔽3)C22×C3112yes—
38C3×C4⋊Q8—32no—
39C3×C4⋊Q8—34no—
40C3×C4⋊Q8—31no—
41C3×#127—43no—
42C3×#127A411yescharacteristic
43C3×#124—32no—
44C3×#123—38no—
45C3×C22⋊Q8—38no—
46C3×#121—34no—
47C3×C4×Q8C22×C312yes—
48C3×C4×D4—34no—
49C3×C4×D4—32no—
50C3×#117—31no—
51C2×C3×C4⋊C4—34no—
52C2×C3×C42—31no—
53C4×SL(2,𝔽3)C22×C316yes—
54C23×C32—44no—
55C32×D4—84no—
56C32×D4—88no—
57C2×C32×C4—41no—
58C2×C32×C4—42no—
59C3×SL(2,𝔽3)C2×D411yescharacteristic
60C2×2−1+4C2×C3212yescomplemented
61D4×Q8C2×C3214yescomplemented
62#547—34no—
63#536—32no—
64#532—31no—
65C2×C4⋊Q8—31no—
66C4×#41C2×C3211yescharacteristic, complemented
67C3×#41—32no—
68C3×#41—64no—
69C3×#41—64no—
70C3×#41—68no—
71C3×#41—68no—
72C3×#41C3×D412yescomplemented
73C2×C3×Q8—24no—
74C2×C3×Q8—32no—
75C2×C3×Q8—34no—
76C2×C3×Q8—64no—
77C2×C3×Q8—64no—
78C2×C3×Q8—64no—
79C2×C3×Q8C23×C311yescharacteristic
80C2×C3×D4—31no—
81C2×C3×D4—34no—
82C2×C3×D4—46no—
83C2×C3×D4—412no—
84C2×C3×D4C2×A414yes—
85C2×C3×D4C2×A412yes—
86C22×C3×C4—31no—
87C22×C3×C4—34no—
88C22×C3×C4—34no—
89C22×C3×C4—32no—
90#41⋊C3C3×D416yescomplemented
91C2×SL(2,𝔽3)—212no—
92C2×SL(2,𝔽3)C23×C313yes—
93C3×C4⋊C4—32no—
94C3×C4⋊C4—38no—
95C3×C4⋊C4—34no—
96C3×C22⋊C4—32no—
97C3×C22⋊C4—38no—
98C3×C42—31no—
99C3×C42—32no—
100C3×C42—43no—
101C3×C42C2×A411yescharacteristic
102C22×C32—41no—
103C22×C32—88no—
104C22×C32—84no—
105C32×C4—42no—
106C32×C4—44no—
1072−1+4—24no—
1082−1+4—64no—
109C2×#41—32no—
110C2×#41—34no—
111C2×#41C22×C3211yescharacteristic
112C22×Q8—32no—
113C22×Q8C22×C3214yes—
114C4⋊Q8—32no—
115C4⋊Q8—34no—
116C4⋊Q8—31no—
117#127C3×A411yescharacteristic
118#124—32no—
119#123—38no—
120C22⋊Q8—38no—
121#121—34no—
122C4×Q8C22×C3212yes—
123C4×D4—34no—
124C4×D4—32no—
125#117—31no—
126C2×C4⋊C4—34no—
127C2×C42—31no—
128C23×C3—31no—
129C23×C3—412no—
130C23×C3C22×A414yes—
131C3×Q8—31no—
132C3×Q8—64no—
133C3×Q8—64no—
134C3×Q8—64no—
135C3×Q8C2×C3×D411yescharacteristic
136C3×D4—24no—
137C3×D4—28no—
138C3×D4—32no—
139C3×D4—32no—
140C3×D4—68no—
141C3×D4—812no—
142C3×D4—824no—
143C2×C3×C4—31no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—34no—
148C2×C3×C4—46no—
149C2×C3×C4—43no—
150C2×C3×C4—64no—
151C2×C3×C4—64no—
152C2×C3×C4—68no—
153C2×C3×C4—64no—
154C2×C3×C4C22×A411yescharacteristic
155C2×C3×C4C22×A412yes—
156SL(2,𝔽3)C2×C3×D413yes—
157C2×C32—41no—
158C2×C32—42no—
159C2×C32—164no—
160#41—32no—
161#41—64no—
162#41—64no—
163#41—68no—
164#41—68no—
165#41C32×D412yescomplemented
166C2×Q8—24no—
167C2×Q8—32no—
168C2×Q8—34no—
169C2×Q8—64no—
170C2×Q8—64no—
171C2×Q8—64no—
172C2×Q8C23×C3211yescharacteristic
173C2×D4—31no—
174C2×D4—34no—
175C2×D4C2×C3×A414yes—
176C2×D4C2×C3×A412yes—
177C22×C4—31no—
178C22×C4—34no—
179C22×C4—34no—
180C22×C4—32no—
181C4⋊C4—32no—
182C4⋊C4—38no—
183C4⋊C4—34no—
184C22⋊C4—32no—
185C22⋊C4—38no—
186C42—31no—
187C42—32no—
188C42C2×C3×A411yescharacteristic
189C22×C3—24no—
190C22×C3—28no—
191C22×C3—32no—
192C22×C3—43no—
193C22×C3—62no—
194C22×C3—824no—
195C22×C3—812no—
196C22×C3C23×A411yescharacteristic
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—46no—
200C3×C4—412no—
201C3×C4—64no—
202C3×C4—62no—
203C3×C4D4×A412yes—
204C3×C42−1+4⋊C314yescomplemented
205C32—41noSylow
206C23—31no—
207C23C22×C3×A414yes—
208Q8—31no—
209Q8—64no—
210Q8—64no—
211Q8—64no—
212Q8C2×C32×D411yescharacteristic
213D4—24no—
214D4—28no—
215D4—32no—
216D4—32no—
217D4—68no—
218C2×C4—31no—
219C2×C4—32no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—34no—
223C2×C4—64no—
224C2×C4—64no—
225C2×C4—68no—
226C2×C4—64no—
227C2×C4C22×C3×A411yescharacteristic
228C2×C4C22×C3×A412yes—
229C2×C3—44no—
230C2×C3—43no—
231C2×C3—46no—
232C2×C3—62no—
233C2×C3—1612no—
234C2×C3C2×D4×A411yescharacteristic
235C2×C3C2×2−1+4⋊C312yes—
236C22—24no—
237C22—28no—
238C22—32no—
239C22—62no—
240C22C23×C3×A411yescharacteristic
241C4—31no—
242C4—31no—
243C4—64no—
244C4—62no—
245C4C3×D4×A412yes—
246C4C3×2−1+4⋊C314yescomplemented
247C3—43no—
248C3#3499⋊C311yescharacteristic, direct product factor
249C2—44no—
250C2—62no—
251C2C2×C3×D4×A411yescharacteristic
252C2C2×C3×2−1+4⋊C312yes—
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3499C3
hypercenterC3×#127A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C3×D4×SL(2,𝔽3)C2semidirect
3. C3×C4×#41⋊C3C2semidirect
4. C3×#3499C3semidirect
5. #3499⋊C3C3direct (N × H)
11. C2×C3×2−1+4C2×C3semidirect
12. C2×2−1+4⋊C3C2×C3semidirect
13. C3×D4×Q8C2×C3semidirect
18. C3×C4×#41C2×C3semidirect
19. D4×SL(2,𝔽3)C2×C3semidirect
20. C4×#41⋊C3C2×C3semidirect
23. C3×#41⋊C3D4semidirect
27. #3499C32semidirect
60. C2×2−1+4C2×C32semidirect
61. D4×Q8C2×C32semidirect
66. C4×#41C2×C32semidirect
72. C3×#41C3×D4semidirect
90. #41⋊C3C3×D4semidirect
165. #41C32×D4semidirect
204. C3×C42−1+4⋊C3semidirect
246. C4C3×2−1+4⋊C3semidirect
248. C3#3499⋊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×D4×SL(2,𝔽3)2
C3×C4×#41⋊C32
C3×#34993
#3499⋊C33
C32×#1274