Finite Group Databasebeta

← #13138310

Subgroups — #13138310

Order 1728.

434 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#88524C211yesmaximal, characteristic
2#11720992C211yesmaximal, characteristic
3#11720913C211yesmaximal, characteristic, complemented
4C9×#817C211yesmaximal, characteristic, complemented
5C3×C8×Dic9C211yesmaximal, characteristic, complemented
6#11720269C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic
8#10590758—31nomaximal
9#10588532—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×(C3⋊D9)⋊C4—21no—
13C2×C3×C8×C9C2211yescharacteristic
14#88524C414yescomplemented
15C9×C4⋊Dic3C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic9C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20#88442—21no—
21Dic3×Dic9—21no—
22C2×#3744—31no—
23#63894—31no—
24#63844—31no—
25C3×#817D311yescharacteristic
26C3×C8×Dic3—31no—
27#63567—31no—
28#63564—31no—
29C2×#3664—31no—
30C2×#3664—61no—
31#63455—31no—
32#63449—31no—
33C9×#106—31no—
34C8×Dic9D311yescharacteristic, complemented
35#63369—31no—
36#63368—31no—
37C22×C3⋊D9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)C2×C412yes—
42#6226—21no—
43#6226C2×C412yescomplemented
44C2×C9×Dic3—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9D411yescharacteristic
47C2×C3×Dic9D411yescharacteristic
48(C3⋊D9)⋊C4—42no—
49C3×C8×C9C2×C412yescomplemented
50C3×C9⋊C8—21no—
51#4852—91no—
52C4×#204—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×#634—31no—
55C2×#624—61no—
56C2×C32×C8C2×D311yescharacteristic
57#3744—34no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C3×C4⋊Dic3C2×D311yescharacteristic
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8—31no—
62#3724—61no—
63#3722—61no—
64Dic32—61no—
65C2×D36—31no—
66C2×D36—61no—
67C2×C4×D9—61no—
68C2×Dic18—31no—
69C2×Dic18—61no—
70C2×C8×C9—21no—
71C2×C8×C9C2×D311yescharacteristic
72C9×C4⋊C4—31no—
73C9×C4⋊C4—31no—
74(C2×D9)⋊C4—61no—
75Dic9⋊C4—61no—
76C4×Dic9—61no—
77C4×Dic9C2×D311yescharacteristic
78C2×C9⋊C8C2×D311yescharacteristic
79#3664—34no—
80#3664—64no—
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—22no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C3×C4×C9C22×C411yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3⋊Dic9—22no—
88C9×Dic3—41no—
89C9×Dic3—41no—
90C3×Dic9—21no—
91C3×Dic9SD1612yescomplemented
92C2×#204—91no—
93C2×#204—91no—
94C2×#204—181no—
95#893—91no—
96#887—91no—
97C4×D12—91no—
98C4×Dic6—91no—
99C3×#106—31no—
100C3×C4×C8—91no—
101#821—91no—
102#817D911yescharacteristic, complemented
103#816—91no—
104C8×Dic3—31no—
105#810—91no—
106#809—91no—
107C22×C3⋊D3—61no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—61no—
110C4⋊(C3⋊D3)—32no—
111C4⋊(C3⋊D3)—61no—
112#634—32no—
113#634—61no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118#624—122no—
119C22×D9—61no—
120C22×D9—121no—
121C32×C8C4×D312yes—
122C3×C3⋊C8—61no—
123C2×C4×C9—21no—
124C2×C4×C9—61no—
125C2×C4×C9—61no—
126C2×C4×C9C22×D311yescharacteristic
127C2×Dic9—61no—
128C2×Dic9—121no—
129C2×Dic9D1211yescharacteristic
130C2×Dic9D1211yescharacteristic
131D36—32no—
132D36—62no—
133D36—61no—
134D36—121no—
135C4×D9—122no—
136Dic18—32no—
137Dic18—62no—
138Dic18—61no—
139Dic18—121no—
140C8×C9—22no—
141C8×C9C4×D312yescomplemented
142C9⋊C8—21no—
143C4×SD16—271noSylow
144C2×C3×C9C4×D411yescharacteristic
145C2×C3×C9C2×SD1611yescharacteristic
146C2×C3×C9#13511yescharacteristic
147C3⋊D9—42no—
148C2×D12—91no—
149C2×D12—91no—
150C2×D12—181no—
151C2×C4×D3—181no—
152C2×C4×D3—181no—
153C2×Dic6—91no—
154C2×Dic6—91no—
155C2×Dic6—181no—
156C2×C3×C8—21no—
157C2×C3×C8—91no—
158C2×C3×C8C2×D911yescharacteristic
159C2×C3×C8D3211yescharacteristic
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C42—91no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165C4⋊Dic3C2×D911yescharacteristic
166C4⋊Dic3C2×D911yescharacteristic
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169C4×Dic3—31no—
170C4×Dic3—181no—
171C4×Dic3—181no—
172C2×C3⋊C8—31no—
173#204—94no—
174#204—94no—
175#204—184no—
176C22×C32D3×D411yescharacteristic
177C2×C3⋊D3—62no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C32×C4C2×C4×D311yescharacteristic
181C32×C4#23911yescharacteristic
182C3⋊Dic3—62no—
183C3×Dic3—32no—
184C3×Dic3—41no—
185C3×Dic3—41no—
186C3×Dic3—61no—
187C22×C9—21no—
188C22×C9C2×D1211yescharacteristic
189C2×D9—62no—
190C2×D9—122no—
191C2×D9—121no—
192C2×D9—121no—
193C2×D9—241no—
194C2×D9—241no—
195C4×C9—21no—
196C4×C9—21no—
197C4×C9—121no—
198C4×C9—121no—
199C4×C9C2×C4×D311yescharacteristic
200C4×C9#23511yescharacteristic
201Dic9—21no—
202Dic9—62no—
203Dic9—122no—
204Dic9#20412yescomplemented
205C2×SD16—271no—
206C4×Q8—271no—
207C4×D4—271no—
208#106—31no—
209#103—271no—
210#102—271no—
211C4×C8—91no—
212C3×C9C4×SD1611yescharacteristic, complemented, Sylow
213C22×D3—181no—
214C22×D3—181no—
215C22×D3—361no—
216C2×C3×C4—21no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4C22×D911yescharacteristic
222C2×C3×C4C2×D3211yescharacteristic
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—31no—
226C2×Dic3—31no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C2×Dic3—361no—
230D12—92no—
231D12—92no—
232D12—182no—
233D12—181no—
234D12—181no—
235D12—361no—
236C4×D3—362no—
237C4×D3—362no—
238Dic6—92no—
239Dic6—92no—
240Dic6—182no—
241Dic6—181no—
242Dic6—181no—
243Dic6—361no—
244C3×C8—22no—
245C3×C8—181no—
246C3×C8C4×D912yescomplemented
247C3×C8#62412yes—
248C3⋊C8—61no—
249C2×C32#89311yescharacteristic
250C2×C32D3×SD1611yescharacteristic
251C2×C32#91611yescharacteristic
252C3⋊D3—122no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9C4×D1211yescharacteristic
257C2×C9C2×#20411yescharacteristic
258C2×C9#90411yescharacteristic
259D9—122no—
260D9—242no—
261C2×Q8—271no—
262C2×D4—271no—
263C22×C4—541no—
264SD16—274no—
265C2×C8—91no—
266C2×C8D3×D911yescharacteristic
267C4⋊C4—31no—
268C4⋊C4—31no—
269C4⋊C4—541no—
270C22⋊C4—541no—
271C42—91no—
272C42—541no—
273C22×C3—21no—
274C22×C3D4×D911yescharacteristic
275C22×C3D3×D1211yescharacteristic
276C2×D3—182no—
277C2×D3—182no—
278C2×D3—362no—
279C2×D3—361no—
280C2×D3—361no—
281C2×D3—361no—
282C2×D3—361no—
283C2×D3—721no—
284C2×D3—721no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—92no—
288C3×C4—121no—
289C3×C4—121no—
290C3×C4—181no—
291C3×C4C2×C4×D911yescharacteristic
292C3×C4#370111yescharacteristic
293C3×C4#379911yescharacteristic
294C3×C4C2×#62411yescharacteristic
295Dic3—32no—
296Dic3—41no—
297Dic3—41no—
298Dic3—61no—
299Dic3—182no—
300Dic3—182no—
301Dic3—362no—
302C32#485211yescharacteristic
303C9—21no—
304C9C4×#20411yescharacteristic, complemented
305C23—541no—
306Q8—272no—
307Q8—541no—
308D4—272no—
309D4—541no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—91no—
313C2×C4—91no—
314C2×C4—541no—
315C2×C4—1082no—
316C2×C4C2×D3×D911yescharacteristic
317C8—181no—
318C8(C3⋊D9)⋊C412yescomplemented
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3#6345511yescharacteristic
323C2×C3SD16×D911yescharacteristic
324C2×C3#6347811yescharacteristic
325C2×C3D3×#20411yescharacteristic
326C2×C3#6380611yescharacteristic
327C2×C3#6389411yescharacteristic
328D3—362no—
329D3—362no—
330D3—722no—
331C22—542no—
332C22—1081no—
333C22—1081no—
334C22D9×D1211yescharacteristic
335C4—92no—
336C4—121no—
337C4—121no—
338C4—181no—
339C4—542no—
340C4#8863511yescharacteristic
341C4C2×(C3⋊D9)⋊C411yescharacteristic
342C3—21no—
343C3#1058853211yescharacteristic, complemented
344C3#1059075811yescharacteristic
345C2—1082no—
346C2D9×#20411yescharacteristic
347C2#1172082111yescharacteristic
348C2#1172099211yescharacteristic
subgroupG/H
centerC22D9×D12
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9C4×SD16
LCSG ▷ 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
3. #11720913C2semidirect
4. C9×#817C2semidirect
5. C3×C8×Dic9C2semidirect
6. #11720269C2semidirect
14. #88524C4semidirect
34. C8×Dic9D3semidirect
43. #6226C2×C4semidirect
49. C3×C8×C9C2×C4semidirect
91. C3×Dic9SD16semidirect
102. #817D9semidirect
141. C8×C9C4×D3semidirect
204. Dic9#204semidirect
212. C3×C9C4×SD16semidirect
246. C3×C8C4×D9semidirect
304. C9C4×#204semidirect
318. C8(C3⋊D9)⋊C4semidirect
343. C3#10588532semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
#117209922
#117209132
C9×#8172
C3×C8×Dic92
#117202692
#117202672
#105907583
#105885323