Finite Group Databasebeta

← #13143973

Subgroups — #13143973

Order 1728.

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

normal subgroups only

This group has 550 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
1Q8×C3⋊Dic9C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic
3#11721743C211yesmaximal, characteristic
4C3×C9×#103C211yesmaximal, characteristic
5#11720539C211yesmaximal, characteristic
6C8×C3⋊Dic9C211yesmaximal, characteristic
7#11720525C211yesmaximal, characteristic
8#10592489—31nomaximal
9#10588470—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14#88547C414yescomplemented
15C4⋊(C3⋊Dic9)—21no—
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17#88534—21no—
18C4×C3⋊Dic9—21no—
19C4×C3⋊Dic9—21no—
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22Q8×C3⋊Dic3—31no—
23C2×#3756—31no—
24#64096—31no—
25C32×#103D311yescharacteristic
26#63643—31no—
27C8×C3⋊Dic3—31no—
28#63634—31no—
29Q8×Dic9—33no—
30C2×#3674—33no—
31#63449—33no—
32C9×#103D313yes—
33#63376—33no—
34C8×Dic9—33no—
35#63366—33no—
36C3×Q8×C9—21no—
37C3×Q8×C9C2×C412yes—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45#6226—21no—
46#6226C2×C412yes—
47C3×C8×C9—21no—
48C3⋊(C9⋊C8)C2×C412yes—
49#4790—93no—
50#4790—91no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×#634—31no—
53C2×C32×C8C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55#3756—34no—
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—61no—
58#3750—61no—
59C4×C3⋊Dic3—31no—
60C4×C3⋊Dic3—61no—
61C4×C3⋊Dic3—61no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×Q8×C9C2×D313yes—
64C2×Dic18—33no—
65C2×C8×C9C2×D313yes—
66C9×C4⋊C4C2×D313yes—
67#3674—312no—
68C4⋊Dic9—33no—
69C4⋊Dic9—63no—
70Dic9⋊C4—63no—
71C4×Dic9—33no—
72C4×Dic9—63no—
73C4×Dic9—63no—
74C2×C9⋊C8—33no—
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9—22no—
77C3×C4×C9—41no—
78C3×C4×C9C22×C411yescharacteristic
79C3×C4×C9#4111yescharacteristic
80C3⋊Dic9—22no—
81C3⋊Dic9—21no—
82C3⋊Dic9—41no—
83C3⋊Dic9Q1612yescomplemented
84Q8×Dic3—93no—
85Q8×Dic3—91no—
86C2×#216—93no—
87C2×#216—91no—
88#887—93no—
89#887—91no—
90C3×#103D913yes—
91C3×#103C3⋊D311yescharacteristic
92#816—93no—
93#816—91no—
94C8×Dic3—93no—
95C8×Dic3—91no—
96#807—93no—
97#807—91no—
98C32×Q8—21no—
99C32×Q8C4×D312yes—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—61no—
107#634—32no—
108#634—61no—
109C32×C8—21no—
110C3⋊(C3⋊C8)—32no—
111Q8×C9—23no—
112Q8×C9C4×D316yes—
113C2×C4×C9—23no—
114C2×C4×C9—23no—
115C2×C4×C9C22×D313yes—
116C2×Dic9—33no—
117C2×Dic9—33no—
118C2×Dic9—63no—
119C2×Dic9—63no—
120Dic18—36no—
121Dic18—63no—
122C8×C9—23no—
123C9⋊C8—36no—
124C4×Q16—271noSylow
125C2×C3×C9C4×D411yescharacteristic
126C2×C3×C9C2×Q1611yescharacteristic
127C2×C3×C9#13511yescharacteristic
128C2×C3×Q8C2×D913yes—
129C2×C3×Q8C2×C3⋊D311yescharacteristic
130C2×Dic6—93no—
131C2×Dic6—91no—
132C2×C3×C8C2×D913yes—
133C2×C3×C8C2×C3⋊D311yescharacteristic
134C3×C4⋊C4C2×D913yes—
135C3×C4⋊C4C2×C3⋊D311yescharacteristic
136#216—912no—
137#216—94no—
138C4⋊Dic3—93no—
139C4⋊Dic3—91no—
140C4⋊Dic3—181no—
141C4⋊Dic3—183no—
142Dic3⋊C4—181no—
143Dic3⋊C4—183no—
144C4×Dic3—93no—
145C4×Dic3—91no—
146C4×Dic3—181no—
147C4×Dic3—181no—
148C4×Dic3—183no—
149C4×Dic3—183no—
150C2×C3⋊C8—93no—
151C2×C3⋊C8—91no—
152C22×C32D3×D411yescharacteristic
153C32×C4—22no—
154C32×C4—41no—
155C32×C4C2×C4×D311yescharacteristic
156C32×C4#23711yescharacteristic
157C3⋊Dic3—32no—
158C3⋊Dic3—62no—
159C3⋊Dic3—61no—
160C3⋊Dic3—121no—
161C22×C9D3×D413yes—
162C4×C9—26no—
163C4×C9—43no—
164C4×C9C2×C4×D313yes—
165C4×C9#23713yes—
166Dic9—36no—
167Dic9—66no—
168Dic9—63no—
169Dic9—123no—
170C2×Q16—271no—
171C4×Q8—271no—
172C4×Q8—271no—
173#107—271no—
174#103—271no—
175#103C3⋊D911yescharacteristic
176C4×C8—271no—
177C3×C9C4×Q1611yescharacteristic, complemented, Sylow
178C3×Q8—23no—
179C3×Q8—21no—
180C3×Q8C4×D916yes—
181C3×Q8C4×C3⋊D312yes—
182C2×C3×C4—23no—
183C2×C3×C4—21no—
184C2×C3×C4—23no—
185C2×C3×C4—21no—
186C2×C3×C4C22×D913yes—
187C2×C3×C4C22×C3⋊D311yescharacteristic
188C2×Dic3—93no—
189C2×Dic3—91no—
190C2×Dic3—93no—
191C2×Dic3—91no—
192C2×Dic3—183no—
193C2×Dic3—181no—
194C2×Dic3—183no—
195C2×Dic3—181no—
196Dic6—96no—
197Dic6—92no—
198Dic6—183no—
199Dic6—181no—
200C3×C8—23no—
201C3×C8—21no—
202C3⋊C8—96no—
203C3⋊C8—92no—
204C2×C32#67⋊C411yescharacteristic
205C2×C32#91611yescharacteristic
206C2×C32D3×Q1611yescharacteristic
207C2×C9#67⋊C413yes—
208C2×C9#91613yes—
209C2×C9D3×Q1613yes—
210C2×Q8—271no—
211C2×Q8C2×C3⋊D911yescharacteristic
212Q16—274no—
213C2×C8—271no—
214C2×C8C2×C3⋊D911yescharacteristic
215C4⋊C4—271no—
216C4⋊C4—541no—
217C4⋊C4—541no—
218C4⋊C4C2×C3⋊D911yescharacteristic
219C42—271no—
220C42—541no—
221C42—541no—
222C22×C3D4×D913yes—
223C22×C3D4×C3⋊D311yescharacteristic
224C3×C4—22no—
225C3×C4—26no—
226C3×C4—41no—
227C3×C4—43no—
228C3×C4C2×C4×D913yes—
229C3×C4#369913yes—
230C3×C4C2×C4×C3⋊D311yescharacteristic
231C3×C4#383011yescharacteristic
232Dic3—92no—
233Dic3—96no—
234Dic3—182no—
235Dic3—181no—
236Dic3—186no—
237Dic3—183no—
238Dic3—361no—
239Dic3—363no—
240C32#479011yescharacteristic
241C9#479013yescomplemented
242Q8—21no—
243Q8—272no—
244Q8—541no—
245Q8C4×C3⋊D912yes—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—271no—
249C2×C4—271no—
250C2×C4—541no—
251C2×C4—541no—
252C2×C4C22×C3⋊D911yescharacteristic
253C8—21no—
254C8—272no—
255C2×C3#611⋊C413yes—
256C2×C3#6347813yes—
257C2×C3Q16×D913yes—
258C2×C3#6409011yescharacteristic
259C2×C3#6412511yescharacteristic
260C2×C3Q16×C3⋊D311yescharacteristic
261C22D4×C3⋊D911yescharacteristic
262C4—22no—
263C4—41no—
264C4—272no—
265C4—541no—
266C4—542no—
267C4—1081no—
268C4C2×C4×C3⋊D911yescharacteristic
269C4#8874211yescharacteristic
270C3#1058847013yescomplemented
271C3#1059248911yescharacteristic
272C2#1172172711yescharacteristic
273C2#1172179811yescharacteristic
274C2Q16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socle, prime radicalC22×C32D3×D4
cosocleC2×C3×C4×C9C23
hypercenter#103C3⋊D9
hypocenterC3×C9C4×Q16
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
14. #88547C4semidirect
83. C3⋊Dic9Q16semidirect
177. C3×C9C4×Q16semidirect
241. C9#4790semidirect
270. C3#10588470semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
C2×#885472
#117217432
C3×C9×#1032
#117205392
C8×C3⋊Dic92
#117205252
#105924893
#105884703