Finite Group Databasebeta

← #72209796

Subgroups — #72209796

Order 1536.

401 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 21 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
1#475912C311yesmaximal, characteristic, complemented, Sylow
2#3458⋊C3—42nomaximal
3C2×C4×C42⋊C3—41nomaximal
4C2×#338⋊C3—42nomaximal
5#23560—32no—
6#11243—32no—
7#10839—31no—
8#543⋊C3—42no—
9C22×C42⋊C3—41no—
10C4×C42⋊C3—41no—
11C4×C42⋊C3—41no—
12#338⋊C3—42no—
13#3458A412yes—
14#2911—32no—
15C2×#423—31no—
16#2285—31no—
17C2×C43A411yescharacteristic
18#2061—64no—
19#2060—64no—
20#1742—32no—
21C2×#339—34no—
22C2×#338A412yes—
23C2×C4×A4—161no—
24C4×SL(2,𝔽3)—168no—
25C2×C42⋊C3—41no—
26C2×C42⋊C3—42no—
27C22×M16—32no—
28#544—64no—
29#543—42no—
30#533—122no—
31C2×#117—32no—
32C22×C42—31no—
33C22×C42—41no—
34#428—68no—
35#427—122no—
36#423—32no—
37#423—32no—
38#423—121no—
39C2×C4⋊C8—121no—
40#384—32no—
41#382—32no—
42#380—121no—
43C4×C4⋊C4—121no—
44C43—31no—
45C43—41no—
46C43SL(2,𝔽3)11yescharacteristic, complemented
47#343—64no—
48#339—64no—
49#339—64no—
50#338—42no—
51C22×A4—161no—
52C2×SL(2,𝔽3)—168no—
53C4×A4—161no—
54C4×A4—161no—
55C42⋊C3—41no—
56C23×C4—61no—
57C23×C4C22⋊A411yescharacteristic
58C2×M16—34no—
59C2×M16—62no—
60C2×M16—64no—
61C2×M16—64no—
62C22×C8—62no—
63C4⋊Q8—124no—
64#125—124no—
65C22⋊Q8—128no—
66C4×Q8—48no—
67#117—64no—
68#117—64no—
69C2×C4⋊C4—34no—
70C2×C4⋊C4—122no—
71C2×C22⋊C4—62no—
72C2×C42—31no—
73C2×C42—31no—
74C2×C42—31no—
75C2×C42—31no—
76C2×C42—31no—
77C2×C42—31no—
78C2×C42—41no—
79C2×C42—42no—
80C2×C42—61no—
81C2×C42—61no—
82C2×C42—61no—
83C2×C42—62no—
84C2×C42—121no—
85C2×C42—122no—
86#107—124no—
87#106—124no—
88C4⋊C8—122no—
89C4⋊C8—122no—
90#95—122no—
91C2×A4—162no—
92C2×A4—161no—
93C2×C3×C4—641no—
94SL(2,𝔽3)—168no—
95C24#99711yescharacteristic
96C2×Q8—48no—
97C22×C4—31no—
98C22×C4—31no—
99C22×C4—31no—
100C22×C4—31no—
101C22×C4—61no—
102C22×C4—61no—
103C22×C4—61no—
104C22×C4—61no—
105C22×C4—61no—
106C22×C4—61no—
107C22×C4—64no—
108C22×C4—122no—
109C22×C4—122no—
110C22×C4C22⋊SL(2,𝔽3)11yescharacteristic
111C22×C4#99711yescharacteristic
112M16—68no—
113M16—124no—
114C2×C8—64no—
115C2×C8—122no—
116C2×C8—122no—
117C4⋊C4—68no—
118C4⋊C4—128no—
119C4⋊C4—242no—
120C4⋊C4—242no—
121C22⋊C4—124no—
122C42—31no—
123C42—31no—
124C42—31no—
125C42—31no—
126C42—41no—
127C42—61no—
128C42—61no—
129C42—61no—
130C42—61no—
131C42—61no—
132C42—61no—
133C42—122no—
134C42—121no—
135C42—122no—
136C42—122no—
137C42—122no—
138C22×C3—641no—
139A4—161no—
140C3×C4—641no—
141C3×C4—641no—
142C23—31no—
143C23—31no—
144C23—42no—
145C23#543⋊C311yescharacteristic
146Q8—168no—
147C2×C4—32no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—31no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—62no—
158C2×C4—62no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—121no—
162C2×C4—121no—
163C2×C4—122no—
164C2×C4—121no—
165C2×C4—122no—
166C2×C4—121no—
167C2×C4—122no—
168C2×C4—124no—
169C2×C4—244no—
170C2×C4#546411yescharacteristic
171C8—124no—
172C2×C3—642no—
173C2×C3—641no—
174C22—31no—
175C22—31no—
176C22—41no—
177C22—62no—
178C22—121no—
179C22#2679⋊C311yescharacteristic
180C4—31no—
181C4—31no—
182C4—61no—
183C4—61no—
184C4—62no—
185C4—121no—
186C4—121no—
187C4—244no—
188C4#7329011yescharacteristic
189C4#2844⋊C311yescharacteristic
190C3—641noSylow
191C2—61no—
192C2—61no—
193C2#17072⋊C311yescharacteristic
194C2#31076⋊C312yes—
subgroupG/H
center, hypercenterC2×C4#5464
commutator, Fitting, cosocle, hypocenter#475912C3
FrattiniC23×C4C22⋊A4
socleC22#2679⋊C3
LCSG ▷ #475912
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #475912C3semidirect
46. C43SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
#4759123
#3458⋊C34
C2×C4×C42⋊C34
C2×#338⋊C34