Finite Group Databasebeta

← #13144098

Subgroups — #13144098

Order 1728.

590 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#88523C211yesmaximal, characteristic
2#11721748C211yesmaximal, characteristic
3C4⋊C4×C3⋊D9C211yesmaximal, characteristic
4C3×C9×#106C211yesmaximal, characteristic, complemented
5#11720541C211yesmaximal, characteristic, complemented
6#11720526C211yesmaximal, characteristic, complemented
7#11720525C211yesmaximal, characteristic, complemented
8#10592560—31nomaximal
9#10588541—33nomaximal
10C2×C4×C3⋊D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88536—21no—
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18#88534—21no—
19C4×C3⋊Dic9—21no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#88523C414yescomplemented
22C2×#3743—31no—
23#64101—31no—
24C4⋊C4×C3⋊D3—31no—
25C32×#106D311yescharacteristic
26#63645—31no—
27#63635—31no—
28#63634—31no—
29C2×#3663—33no—
30#63454—33no—
31C4⋊C4×D9—33no—
32C9×#106D313yescomplemented
33#63378—33no—
34#63367—33no—
35#63366—33no—
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9D411yescharacteristic
43C4×C3⋊D9—42no—
44C4×C3⋊D9C2×C412yes—
45C4×C3⋊D9Q812yes—
46C3×C8×C9C2×C412yescomplemented
47C3⋊(C9⋊C8)C2×C412yescomplemented
48#4861—93no—
49#4861—91no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—61no—
52C2×C32×C8C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55#3752—61no—
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—31no—
58#3750—61no—
59C4×C3⋊Dic3—61no—
60C2×C3⋊(C3⋊C8)—31no—
61#3743—34no—
62C2×C4×D9—33no—
63C2×C4×D9—63no—
64C2×C8×C9C2×D313yes—
65C9×C4⋊C4C2×D313yes—
66C9×C4⋊C4C2×D313yes—
67(C2×D9)⋊C4—63no—
68C4⋊Dic9—33no—
69C4⋊Dic9—33no—
70Dic9⋊C4—63no—
71C4×Dic9—63no—
72C2×C9⋊C8—33no—
73#3663—312no—
74C22×C3×C9C2×D411yescharacteristic
75C2×C3⋊D9—21no—
76C2×C3⋊D9—21no—
77C2×C3⋊D9C4⋊C412yes—
78C3×C4×C9—41no—
79C3×C4×C9—41no—
80C3×C4×C9C22×C411yescharacteristic
81C3×C4×C9C2×Q811yescharacteristic
82C3⋊Dic9—41no—
83C3⋊Dic9—41no—
84C3⋊Dic9C4⋊C412yes—
85C2×ΓL(1,𝔽25)—93no—
86C2×ΓL(1,𝔽25)—91no—
87#892—93no—
88#892—91no—
89D3×C4⋊C4—93no—
90D3×C4⋊C4—91no—
91C3×#106D913yescomplemented
92C3×#106C3⋊D311yescharacteristic
93#818—93no—
94#818—91no—
95#808—93no—
96#808—91no—
97#807—93no—
98#807—91no—
99C22×C3⋊D3—31no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—61no—
106C4×C3⋊D3—32no—
107C4×C3⋊D3—32no—
108C4×C3⋊D3—122no—
109C22×D9—33no—
110C32×C8C4×D312yes—
111C3⋊(C3⋊C8)—32no—
112C2×C4×C9—23no—
113C2×C4×C9—23no—
114C2×C4×C9C22×D313yes—
115C2×Dic9—33no—
116C2×Dic9—63no—
117C2×Dic9—63no—
118C4×D9—36no—
119C4×D9—36no—
120C4×D9—126no—
121C8×C9C4×D316yescomplemented
122C9⋊C8—36no—
123#428—271noSylow
124C2×C3×C9C2×C4⋊C411yescharacteristic
125C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
126C2×C3×C9#13711yescharacteristic
127C3⋊D9—22no—
128C2×C4×D3—93no—
129C2×C4×D3—91no—
130C2×C4×D3—181no—
131C2×C4×D3—183no—
132C2×C3×C8C2×D913yes—
133C2×C3×C8C2×C3⋊D311yescharacteristic
134C3×C4⋊C4C2×D913yes—
135C3×C4⋊C4C2×D913yes—
136C3×C4⋊C4C2×C3⋊D311yescharacteristic
137C3×C4⋊C4C2×C3⋊D311yescharacteristic
138(C2×D3)⋊C4—181no—
139(C2×D3)⋊C4—183no—
140C4⋊Dic3—93no—
141C4⋊Dic3—93no—
142C4⋊Dic3—91no—
143C4⋊Dic3—91no—
144Dic3⋊C4—181no—
145Dic3⋊C4—183no—
146C4×Dic3—181no—
147C4×Dic3—183no—
148C2×C3⋊C8—93no—
149C2×C3⋊C8—91no—
150ΓL(1,𝔽25)—912no—
151ΓL(1,𝔽25)—94no—
152C22×C32D3×D411yescharacteristic
153C2×C3⋊D3—32no—
154C2×C3⋊D3—61no—
155C2×C3⋊D3—61no—
156C32×C4—41no—
157C32×C4—41no—
158C32×C4C2×C4×D311yescharacteristic
159C32×C4D3×Q811yescharacteristic
160C3⋊Dic3—32no—
161C3⋊Dic3—121no—
162C3⋊Dic3—121no—
163C22×C9D3×D413yes—
164C2×D9—36no—
165C2×D9—63no—
166C2×D9—63no—
167C4×C9—43no—
168C4×C9—43no—
169C4×C9C2×C4×D313yes—
170C4×C9D3×Q813yes—
171Dic9—36no—
172Dic9—123no—
173Dic9—123no—
174C2×M16—271no—
175#117—271no—
176C2×C4⋊C4—271no—
177#107—271no—
178#107—271no—
179#106—271no—
180#106C3⋊D911yescharacteristic, complemented
181C3×C9#42811yescharacteristic, complemented, Sylow
182C22×D3—93no—
183C22×D3—91no—
184C2×C3×C4—23no—
185C2×C3×C4—23no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4C22×D913yes—
189C2×C3×C4C22×C3⋊D311yescharacteristic
190C2×Dic3—93no—
191C2×Dic3—91no—
192C2×Dic3—183no—
193C2×Dic3—183no—
194C2×Dic3—181no—
195C2×Dic3—181no—
196C4×D3—96no—
197C4×D3—96no—
198C4×D3—92no—
199C4×D3—92no—
200C4×D3—362no—
201C4×D3—366no—
202C3×C8C4×D916yescomplemented
203C3×C8C4×C3⋊D312yes—
204C3⋊C8—96no—
205C3⋊C8—92no—
206C2×C32D3×C4⋊C411yescharacteristic
207C2×C32#91411yescharacteristic
208C2×C32#91511yescharacteristic
209C3⋊D3—62no—
210C2×C9D3×C4⋊C413yes—
211C2×C9#91413yes—
212C2×C9#91513yes—
213D9—66no—
214C22×C4—271no—
215C22×C4—541no—
216M16—274no—
217C2×C8—271no—
218C2×C8C2×C3⋊D911yescharacteristic
219C4⋊C4—271no—
220C4⋊C4—271no—
221C4⋊C4—541no—
222C4⋊C4C2×C3⋊D911yescharacteristic
223C4⋊C4C2×C3⋊D911yescharacteristic
224C22⋊C4—541no—
225C42—541no—
226C22×C3D4×D913yes—
227C22×C3D4×C3⋊D311yescharacteristic
228C2×D3—92no—
229C2×D3—96no—
230C2×D3—181no—
231C2×D3—181no—
232C2×D3—183no—
233C2×D3—183no—
234C3×C4—41no—
235C3×C4—41no—
236C3×C4—43no—
237C3×C4—43no—
238C3×C4C2×C4×D913yes—
239C3×C4Q8×D913yes—
240C3×C4C2×C4×C3⋊D311yescharacteristic
241C3×C4Q8×C3⋊D311yescharacteristic
242Dic3—92no—
243Dic3—96no—
244Dic3—361no—
245Dic3—361no—
246Dic3—363no—
247Dic3—363no—
248C32#486111yescharacteristic
249C9#486113yescomplemented
250C23—271no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—271no—
254C2×C4—272no—
255C2×C4—272no—
256C2×C4—541no—
257C2×C4—541no—
258C2×C4—1082no—
259C2×C4C22×C3⋊D911yescharacteristic
260C8—272no—
261C8C4×C3⋊D912yescomplemented
262C2×C3C4⋊C4×D913yes—
263C2×C3#6347613yes—
264C2×C3#6347713yes—
265C2×C3C4⋊C4×C3⋊D311yescharacteristic
266C2×C3#6412311yescharacteristic
267C2×C3#6412411yescharacteristic
268D3—182no—
269D3—186no—
270C22—272no—
271C22—541no—
272C22—541no—
273C22D4×C3⋊D911yescharacteristic
274C4—41no—
275C4—41no—
276C4—272no—
277C4—1081no—
278C4—1081no—
279C4C2×C4×C3⋊D911yescharacteristic
280C4Q8×C3⋊D911yescharacteristic
281C3#1058854113yescomplemented
282C3#1059256011yescharacteristic
283C2—542no—
284C2C4⋊C4×C3⋊D911yescharacteristic
285C2#1172179611yescharacteristic
286C2#1172179711yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenter#106C3⋊D9
hypocenterC3×C9#428
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
4. C3×C9×#106C2semidirect
5. #11720541C2semidirect
6. #11720526C2semidirect
7. #11720525C2semidirect
21. #88523C4semidirect
32. C9×#106D3semidirect
46. C3×C8×C9C2×C4semidirect
47. C3⋊(C9⋊C8)C2×C4semidirect
91. C3×#106D9semidirect
121. C8×C9C4×D3semidirect
180. #106C3⋊D9semidirect
181. C3×C9#428semidirect
202. C3×C8C4×D9semidirect
249. C9#4861semidirect
261. C8C4×C3⋊D9semidirect
281. C3#10588541semidirect

Maximal subgroups

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

subgroupindex
C2×#885232
#117217482
C4⋊C4×C3⋊D92
C3×C9×#1062
#117205412
#117205262
#117205252
#105925603
#105885413