Finite Group Databasebeta

← #13147188

Subgroups — #13147188

Order 1728.

241 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C9×#994C211yesmaximal, characteristic, complemented
2#11722416C211yesmaximal, characteristic, complemented
3#11720753C211yesmaximal, characteristic, complemented
4#5836⋊C3—31nomaximal
5#10589938C311yesmaximal, characteristic, complemented
6C9×#41⋊C3—21no—
7C9×#41⋊C3C2211yescharacteristic
8D9×SL(2,𝔽3)—21no—
9C3×#3675—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—
21C3×Q8×C9—41no—
22C3×D36—41no—
23C9×SL(2,𝔽3)D411yescharacteristic
24C3×C9⋊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×C3×D9—81no—
50C3×C4×C9—41no—
51C3×C4×C9—81no—
52C3×H(5,𝔽2)C3×D311yescharacteristic
53#1010—31no—
542−1+4⋊C3—91no—
55#994—21no—
56#994D911yescharacteristic, complemented
57#951—91no—
58#950—91no—
59#948—31no—
60C2×#214—91no—
61#932—91no—
62#131⋊C3—91no—
63C32×Q8—41no—
64C3×D12—121no—
65C3×SL(2,𝔽3)#6711yescharacteristic
66C23×C9—61no—
67C3×C3⋊C8—121no—
68Q8×C9—81no—
69Q8×C9C3×D411yescharacteristic
70Q8×C9C2×A411yescharacteristic
71D4×C9—31no—
72D4×C9—31no—
73D4×C9—61no—
74D4×C9—61no—
75C2×C4×C9—31no—
76C2×C4×C9—61no—
77#611—61no—
78C2×Dic9—61no—
79D36C2×A411yescharacteristic
80C4×D9—61no—
81Dic18—31no—
82Dic18—61no—
83Q8⋊C9—21no—
84C9⋊C8—31no—
85C9⋊C8C2×A411yescharacteristic
86#577—91noSylow
87C2×C3×C9—41no—
88C3×D9—161no—
89C3×#41—21no—
90C3×#41—31no—
91C3×#41C2×C3×D311yescharacteristic
92C2×C3×D4—31no—
93C2×C3×D4—61no—
94D3×Q8—61no—
95#237—181no—
96#235—91no—
97C2×Dic6—91no—
98#41⋊C3—22no—
99#41⋊C3—21no—
100#41⋊C3—21no—
101#41⋊C3C2×D911yescharacteristic
102C2×SL(2,𝔽3)—181no—
103C3×SD16—361no—
104#216—91no—
105#215—31no—
106#214—91no—
107#214—92no—
108#213—91no—
109#208—91no—
110C2×C3⋊C8—91no—
111C2×C3×D3—241no—
112C32×C4—41no—
113C32×C4—81no—
114C22×C9—31no—
115C22×C9—61no—
116C22×C9—241no—
117C2×D9—21no—
118C4×C9—21no—
119C4×C9—31no—
120C4×C9—82no—
121C4×C9—81no—
122C4×C9C22×A411yescharacteristic
123Dic9—61no—
1242−1+4—91no—
125H(5,𝔽2)C3×D911yescharacteristic, complemented
126#137—271no—
127AGL(1,ℤ/8)—271no—
128#135—271no—
129C2×SD16—271no—
130#131—91no—
131C3×C9—41noSylow
132C23×C3—61no—
133C3×Q8—41no—
134C3×Q8—81no—
135C3×Q8C3×#6711yescharacteristic
136C3×Q8D3×A411yescharacteristic
137C3×D4—31no—
138C3×D4—31no—
139C3×D4—61no—
140C3×D4—61no—
141C3×D4—361no—
142C2×C3×C4—31no—
143C2×C3×C4—61no—
144#67—181no—
145C2×Dic3—181no—
146D12—31no—
147C4×D3—181no—
148Dic6—91no—
149Dic6—181no—
150SL(2,𝔽3)—21no—
151SL(2,𝔽3)#61111yescharacteristic
152C3×C8—361no—
153C3⋊C8—31no—
154C3⋊C8—91no—
155C2×C32—41no—
156C3×D3—481no—
157C2×C9—61no—
158C2×C9—81no—
159C2×C9—121no—
160C2×C9D4×A411yescharacteristic
161D9—41no—
162#41—21no—
163#41—31no—
164#41—271no—
165#41—541no—
166#41C2×C3×D911yescharacteristic
167C2×Q8—181no—
168C2×Q8—271no—
169C2×D4—31no—
170C2×D4—61no—
171Q16—271no—
172SD16—91no—
173SD16—271no—
174SD16—272no—
175D8—271no—
176M16—271no—
177C2×C8—271no—
178C22×C3—31no—
179C22×C3—61no—
180C22×C3—241no—
181C22×C3—721no—
182C2×D3—61no—
183C3×C4—21no—
184C3×C4—31no—
185C3×C4—41no—
186C3×C4—81no—
187C3×C4—82no—
188C3×C4—81no—
189C3×C4C2×D3×A411yescharacteristic
190Dic3—181no—
191C32—41no—
192C9—81no—
193C9#577⋊C311yescharacteristic, complemented
194C23—61no—
195Q8—271no—
196Q8—541no—
197Q8C3×#61111yescharacteristic
198Q8A4×D911yescharacteristic
199D4—31no—
200D4—31no—
201D4—61no—
202D4—61no—
203D4—91no—
204D4—541no—
205C2×C4—31no—
206C2×C4—61no—
207C2×C4—541no—
208C2×C4—541no—
209C8—91no—
210C8—271no—
211C2×C3—41no—
212C2×C3—61no—
213C2×C3—81no—
214C2×C3—121no—
215C2×C3—1441no—
216C2×C3A4×#6711yescharacteristic
217D3—121no—
218C22—31no—
219C22—61no—
220C22—181no—
221C22—241no—
222C4—21no—
223C4—31no—
224C4—541no—
225C4C2×A4×D911yescharacteristic
226C3—41no—
227C3—81no—
228C3#5836⋊C311yescharacteristic
229C2—61no—
230C2—121no—
231C2—361no—
232C2A4×#61111yescharacteristic
subgroupG/H
centerC2A4×#611
commutator, cosocleC9×#41C22×C3
FrattiniC3×C4C2×D3×A4
FittingC9×H(5,𝔽2)C2×C3
socleC2×C3A4×#67
hypercenterQ8A4×D9
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. C9×#994C2semidirect
2. #11722416C2semidirect
3. #11720753C2semidirect
5. #10589938C3semidirect
13. C9×H(5,𝔽2)C2×C3semidirect
18. #63510C2×C3semidirect
56. #994D9semidirect
125. H(5,𝔽2)C3×D9semidirect
193. C9#577⋊C3semidirect

Maximal subgroups

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

subgroupindex
C9×#9942
#117224162
#117207532
#5836⋊C33
#105899383
C3×#36754