Finite Group Databasebeta

← #84112⋊C3 #11921380

Subgroups — #84112⋊C3 #11921380

Order 1152.

411 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 59 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⋊C3C211yesmaximal, characteristic
2#5819⋊C3C212yesmaximal, complemented
3#84112C311yesmaximal, characteristic, complemented
4#2889⋊C3—31nomaximal
5C3×2−1+4⋊C3—22no—
6C2×C3×#41⋊C3C2211yescharacteristic
7C22×C3×SL(2,𝔽3)C412yes—
8C3⋊(#131⋊C3)—22no—
9C3×#833—41nomaximal
10C2×C3×2−1+4C2×C311yescharacteristic
11C2×2−1+4⋊C3—21no—
12C2×2−1+4⋊C3D311yescharacteristic
13#5819C2×C312yescomplemented
14#568⋊C3—32no—
15#5237—31no—
16#5237—31no—
17C2×#836—32no—
18C2×C32×D4—41no—
19C3×#41⋊C3D412yes—
20C2×C3×SL(2,𝔽3)—22no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3×#208—42no—
23#2889—31noSylow
24C3×2−1+4—22no—
25C3×2−1+4—62no—
26C2×C3×#41—31no—
27C2×C3×#41—32no—
28C2×C3×#41C22×C311yescharacteristic
29C22×C3×Q8—31no—
30C22×C3×Q8C3×C412yes—
312−1+4⋊C3—24no—
322−1+4⋊C3—22no—
33C2×#41⋊C3—21no—
34C2×#41⋊C3C2×D311yescharacteristic
35C22×SL(2,𝔽3)—22no—
36C22×SL(2,𝔽3)Dic312yes—
37#948—22no—
38#948—62no—
39C2×#208—32no—
40C2×#208—32no—
41#131⋊C3—62no—
42C3×#100—121no—
43#836—32no—
44#836—32no—
45#833—31no—
46#833A411yescharacteristic
47C23×C32—42no—
48C32×D4—82no—
49C2×C32×C4—41no—
50C3×SL(2,𝔽3)C22⋊C411yescharacteristic
51C3×C3⋊C8—82no—
52C2×2−1+4C3×D311yescharacteristic
53#568—32no—
54#413—91no—
55#413—91no—
56C2×#101—92no—
57C3×#41—32no—
58C3×#41—62no—
59C3×#41—62no—
60C3×#41—64no—
61C3×#41—64no—
62C3×#41C3×D412yes—
63C2×C3×Q8—22no—
64C2×C3×Q8—31no—
65C2×C3×Q8—32no—
66C2×C3×Q8—62no—
67C2×C3×Q8—62no—
68C2×C3×Q8—62no—
69C2×C3×Q8C2×C3×C411yescharacteristic
70C2×C3×D4—31no—
71C2×C3×D4—32no—
72C2×C3×D4—41no—
73C2×C3×D4—81no—
74C2×C3×D4C2×A411yescharacteristic
75C22×C3×C4—31no—
76C22×C3×C4—32no—
77C22×C3×C4—32no—
78#41⋊C3—22no—
79#41⋊C3#6712yes—
80C2×SL(2,𝔽3)—24no—
81C2×SL(2,𝔽3)—22no—
82C2×SL(2,𝔽3)—21no—
83C2×SL(2,𝔽3)C2×Dic311yescharacteristic
84C3×M16—122no—
85#208—32no—
86#208—62no—
87#208—62no—
88#208C2×A412yes—
89C2×C3⋊C8—62no—
90C2×C3⋊C8—62no—
91C22×C32—41no—
92C22×C32—82no—
93C22×C32—162no—
94C32×C4—42no—
952−1+4—22no—
962−1+4—62no—
97C2×#41—31no—
98C2×#41—32no—
99C2×#41C2×C3×D311yescharacteristic
100C22×Q8—31no—
101C22×Q8C3×Dic312yes—
102#131—62no—
103#131—182no—
104C2×M16—92no—
105C2×M16—92no—
106#101—92no—
107#101—92no—
108#100—31no—
109#100—91no—
110C23×C3—31no—
111C23×C3—42no—
112C23×C3—82no—
113C23×C3C4×A412yes—
114C3×Q8—31no—
115C3×Q8—62no—
116C3×Q8—62no—
117C3×Q8—62no—
118C3×Q8C3×C22⋊C411yescharacteristic
119C3×D4—22no—
120C3×D4—32no—
121C3×D4—32no—
122C3×D4—64no—
123C3×D4—82no—
124C3×D4—84no—
125C2×C3×C4—31no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—32no—
129C2×C3×C4—41no—
130C2×C3×C4—62no—
131C2×C3×C4—62no—
132C2×C3×C4—64no—
133C2×C3×C4—81no—
134C2×C3×C4C22×A411yescharacteristic
135SL(2,𝔽3)—21no—
136SL(2,𝔽3)#21711yescharacteristic
137C3×C8—242no—
138C3⋊C8—22no—
139C3⋊C8—62no—
140C2×C32—41no—
141C2×C32—81no—
142C2×C32—162no—
143#41—32no—
144#41—62no—
145#41—62no—
146#41—64no—
147#41—64no—
148#41C3×#6712yes—
149C2×Q8—22no—
150C2×Q8—31no—
151C2×Q8—32no—
152C2×Q8—62no—
153C2×Q8—62no—
154C2×Q8—62no—
155C2×Q8C2×C3×Dic311yescharacteristic
156C2×D4—31no—
157C2×D4—32no—
158C2×D4D3×A411yescharacteristic
159C22×C4—31no—
160C22×C4—32no—
161C22×C4—32no—
162M16—32no—
163M16—92no—
164M16—182no—
165M16—182no—
166C2×C8—182no—
167C2×C8—182no—
168C22×C3—22no—
169C22×C3—32no—
170C22×C3—42no—
171C22×C3—41no—
172C22×C3—82no—
173C22×C3—84no—
174C22×C3—81no—
175C22×C3—121no—
176C22×C3—162no—
177C22×C3—164no—
178C22×C3C2×C4×A411yescharacteristic
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—42no—
182C3×C4—62no—
183C3×C4—82no—
184C3×C4D4×A412yes—
185C32—41noSylow
186C23—31no—
187C23Dic3×A412yes—
188Q8—31no—
189Q8—62no—
190Q8—62no—
191Q8—62no—
192Q8C3×#21711yescharacteristic
193D4—22no—
194D4—32no—
195D4—32no—
196D4—64no—
197C2×C4—31no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—32no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—64no—
204C2×C4C2×D3×A411yescharacteristic
205C8—62no—
206C8—182no—
207C2×C3—21no—
208C2×C3—42no—
209C2×C3—41no—
210C2×C3—62no—
211C2×C3—81no—
212C2×C3—82no—
213C2×C3—81no—
214C2×C3—162no—
215C2×C3—164no—
216C2×C3A4×C22⋊C411yescharacteristic
217C22—22no—
218C22—32no—
219C22—42no—
220C22—121no—
221C22C2×Dic3×A411yescharacteristic
222C4—31no—
223C4—31no—
224C4—62no—
225C4A4×#6712yes—
226C3—41no—
227C3—81no—
228C3#2889⋊C311yescharacteristic, complemented
229C2—21no—
230C2—42no—
231C2—62no—
232C2A4×#21711yescharacteristic
subgroupG/H
centerC2A4×#217
commutatorC2×C3×Q8C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC2×C3×2−1+4C2×C3
socleC2×C3A4×C22⋊C4
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×C3×#41C22×C3
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #5819⋊C3C2semidirect
3. #84112C3semidirect
13. #5819C2×C3semidirect
228. C3#2889⋊C3semidirect

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
#5819⋊C32
#841123
#2889⋊C33
C3×#8334