Finite Group Databasebeta

← #13143889

Subgroups — #13143889

Order 1728.

520 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 115 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#11721853C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3#11721783C211yesmaximal, characteristic
4#11721782C211yesmaximal, characteristic, complemented
5C3×C9×#101C211yesmaximal, characteristic, complemented
6#11720557C211yesmaximal, characteristic, complemented
7#11720555C211yesmaximal, characteristic, complemented
8#10592453—31nomaximal
9#10588434—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11#88736C2211yescharacteristic
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14C3×C9×M16C2211yescharacteristic
15#88548—21no—
16#88548—21no—
17#88548—21no—
18#88547—21no—
19#88532C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#88525—21no—
22#88524—21no—
23#88523—21no—
24C8×C3⋊D9—21no—
25#64151—31no—
26C2×#3755—31no—
27#64117—31no—
28#64116—31no—
29C32×#101D311yescharacteristic
30#63650—31no—
31#63649—31no—
32#63505—33no—
33C2×#3675—33no—
34#63470—33no—
35#63469—33no—
36C9×#101D313yescomplemented
37#63383—33no—
38#63381—33no—
39C22×C3⋊D9—21no—
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44#6233—21no—
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)D411yescharacteristic
48C4×C3⋊D9—21no—
49#6226D411yescharacteristic
50C3×C8×C9—21no—
51C3×C8×C9—21no—
52C3⋊(C9⋊C8)—21no—
53C3⋊(C9⋊C8)D411yescharacteristic
54C3⋊(C9⋊C8)D411yescharacteristic
55#4754—93no—
56#4754—91no—
57C2×C32×Q8C2×D311yescharacteristic
58#3828—31no—
59C2×C4⋊(C3⋊D3)—31no—
60C32×M16C2×D311yescharacteristic
61C32×M16C2×D311yescharacteristic
62#3756—61no—
63#3755—61no—
64#3755—61no—
65#3755—61no—
66#3748—31no—
67C2×C3⋊(C3⋊C8)—31no—
68#3745—61no—
69#3744—61no—
70#3743—61no—
71C8×C3⋊D3—61no—
72C2×Q8×C9C2×D313yes—
73#3697—33no—
74C2×D36—33no—
75C9×M16C2×D313yes—
76C9×M16C2×D313yes—
77#3675—63no—
78#3675—63no—
79#3675—63no—
80#3674—63no—
81#3667—33no—
82C2×C9⋊C8—33no—
83D72—63no—
84#3664—63no—
85#3663—63no—
86C8×D9—63no—
87C22×C3×C9#4111yescharacteristic
88C2×C3⋊D9—21no—
89C2×C3⋊D9—41no—
90C2×C3⋊D9—81no—
91C3×C4×C9—41no—
92C3×C4×C9C2×D411yescharacteristic
93C3×C4×C9C2×D411yescharacteristic
94C3⋊Dic9—21no—
95#942—93no—
96#942—91no—
97C2×#215—93no—
98C2×#215—91no—
99#908—93no—
100#908—91no—
101#907—93no—
102#907—91no—
103C3×#101D913yescomplemented
104C3×#101C3⋊D311yescharacteristic
105#823—93no—
106#823—91no—
107#822—93no—
108#822—91no—
109C22×C3⋊D3—61no—
110C32×Q8—21no—
111C32×Q8—21no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114#638—61no—
115C4⋊(C3⋊D3)—31no—
116C4⋊(C3⋊D3)—61no—
117C4⋊(C3⋊D3)—61no—
118C4×C3⋊D3—61no—
119#634—31no—
120C22×D9—63no—
121C32×C8—21no—
122C32×C8—21no—
123C3⋊(C3⋊C8)—31no—
124C3⋊(C3⋊C8)—31no—
125C3⋊(C3⋊C8)—61no—
126Q8×C9—23no—
127Q8×C9—23no—
128C2×C4×C9—23no—
129C2×C4×C9C22×D313yes—
130#611—63no—
131D36—33no—
132D36—63no—
133D36—63no—
134C4×D9—63no—
135Dic18—33no—
136C8×C9—23no—
137C8×C9—23no—
138C9⋊C8—33no—
139C9⋊C8—33no—
140C9⋊C8—63no—
141#471—271noSylow
142C2×C3×C9—21no—
143C2×C3×C9#12111yescharacteristic
144C3⋊D9—41no—
145C3⋊D9—81no—
146C2×C3×Q8C2×D913yes—
147C2×C3×Q8C2×C3⋊D311yescharacteristic
148#235—93no—
149#235—91no—
150C2×D12—93no—
151C2×D12—91no—
152C3×M16C2×D913yes—
153C3×M16C2×D913yes—
154C3×M16C2×C3⋊D311yescharacteristic
155C3×M16C2×C3⋊D311yescharacteristic
156#216—181no—
157#216—183no—
158#215—181no—
159#215—181no—
160#215—183no—
161#215—183no—
162#215—181no—
163#215—183no—
164#208—93no—
165#208—91no—
166C2×C3⋊C8—93no—
167C2×C3⋊C8—91no—
168D24—181no—
169D24—183no—
170#204—181no—
171#204—183no—
172ΓL(1,𝔽25)—181no—
173ΓL(1,𝔽25)—183no—
174D3×C8—181no—
175D3×C8—183no—
176C22×C32#23511yescharacteristic
177C2×C3⋊D3—61no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—241no—
180C32×C4—41no—
181C32×C4D3×D411yescharacteristic
182C32×C4D3×D411yescharacteristic
183C3⋊Dic3—61no—
184C22×C9#23513yes—
185C2×D9—63no—
186C2×D9—123no—
187C2×D9—243no—
188C4×C9—43no—
189C4×C9D3×D413yes—
190C4×C9D3×D413yes—
191Dic9—63no—
192#137—271no—
193AGL(1,ℤ/8)—271no—
194C2×SD16—271no—
195#131—271no—
196#108—271no—
197#101C3⋊D911yescharacteristic, complemented
198#100—271no—
199C3×C9#47111yescharacteristic, complemented, Sylow
200C22×D3—183no—
201C22×D3—181no—
202C3×Q8—23no—
203C3×Q8—23no—
204C3×Q8—21no—
205C3×Q8—21no—
206C2×C3×C4—23no—
207C2×C3×C4—21no—
208C2×C3×C4C22×D913yes—
209C2×C3×C4C22×C3⋊D311yescharacteristic
210#67—183no—
211#67—181no—
212D12—93no—
213D12—91no—
214D12—183no—
215D12—183no—
216D12—181no—
217D12—181no—
218C4×D3—183no—
219C4×D3—181no—
220Dic6—93no—
221Dic6—91no—
222C3×C8—23no—
223C3×C8—23no—
224C3×C8—21no—
225C3×C8—21no—
226C3⋊C8—93no—
227C3⋊C8—93no—
228C3⋊C8—91no—
229C3⋊C8—91no—
230C3⋊C8—183no—
231C3⋊C8—181no—
232C2×C32—21no—
233C2×C32#88411yescharacteristic
234C3⋊D3—121no—
235C3⋊D3—241no—
236C2×C9—23no—
237C2×C9#88413yes—
238D9—123no—
239D9—243no—
240#41—271no—
241C2×Q8C2×C3⋊D911yescharacteristic
242C2×D4—271no—
243Q16—541no—
244SD16—541no—
245SD16—541no—
246SD16—541no—
247SD16—541no—
248D8—541no—
249M16—271no—
250M16—541no—
251M16C2×C3⋊D911yescharacteristic
252M16C2×C3⋊D911yescharacteristic
253C2×C8—271no—
254C2×C8—541no—
255C22×C3#369713yes—
256C22×C3#382811yescharacteristic
257C2×D3—181no—
258C2×D3—183no—
259C2×D3—361no—
260C2×D3—363no—
261C2×D3—721no—
262C2×D3—723no—
263C3×C4—41no—
264C3×C4—43no—
265C3×C4D4×D913yes—
266C3×C4D4×D913yes—
267C3×C4D4×C3⋊D311yescharacteristic
268C3×C4D4×C3⋊D311yescharacteristic
269Dic3—181no—
270Dic3—183no—
271C32#475411yescharacteristic
272C9#475413yescomplemented
273C23—541no—
274Q8—21no—
275Q8—21no—
276Q8—271no—
277D4—271no—
278D4—541no—
279D4—541no—
280D4—541no—
281C2×C4—21no—
282C2×C4—541no—
283C2×C4C22×C3⋊D911yescharacteristic
284C8—21no—
285C8—21no—
286C8—271no—
287C8—271no—
288C8—541no—
289C2×C3—23no—
290C2×C3—21no—
291C2×C3#6344613yes—
292C2×C3#6409311yescharacteristic
293D3—363no—
294D3—361no—
295D3—723no—
296D3—721no—
297C22—541no—
298C22—1081no—
299C22—2161no—
300C22#8873611yescharacteristic
301C4—41no—
302C4—541no—
303C4D4×C3⋊D911yescharacteristic
304C4D4×C3⋊D911yescharacteristic
305C3#1058843413yescomplemented
306C3#1059245311yescharacteristic
307C2—21no—
308C2—1081no—
309C2—2161no—
310C2#1172173011yescharacteristic
subgroupG/H
centerC2#11721730
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#101C2
socleC2×C32#884
prime radical#11721783C2
hypercenter#101C3⋊D9
hypocenterC3×C9#471
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #11721853C2semidirect
2. C2×#88548C2semidirect
4. #11721782C2semidirect
5. C3×C9×#101C2semidirect
6. #11720557C2semidirect
7. #11720555C2semidirect
36. C9×#101D3semidirect
103. C3×#101D9semidirect
197. #101C3⋊D9semidirect
199. C3×C9#471semidirect
272. C9#4754semidirect
305. C3#10588434semidirect

Maximal subgroups

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

subgroupindex
#117218532
C2×#885482
#117217832
#117217822
C3×C9×#1012
#117205572
#117205552
#105924533
#105884343