Finite Group Databasebeta

← C2×C3×#557⋊C3 #11919722

Subgroups — C2×C3×#557⋊C3 #11919722

Order 1152.

696 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 190 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
1C3×#557⋊C3C214yesmaximal, direct product factor
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×C3×C4×SL(2,𝔽3)C212yesmaximal
4C2×C3×#557C311yesmaximal, characteristic, complemented
5C2×#557⋊C3C313yesmaximal, direct product factor
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C32×C4⋊C4—41nomaximal
8C3×C4×SL(2,𝔽3)C2212yes—
9C3×C4×SL(2,𝔽3)C2214yes—
10C3×#557—34no—
11C3×#557C2×C314yescomplemented
12C2×C3×C4⋊Q8—31no—
13C2×C3×#125—32no—
14C2×C3×C4×Q8—31no—
15C2×C3×C4×Q8C2×C312yes—
16C2×C3×C4×Q8C2×C311yescharacteristic
17#557⋊C3C2×C3112yesdirect product factor
18C2×C4×SL(2,𝔽3)C2×C313yes—
19C2×C4×SL(2,𝔽3)C2×C316yes—
20C22×C32×C4—41no—
21C22×C32×C4—42no—
22C2×C3×SL(2,𝔽3)Q812yescomplemented
23C2×C3×SL(2,𝔽3)C2311yescharacteristic
24C32×C4⋊C4—44no—
25C2×#557C3211yescharacteristic, complemented, Sylow
26C22×C3×Q8—31no—
27C22×C3×Q8C22×C311yescharacteristic
28C22×SL(2,𝔽3)C22×C313yes—
29C3×C4⋊Q8—34no—
30C3×C4⋊Q8—34no—
31C3×#125—38no—
32C3×#125—38no—
33C3×C4×Q8—34no—
34C3×C4×Q8—34no—
35C3×C4×Q8—34no—
36C3×C4×Q8—32no—
37C3×C4×Q8C22×C314yes—
38C3×C4×Q8C22×C312yes—
39C2×C3×C4⋊C4—31no—
40C2×C3×C4⋊C4—32no—
41C2×C3×C4⋊C4—32no—
42C2×C3×C4⋊C4—32no—
43C2×C3×C4⋊C4—43no—
44C2×C3×C4⋊C4A411yescharacteristic
45C2×C3×C42—32no—
46C2×C3×C42—31no—
47C4×SL(2,𝔽3)C22×C316yes—
48C4×SL(2,𝔽3)C22×C3112yes—
49C23×C32—41no—
50C2×C32×C4—44no—
51C2×C32×C4—42no—
52C2×C32×C4—44no—
53C2×C32×C4—84no—
54C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
55#557—34no—
56#557C2×C3214yescomplemented
57C2×C4⋊Q8—31no—
58C2×#125—32no—
59C2×C4×Q8—31no—
60C2×C4×Q8C2×C3212yes—
61C2×C4×Q8C2×C3211yescharacteristic
62C2×C3×Q8—34no—
63C2×C3×Q8—31no—
64C2×C3×Q8—32no—
65C2×C3×Q8—64no—
66C2×C3×Q8C3×Q812yes—
67C2×C3×Q8C23×C311yescharacteristic
68C22×C3×C4—31no—
69C22×C3×C4—31no—
70C22×C3×C4—32no—
71C22×C3×C4—43no—
72C22×C3×C4—46no—
73C22×C3×C4C2×A412yes—
74C22×C3×C4C2×A411yescharacteristic
75C2×SL(2,𝔽3)C3×Q816yescomplemented
76C2×SL(2,𝔽3)C23×C313yes—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—34no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4—34no—
81C3×C4⋊C4—38no—
82C3×C4⋊C4—38no—
83C3×C4⋊C4—412no—
84C3×C4⋊C4C2×A414yes—
85C3×C42—34no—
86C3×C42—32no—
87C3×C42—34no—
88C3×C42—32no—
89C22×C32—42no—
90C22×C32—42no—
91C22×C32—42no—
92C22×C32—41no—
93C32×C4—44no—
94C32×C4—84no—
95C22×Q8—31no—
96C22×Q8C22×C3211yescharacteristic
97C4⋊Q8—34no—
98C4⋊Q8—34no—
99#125—38no—
100#125—38no—
101C4×Q8—34no—
102C4×Q8—34no—
103C4×Q8—34no—
104C4×Q8—32no—
105C4×Q8C22×C3214yes—
106C4×Q8C22×C3212yes—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4C3×A411yescharacteristic
112C2×C42—32no—
113C2×C42—31no—
114C23×C3—43no—
115C23×C3C22×A411yescharacteristic
116C3×Q8—32no—
117C3×Q8—31no—
118C3×Q8—32no—
119C3×Q8—68no—
120C3×Q8C2×C3×Q811yescharacteristic
121C2×C3×C4—24no—
122C2×C3×C4—32no—
123C2×C3×C4—32no—
124C2×C3×C4—31no—
125C2×C3×C4—31no—
126C2×C3×C4—32no—
127C2×C3×C4—34no—
128C2×C3×C4—46no—
129C2×C3×C4—412no—
130C2×C3×C4—412no—
131C2×C3×C4—64no—
132C2×C3×C4—62no—
133C2×C3×C4—812no—
134C2×C3×C4#41⋊C314yes—
135C2×C3×C4C22×A414yes—
136C2×C3×C4C22×A412yes—
137SL(2,𝔽3)C2×C3×Q813yescomplemented
138C2×C32—41no—
139C2×C32—41no—
140C2×C32—41no—
141C2×C32—44no—
142C2×Q8—34no—
143C2×Q8—31no—
144C2×Q8—32no—
145C2×Q8—64no—
146C2×Q8C32×Q812yes—
147C2×Q8C23×C3211yescharacteristic
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—32no—
151C22×C4C2×C3×A412yes—
152C22×C4C2×C3×A411yescharacteristic
153C4⋊C4—32no—
154C4⋊C4—34no—
155C4⋊C4—32no—
156C4⋊C4—34no—
157C4⋊C4—38no—
158C4⋊C4—38no—
159C4⋊C4C2×C3×A414yes—
160C42—34no—
161C42—32no—
162C42—34no—
163C42—32no—
164C22×C3—46no—
165C22×C3—46no—
166C22×C3—46no—
167C22×C3—43no—
168C22×C3Q8×A412yes—
169C22×C3C2×#41⋊C312yes—
170C22×C32−1+4⋊C312yes—
171C22×C3C23×A411yescharacteristic
172C3×C4—24no—
173C3×C4—31no—
174C3×C4—32no—
175C3×C4—31no—
176C3×C4—412no—
177C3×C4—64no—
178C3×C4—62no—
179C3×C4—812no—
180C3×C4C2×#41⋊C314yes—
181C32—41noSylow
182C23C22×C3×A411yescharacteristic
183Q8—32no—
184Q8—31no—
185Q8—32no—
186Q8—68no—
187Q8C2×C32×Q811yescharacteristic
188C2×C4—24no—
189C2×C4—32no—
190C2×C4—32no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—32no—
194C2×C4—34no—
195C2×C4—64no—
196C2×C4—62no—
197C2×C4C3×#41⋊C314yes—
198C2×C4C22×C3×A414yes—
199C2×C4C22×C3×A412yes—
200C2×C3—43no—
201C2×C3—43no—
202C2×C3—43no—
203C2×C3—412no—
204C2×C3#557⋊C314yesdirect product factor
205C2×C3C2×Q8×A411yescharacteristic
206C2×C3C22×#41⋊C311yescharacteristic
207C2×C3C2×2−1+4⋊C311yescharacteristic
208C22C3×Q8×A412yes—
209C22C2×C3×#41⋊C312yes—
210C22C3×2−1+4⋊C312yes—
211C22C23×C3×A411yescharacteristic
212C4—24no—
213C4—31no—
214C4—32no—
215C4—31no—
216C4—64no—
217C4—62no—
218C4C2×C3×#41⋊C314yes—
219C3—43no—
220C3C2×#557⋊C311yescharacteristic, direct product factor
221C2C3×#557⋊C314yesdirect product factor
222C2C2×C3×Q8×A411yescharacteristic
223C2C22×C3×#41⋊C311yescharacteristic
224C2C2×C3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC2×C3×#557C3
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC2×C3×C4⋊C4A4
hypocenterQ8C2×C32×Q8
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2direct (N × H)
4. C2×C3×#557C3semidirect
5. C2×#557⋊C3C3direct (N × H)
11. C3×#557C2×C3semidirect
17. #557⋊C3C2×C3direct (N × H)
22. C2×C3×SL(2,𝔽3)Q8semidirect
25. C2×#557C32semidirect
54. C3×SL(2,𝔽3)C2×Q8semidirect
56. #557C2×C32semidirect
75. C2×SL(2,𝔽3)C3×Q8semidirect
137. SL(2,𝔽3)C2×C3×Q8semidirect
204. C2×C3#557⋊C3direct (N × H)
220. C3C2×#557⋊C3direct (N × H)
221. C2C3×#557⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#5573
C2×#557⋊C33
C2×C32×C4⋊C44