Finite Group Databasebeta

← C3×#10588686 #13142681

Subgroups — C3×#10588686 #13142681

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C3×C9×#117C211yesmaximal, characteristic, complemented
3C3×#63488C211yesmaximal, characteristic, complemented
4C3×#63365C214yesmaximal, complemented
5C3×#5006—31nomaximal
6#10588686C311yesmaximal, characteristic, direct product factor
7C22×C3×C4×C9C2211yescharacteristic
8C2×C3×#611C2212yes—
9C3×#3697—21no—
10C3×#3697—21no—
11C3×#3697—21no—
12C3×#3697—21no—
13C2×C3×D36C411yescharacteristic, complemented
14C2×C3×C4×D9C412yescomplemented
15C2×C3×Dic18C411yescharacteristic, complemented
16C3×C9×C4⋊C4—21no—
17C3×C9×C22⋊C4C2212yes—
18C3×C42×C9—21no—
19C3×#3676C2212yes—
20C3×C4⋊Dic9—21no—
21C3×C4×Dic9—21no—
22C2×C3×#235—31no—
23C32×#117D311yescharacteristic
24C3×#926—31no—
25C2×#3697C2×C311yescharacteristic
26C3×#806—34no—
27C9×#117—21no—
28C9×#117C2×C311yescharacteristic, complemented
29#63488C2×C311yescharacteristic, complemented
30#63365C2×C314yescomplemented
31C23×C3×C9C2311yescharacteristic
32C22×C3×D9C2×C412yes—
33C2×C3×C4×C9—22no—
34C2×C3×C4×C9C2×C411yescharacteristic
35C2×C3×C4×C9C2×C411yescharacteristic
36C2×C3×C4×C9D411yescharacteristic
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9D411yescharacteristic
39C2×C3×C4×C9D411yescharacteristic
40C3×#611—22no—
41C3×#611—22no—
42C2×C3×Dic9—22no—
43C2×C3×Dic9C2×C412yes—
44C3×D36—21no—
45C3×D36—21no—
46C3×C4×D9—22no—
47C3×C4×D9—22no—
48C3×Dic18—21no—
49C3×Dic18—21no—
50C3×#410—91no—
51#5006—31no—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67—32no—
54C3×#235—61no—
55C3×#235—61no—
56C3×#235—61no—
57C3×#235—61no—
58C2×C3×D12—31no—
59C2×C3×C4×D3—32no—
60C2×C3×Dic6—31no—
61C32×C4⋊C4—21no—
62C32×C22⋊C4C2×D312yes—
63C32×C42—21no—
64C3×#217—32no—
65C3×C4⋊Dic3—61no—
66C3×C4×Dic3—61no—
67C22×C4×C9—21no—
68C22×C4×C9C22×C311yescharacteristic
69C2×#611C22×C312yes—
70#3697—21no—
71#3697—21no—
72#3697—21no—
73#3697—21no—
74C2×D36C3×C411yescharacteristic, complemented
75C2×C4×D9C3×C412yescomplemented
76C2×Dic18C3×C411yescharacteristic, complemented
77C9×C4⋊C4—22no—
78C9×C4⋊C4—21no—
79C9×C22⋊C4—22no—
80C9×C22⋊C4C22×C312yes—
81C42×C9—22no—
82C42×C9—21no—
83#3676C22×C312yes—
84C4⋊Dic9—21no—
85C4×Dic9—21no—
86C22×C3×C9—41no—
87C22×C3×C9C22×C411yescharacteristic
88C22×C3×C9C2×D411yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—22no—
91C2×C3×D9—42no—
92C3×C4×C9—42no—
93C3×C4×C9C22⋊C411yescharacteristic
94C3×C4×C9C22⋊C411yescharacteristic
95C3×C4×C9C22⋊C411yescharacteristic
96C3×C4×C9C22⋊C411yescharacteristic
97C3×Dic9—22no—
98C3×Dic9—42no—
99C2×C3×#41—91no—
100C2×#235—31no—
101C3×#117—21no—
102C3×#117—91no—
103C3×#117D911yescharacteristic, complemented
104C3×#117C3×D311yescharacteristic
105#926—31no—
106C3×#99—94no—
107#806—34no—
108C23×C32C22×D311yescharacteristic
109C22×C3×D3—32no—
110C2×C32×C4—22no—
111C2×C32×C4C4×D311yescharacteristic
112C2×C32×C4C4×D311yescharacteristic
113C2×C32×C4D1211yescharacteristic
114C2×C32×C4D1211yescharacteristic
115C2×C32×C4#6711yescharacteristic
116C2×C32×C4#6711yescharacteristic
117C3×#67—62no—
118C3×#67—62no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—62no—
121C3×D12—61no—
122C3×D12—61no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C23×C9—21no—
128C23×C9C23×C311yescharacteristic
129C22×D9C2×C3×C412yes—
130C2×C4×C9—24no—
131C2×C4×C9—22no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9C2×C3×C411yescharacteristic
139C2×C4×C9C2×C3×C411yescharacteristic
140C2×C4×C9C3×D411yescharacteristic
141C2×C4×C9C3×D411yescharacteristic
142C2×C4×C9C3×D411yescharacteristic
143C2×C4×C9C3×D411yescharacteristic
144#611—22no—
145#611—22no—
146C2×Dic9—22no—
147C2×Dic9C2×C3×C412yes—
148D36—21no—
149D36—21no—
150C4×D9—22no—
151C4×D9—22no—
152Dic18—21no—
153Dic18—21no—
154#410—91noSylow
155C2×C3×C9—21no—
156C2×C3×C9—21no—
157C2×C3×C9—21no—
158C2×C3×C9C2×C22⋊C411yescharacteristic
159C3×D9—42no—
160C3×#41—181no—
161C3×#41—181no—
162C3×#41—181no—
163C3×#41—181no—
164C2×C3×Q8—91no—
165C2×C3×D4—92no—
166C2×C3×D4—91no—
167C22×C3×C4—21no—
168C22×C3×C4—92no—
169C22×C3×C4C2×D911yescharacteristic
170C22×C3×C4C2×C3×D311yescharacteristic
171C2×#67—32no—
172#235—61no—
173#235—61no—
174#235—61no—
175#235—61no—
176C2×D12—31no—
177C2×C4×D3—32no—
178C2×Dic6—31no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—181no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—92no—
185C3×C22⋊C4C2×D912yes—
186C3×C22⋊C4C2×C3×D312yes—
187C3×C42—22no—
188C3×C42—21no—
189C3×C42—21no—
190C3×C42—181no—
191#217—32no—
192C4⋊Dic3—61no—
193C4×Dic3—61no—
194C22×C32—41no—
195C22×C32C2×C4×D311yescharacteristic
196C22×C32C2×D1211yescharacteristic
197C22×C32C2×#6711yescharacteristic
198C2×C3×D3—62no—
199C2×C3×D3—122no—
200C32×C4—42no—
201C32×C4(C2×D3)⋊C411yescharacteristic
202C32×C4(C2×D3)⋊C411yescharacteristic
203C32×C4(C2×D3)⋊C411yescharacteristic
204C32×C4(C2×D3)⋊C411yescharacteristic
205C3×Dic3—62no—
206C3×Dic3—122no—
207C22×C9—21no—
208C22×C9—21no—
209C22×C9—21no—
210C22×C9—41no—
211C22×C9—42no—
212C22×C9C22×C3×C411yescharacteristic
213C22×C9C2×C3×D411yescharacteristic
214C22×C9C2×C3×D411yescharacteristic
215C2×D9—22no—
216C2×D9—42no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—44no—
222C4×C9—42no—
223C4×C9C3×C22⋊C411yescharacteristic
224C4×C9C3×C22⋊C411yescharacteristic
225C4×C9C3×C22⋊C411yescharacteristic
226C4×C9C3×C22⋊C411yescharacteristic
227Dic9—22no—
228Dic9—42no—
229C2×#41—91no—
230#117—91no—
231#117C3×D911yescharacteristic, complemented
232#99—94no—
233C3×C9#41011yescharacteristic, complemented, Sylow
234C23×C3—21no—
235C23×C3—92no—
236C23×C3C22×D911yescharacteristic
237C23×C3C22×C3×D311yescharacteristic
238C22×D3—32no—
239C3×Q8—181no—
240C3×Q8—181no—
241C3×D4—182no—
242C3×D4—182no—
243C3×D4—181no—
244C3×D4—181no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—92no—
255C2×C3×C4—182no—
256C2×C3×C4—182no—
257C2×C3×C4—182no—
258C2×C3×C4C4×D911yescharacteristic
259C2×C3×C4C4×D911yescharacteristic
260C2×C3×C4D3611yescharacteristic
261C2×C3×C4D3611yescharacteristic
262C2×C3×C4#61111yescharacteristic
263C2×C3×C4#61111yescharacteristic
264C2×C3×C4C3×C4×D311yescharacteristic
265C2×C3×C4C3×C4×D311yescharacteristic
266C2×C3×C4C3×D1211yescharacteristic
267C2×C3×C4C3×D1211yescharacteristic
268C2×C3×C4C3×#6711yescharacteristic
269C2×C3×C4C3×#6711yescharacteristic
270#67—62no—
271#67—62no—
272C2×Dic3—32no—
273C2×Dic3—62no—
274D12—61no—
275D12—61no—
276C4×D3—62no—
277C4×D3—62no—
278Dic6—61no—
279Dic6—61no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32C2×(C2×D3)⋊C411yescharacteristic
284C3×D3—122no—
285C2×C9—22no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9—41no—
291C2×C9—41no—
292C2×C9C2×C3×C22⋊C411yescharacteristic
293D9—42no—
294#41—181no—
295#41—181no—
296#41—181no—
297#41—181no—
298C2×Q8—91no—
299C2×D4—92no—
300C2×D4—91no—
301C22×C4—92no—
302C22×C4C2×C3×D911yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—181no—
305C22⋊C4—92no—
306C22⋊C4C2×C3×D912yes—
307C42—21no—
308C42—181no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—42no—
315C22×C3—182no—
316C22×C3—362no—
317C22×C3C2×C4×D911yescharacteristic
318C22×C3C2×D3611yescharacteristic
319C22×C3C2×#61111yescharacteristic
320C22×C3C2×C3×C4×D311yescharacteristic
321C22×C3C2×C3×D1211yescharacteristic
322C22×C3C2×C3×#6711yescharacteristic
323C2×D3—62no—
324C2×D3—122no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—42no—
330C3×C4—44no—
331C3×C4—42no—
332C3×C4—182no—
333C3×C4—362no—
334C3×C4(C2×D9)⋊C411yescharacteristic
335C3×C4(C2×D9)⋊C411yescharacteristic
336C3×C4(C2×D9)⋊C411yescharacteristic
337C3×C4(C2×D9)⋊C411yescharacteristic
338C3×C4C3×(C2×D3)⋊C411yescharacteristic
339C3×C4C3×(C2×D3)⋊C411yescharacteristic
340C3×C4C3×(C2×D3)⋊C411yescharacteristic
341C3×C4C3×(C2×D3)⋊C411yescharacteristic
342Dic3—62no—
343Dic3—122no—
344C32#500611yescharacteristic
345C9—21no—
346C9C3×#41011yescharacteristic, complemented
347C23—92no—
348C23C22×C3×D911yescharacteristic
349Q8—181no—
350Q8—181no—
351D4—182no—
352D4—182no—
353D4—181no—
354D4—181no—
355C2×C4—22no—
356C2×C4—92no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4C3×C4×D911yescharacteristic
361C2×C4C3×C4×D911yescharacteristic
362C2×C4C3×D3611yescharacteristic
363C2×C4C3×D3611yescharacteristic
364C2×C4C3×#61111yescharacteristic
365C2×C4C3×#61111yescharacteristic
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—362no—
377C2×C3C2×(C2×D9)⋊C411yescharacteristic
378C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
379D3—122no—
380C22—41no—
381C22—182no—
382C22—362no—
383C22C2×C3×C4×D911yescharacteristic
384C22C2×C3×D3611yescharacteristic
385C22C2×C3×#61111yescharacteristic
386C4—42no—
387C4—182no—
388C4—362no—
389C4C3×(C2×D9)⋊C411yescharacteristic
390C4C3×(C2×D9)⋊C411yescharacteristic
391C4C3×(C2×D9)⋊C411yescharacteristic
392C4C3×(C2×D9)⋊C411yescharacteristic
393C3—21no—
394C3#1058868611yescharacteristic, direct product factor
395C3C3×#500611yescharacteristic
396C2—21no—
397C2—21no—
398C2—21no—
399C2—362no—
400C2C2×C3×(C2×D9)⋊C411yescharacteristic
subgroupG/H
centerC3×C4(C2×D9)⋊C4
commutatorC22×C9C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#117C2
socleC2×C32C2×(C2×D3)⋊C4
prime radicalC2×C3×#3697C2
cosocleC23×C9C23×C3
hypercenterC3×#117D9
hypocenterC9C3×#410
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C3×C9×#117C2semidirect
3. C3×#63488C2semidirect
4. C3×#63365C2semidirect
6. #10588686C3direct (N × H)
13. C2×C3×D36C4semidirect
14. C2×C3×C4×D9C4semidirect
15. C2×C3×Dic18C4semidirect
28. C9×#117C2×C3semidirect
29. #63488C2×C3semidirect
30. #63365C2×C3semidirect
74. C2×D36C3×C4semidirect
75. C2×C4×D9C3×C4semidirect
76. C2×Dic18C3×C4semidirect
103. C3×#117D9semidirect
231. #117C3×D9semidirect
233. C3×C9#410semidirect
346. C9C3×#410semidirect
394. C3#10588686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C3×C9×#1172
C3×#634882
C3×#633652
C3×#50063
#105886863