Finite Group Databasebeta

← #13143909

Subgroups — #13143909

Order 1728.

610 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 163 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
1D4×C3⋊Dic9C211yesmaximal, characteristic
2C2×#88543C211yesmaximal, characteristic
3#11721743C211yesmaximal, characteristic, complemented
4C3×C9×#102C211yesmaximal, characteristic
5#11720539C211yesmaximal, characteristic, complemented
6C8×C3⋊Dic9C211yesmaximal, characteristic, complemented
7#11720526C211yesmaximal, characteristic, complemented
8#10592457—31nomaximal
9#10588438—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12C2×#6226C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88551—21no—
16#88543C414yescomplemented
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18#88534—21no—
19C4×C3⋊Dic9—21no—
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22D4×C3⋊Dic3—31no—
23C2×#3754—31no—
24#64096—31no—
25C32×#102D311yescharacteristic
26#63643—31no—
27C8×C3⋊Dic3—31no—
28#63635—31no—
29D4×Dic9—33no—
30C2×#3672—33no—
31#63449—33no—
32C9×#102D313yes—
33#63376—33no—
34C8×Dic9—33no—
35#63367—33no—
36C23×C3×C9—21no—
37C3×D4×C9—21no—
38C3×D4×C9C2×C412yes—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—42no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46#6226—21no—
47#6226C2×C412yescomplemented
48C3×C8×C9—21no—
49C3⋊(C9⋊C8)C2×C412yescomplemented
50#4758—93no—
51#4758—91no—
52C2×C32×D4C2×D311yescharacteristic
53C22×C3⋊Dic3—61no—
54C2×#634—31no—
55C2×C32×C8C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57#3757—61no—
58#3754—34no—
59C4⋊(C3⋊Dic3)—31no—
60#3750—61no—
61C4×C3⋊Dic3—31no—
62C4×C3⋊Dic3—61no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×D4×C9C2×D313yes—
65C22×Dic9—63no—
66C2×Dic18—33no—
67C2×C8×C9C2×D313yes—
68C9×C4⋊C4C2×D313yes—
69#3676—63no—
70#3672—312no—
71C4⋊Dic9—33no—
72Dic9⋊C4—63no—
73C4×Dic9—33no—
74C4×Dic9—63no—
75C2×C9⋊C8—33no—
76C22×C3×C9—22no—
77C22×C3×C9—41no—
78C22×C3×C9—41no—
79C22×C3×C9C2×D411yescharacteristic
80C3×C4×C9—41no—
81C3×C4×C9C22×C411yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—22no—
84C3⋊Dic9—21no—
85C3⋊Dic9—41no—
86C3⋊Dic9SD1612yescomplemented
87D4×Dic3—93no—
88D4×Dic3—91no—
89C2×#214—93no—
90C2×#214—91no—
91#887—93no—
92#887—91no—
93C3×#102D913yes—
94C3×#102C3⋊D311yescharacteristic
95#816—93no—
96#816—91no—
97C8×Dic3—93no—
98C8×Dic3—91no—
99#808—93no—
100#808—91no—
101C23×C32—21no—
102C32×D4—21no—
103C32×D4C4×D312yes—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—122no—
111#634—32no—
112#634—61no—
113C23×C9—23no—
114C32×C8—21no—
115C3⋊(C3⋊C8)—32no—
116D4×C9—23no—
117D4×C9C4×D316yes—
118C2×C4×C9—23no—
119C2×C4×C9C22×D313yes—
120C2×Dic9—33no—
121C2×Dic9—33no—
122C2×Dic9—63no—
123C2×Dic9—63no—
124C2×Dic9—126no—
125Dic18—36no—
126Dic18—63no—
127C8×C9—23no—
128C9⋊C8—36no—
129C4×SD16—271noSylow
130C2×C3×C9—42no—
131C2×C3×C9C4×D411yescharacteristic
132C2×C3×C9C2×SD1611yescharacteristic
133C2×C3×C9#13511yescharacteristic
134C2×C3×D4C2×D913yes—
135C2×C3×D4C2×C3⋊D311yescharacteristic
136C22×Dic3—181no—
137C22×Dic3—183no—
138C2×Dic6—93no—
139C2×Dic6—91no—
140C2×C3×C8C2×D913yes—
141C2×C3×C8C2×C3⋊D311yescharacteristic
142C3×C4⋊C4C2×D913yes—
143C3×C4⋊C4C2×C3⋊D311yescharacteristic
144#217—181no—
145#217—183no—
146#214—912no—
147#214—94no—
148C4⋊Dic3—93no—
149C4⋊Dic3—91no—
150Dic3⋊C4—181no—
151Dic3⋊C4—183no—
152C4×Dic3—93no—
153C4×Dic3—91no—
154C4×Dic3—181no—
155C4×Dic3—183no—
156C2×C3⋊C8—93no—
157C2×C3⋊C8—91no—
158C22×C32—22no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32D3×D411yescharacteristic
162C32×C4—41no—
163C32×C4C2×C4×D311yescharacteristic
164C32×C4#23711yescharacteristic
165C3⋊Dic3—32no—
166C3⋊Dic3—62no—
167C3⋊Dic3—61no—
168C3⋊Dic3—121no—
169C22×C9—26no—
170C22×C9—43no—
171C22×C9—43no—
172C22×C9D3×D413yes—
173C4×C9—43no—
174C4×C9C2×C4×D313yes—
175C4×C9#23713yes—
176Dic9—36no—
177Dic9—66no—
178Dic9—63no—
179Dic9—123no—
180C2×SD16—271no—
181C4×Q8—271no—
182C4×D4—271no—
183#106—271no—
184#103—271no—
185#102C3⋊D911yescharacteristic
186C4×C8—271no—
187C3×C9C4×SD1611yescharacteristic, complemented, Sylow
188C23×C3—23no—
189C23×C3—21no—
190C3×D4—23no—
191C3×D4—21no—
192C3×D4C4×D916yes—
193C3×D4C4×C3⋊D312yes—
194C2×C3×C4—23no—
195C2×C3×C4—21no—
196C2×C3×C4C22×D913yes—
197C2×C3×C4C22×C3⋊D311yescharacteristic
198C2×Dic3—93no—
199C2×Dic3—91no—
200C2×Dic3—93no—
201C2×Dic3—91no—
202C2×Dic3—183no—
203C2×Dic3—181no—
204C2×Dic3—183no—
205C2×Dic3—181no—
206C2×Dic3—362no—
207C2×Dic3—366no—
208Dic6—96no—
209Dic6—92no—
210Dic6—183no—
211Dic6—181no—
212C3×C8—23no—
213C3×C8—21no—
214C3⋊C8—96no—
215C3⋊C8—92no—
216C2×C32—42no—
217C2×C32#67⋊C411yescharacteristic
218C2×C32#91211yescharacteristic
219C2×C32D3×SD1611yescharacteristic
220C2×C9—46no—
221C2×C9#67⋊C413yes—
222C2×C9#91213yes—
223C2×C9D3×SD1613yes—
224C2×Q8—271no—
225C2×D4C2×C3⋊D911yescharacteristic
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×C3—22no—
237C22×C3—26no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—43no—
241C22×C3—43no—
242C22×C3D4×D913yes—
243C22×C3D4×C3⋊D311yescharacteristic
244C3×C4—41no—
245C3×C4—43no—
246C3×C4C2×C4×D913yes—
247C3×C4#369913yes—
248C3×C4C2×C4×C3⋊D311yescharacteristic
249C3×C4#383011yescharacteristic
250Dic3—92no—
251Dic3—96no—
252Dic3—182no—
253Dic3—181no—
254Dic3—186no—
255Dic3—183no—
256Dic3—361no—
257Dic3—363no—
258C32#475811yescharacteristic
259C9#475813yescomplemented
260C23—21no—
261Q8—272no—
262Q8—541no—
263D4—21no—
264D4C4×C3⋊D912yes—
265C2×C4—21no—
266C2×C4—271no—
267C2×C4—271no—
268C2×C4—541no—
269C2×C4—541no—
270C2×C4—1082no—
271C2×C4C22×C3⋊D911yescharacteristic
272C8—21no—
273C8—272no—
274C2×C3—46no—
275C2×C3—42no—
276C2×C3#611⋊C413yes—
277C2×C3#6347413yes—
278C2×C3SD16×D913yes—
279C2×C3#6409011yescharacteristic
280C2×C3#6412111yescharacteristic
281C2×C3SD16×C3⋊D311yescharacteristic
282C22—22no—
283C22—41no—
284C22—41no—
285C22D4×C3⋊D911yescharacteristic
286C4—41no—
287C4—272no—
288C4—541no—
289C4—542no—
290C4—1081no—
291C4C2×C4×C3⋊D911yescharacteristic
292C4#8874211yescharacteristic
293C3#1058843813yescomplemented
294C3#1059245711yescharacteristic
295C2—42no—
296C2#1172172711yescharacteristic
297C2#1172179111yescharacteristic
298C2SD16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
cosocleC2×C3×C4×C9C23
hypercenter#102C3⋊D9
hypocenterC3×C9C4×SD16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
3. #11721743C2semidirect
5. #11720539C2semidirect
6. C8×C3⋊Dic9C2semidirect
7. #11720526C2semidirect
16. #88543C4semidirect
47. #6226C2×C4semidirect
49. C3⋊(C9⋊C8)C2×C4semidirect
86. C3⋊Dic9SD16semidirect
187. C3×C9C4×SD16semidirect
259. C9#4758semidirect
293. C3#10588438semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
C2×#885432
#117217432
C3×C9×#1022
#117205392
C8×C3⋊Dic92
#117205262
#105924573
#105884383