Finite Group Databasebeta

← #85829⋊C3 #11923197

Subgroups — #85829⋊C3 #11923197

Order 1152.

509 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 87 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×C4×#41⋊C3C211yesmaximal, characteristic
2#5827⋊C3C212yesmaximal
3#5569⋊C3C214yesmaximal, complemented
4#85829C311yesmaximal, characteristic, complemented
5#3504⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#870—41nomaximal
8C3×C4×SL(2,𝔽3)C2212yes—
9Dic3×SL(2,𝔽3)C2214yes—
10C3×C4×#41C2×C311yescharacteristic
11#5827C2×C312yes—
12#5569C2×C314yes—
13#5539—34no—
14#5519—32no—
15#5470—31no—
16C2×#870—31no—
17#557⋊C3—34no—
18#520⋊C3—32no—
19C4×#41⋊C3—21no—
20C4×#41⋊C3D311yescharacteristic
21C3×#41⋊C3Q812yescomplemented
22C2×C3×SL(2,𝔽3)C2311yescharacteristic
23C32×C42—41no—
24C3×C4⋊Dic3—42no—
25C3×Dic3⋊C4—44no—
26#3504—31noSylow
27C2×C3×#41C22×C311yescharacteristic
28C2×#41⋊C3—21no—
29C2×#41⋊C3C2×D311yescharacteristic
30C3×#125—121no—
31C3×C4×Q8C22×C312yes—
32C3×C4×D4—32no—
33C3×#117—31no—
34C2×C3×C42—31no—
35Q8×Dic3C22×C314yes—
36D4×Dic3—34no—
37#926—32no—
38C2×C4⋊Dic3—32no—
39#924—34no—
40C2×Dic3⋊C4—34no—
41#890—34no—
42#888—34no—
43#878—38no—
44#870—32no—
45#870—31no—
46#870—34no—
47#870A411yescharacteristic
48#869—32no—
49C4×Dic6—34no—
50C4×SL(2,𝔽3)—22no—
51C4×SL(2,𝔽3)—34no—
52C4×SL(2,𝔽3)C2×D312yes—
53C2×C32×C4—41no—
54C2×C32×C4—42no—
55C2×C3×Dic3—44no—
56C3×SL(2,𝔽3)C2×Q811yescharacteristic
57#557—34no—
58#554—94no—
59#544—92no—
60#533—91no—
61C2×#125—91no—
62#520—32no—
63C4×#41C3×D311yescharacteristic
64C3×#41—32no—
65C3×#41C3×Q812yes—
66C2×C3×Q8C23×C311yescharacteristic
67C2×C3×D4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—32no—
70C22×Dic3—34no—
71C2×Dic6—34no—
72#41⋊C3—22no—
73#41⋊C3Dic612yescomplemented
74C2×SL(2,𝔽3)—21no—
75C2×SL(2,𝔽3)C22×D311yescharacteristic
76C3×C4⋊C4—32no—
77C3×C4⋊C4—122no—
78C3×C4⋊C4—124no—
79C3×C22⋊C4—32no—
80C3×C42—32no—
81C3×C42—31no—
82C3×C42—41no—
83C3×C42—81no—
84C3×C42C2×A411yescharacteristic
85#217—34no—
86C4⋊Dic3—32no—
87C4⋊Dic3—34no—
88C4⋊Dic3—34no—
89C4⋊Dic3C2×A412yes—
90Dic3⋊C4—38no—
91Dic3⋊C4—34no—
92Dic3⋊C4C2×A414yes—
93C4×Dic3—34no—
94C22×C32—41no—
95C32×C4—42no—
96C32×C4—82no—
97C3×Dic3—84no—
98C2×#41C2×C3×D311yescharacteristic
99C4⋊Q8—92no—
100C4⋊Q8—94no—
101#125—31no—
102#125—94no—
103#125—92no—
104#125—94no—
105#125—91no—
106C22⋊Q8—98no—
107C22⋊Q8—94no—
108C4×Q8—34no—
109C4×Q8—94no—
110C4×Q8C2×C3×D312yes—
111C4×D4—32no—
112C4×D4—94no—
113#117—31no—
114#117—92no—
115C2×C4⋊C4—94no—
116C2×C4⋊C4—92no—
117C2×C42—31no—
118C23×C3—31no—
119C3×Q8—31no—
120C3×Q8C2×C3×Q811yescharacteristic
121C3×D4—32no—
122C3×D4—32no—
123C2×C3×C4—31no—
124C2×C3×C4—32no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4—41no—
128C2×C3×C4—42no—
129C2×C3×C4—82no—
130C2×C3×C4—81no—
131C2×C3×C4—124no—
132C2×C3×C4—122no—
133C2×C3×C4C22×A412yes—
134C2×C3×C4C22×A411yescharacteristic
135C2×Dic3—34no—
136C2×Dic3—124no—
137C2×Dic3C22×A414yes—
138Dic6—124no—
139SL(2,𝔽3)—21no—
140SL(2,𝔽3)C2×Dic611yescharacteristic
141C2×C32—41no—
142C2×C32—42no—
143#41—32no—
144#41C3×Dic612yes—
145C2×Q8—94no—
146C2×Q8C22×C3×D311yescharacteristic
147C2×D4—31no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—94no—
151C4⋊C4—32no—
152C4⋊C4—32no—
153C4⋊C4—34no—
154C4⋊C4—94no—
155C4⋊C4—94no—
156C4⋊C4—98no—
157C4⋊C4—94no—
158C4⋊C4—92no—
159C22⋊C4—32no—
160C22⋊C4—94no—
161C42—31no—
162C42—32no—
163C42—94no—
164C42D3×A411yescharacteristic
165C22×C3—32no—
166C22×C3—41no—
167C22×C3—62no—
168C22×C3—81no—
169C22×C3C23×A411yescharacteristic
170C3×C4—22no—
171C3×C4—31no—
172C3×C4—31no—
173C3×C4—42no—
174C3×C4—62no—
175C3×C4—82no—
176C3×C4—84no—
177C3×C4—82no—
178C3×C4—244no—
179C3×C4Q8×A412yes—
180Dic3—24no—
181Dic3—64no—
182C32—41noSylow
183C23—31no—
184Q8—31no—
185Q8—364no—
186Q8C2×C3×Dic611yescharacteristic
187D4—32no—
188D4—32no—
189C2×C4—31no—
190C2×C4—32no—
191C2×C4—32no—
192C2×C4—32no—
193C2×C4—34no—
194C2×C4—94no—
195C2×C4—122no—
196C2×C4—364no—
197C2×C4C2×D3×A412yes—
198C2×C4C2×D3×A411yescharacteristic
199C2×C3—42no—
200C2×C3—41no—
201C2×C3—62no—
202C2×C3—82no—
203C2×C3—81no—
204C2×C3C2×Q8×A411yescharacteristic
205C2×C3#594512yes—
206C22—32no—
207C22—62no—
208C22C22×D3×A411yescharacteristic
209C4—22no—
210C4—31no—
211C4—31no—
212C4—62no—
213C4—64no—
214C4—184no—
215C4A4×Dic612yes—
216C3—41no—
217C3—81no—
218C3#3504⋊C311yescharacteristic, complemented
219C2—62no—
220C2C2×A4×Dic611yescharacteristic
221C2#5964⋊C312yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
prime radicalC2×C3×#41⋊C3C22
hypercenterC42D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #5569⋊C3C2semidirect
4. #85829C3semidirect
21. C3×#41⋊C3Q8semidirect
73. #41⋊C3Dic6semidirect
218. C3#3504⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#5827⋊C32
#5569⋊C32
#858293
#3504⋊C33
C3×#8704