Finite Group Databasebeta

← #13144100

Subgroups — #13144100

Order 1728.

570 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 135 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#11721751C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×C9×#106C211yesmaximal, characteristic, complemented
4#11720543C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic, complemented
6#11720528C211yesmaximal, characteristic, complemented
7#11720526C211yesmaximal, characteristic, complemented
8#10592562—31nomaximal
9#10588543—33nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9—21no—
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C4⋊C4C2211yescharacteristic
17#88536—21no—
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19#88534—21no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64104—31no—
22C4⋊C4×C3⋊D3—31no—
23C32×#106D311yescharacteristic
24#63648—31no—
25#63647—31no—
26#63636—31no—
27#63635—31no—
28#63457—33no—
29C4⋊C4×D9—33no—
30C9×#106D313yescomplemented
31#63380—33no—
32(C2×D9)⋊C8—33no—
33#63369—33no—
34#63367—33no—
35C22×C3⋊D9—21no—
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—42no—
45C4×C3⋊D9—21no—
46C4×C3⋊D9—21no—
47C4×C3⋊D9—42no—
48C3×C8×C9—21no—
49C3⋊(C9⋊C8)—21no—
50#4863—93no—
51#4863—91no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4⋊(C3⋊D3)—61no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—61no—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59#3752—61no—
60C4⋊(C3⋊Dic3)—31no—
61#3750—61no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×D36—33no—
64C2×D36—63no—
65C2×C4×D9—33no—
66C2×C4×D9—63no—
67C2×C8×C9C2×D313yes—
68C9×C4⋊C4C2×D313yes—
69C9×C4⋊C4C2×D313yes—
70(C2×D9)⋊C4—63no—
71C4⋊Dic9—33no—
72Dic9⋊C4—63no—
73C2×C9⋊C8—33no—
74C22×C3×C9C2×D411yescharacteristic
75C2×C3⋊D9—21no—
76C2×C3⋊D9—21no—
77C2×C3⋊D9—41no—
78C2×C3⋊D9—41no—
79C2×C3⋊D9—41no—
80C2×C3⋊D9SD1612yes—
81C3×C4×C9—41no—
82C3×C4×C9—41no—
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—41no—
87#895—93no—
88#895—91no—
89D3×C4⋊C4—93no—
90D3×C4⋊C4—91no—
91C3×#106D913yescomplemented
92C3×#106C3⋊D311yescharacteristic
93#821—93no—
94#821—91no—
95(C2×D3)⋊C8—93no—
96(C2×D3)⋊C8—91no—
97#809—93no—
98#809—91no—
99#808—93no—
100#808—91no—
101C22×C3⋊D3—31no—
102C22×C3⋊D3—61no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—61no—
108C4⋊(C3⋊D3)—61no—
109C4⋊(C3⋊D3)—61no—
110C4⋊(C3⋊D3)—122no—
111C4×C3⋊D3—61no—
112C4×C3⋊D3—61no—
113C4×C3⋊D3—122no—
114C22×D9—33no—
115C22×D9—63no—
116C32×C8—21no—
117C3⋊(C3⋊C8)—61no—
118C2×C4×C9—23no—
119C2×C4×C9—23no—
120C2×C4×C9C22×D313yes—
121C2×Dic9—33no—
122C2×Dic9—63no—
123D36—63no—
124D36—63no—
125D36—126no—
126C4×D9—63no—
127C4×D9—63no—
128C4×D9—126no—
129C8×C9—23no—
130C9⋊C8—63no—
131#481—271noSylow
132C2×C3×C9#12311yescharacteristic
133C2×C3×C9C2×SD1611yescharacteristic
134C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
135C3⋊D9—22no—
136C3⋊D9—81no—
137C2×D12—93no—
138C2×D12—91no—
139C2×D12—181no—
140C2×D12—183no—
141C2×C4×D3—93no—
142C2×C4×D3—91no—
143C2×C4×D3—181no—
144C2×C4×D3—183no—
145C2×C3×C8C2×D913yes—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4C2×D913yes—
148C3×C4⋊C4C2×D913yes—
149C3×C4⋊C4C2×C3⋊D311yescharacteristic
150C3×C4⋊C4C2×C3⋊D311yescharacteristic
151(C2×D3)⋊C4—181no—
152(C2×D3)⋊C4—183no—
153C4⋊Dic3—93no—
154C4⋊Dic3—91no—
155Dic3⋊C4—181no—
156Dic3⋊C4—183no—
157C2×C3⋊C8—93no—
158C2×C3⋊C8—91no—
159C22×C32D3×D411yescharacteristic
160C2×C3⋊D3—32no—
161C2×C3⋊D3—61no—
162C2×C3⋊D3—61no—
163C2×C3⋊D3—121no—
164C2×C3⋊D3—121no—
165C2×C3⋊D3—121no—
166C32×C4—41no—
167C32×C4—41no—
168C32×C4#23511yescharacteristic
169C32×C4#23911yescharacteristic
170C3⋊Dic3—61no—
171C3⋊Dic3—121no—
172C22×C9D3×D413yes—
173C2×D9—36no—
174C2×D9—63no—
175C2×D9—63no—
176C2×D9—123no—
177C2×D9—123no—
178C2×D9—123no—
179C4×C9—43no—
180C4×C9—43no—
181C4×C9#23513yes—
182C4×C9#23913yes—
183Dic9—63no—
184Dic9—123no—
185#121—271no—
186C2×C4⋊C4—271no—
187#106—271no—
188#106C3⋊D911yescharacteristic, complemented
189#102—271no—
190#102—271no—
191C22⋊C8—271no—
192C3×C9#48111yescharacteristic, complemented, Sylow
193C22×D3—93no—
194C22×D3—91no—
195C22×D3—183no—
196C22×D3—181no—
197C2×C3×C4—23no—
198C2×C3×C4—23no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4C22×D913yes—
202C2×C3×C4C22×C3⋊D311yescharacteristic
203C2×Dic3—93no—
204C2×Dic3—91no—
205C2×Dic3—183no—
206C2×Dic3—181no—
207D12—183no—
208D12—183no—
209D12—181no—
210D12—181no—
211D12—362no—
212D12—366no—
213C4×D3—183no—
214C4×D3—183no—
215C4×D3—181no—
216C4×D3—181no—
217C4×D3—362no—
218C4×D3—366no—
219C3×C8—23no—
220C3×C8—21no—
221C3⋊C8—183no—
222C3⋊C8—181no—
223C2×C32#89411yescharacteristic
224C2×C32D3×SD1611yescharacteristic
225C2×C32#91411yescharacteristic
226C3⋊D3—62no—
227C3⋊D3—241no—
228C2×C9#89413yes—
229C2×C9D3×SD1613yes—
230C2×C9#91413yes—
231D9—66no—
232D9—243no—
233C2×D4—271no—
234C2×D4—541no—
235C22×C4—271no—
236C22×C4—541no—
237C2×C8—271no—
238C2×C8C2×C3⋊D911yescharacteristic
239C4⋊C4—271no—
240C4⋊C4—541no—
241C4⋊C4C2×C3⋊D911yescharacteristic
242C4⋊C4C2×C3⋊D911yescharacteristic
243C22⋊C4—541no—
244C22×C3D4×D913yes—
245C22×C3D4×C3⋊D311yescharacteristic
246C2×D3—92no—
247C2×D3—96no—
248C2×D3—181no—
249C2×D3—181no—
250C2×D3—183no—
251C2×D3—183no—
252C2×D3—361no—
253C2×D3—361no—
254C2×D3—361no—
255C2×D3—363no—
256C2×D3—363no—
257C2×D3—363no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—43no—
261C3×C4—43no—
262C3×C4#369713yes—
263C3×C4#370113yes—
264C3×C4#382811yescharacteristic
265C3×C4#383211yescharacteristic
266Dic3—181no—
267Dic3—183no—
268Dic3—361no—
269Dic3—363no—
270C32#486311yescharacteristic
271C9#486313yescomplemented
272C23—271no—
273C23—541no—
274D4—541no—
275D4—541no—
276D4—1082no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—271no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4—541no—
283C2×C4—1082no—
284C2×C4C22×C3⋊D911yescharacteristic
285C8—21no—
286C8—541no—
287C2×C3#6345613yes—
288C2×C3SD16×D913yes—
289C2×C3#6347613yes—
290C2×C3#6410311yescharacteristic
291C2×C3SD16×C3⋊D311yescharacteristic
292C2×C3#6412311yescharacteristic
293D3—182no—
294D3—186no—
295D3—721no—
296D3—723no—
297C22—272no—
298C22—541no—
299C22—541no—
300C22—1081no—
301C22—1081no—
302C22—1081no—
303C22D4×C3⋊D911yescharacteristic
304C4—41no—
305C4—41no—
306C4—541no—
307C4—1081no—
308C4#8873611yescharacteristic
309C4#8874611yescharacteristic
310C3#1058854313yescomplemented
311C3#1059256211yescharacteristic
312C2—542no—
313C2—2161no—
314C2#1172175011yescharacteristic
315C2SD16×C3⋊D911yescharacteristic
316C2#1172179611yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socleC22×C32D3×D4
prime radical#11721751C2
hypercenter#106C3⋊D9
hypocenterC3×C9#481
LCSG ▷ C2×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
2. C4⋊C4×C3⋊D9C2semidirect
3. C3×C9×#106C2semidirect
4. #11720543C2semidirect
5. #11720542C2semidirect
6. #11720528C2semidirect
7. #11720526C2semidirect
30. C9×#106D3semidirect
91. C3×#106D9semidirect
188. #106C3⋊D9semidirect
192. C3×C9#481semidirect
271. C9#4863semidirect
310. C3#10588543semidirect

Maximal subgroups

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

subgroupindex
#117217512
C4⋊C4×C3⋊D92
C3×C9×#1062
#117205432
#117205422
#117205282
#117205262
#105925623
#105885433