Finite Group Databasebeta

← #13138259

Subgroups — #13138259

Order 1728.

306 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 97 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#11720969C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic
3#11720540C211yesmaximal, characteristic
4C9×#818C211yesmaximal, characteristic
5C3×#63375C211yesmaximal, characteristic
6#11720264C211yesmaximal, characteristic
7#11720262C211yesmaximal, characteristic
8#10590894—31nomaximal
9#10588556—31nomaximal
10C2×#6189—21no—
11C2×C3×C8×C9C2211yescharacteristic
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4×Dic9C2211yescharacteristic
17C2×C3×C9⋊C8C2211yescharacteristic
18Dic3⋊Dic9—21no—
19#88442—21no—
20Dic9⋊Dic3—21no—
21#63918—31no—
22#63869—31no—
23#63644—31no—
24C3×#818D311yescharacteristic
25C3×#815—31no—
26#63575—31no—
27#63573—31no—
28#63452—31no—
29#63450—31no—
30C9×#107—31no—
31#63377—31no—
32#63377—61no—
33#63375D311yescharacteristic
34#63367—31no—
35#63366—31no—
36C2×C3×C4×C9C2311yescharacteristic
37C2×C3⋊Dic9—21no—
38C2×C3⋊Dic9—21no—
39C2×C9×Dic3—21no—
40C2×C9×Dic3—21no—
41C2×C3×Dic9D411yescharacteristic
42C2×C3×Dic9D411yescharacteristic
43#6189—42no—
44C3×C8×C9Q812yescomplemented
45C3×C9⋊C8Q812yescomplemented
46#4876—91no—
47#4702—91no—
48C2×#627—61no—
49C2×C32×C8C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—31no—
51C4⋊(C3⋊Dic3)—31no—
52C3×C4⋊Dic3C2×D311yescharacteristic
53C3×C4⋊Dic3C2×D311yescharacteristic
54C3×C4×Dic3—31no—
55C2×C3×C3⋊C8—31no—
56#3724—61no—
57Dic3⋊Dic3—61no—
58Dic3⋊Dic3—61no—
59C2×Dic18—61no—
60C2×C8×C9—21no—
61C2×C8×C9C2×D311yescharacteristic
62C9×C4⋊C4—31no—
63C9×C4⋊C4—31no—
64C4⋊Dic9—31no—
65C4⋊Dic9—31no—
66C4⋊Dic9—61no—
67C4⋊Dic9—61no—
68C4⋊Dic9—61no—
69Dic9⋊C4—61no—
70Dic9⋊C4—61no—
71C4×Dic9C2×D311yescharacteristic
72C2×C9⋊C8C2×D311yescharacteristic
73C22×C3×C9C2×D411yescharacteristic
74C3×C4×C9C2×Q811yescharacteristic
75C3×C4×C9C2×Q811yescharacteristic
76C3⋊Dic9—41no—
77C3⋊Dic9—41no—
78C9×Dic3—41no—
79C9×Dic3—41no—
80C3×Dic9—21no—
81C3×Dic9—21no—
82#890—91no—
83#888—91no—
84#870—91no—
85#869—91no—
86C3×#107—31no—
87C3×#97—91no—
88#818—91no—
89#818D911yescharacteristic
90#817—91no—
91#817—91no—
92#817—91no—
93#817—181no—
94#815—31no—
95#808—91no—
96#807—91no—
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—61no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—31no—
104#627—122no—
105C32×C8Dic612yes—
106C3×C3⋊C8—32no—
107C2×C4×C9—21no—
108C2×C4×C9—61no—
109C2×C4×C9—61no—
110C2×C4×C9C22×D311yescharacteristic
111C2×Dic9—61no—
112C2×Dic9—61no—
113C2×Dic9—121no—
114C2×Dic9—121no—
115C2×Dic9D1211yescharacteristic
116C2×Dic9D1211yescharacteristic
117Dic18—122no—
118C8×C9—22no—
119C8×C9Dic612yescomplemented
120C9⋊C8Dic612yescomplemented
121#501—271noSylow
122C2×C3×C9C4⋊Q811yescharacteristic
123C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
124C2×C3×C9#13711yescharacteristic
125C2×Dic6—181no—
126C2×Dic6—181no—
127C2×C3×C8—21no—
128C2×C3×C8—91no—
129C2×C3×C8C2×D911yescharacteristic
130C2×C3×C8D3211yescharacteristic
131C3×C4⋊C4—31no—
132C3×C4⋊C4—31no—
133C3×C42—91no—
134C4⋊Dic3—91no—
135C4⋊Dic3—91no—
136C4⋊Dic3—91no—
137C4⋊Dic3—91no—
138C4⋊Dic3—181no—
139C4⋊Dic3—181no—
140C4⋊Dic3—181no—
141C4⋊Dic3—181no—
142C4⋊Dic3C2×D911yescharacteristic
143C4⋊Dic3C2×D911yescharacteristic
144Dic3⋊C4—181no—
145Dic3⋊C4—181no—
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148C4×Dic3—31no—
149C2×C3⋊C8—31no—
150C22×C32D3×D411yescharacteristic
151C32×C4C2×Dic611yescharacteristic
152C32×C4D3×Q811yescharacteristic
153C3⋊Dic3—121no—
154C3⋊Dic3—121no—
155C3×Dic3—41no—
156C3×Dic3—41no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C22×C9—21no—
160C22×C9C2×D1211yescharacteristic
161C4×C9—21no—
162C4×C9—21no—
163C4×C9—121no—
164C4×C9—121no—
165C4×C9C2×Dic611yescharacteristic
166C4×C9C2×Dic611yescharacteristic
167Dic9—21no—
168Dic9—21no—
169Dic9—121no—
170Dic9—121no—
171Dic9—241no—
172Dic9—241no—
173C4⋊Q8—271no—
174#125—271no—
175#107—31no—
176#107—271no—
177#106—271no—
178#106—271no—
179#97—91no—
180C3×C9#50111yescharacteristic, complemented, Sylow
181C2×C3×C4—21no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4C22×D911yescharacteristic
187C2×C3×C4C2×D3211yescharacteristic
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—31no—
191C2×Dic3—31no—
192C2×Dic3—181no—
193C2×Dic3—181no—
194C2×Dic3—181no—
195C2×Dic3—181no—
196C2×Dic3—361no—
197C2×Dic3—361no—
198Dic6—362no—
199Dic6—362no—
200C3×C8—22no—
201C3×C8—92no—
202C3×C8Dic1812yescomplemented
203C3×C8#62712yes—
204C3⋊C8—32no—
205C2×C32#88811yescharacteristic
206C2×C32#91111yescharacteristic
207C2×C32#91811yescharacteristic
208C2×C9—21no—
209C2×C9—21no—
210C2×C9—21no—
211C2×C9#86911yescharacteristic
212C2×C9#90811yescharacteristic
213C2×C9#90911yescharacteristic
214C2×Q8—541no—
215C2×C8—91no—
216C2×C8D3×D911yescharacteristic
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—271no—
220C4⋊C4—271no—
221C4⋊C4—541no—
222C4⋊C4—541no—
223C4⋊C4—541no—
224C42—91no—
225C22×C3—21no—
226C22×C3D4×D911yescharacteristic
227C22×C3D3×D1211yescharacteristic
228C3×C4—21no—
229C3×C4—21no—
230C3×C4—121no—
231C3×C4—121no—
232C3×C4—181no—
233C3×C4—181no—
234C3×C4C2×Dic1811yescharacteristic
235C3×C4Q8×D911yescharacteristic
236C3×C4D3×Dic611yescharacteristic
237C3×C4C2×#62711yescharacteristic
238Dic3—41no—
239Dic3—41no—
240Dic3—61no—
241Dic3—61no—
242Dic3—361no—
243Dic3—361no—
244Dic3—361no—
245Dic3—361no—
246Dic3—721no—
247Dic3—721no—
248C32#487611yescharacteristic
249C9—21no—
250C9#470211yescharacteristic, complemented
251Q8—1082no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—541no—
257C2×C4—541no—
258C2×C4C2×D3×D911yescharacteristic
259C8—92no—
260C8#618912yescomplemented
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3#6345011yescharacteristic
265C2×C3#6347311yescharacteristic
266C2×C3#6348011yescharacteristic
267C2×C3#6379511yescharacteristic
268C2×C3#6380111yescharacteristic
269C2×C3#6391811yescharacteristic
270C22D9×D1211yescharacteristic
271C4—121no—
272C4—121no—
273C4—181no—
274C4—181no—
275C4—1081no—
276C4—1081no—
277C4D9×Dic611yescharacteristic
278C4C2×#618911yescharacteristic
279C3—21no—
280C3#1058855611yescharacteristic, complemented
281C3#1059089411yescharacteristic
282C2#1172081611yescharacteristic
283C2#1172082811yescharacteristic
284C2#1172096911yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#501
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
44. C3×C8×C9Q8semidirect
45. C3×C9⋊C8Q8semidirect
119. C8×C9Dic6semidirect
120. C9⋊C8Dic6semidirect
180. C3×C9#501semidirect
202. C3×C8Dic18semidirect
250. C9#4702semidirect
260. C8#6189semidirect
280. C3#10588556semidirect

Maximal subgroups

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

subgroupindex
#117209692
#117209352
#117205402
C9×#8182
C3×#633752
#117202642
#117202622
#105908943
#105885563