Finite Group Databasebeta

← C3×#10588437 #13141732

Subgroups — C3×#10588437 #13141732

Order 1728.

410 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×Dic9C211yesmaximal, characteristic
2C2×C3×#3672C211yesmaximal, characteristic
3C3×#63449C211yesmaximal, characteristic, complemented
4C3×C9×#102C211yesmaximal, characteristic
5C3×#63376C211yesmaximal, characteristic, complemented
6C3×#63375C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#4757—31nomaximal
9#10588437C311yesmaximal, characteristic, direct product factor
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3×Dic9—21no—
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×#3676—21no—
16C3×#3672C414yescomplemented
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C3×C4×Dic9—21no—
20C3×C4×Dic9C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22C3×D4×Dic3—31no—
23C2×C3×#214—31no—
24C3×#887—31no—
25C32×#102D311yescharacteristic
26C3×#816—31no—
27C3×#815—31no—
28C3×#807—31no—
29D4×Dic9C2×C311yescharacteristic
30C2×#3672C2×C311yescharacteristic
31#63449C2×C311yescharacteristic, complemented
32C9×#102—21no—
33C9×#102C2×C311yescharacteristic
34#63376C2×C311yescharacteristic, complemented
35#63375C2×C311yescharacteristic, complemented
36#63366C2×C311yescharacteristic, complemented
37C23×C3×C9—21no—
38C3×D4×C9—21no—
39C3×D4×C9C2×C412yes—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3×Dic9—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9—42no—
45C2×C3×Dic9D411yescharacteristic
46C2×C3×Dic9D411yescharacteristic
47C3×Dic18—21no—
48C3×Dic18C2×C412yescomplemented
49C3×C8×C9—21no—
50C3×C9⋊C8C2×C412yescomplemented
51C3×#440—91no—
52#4757—31no—
53C2×C32×D4C2×D311yescharacteristic
54C22×C3×Dic3—61no—
55C2×C3×Dic6—31no—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C3×#217—61no—
59C3×#214—34no—
60C3×C4⋊Dic3—31no—
61C3×Dic3⋊C4—61no—
62C3×C4×Dic3—31no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—31no—
65C2×D4×C9—21no—
66C2×D4×C9C22×C311yescharacteristic
67C22×Dic9—21no—
68C2×Dic18C22×C311yescharacteristic
69C2×C8×C9—21no—
70C2×C8×C9C22×C311yescharacteristic
71C9×C4⋊C4—21no—
72C9×C4⋊C4C22×C311yescharacteristic
73#3676—21no—
74#3672C3×C414yescomplemented
75C4⋊Dic9C22×C311yescharacteristic
76Dic9⋊C4—21no—
77C4×Dic9—21no—
78C4×Dic9C22×C311yescharacteristic
79C2×C9⋊C8C22×C311yescharacteristic
80C22×C3×C9—22no—
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9C2×D411yescharacteristic
84C3×C4×C9—41no—
85C3×C4×C9C22×C411yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3×Dic9—22no—
88C3×Dic9—21no—
89C3×Dic9—21no—
90C3×Dic9—41no—
91C2×C3×SD16—91no—
92C3×C4×Q8—91no—
93C3×C4×D4—91no—
94D4×Dic3—31no—
95C2×#214—31no—
96#887—31no—
97C3×#107—91no—
98C3×#103—91no—
99C3×#102—21no—
100C3×#102D911yescharacteristic
101C3×#102C3×D311yescharacteristic
102C3×#97—91no—
103#816—31no—
104#815—31no—
105#807—31no—
106C23×C32—21no—
107C32×D4—21no—
108C32×D4C4×D312yes—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3×Dic3—31no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—122no—
116C3×Dic6—32no—
117C3×Dic6—61no—
118C23×C9—21no—
119C23×C9—41no—
120C32×C8—21no—
121C3×C3⋊C8—32no—
122D4×C9—21no—
123D4×C9—22no—
124D4×C9—41no—
125D4×C9C2×C3×C412yes—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—41no—
129C2×C4×C9C23×C311yescharacteristic
130C2×Dic9—21no—
131C2×Dic9—21no—
132C2×Dic9—42no—
133C2×Dic9C3×D411yescharacteristic
134C2×Dic9C3×D411yescharacteristic
135Dic18—21no—
136Dic18C2×C3×C412yescomplemented
137C8×C9—21no—
138C8×C9—41no—
139C9⋊C8C2×C3×C412yescomplemented
140#440—91noSylow
141C2×C3×C9—42no—
142C2×C3×C9C4×D411yescharacteristic
143C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
144C2×C3×C9#13711yescharacteristic
145C2×C3×Q8—91no—
146C2×C3×D4—21no—
147C2×C3×D4C2×D911yescharacteristic
148C2×C3×D4C2×C3×D311yescharacteristic
149C22×C3×C4—181no—
150C22×Dic3—61no—
151C2×Dic6—31no—
152C3×SD16—94no—
153C2×C3×C8—21no—
154C2×C3×C8—91no—
155C2×C3×C8C2×D911yescharacteristic
156C2×C3×C8C2×C3×D311yescharacteristic
157C3×C4⋊C4—21no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4C2×D911yescharacteristic
161C3×C4⋊C4C2×C3×D311yescharacteristic
162C3×C22⋊C4—181no—
163C3×C42—91no—
164C3×C42—181no—
165#217—61no—
166#214—34no—
167C4⋊Dic3—31no—
168Dic3⋊C4—61no—
169C4×Dic3—31no—
170C4×Dic3—61no—
171C2×C3⋊C8—31no—
172C22×C32—22no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32D3×D411yescharacteristic
176C32×C4—41no—
177C32×C4C2×C4×D311yescharacteristic
178C32×C4#23711yescharacteristic
179C3×Dic3—62no—
180C3×Dic3—61no—
181C3×Dic3—61no—
182C3×Dic3—121no—
183C22×C9—22no—
184C22×C9—21no—
185C22×C9—42no—
186C22×C9—41no—
187C22×C9—41no—
188C22×C9—81no—
189C22×C9—81no—
190C22×C9C2×C3×D411yescharacteristic
191C4×C9—21no—
192C4×C9—21no—
193C4×C9—41no—
194C4×C9—81no—
195C4×C9C22×C3×C411yescharacteristic
196C4×C9C3×#4111yescharacteristic
197Dic9—22no—
198Dic9—21no—
199Dic9—21no—
200Dic9—41no—
201C2×SD16—91no—
202C4×Q8—91no—
203C4×D4—91no—
204#107—91no—
205#103—91no—
206#102C3×D911yescharacteristic
207#97—91no—
208C3×C9#44011yescharacteristic, complemented, Sylow
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—41no—
212C3×Q8—92no—
213C3×Q8—181no—
214C3×D4—21no—
215C3×D4—21no—
216C3×D4—22no—
217C3×D4—41no—
218C3×D4C4×D912yes—
219C3×D4C3×C4×D312yes—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—362no—
229C2×C3×C4C22×D911yescharacteristic
230C2×C3×C4C22×C3×D311yescharacteristic
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—122no—
236Dic6—32no—
237Dic6—61no—
238C3×C8—21no—
239C3×C8—21no—
240C3×C8—41no—
241C3×C8—92no—
242C3⋊C8—32no—
243C2×C32—42no—
244C2×C32#67⋊C411yescharacteristic
245C2×C32#91111yescharacteristic
246C2×C32#91511yescharacteristic
247C2×C9—21no—
248C2×C9—21no—
249C2×C9—21no—
250C2×C9—42no—
251C2×C9—82no—
252C2×C9C3×C4×D411yescharacteristic
253C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
254C2×C9C3×#13711yescharacteristic
255C2×Q8—91no—
256C2×D4C2×C3×D911yescharacteristic
257C22×C4—181no—
258SD16—94no—
259C2×C8—91no—
260C2×C8C2×C3×D911yescharacteristic
261C4⋊C4—91no—
262C4⋊C4—181no—
263C4⋊C4C2×C3×D911yescharacteristic
264C22⋊C4—181no—
265C42—91no—
266C42—181no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—42no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3D4×D911yescharacteristic
278C22×C3C3×D3×D411yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—81no—
284C3×C4—182no—
285C3×C4—181no—
286C3×C4—181no—
287C3×C4—361no—
288C3×C4C2×C4×D911yescharacteristic
289C3×C4#369911yescharacteristic
290C3×C4C2×C3×C4×D311yescharacteristic
291C3×C4C3×#23711yescharacteristic
292Dic3—62no—
293Dic3—61no—
294Dic3—61no—
295Dic3—121no—
296C32#475711yescharacteristic
297C9—21no—
298C9C3×#44011yescharacteristic, complemented
299C23—21no—
300Q8—92no—
301Q8—181no—
302D4—21no—
303D4C3×C4×D912yes—
304C2×C4—21no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4—362no—
310C2×C4C22×C3×D911yescharacteristic
311C8—21no—
312C8—92no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—42no—
317C2×C3—42no—
318C2×C3—82no—
319C2×C3#611⋊C411yescharacteristic
320C2×C3#6347311yescharacteristic
321C2×C3#6347711yescharacteristic
322C2×C3C3×#67⋊C411yescharacteristic
323C2×C3C3×#91111yescharacteristic
324C2×C3C3×#91511yescharacteristic
325C22—22no—
326C22—41no—
327C22—41no—
328C22C3×D4×D911yescharacteristic
329C4—41no—
330C4—181no—
331C4—181no—
332C4—182no—
333C4—361no—
334C4C2×C3×C4×D911yescharacteristic
335C4C3×#369911yescharacteristic
336C3—21no—
337C3#1058843711yescharacteristic, direct product factor
338C3C3×#475711yescharacteristic
339C2—42no—
340C2C3×#611⋊C411yescharacteristic
341C2C3×#6347311yescharacteristic
342C2C3×#6347711yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
cosocleC2×C4×C9C23×C3
hypercenterC3×#102D9
hypocenterC9C3×#440
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
3. C3×#63449C2semidirect
5. C3×#63376C2semidirect
6. C3×#63375C2semidirect
7. C3×#63366C2semidirect
9. #10588437C3direct (N × H)
16. C3×#3672C4semidirect
31. #63449C2×C3semidirect
34. #63376C2×C3semidirect
35. #63375C2×C3semidirect
36. #63366C2×C3semidirect
48. C3×Dic18C2×C4semidirect
50. C3×C9⋊C8C2×C4semidirect
74. #3672C3×C4semidirect
136. Dic18C2×C3×C4semidirect
139. C9⋊C8C2×C3×C4semidirect
208. C3×C9#440semidirect
298. C9C3×#440semidirect
337. C3#10588437direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×Dic92
C2×C3×#36722
C3×#634492
C3×C9×#1022
C3×#633762
C3×#633752
C3×#633662
C3×#47573
#105884373