Finite Group Databasebeta

← #13147192

Subgroups — #13147192

Order 1728.

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

normal subgroups only

This group has 245 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
1C9×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#11722416C211yesmaximal, characteristic, complemented
3#11720753C211yesmaximal, characteristic, complemented
4#5838⋊C3—31nomaximal
5#10589940C311yesmaximal, characteristic, complemented
6C9×#41⋊C3C2211yescharacteristic
7C2×C9×SL(2,𝔽3)—21no—
8D9×SL(2,𝔽3)—21no—
9C3×#3673—41nomaximal
10C3×2−1+4⋊C3D311yescharacteristic
11#1010⋊C3—31no—
12C3⋊(#131⋊C3)—31no—
13C9×2−1+4C2×C311yescharacteristic, complemented
14#63716C2×C311yescharacteristic, complemented
152−1+4⋊C9—21no—
16#63513—31no—
17#63511—31no—
18#63510C2×C311yescharacteristic, complemented
19#63505—31no—
20C2×#3674—31no—
21C3×D4×C9—41no—
22C3×D36—41no—
23C9×SL(2,𝔽3)D411yescharacteristic
24C3×C9⋊C8—41no—
25#5838—31no—
26#578⋊C3—91no—
27C3×#41⋊C3C2×D311yescharacteristic
28C2×C3×SL(2,𝔽3)—21no—
29D3×SL(2,𝔽3)—61no—
30C3×#213—121no—
31C9×#41—31no—
32C9×#41—61no—
33C9×#41C22×C311yescharacteristic
34C2×Q8×C9—21no—
35C2×Q8×C9—31no—
36Q8×D9—21no—
37#3699—61no—
38#3697—31no—
39C2×Dic18—31no—
40#41⋊C9—21no—
41C2×Q8⋊C9—22no—
42#3675—31no—
43#3674—32no—
44#3674—31no—
45#3673A411yescharacteristic
46#3672—31no—
47#3667—31no—
48C2×C9⋊C8—31no—
49C22×C3×C9—81no—
50C2×C3×D9—81no—
51C3×C4×C9—41no—
52C3×2−1+4C3×D311yescharacteristic
53#1010—31no—
542−1+4⋊C3—21no—
552−1+4⋊C3—91no—
562−1+4⋊C3D911yescharacteristic, complemented
57#951—91no—
58#950—91no—
59#948—31no—
60C2×#216—91no—
61#942—91no—
62#131⋊C3—91no—
63C32×D4—41no—
64C3×D12—121no—
65C3×SL(2,𝔽3)#6711yescharacteristic
66C3×C3⋊C8—121no—
67Q8×C9—31no—
68Q8×C9—61no—
69Q8×C9C3×D411yescharacteristic
70D4×C9—31no—
71D4×C9—61no—
72D4×C9—81no—
73D4×C9C2×A411yescharacteristic
74C2×C4×C9—31no—
75C2×C4×C9—61no—
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#578—91noSylow
87C2×C3×C9—41no—
88C2×C3×C9—161no—
89C3×D9—161no—
90C3×#41—31no—
91C3×#41—61no—
92C3×#41C2×C3×D311yescharacteristic
93C2×C3×Q8—21no—
94C2×C3×Q8—31no—
95D3×Q8—61no—
96#237—181no—
97#235—91no—
98C2×Dic6—91no—
99#41⋊C3—21no—
100#41⋊C3C2×D911yescharacteristic
101C2×SL(2,𝔽3)—22no—
102C2×SL(2,𝔽3)—21no—
103C2×SL(2,𝔽3)—181no—
104C3×D8—361no—
105#216—91no—
106#216—92no—
107#215—91no—
108#214—91no—
109#213—31no—
110#208—91no—
111C2×C3⋊C8—91no—
112C22×C32—81no—
113C2×C3×D3—241no—
114C32×C4—41no—
115C22×C9—21no—
116C22×C9—31no—
117C22×C9—82no—
118C2×D9—21no—
119C4×C9—31no—
120C4×C9—61no—
121C4×C9—81no—
122C4×C9C22×A411yescharacteristic
123Dic9—61no—
1242−1+4—91no—
1252−1+4C3×D911yescharacteristic, complemented
126#137—271no—
127#137—271no—
128#135—271no—
129C2×Q16—271no—
130#131—91no—
131C3×C9—41noSylow
132C3×Q8—31no—
133C3×Q8—61no—
134C3×Q8C3×#6711yescharacteristic
135C3×D4—31no—
136C3×D4—41no—
137C3×D4—61no—
138C3×D4—81no—
139C3×D4—361no—
140C3×D4D3×A411yescharacteristic
141C2×C3×C4—31no—
142C2×C3×C4—61no—
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—
156C2×C32—161no—
157C3×D3—481no—
158C2×C9—41no—
159C2×C9—61no—
160C2×C9—81no—
161C2×C9—162no—
162C2×C9D4×A411yescharacteristic
163D9—41no—
164#41—31no—
165#41—61no—
166#41—271no—
167#41—541no—
168#41C2×C3×D911yescharacteristic
169C2×Q8—21no—
170C2×Q8—31no—
171C2×Q8—181no—
172C2×Q8—271no—
173Q16—271no—
174Q16—272no—
175SD16—271no—
176SD16—271no—
177D8—91no—
178M16—271no—
179C2×C8—271no—
180C22×C3—21no—
181C22×C3—31no—
182C22×C3—81no—
183C22×C3—82no—
184C22×C3—721no—
185C2×D3—61no—
186C3×C4—31no—
187C3×C4—41no—
188C3×C4—61no—
189C3×C4—81no—
190C3×C4C2×D3×A411yescharacteristic
191Dic3—181no—
192C32—41no—
193C9—81no—
194C9#578⋊C311yescharacteristic, complemented
195Q8—31no—
196Q8—61no—
197Q8—271no—
198Q8—541no—
199Q8C3×#61111yescharacteristic
200D4—31no—
201D4—61no—
202D4—91no—
203D4—541no—
204D4A4×D911yescharacteristic
205C2×C4—31no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—541no—
209C2×C4—541no—
210C8—91no—
211C8—271no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—61no—
215C2×C3—81no—
216C2×C3—161no—
217C2×C3—162no—
218C2×C3—1441no—
219C2×C3A4×#6711yescharacteristic
220D3—121no—
221C22—21no—
222C22—31no—
223C22—181no—
224C4—31no—
225C4—61no—
226C4—541no—
227C4C2×A4×D911yescharacteristic
228C3—41no—
229C3—81no—
230C3#5838⋊C311yescharacteristic
231C2—41no—
232C2—61no—
233C2—361no—
234C2A4×#61111yescharacteristic
subgroupG/H
centerC2A4×#611
commutator, cosocleC9×#41C22×C3
FrattiniC3×C4C2×D3×A4
FittingC9×2−1+4C2×C3
socleC2×C3A4×#67
hypercenterD4A4×D9
hypocenterQ8×C9C3×D4
LCSG ▷ C9×#41 ▷ Q8×C9
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C9×2−1+4⋊C3C2semidirect
2. #11722416C2semidirect
3. #11720753C2semidirect
5. #10589940C3semidirect
13. C9×2−1+4C2×C3semidirect
14. #63716C2×C3semidirect
18. #63510C2×C3semidirect
56. 2−1+4⋊C3D9semidirect
125. 2−1+4C3×D9semidirect
194. C9#578⋊C3semidirect

Maximal subgroups

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

subgroupindex
C9×2−1+4⋊C32
#117224162
#117207532
#5838⋊C33
#105899403
C3×#36734