Finite Group Databasebeta

← #13137604

Subgroups — #13137604

Order 1728.

384 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#63459C211yesmaximal, characteristic
2#11720945C211yesmaximal, characteristic
3C2×#88418C211yesmaximal, characteristic, complemented
4C2×D9×C3⋊C8C211yesmaximal, characteristic
5#11720527C211yesmaximal, characteristic, complemented
6C9×#807C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590579—31nomaximal
9#10588567—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20Dic3⋊Dic9—21no—
21#88418—22no—
22D9×C3⋊C8—22no—
23C3×#897—31no—
24#63857—31no—
25C2×#3719—31no—
26C2×D3×C3⋊C8—31no—
27#63637—31no—
28C3×#807D311yescharacteristic
29#63565—31no—
30C2×C8×D9—31no—
31C2×Dic36—31no—
32#63459—31no—
33#63459D311yescharacteristic
34C9×#107—31no—
35#63368—31no—
36#63368—31no—
37#63368—61no—
38C22×C3×D9D411yescharacteristic
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42#6226—21no—
43#6226—21no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D411yescharacteristic
47C3×C4×D9—21no—
48C3×C4×D9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8D412yes—
53#5051—91no—
54#4887—91no—
55C2×#634—31no—
56C2×C3×C4×D3—31no—
57C2×C3×Dic6—31no—
58C32×C4⋊C4C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)—31no—
60C3×(C2×D3)⋊C4—61no—
61C3×C4⋊Dic3—61no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63C2×C3×C3⋊C8C2×D311yescharacteristic
64Dic3⋊Dic3—61no—
65(C2×D3)⋊Dic3—61no—
66#3719—62no—
67D3×C3⋊C8—62no—
68C2×C4×D9C2×D311yescharacteristic
69C2×Dic18—31no—
70C2×Dic18—61no—
71C2×Dic18C2×D311yescharacteristic
72C2×C8×C9—31no—
73C9×C4⋊C4—21no—
74C9×C4⋊C4—31no—
75C9×C4⋊C4C2×D311yescharacteristic
76(C2×D9)⋊C4—21no—
77(C2×D9)⋊C4—61no—
78C4⋊Dic9—21no—
79C4⋊Dic9—61no—
80C2×C9⋊C8—31no—
81C2×C9⋊C8—61no—
82C8×D9—62no—
83Dic36—62no—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3×D9—21no—
86C2×C3×D9—21no—
87C2×C3×D9Q1612yescomplemented
88C3×C4×C9—41no—
89C3×C4×C9C2×D411yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C3⋊Dic9—41no—
92C9×Dic3—41no—
93C3×Dic9—21no—
94C3×Dic9—41no—
95C3×C22⋊Q8—91no—
96C2×#216—91no—
97#924—91no—
98C22×C3⋊C8—91no—
99C2×Dic12—91no—
100C2×D3×C8—91no—
101#897—31no—
102#897—91no—
103C3×#107—31no—
104#835—91no—
105#810—91no—
106#810—91no—
107#810—91no—
108#810—181no—
109#807D911yescharacteristic, complemented
110C22×C3×D3—31no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—61no—
114#634—61no—
115#634—61no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—61no—
119C3×C4×D3—61no—
120C3×C4×D3—61no—
121C3×Dic6—61no—
122C3×Dic6—61no—
123C22×D9#6711yescharacteristic
124C3⋊(C3⋊C8)—61no—
125C3×C3⋊C8D1212yes—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—41no—
129C2×C4×C9—61no—
130C2×C4×C9C22×D311yescharacteristic
131C2×Dic9—21no—
132C2×Dic9—61no—
133C2×Dic9—121no—
134C2×Dic9#6711yescharacteristic
135C4×D9—21no—
136C4×D9—21no—
137Dic18—21no—
138Dic18—21no—
139Dic18—62no—
140Dic18—61no—
141Dic18—61no—
142Dic18—121no—
143C8×C9—32no—
144C9⋊C8—61no—
145C9⋊C8—121no—
146#467—271noSylow
147C2×C3×C9#12111yescharacteristic
148C2×C3×C9C2×Q1611yescharacteristic
149C2×C3×C9#13511yescharacteristic
150C3×D9—22no—
151C2×C3×Q8—91no—
152C22×C3×C4—91no—
153C2×C4×D3—31no—
154C2×Dic6—31no—
155C2×Dic6—91no—
156C2×Dic6—91no—
157C2×Dic6—181no—
158C2×C3×C8—31no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4C2×D911yescharacteristic
163C3×C4⋊C4D3211yescharacteristic
164C3×C22⋊C4—181no—
165#217—181no—
166#216—182no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—61no—
170C4⋊Dic3—181no—
171C4⋊Dic3C2×D911yescharacteristic
172Dic3⋊C4—181no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—181no—
176C2×C3⋊C8—182no—
177C2×C3⋊C8C2×D911yescharacteristic
178Dic12—182no—
179D3×C8—182no—
180C22×C32D3×D411yescharacteristic
181C2×C3×D3—32no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C32×C4—41no—
185C32×C4C2×D1211yescharacteristic
186C32×C4#23911yescharacteristic
187C3⋊Dic3—121no—
188C3×Dic3—41no—
189C3×Dic3—61no—
190C3×Dic3—121no—
191C22×C9—21no—
192C22×C9C2×#6711yescharacteristic
193C2×D9—21no—
194C2×D9—21no—
195C2×D9#21612yescomplemented
196C4×C9—21no—
197C4×C9—21no—
198C4×C9—41no—
199C4×C9—81no—
200C4×C9—121no—
201C4×C9D3×D411yescharacteristic
202C4×C9#23711yescharacteristic
203Dic9—21no—
204Dic9—41no—
205Dic9—122no—
206Dic9—121no—
207C2×Q16—271no—
208C22×C8—271no—
209C22⋊Q8—91no—
210C22⋊Q8—271no—
211#107—31no—
212#103—271no—
213#103—271no—
214C3×C9#46711yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—31no—
217C3×Q8—181no—
218C3×Q8—181no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—31no—
232C2×Dic3—61no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—361no—
236C4×D3—61no—
237C4×D3—61no—
238Dic6—61no—
239Dic6—61no—
240Dic6—182no—
241Dic6—181no—
242Dic6—181no—
243Dic6—181no—
244Dic6—181no—
245Dic6—361no—
246C3×C8—32no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—361no—
250C3⋊C8D3612yes—
251C2×C32#89511yescharacteristic
252C2×C32#91211yescharacteristic
253C2×C32D3×Q1611yescharacteristic
254C3×D3—62no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9#93911yescharacteristic
259C2×C9C2×#21611yescharacteristic
260C2×C9#95011yescharacteristic
261D9—22no—
262C2×Q8—91no—
263C2×Q8—271no—
264C22×C4—91no—
265Q16—542no—
266C2×C8—31no—
267C2×C8—271no—
268C2×C8—542no—
269C4⋊C4—31no—
270C4⋊C4—181no—
271C4⋊C4—541no—
272C4⋊C4D3×D911yescharacteristic
273C22⋊C4—181no—
274C22⋊C4—541no—
275C22×C3—21no—
276C22×C3—92no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3D4×D911yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—32no—
282C2×D3—61no—
283C2×D3—61no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—81no—
289C3×C4—121no—
290C3×C4—181no—
291C3×C4—361no—
292C3×C4C2×D3611yescharacteristic
293C3×C4#370111yescharacteristic
294C3×C4#379611yescharacteristic
295C3×C4D3×D1211yescharacteristic
296Dic3—41no—
297Dic3—61no—
298Dic3—121no—
299Dic3—362no—
300Dic3—361no—
301Dic3—361no—
302C32#488711yescharacteristic
303C9—21no—
304C9#505111yescharacteristic, complemented
305C23—91no—
306Q8—181no—
307Q8—181no—
308Q8—541no—
309Q8—541no—
310C2×C4—21no—
311C2×C4—61no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—541no—
317C2×C4C2×D3×D911yescharacteristic
318C8—32no—
319C8—541no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—182no—
324C2×C3#6345711yescharacteristic
325C2×C3#6347411yescharacteristic
326C2×C3Q16×D911yescharacteristic
327C2×C3#6388611yescharacteristic
328C2×C3#6393311yescharacteristic
329C2×C3D3×#21611yescharacteristic
330D3—62no—
331C22—92no—
332C22—181no—
333C22—181no—
334C22D9×#6711yescharacteristic
335C4—41no—
336C4—121no—
337C4—181no—
338C4—361no—
339C4—1081no—
340C4#8863411yescharacteristic
341C4D3×D3611yescharacteristic
342C3—21no—
343C3#1058856711yescharacteristic, complemented
344C3#1059057911yescharacteristic
345C2—182no—
346C2#1172098311yescharacteristic
347C2#1172117711yescharacteristic
348C2D9×#21611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#467
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#88418C2semidirect
5. #11720527C2semidirect
6. C9×#807C2semidirect
7. #11720268C2semidirect
87. C2×C3×D9Q16semidirect
109. #807D9semidirect
195. C2×D9#216semidirect
214. C3×C9#467semidirect
304. C9#5051semidirect
343. C3#10588567semidirect

Maximal subgroups

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

subgroupindex
C3×#634592
#117209452
C2×#884182
C2×D9×C3⋊C82
#117205272
C9×#8072
#117202682
#105905793
#105885673