Finite Group Databasebeta

← C3×#10589940 #13153752

Subgroups — C3×#10589940 #13153752

Order 1728.

890 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×2−1+4C211yesmaximal, characteristic, complemented
2C3×#63716C211yesmaximal, characteristic, complemented
3C3×#63513C213yesmaximal, complemented
4C3×#63511C213yesmaximal, complemented
5C3×#63510C211yesmaximal, characteristic, complemented
6C3×#63505C213yesmaximal, complemented
7C2×C3×#3674C213yesmaximal, complemented
8C3×#5838—31nomaximal
9#10589940C311yesmaximal, characteristic, direct product factor
10C3×C9×#41—23no—
11C3×C9×#41C2213yes—
12C3×C9×#41C2211yescharacteristic
13C2×C3×Q8×C9—21no—
14C2×C3×Q8×C9C2213yes—
15C3×Q8×D9—21no—
16C3×#3699—23no—
17C3×#3697C2213yes—
18C2×C3×Dic18C2213yes—
19C3×#3675C2213yes—
20C3×#3674C2213yes—
21C3×#3674C2216yes—
22C3×#3673C2211yescharacteristic
23C3×#3672C2213yes—
24C3×#3667C2213yes—
25C2×C3×C9⋊C8C2213yes—
26C32×2−1+4D311yescharacteristic
27C3×#1010—31no—
28C3×#951—33no—
29C3×#950—33no—
30C3×#948—31no—
31C2×C3×#216—33no—
32C3×#942—33no—
33C9×2−1+4—21no—
34C9×2−1+4C2×C311yescharacteristic, complemented
35#63716C2×C311yescharacteristic, complemented
36#63513C2×C313yescomplemented
37#63511C2×C313yescomplemented
38#63510C2×C311yescharacteristic, complemented
39#63505C2×C313yescomplemented
40C2×#3674C2×C313yescomplemented
41C3×Q8×C9—23no—
42C3×Q8×C9D411yescharacteristic
43C3×Q8×C9C2313yes—
44C3×D4×C9—23no—
45C3×D4×C9D413yes—
46C3×D4×C9C2311yescharacteristic
47C2×C3×C4×C9—23no—
48C2×C3×C4×C9—23no—
49C2×C3×C4×C9C2313yes—
50C3×#611—23no—
51C2×C3×Dic9—23no—
52C3×D36C2311yescharacteristic
53C3×C4×D9—23no—
54C3×Dic18—23no—
55C3×Dic18C2313yes—
56C3×C9⋊C8C2311yescharacteristic
57C3×C9⋊C8C2313yes—
58C3×#578—91no—
59#5838—31no—
60C32×#41—23no—
61C32×#41C2×D313yes—
62C32×#41C2×D311yescharacteristic
63C2×C32×Q8—21no—
64C2×C32×Q8C2×D313yes—
65C3×D3×Q8—61no—
66C3×#237—63no—
67C3×#235—33no—
68C2×C3×Dic6—33no—
69C3×#216—36no—
70C3×#216—33no—
71C3×#215—33no—
72C3×#214—33no—
73C3×#213—31no—
74C3×#208—33no—
75C2×C3×C3⋊C8—33no—
76C9×#41—26no—
77C9×#41—23no—
78C9×#41—21no—
79C9×#41—23no—
80C9×#41C22×C313yes—
81C9×#41C22×C311yescharacteristic
82C2×Q8×C9—22no—
83C2×Q8×C9—23no—
84C2×Q8×C9—21no—
85C2×Q8×C9C22×C313yes—
86Q8×D9—21no—
87#3699—23no—
88#3697C22×C313yes—
89C2×Dic18C22×C313yes—
90#3675C22×C313yes—
91#3674C22×C313yes—
92#3674C22×C316yes—
93#3673C22×C311yescharacteristic
94#3672C22×C313yes—
95#3667C22×C313yes—
96C2×C9⋊C8C22×C313yes—
97C22×C3×C9—21no—
98C22×C3×C9C2×D413yes—
99C2×C3×D9—21no—
100C3×C4×C9—23no—
101C3×C4×C9C2×D413yes—
102C3×C4×C9C2411yescharacteristic
103C3×Dic9—23no—
104C3×2−1+4—21no—
105C3×2−1+4—91no—
106C3×2−1+4D911yescharacteristic, complemented
107C3×2−1+4C3×D311yescharacteristic
108#1010—31no—
109C3×#137—93no—
110C3×#137—93no—
111C3×#135—93no—
112C2×C3×Q16—93no—
113C3×#131—91no—
114#951—33no—
115#950—33no—
116#948—31no—
117C2×#216—33no—
118#942—33no—
119C32×Q8—23no—
120C32×Q8#6711yescharacteristic
121C32×Q8C22×D313yes—
122C32×D4—23no—
123C32×D4#6713yes—
124C32×D4C22×D311yescharacteristic
125C2×C32×C4—23no—
126C2×C32×C4—23no—
127C2×C32×C4C22×D313yes—
128C3×#67—63no—
129C2×C3×Dic3—63no—
130C3×D12—31no—
131C3×C4×D3—63no—
132C3×Dic6—33no—
133C3×Dic6—63no—
134C3×C3⋊C8—31no—
135C3×C3⋊C8—33no—
136Q8×C9—26no—
137Q8×C9—23no—
138Q8×C9—21no—
139Q8×C9—23no—
140Q8×C9C3×D411yescharacteristic
141Q8×C9C23×C313yes—
142D4×C9—26no—
143D4×C9—23no—
144D4×C9—23no—
145D4×C9—21no—
146D4×C9C3×D413yes—
147D4×C9C23×C311yescharacteristic
148C2×C4×C9—26no—
149C2×C4×C9—26no—
150C2×C4×C9—23no—
151C2×C4×C9—23no—
152C2×C4×C9—23no—
153C2×C4×C9C23×C313yes—
154#611—23no—
155C2×Dic9—23no—
156D36C23×C311yescharacteristic
157C4×D9—23no—
158Dic18—23no—
159Dic18C23×C313yes—
160C9⋊C8C23×C311yescharacteristic
161C9⋊C8C23×C313yes—
162#578—91noSylow
163C2×C3×C9—23no—
164C2×C3×C9—41no—
165C2×C3×C9C22×D411yescharacteristic
166C3×D9—41no—
167C3×#41—26no—
168C3×#41—23no—
169C3×#41—23no—
170C3×#41—23no—
171C3×#41—21no—
172C3×#41—93no—
173C3×#41—183no—
174C3×#41C2×D913yes—
175C3×#41C2×D911yescharacteristic
176C3×#41C2×C3×D313yes—
177C3×#41C2×C3×D311yescharacteristic
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—23no—
182C2×C3×Q8—93no—
183C2×C3×Q8—181no—
184C2×C3×Q8C2×D913yes—
185C2×C3×Q8C2×C3×D313yes—
186D3×Q8—61no—
187#237—63no—
188#235—33no—
189C2×Dic6—33no—
190C3×Q16—93no—
191C3×Q16—96no—
192C3×SD16—93no—
193C3×SD16—93no—
194C3×D8—91no—
195C3×M16—93no—
196C2×C3×C8—93no—
197#216—36no—
198#216—33no—
199#215—33no—
200#214—33no—
201#213—31no—
202#208—33no—
203C2×C3⋊C8—33no—
204C22×C32—21no—
205C22×C32C2×#6713yes—
206C2×C3×D3—61no—
207C32×C4—23no—
208C32×C4C2×#6713yes—
209C32×C4C23×D311yescharacteristic
210C3×Dic3—63no—
211C22×C9—22no—
212C22×C9—21no—
213C22×C9—23no—
214C22×C9C2×C3×D413yes—
215C2×D9—21no—
216C4×C9—26no—
217C4×C9—23no—
218C4×C9—21no—
219C4×C9—23no—
220C4×C9C2×C3×D413yes—
221C4×C9C24×C311yescharacteristic
222Dic9—23no—
2232−1+4—91no—
2242−1+4C3×D911yescharacteristic, complemented
225#137—93no—
226#137—93no—
227#135—93no—
228C2×Q16—93no—
229#131—91no—
230C3×C9#57811yescharacteristic, complemented, Sylow
231C3×Q8—26no—
232C3×Q8—23no—
233C3×Q8—23no—
234C3×Q8—21no—
235C3×Q8—23no—
236C3×Q8—93no—
237C3×Q8—183no—
238C3×Q8#61111yescharacteristic
239C3×Q8C22×D913yes—
240C3×Q8C3×#6711yescharacteristic
241C3×Q8C22×C3×D313yes—
242C3×D4—26no—
243C3×D4—23no—
244C3×D4—23no—
245C3×D4—23no—
246C3×D4—21no—
247C3×D4—91no—
248C3×D4—183no—
249C3×D4#61113yes—
250C3×D4C22×D911yescharacteristic
251C3×D4C3×#6713yes—
252C3×D4C22×C3×D311yescharacteristic
253C2×C3×C4—26no—
254C2×C3×C4—26no—
255C2×C3×C4—23no—
256C2×C3×C4—23no—
257C2×C3×C4—23no—
258C2×C3×C4—23no—
259C2×C3×C4—23no—
260C2×C3×C4—183no—
261C2×C3×C4—183no—
262C2×C3×C4C22×D913yes—
263C2×C3×C4C22×C3×D313yes—
264#67—63no—
265C2×Dic3—63no—
266D12—31no—
267C4×D3—63no—
268Dic6—33no—
269Dic6—63no—
270C3×C8—91no—
271C3×C8—93no—
272C3⋊C8—31no—
273C3⋊C8—33no—
274C2×C32—23no—
275C2×C32—41no—
276C2×C32C22×#6711yescharacteristic
277C3×D3—121no—
278C2×C9—23no—
279C2×C9—21no—
280C2×C9—43no—
281C2×C9—42no—
282C2×C9—41no—
283C2×C9C22×C3×D411yescharacteristic
284D9—41no—
285#41—23no—
286#41—93no—
287#41—183no—
288#41C2×C3×D913yes—
289#41C2×C3×D911yescharacteristic
290C2×Q8—21no—
291C2×Q8—93no—
292C2×Q8—181no—
293C2×Q8C2×C3×D913yes—
294Q16—93no—
295Q16—96no—
296SD16—93no—
297SD16—93no—
298D8—91no—
299M16—93no—
300C2×C8—93no—
301C22×C3—22no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—23no—
305C22×C3—181no—
306C22×C3C2×#61113yes—
307C22×C3C2×C3×#6713yes—
308C2×D3—61no—
309C3×C4—26no—
310C3×C4—23no—
311C3×C4—23no—
312C3×C4—21no—
313C3×C4—23no—
314C3×C4—183no—
315C3×C4C2×#61113yes—
316C3×C4C23×D911yescharacteristic
317C3×C4C2×C3×#6713yes—
318C3×C4C23×C3×D311yescharacteristic
319Dic3—63no—
320C32#583811yescharacteristic
321C9—21no—
322C9C3×#57811yescharacteristic, complemented
323Q8—23no—
324Q8—93no—
325Q8—183no—
326Q8C3×#61111yescharacteristic
327Q8C22×C3×D913yes—
328D4—23no—
329D4—91no—
330D4—183no—
331D4C3×#61113yes—
332D4C22×C3×D911yescharacteristic
333C2×C4—23no—
334C2×C4—23no—
335C2×C4—183no—
336C2×C4—183no—
337C2×C4C22×C3×D913yes—
338C8—91no—
339C8—93no—
340C2×C3—23no—
341C2×C3—23no—
342C2×C3—21no—
343C2×C3—43no—
344C2×C3—42no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—361no—
348C2×C3C22×#61111yescharacteristic
349C2×C3C22×C3×#6711yescharacteristic
350D3—121no—
351C22—21no—
352C22—181no—
353C22C2×C3×#61113yes—
354C4—23no—
355C4—183no—
356C4C2×C3×#61113yes—
357C4C23×C3×D911yescharacteristic
358C3—21no—
359C3#1058994011yescharacteristic, direct product factor
360C3C3×#583811yescharacteristic
361C2—23no—
362C2—41no—
363C2—361no—
364C2C22×C3×#61111yescharacteristic
subgroupG/H
centerC2×C3C22×#611
commutator, cosocleC4×C9C24×C3
FrattiniC3×C4C23×C3×D3
FittingC3×C9×2−1+4C2
socleC2×C32C22×#67
hypercenterC3×2−1+4D9
hypocenterC9C3×#578
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×C9×2−1+4C2semidirect
2. C3×#63716C2semidirect
3. C3×#63513C2semidirect
4. C3×#63511C2semidirect
5. C3×#63510C2semidirect
6. C3×#63505C2semidirect
7. C2×C3×#3674C2semidirect
9. #10589940C3direct (N × H)
34. C9×2−1+4C2×C3semidirect
35. #63716C2×C3semidirect
36. #63513C2×C3semidirect
37. #63511C2×C3semidirect
38. #63510C2×C3semidirect
39. #63505C2×C3semidirect
40. C2×#3674C2×C3semidirect
106. C3×2−1+4D9semidirect
224. 2−1+4C3×D9semidirect
230. C3×C9#578semidirect
322. C9C3×#578semidirect
359. C3#10589940direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×2−1+42
C3×#637162
C3×#635132
C3×#635112
C3×#635102
C3×#635052
C2×C3×#36742
C3×#58383
#105899403