Finite Group Databasebeta

← #13147189

Subgroups — #13147189

Order 1728.

230 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 36 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#11722557C211yesmaximal, characteristic, complemented
2#11722414C211yesmaximal, characteristic, complemented
3#11720752C211yesmaximal, characteristic, complemented
4#5836⋊C3—31nomaximal
5#10589938C311yesmaximal, characteristic, complemented
6#88683—21no—
7#88683C2211yescharacteristic
8D9⋊SL(2,𝔽3)—21no—
9#3675⋊C3—41nomaximal
10C3×#994D311yescharacteristic
11#1010⋊C3—31no—
12C3⋊(#131⋊C3)—31no—
13C9×H(5,𝔽2)C2×C311yescharacteristic, complemented
14#63716C2×C311yescharacteristic
15#63700—21no—
16#63513—31no—
17#63511—31no—
18#63510C2×C311yescharacteristic, complemented
19#63495—31no—
20C2×#3672—31no—
21Q8×3−1+2—41no—
22C4⋊AGL(1,ℤ/9)—41no—
23C9⋊SL(2,𝔽3)D411yescharacteristic
243−1+2⋊C8—41no—
25#5836—31no—
26#577⋊C3—91no—
27C3×#41⋊C3—21no—
28C3×#41⋊C3C2×D311yescharacteristic
29D3×SL(2,𝔽3)—61no—
30C3×#215—121no—
31C9×#41—21no—
32C9×#41—31no—
33C9×#41C22×C311yescharacteristic
34C2×D4×C9—31no—
35C2×D4×C9—61no—
36Q8×D9—21no—
37#3699—61no—
38#3697—31no—
39C2×Dic18—31no—
40#41⋊C9—22no—
41#41⋊C9—21no—
42#3675A411yescharacteristic
43#3674—31no—
44#3673—31no—
45#3672—32no—
46#3672—31no—
47#3667—31no—
48C2×C9⋊C8—31no—
49C2×AGL(1,ℤ/9)—81no—
50C4×3−1+2—41no—
51C4×3−1+2—81no—
52C3×H(5,𝔽2)C3×D311yescharacteristic
53#1010—31no—
542−1+4⋊C3—91no—
55#994—31no—
56#951—91no—
57#950—91no—
58#948—31no—
59C2×#214—91no—
60#932—91no—
61#131⋊C3—91no—
62C32×Q8—41no—
63C3×D12—121no—
64C3×SL(2,𝔽3)#6711yescharacteristic
65C23×C9—61no—
66C3×C3⋊C8—121no—
67Q8×C9—81no—
68Q8×C9C3×D411yescharacteristic
69Q8×C9C2×A411yescharacteristic
70D4×C9—31no—
71D4×C9—31no—
72D4×C9—61no—
73D4×C9—61no—
74C2×C4×C9—31no—
75C2×C4×C9—61no—
76#611—61no—
77C2×Dic9—61no—
78D36C2×A411yescharacteristic
79C4×D9—61no—
80Dic18—31no—
81Dic18—61no—
82Q8⋊C9—21no—
83C9⋊C8—31no—
84C9⋊C8C2×A411yescharacteristic
85#577—91noSylow
86C2×3−1+2—41no—
87AGL(1,ℤ/9)—161no—
88C3×#41—21no—
89C3×#41—31no—
90C3×#41C2×C3×D311yescharacteristic
91C2×C3×D4—31no—
92C2×C3×D4—61no—
93D3×Q8—61no—
94#237—181no—
95#235—91no—
96C2×Dic6—91no—
97#41⋊C3—31no—
98#41⋊C3—61no—
99C2×SL(2,𝔽3)—181no—
100C3×SD16—361no—
101#216—91no—
102#215—31no—
103#214—91no—
104#214—92no—
105#213—91no—
106#208—91no—
107C2×C3⋊C8—91no—
108C2×C3×D3—241no—
109C32×C4—41no—
110C32×C4—81no—
111C22×C9—31no—
112C22×C9—61no—
113C22×C9—241no—
114C2×D9—21no—
115C4×C9—21no—
116C4×C9—31no—
117C4×C9—82no—
118C4×C9—81no—
119C4×C9C22×A411yescharacteristic
120Dic9—61no—
1212−1+4—91no—
122H(5,𝔽2)AGL(1,ℤ/9)11yescharacteristic, complemented
123#137—271no—
124AGL(1,ℤ/8)—271no—
125#135—271no—
126C2×SD16—271no—
127#131—91no—
1283−1+2—41noSylow
129C23×C3—61no—
130C3×Q8—121no—
131C3×Q8C3×#6711yescharacteristic
132C3×Q8D3×A411yescharacteristic
133C3×D4—31no—
134C3×D4—31no—
135C3×D4—61no—
136C3×D4—61no—
137C3×D4—361no—
138C2×C3×C4—31no—
139C2×C3×C4—61no—
140#67—181no—
141C2×Dic3—181no—
142D12—31no—
143C4×D3—181no—
144Dic6—91no—
145Dic6—181no—
146SL(2,𝔽3)—31no—
147C3×C8—361no—
148C3⋊C8—31no—
149C3⋊C8—91no—
150C2×C32—41no—
151C3×D3—481no—
152C2×C9—61no—
153C2×C9—81no—
154C2×C9—121no—
155C2×C9D4×A411yescharacteristic
156D9—41no—
157#41—21no—
158#41—31no—
159#41—271no—
160#41—541no—
161#41C2×AGL(1,ℤ/9)11yescharacteristic
162C2×Q8—181no—
163C2×Q8—271no—
164C2×D4—31no—
165C2×D4—61no—
166Q16—271no—
167SD16—91no—
168SD16—271no—
169SD16—272no—
170D8—271no—
171M16—271no—
172C2×C8—271no—
173C22×C3—31no—
174C22×C3—61no—
175C22×C3—241no—
176C22×C3—721no—
177C2×D3—61no—
178C3×C4—21no—
179C3×C4—31no—
180C3×C4—121no—
181C3×C4—241no—
182C3×C4C2×D3×A411yescharacteristic
183Dic3—181no—
184C32—41no—
185C9—81no—
186C9#577⋊C311yescharacteristic, complemented
187C23—61no—
188Q8—271no—
189Q8—541no—
190Q8#611⋊C311yescharacteristic
191Q8D9⋊A411yescharacteristic
192D4—31no—
193D4—31no—
194D4—61no—
195D4—61no—
196D4—91no—
197D4—541no—
198C2×C4—31no—
199C2×C4—61no—
200C2×C4—541no—
201C2×C4—541no—
202C8—91no—
203C8—271no—
204C2×C3—61no—
205C2×C3—121no—
206C2×C3—121no—
207C2×C3—1441no—
208C2×C3A4×#6711yescharacteristic
209D3—121no—
210C22—31no—
211C22—61no—
212C22—181no—
213C22—241no—
214C4—21no—
215C4—31no—
216C4—541no—
217C4C2×D9⋊A411yescharacteristic
218C3—121no—
219C3#5836⋊C311yescharacteristic
220C2—61no—
221C2—121no—
222C2—361no—
223C2#611⋊A411yescharacteristic
subgroupG/H
centerC2#611⋊A4
commutator, cosocleC9×#41C22×C3
FrattiniC3×C4C2×D3×A4
FittingC9×H(5,𝔽2)C2×C3
socleC2×C3A4×#67
hypercenterQ8D9⋊A4
hypocenterQ8×C9C3×D4
LCSG ▷ C9×#41 ▷ Q8×C9
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #11722557C2semidirect
2. #11722414C2semidirect
3. #11720752C2semidirect
5. #10589938C3semidirect
13. C9×H(5,𝔽2)C2×C3semidirect
18. #63510C2×C3semidirect
122. H(5,𝔽2)AGL(1,ℤ/9)semidirect
186. C9#577⋊C3semidirect

Maximal subgroups

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

subgroupindex
#117225572
#117224142
#117207522
#5836⋊C33
#105899383
#3675⋊C34