Finite Group Databasebeta

← #13143988

Subgroups — #13143988

Order 1728.

600 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 161 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
1Q8×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic
3#11721749C211yesmaximal, characteristic
4C3×C9×#103C211yesmaximal, characteristic, complemented
5#11720543C211yesmaximal, characteristic
6#11720538C211yesmaximal, characteristic, complemented
7#11720525C211yesmaximal, characteristic, complemented
8#10592516—31nomaximal
9#10588497—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88548C414yescomplemented
16#88536—21no—
17C4⋊(C3⋊Dic9)—21no—
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C4×C3⋊Dic9—21no—
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22Q8×C3⋊Dic3—31no—
23C2×#3755—31no—
24#64102—31no—
25C32×#103D311yescharacteristic
26#63648—31no—
27#63642—31no—
28#63634—31no—
29Q8×Dic9—33no—
30C2×#3675—33no—
31#63455—33no—
32C9×#103D313yescomplemented
33#63380—33no—
34#63375—33no—
35#63366—33no—
36C22×C3⋊D9—21no—
37C3×Q8×C9—21no—
38C3×Q8×C9C2×C412yescomplemented
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)C2×C412yes—
47C4×C3⋊D9—42no—
48C3×C8×C9—21no—
49C3⋊(C9⋊C8)C2×C412yescomplemented
50#4817—93no—
51#4817—91no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4×C3⋊D3—61no—
55C2×C32×C8C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57#3755—34no—
58#3752—61no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—61no—
61C4×C3⋊Dic3—31no—
62C4×C3⋊Dic3—61no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×Q8×C9C2×D313yes—
65C2×D36—33no—
66C2×C4×D9—63no—
67C2×C8×C9C2×D313yes—
68C9×C4⋊C4C2×D313yes—
69#3675—312no—
70(C2×D9)⋊C4—63no—
71C4⋊Dic9—33no—
72C4⋊Dic9—63no—
73C4×Dic9—33no—
74C4×Dic9—63no—
75C2×C9⋊C8—33no—
76C22×C3×C9C2×D411yescharacteristic
77C2×C3⋊D9—22no—
78C2×C3⋊D9—41no—
79C2×C3⋊D9—41no—
80C3×C4×C9—22no—
81C3×C4×C9—41no—
82C3×C4×C9C22×C411yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—21no—
85C3⋊Dic9—21no—
86C3⋊Dic9—41no—
87Q8×Dic3—93no—
88Q8×Dic3—91no—
89C2×#215—93no—
90C2×#215—91no—
91#893—93no—
92#893—91no—
93C3×#103D913yescomplemented
94C3×#103C3⋊D311yescharacteristic
95#821—93no—
96#821—91no—
97#815—93no—
98#815—91no—
99#807—93no—
100#807—91no—
101C22×C3⋊D3—61no—
102C32×Q8—21no—
103C32×Q8C4×D312yes—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110C4⋊(C3⋊D3)—32no—
111C4⋊(C3⋊D3)—61no—
112C4×C3⋊D3—122no—
113C22×D9—63no—
114C32×C8—21no—
115C3⋊(C3⋊C8)—32no—
116Q8×C9—23no—
117Q8×C9C4×D316yescomplemented
118C2×C4×C9—23no—
119C2×C4×C9—23no—
120C2×C4×C9C22×D313yes—
121C2×Dic9—33no—
122C2×Dic9—33no—
123C2×Dic9—63no—
124D36—36no—
125D36—63no—
126C4×D9—126no—
127C8×C9—23no—
128C9⋊C8—36no—
129#440—271noSylow
130C2×C3×C9C4×D411yescharacteristic
131C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
132C2×C3×C9#13711yescharacteristic
133C3⋊D9—42no—
134C2×C3×Q8C2×D913yes—
135C2×C3×Q8C2×C3⋊D311yescharacteristic
136C2×D12—93no—
137C2×D12—91no—
138C2×C4×D3—181no—
139C2×C4×D3—183no—
140C2×C3×C8C2×D913yes—
141C2×C3×C8C2×C3⋊D311yescharacteristic
142C3×C4⋊C4C2×D913yes—
143C3×C4⋊C4C2×C3⋊D311yescharacteristic
144#215—912no—
145#215—94no—
146(C2×D3)⋊C4—181no—
147(C2×D3)⋊C4—183no—
148C4⋊Dic3—93no—
149C4⋊Dic3—91no—
150C4⋊Dic3—181no—
151C4⋊Dic3—183no—
152C4×Dic3—93no—
153C4×Dic3—91no—
154C4×Dic3—181no—
155C4×Dic3—183no—
156C2×C3⋊C8—93no—
157C2×C3⋊C8—91no—
158C22×C32D3×D411yescharacteristic
159C2×C3⋊D3—62no—
160C2×C3⋊D3—121no—
161C2×C3⋊D3—121no—
162C32×C4—22no—
163C32×C4—41no—
164C32×C4C2×C4×D311yescharacteristic
165C32×C4#23711yescharacteristic
166C3⋊Dic3—61no—
167C3⋊Dic3—61no—
168C3⋊Dic3—121no—
169C22×C9D3×D413yes—
170C2×D9—66no—
171C2×D9—123no—
172C2×D9—123no—
173C4×C9—26no—
174C4×C9—43no—
175C4×C9C2×C4×D313yes—
176C4×C9#23713yes—
177Dic9—63no—
178Dic9—63no—
179Dic9—123no—
180C2×SD16—271no—
181C4×Q8—271no—
182C4×D4—271no—
183#107—271no—
184#103C3⋊D911yescharacteristic, complemented
185#102—271no—
186#97—271no—
187C3×C9#44011yescharacteristic, complemented, Sylow
188C22×D3—183no—
189C22×D3—181no—
190C3×Q8—23no—
191C3×Q8—21no—
192C3×Q8C4×D916yescomplemented
193C3×Q8C4×C3⋊D312yes—
194C2×C3×C4—23no—
195C2×C3×C4—21no—
196C2×C3×C4—23no—
197C2×C3×C4—21no—
198C2×C3×C4C22×D913yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—93no—
201C2×Dic3—93no—
202C2×Dic3—91no—
203C2×Dic3—91no—
204C2×Dic3—183no—
205C2×Dic3—181no—
206D12—96no—
207D12—92no—
208D12—183no—
209D12—181no—
210C4×D3—362no—
211C4×D3—366no—
212C3×C8—23no—
213C3×C8—21no—
214C3⋊C8—96no—
215C3⋊C8—92no—
216C2×C32#67⋊C411yescharacteristic
217C2×C32#91411yescharacteristic
218C2×C32#91811yescharacteristic
219C3⋊D3—122no—
220C2×C9#67⋊C413yes—
221C2×C9#91413yes—
222C2×C9#91813yes—
223D9—126no—
224C2×Q8C2×C3⋊D911yescharacteristic
225C2×D4—271no—
226C22×C4—541no—
227SD16—274no—
228C2×C8—271no—
229C2×C8C2×C3⋊D911yescharacteristic
230C4⋊C4—271no—
231C4⋊C4—541no—
232C4⋊C4C2×C3⋊D911yescharacteristic
233C22⋊C4—541no—
234C42—271no—
235C42—541no—
236C22×C3D4×D913yes—
237C22×C3D4×C3⋊D311yescharacteristic
238C2×D3—182no—
239C2×D3—186no—
240C2×D3—361no—
241C2×D3—361no—
242C2×D3—363no—
243C2×D3—363no—
244C3×C4—22no—
245C3×C4—26no—
246C3×C4—41no—
247C3×C4—43no—
248C3×C4C2×C4×D913yes—
249C3×C4#369913yes—
250C3×C4C2×C4×C3⋊D311yescharacteristic
251C3×C4#383011yescharacteristic
252Dic3—181no—
253Dic3—181no—
254Dic3—183no—
255Dic3—183no—
256Dic3—361no—
257Dic3—363no—
258C32#481711yescharacteristic
259C9#481713yescomplemented
260C23—541no—
261Q8—21no—
262Q8C4×C3⋊D912yescomplemented
263D4—272no—
264D4—541no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—271no—
268C2×C4—271no—
269C2×C4—541no—
270C2×C4—1082no—
271C2×C4C22×C3⋊D911yescharacteristic
272C8—21no—
273C8—272no—
274C2×C3#611⋊C413yes—
275C2×C3#6347613yes—
276C2×C3#6348013yes—
277C2×C3#6409011yescharacteristic
278C2×C3#6412311yescharacteristic
279C2×C3#6412711yescharacteristic
280D3—362no—
281D3—366no—
282C22—542no—
283C22—1081no—
284C22—1081no—
285C22D4×C3⋊D911yescharacteristic
286C4—22no—
287C4—41no—
288C4—541no—
289C4—541no—
290C4—1081no—
291C4C2×C4×C3⋊D911yescharacteristic
292C4#8874211yescharacteristic
293C3#1058849713yescomplemented
294C3#1059251611yescharacteristic
295C2—1082no—
296C2#1172172711yescharacteristic
297C2#1172179611yescharacteristic
298C2#1172180411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
cosocleC2×C3×C4×C9C23
hypercenter#103C3⋊D9
hypocenterC3×C9#440
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊Dic9C2semidirect
4. C3×C9×#103C2semidirect
6. #11720538C2semidirect
7. #11720525C2semidirect
15. #88548C4semidirect
32. C9×#103D3semidirect
38. C3×Q8×C9C2×C4semidirect
49. C3⋊(C9⋊C8)C2×C4semidirect
93. C3×#103D9semidirect
117. Q8×C9C4×D3semidirect
184. #103C3⋊D9semidirect
187. C3×C9#440semidirect
192. C3×Q8C4×D9semidirect
259. C9#4817semidirect
262. Q8C4×C3⋊D9semidirect
293. C3#10588497semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
C2×#885482
#117217492
C3×C9×#1032
#117205432
#117205382
#117205252
#105925163
#105884973