Finite Group Databasebeta

← C3×#10588721 #13142816

Subgroups — C3×#10588721 #13142816

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#121C211yesmaximal, characteristic, complemented
2C2×C3×#3673C211yesmaximal, characteristic, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4C22×C3×C9⋊C8C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#5038—31nomaximal
9#10588721C311yesmaximal, characteristic, direct product factor
10C2×C3×D4×C9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C2×C3×#611—21no—
14C2×C3×D36C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C22⋊C4—21no—
17C3×#3673—22no—
18C3×(C2×D9)⋊C4—21no—
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C3×C9⋊C8—22no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C32×#121D311yescharacteristic
24C2×C3×#213—31no—
25C3×#930—31no—
26C22×C3×C3⋊C8—31no—
27C3×#832—31no—
28C3×#809—31no—
29C3×#807—31no—
30C9×#121—21no—
31C9×#121C2×C311yescharacteristic, complemented
32C2×#3673C2×C311yescharacteristic, complemented
33#63492C2×C311yescharacteristic, complemented
34C22×C9⋊C8C2×C311yescharacteristic, complemented
35#63392C2×C311yescharacteristic, complemented
36#63369C2×C311yescharacteristic, complemented
37#63366C2×C311yescharacteristic, complemented
38C23×C3×C9—21no—
39C23×C3×C9D411yescharacteristic
40C22×C3×D9—21no—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C3×D4×C9—42no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9D411yescharacteristic
48C2×C3×C4×C9C2311yescharacteristic
49C3×#611—42no—
50C2×C3×Dic9—21no—
51C3×D36—21no—
52C3×D36—21no—
53C3×C9⋊C8—21no—
54C3×C9⋊C8D412yescomplemented
55C3×#466—91no—
56#5038—31no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C2×C3×#67—61no—
61C2×C3×D12—31no—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C22⋊C4—21no—
64C3×#213—62no—
65C3×(C2×D3)⋊C4—61no—
66C3×C4⋊Dic3—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—62no—
70C2×D4×C9—22no—
71C2×D4×C9—21no—
72C2×D4×C9—21no—
73C2×D4×C9C22×C311yescharacteristic
74C22×C4×C9—21no—
75C22×C4×C9C22×C311yescharacteristic
76C2×#611—21no—
77C2×D36C22×C311yescharacteristic
78C9×C4⋊C4—21no—
79C9×C4⋊C4C22×C311yescharacteristic
80C9×C22⋊C4—22no—
81C9×C22⋊C4—21no—
82#3673—22no—
83(C2×D9)⋊C4—21no—
84C4⋊Dic9C22×C311yescharacteristic
85C2×C9⋊C8—22no—
86C2×C9⋊C8C22×C311yescharacteristic
87C2×C9⋊C8C22×C311yescharacteristic
88C22×C3×C9—21no—
89C22×C3×C9—21no—
90C22×C3×C9—41no—
91C22×C3×C9—41no—
92C22×C3×C9—41no—
93C22×C3×C9D812yescomplemented
94C22×C3×C9C2×D411yescharacteristic
95C2×C3×D9—41no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C3×C4×C9—21no—
99C3×C4×C9—41no—
100C3×C4×C9C2×D411yescharacteristic
101C3×C4×C9#4111yescharacteristic
102C3×Dic9—41no—
103C2×C3×D8—91no—
104C22×C3×C8—91no—
105C3×#121—21no—
106C3×#121—91no—
107C3×#121D911yescharacteristic, complemented
108C3×#121C3×D311yescharacteristic
109C2×#213—31no—
110#930—31no—
111C22×C3⋊C8—31no—
112C3×#107—91no—
113C3×#102—91no—
114C3×#102—91no—
115#832—31no—
116#809—31no—
117#807—31no—
118C23×C32—21no—
119C23×C32#6711yescharacteristic
120C22×C3×D3—61no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—42no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C3×#67—122no—
130C2×C3×Dic3—61no—
131C3×D12—61no—
132C3×D12—61no—
133C23×C9—22no—
134C23×C9—21no—
135C23×C9—21no—
136C23×C9C3×D411yescharacteristic
137C22×D9—21no—
138C3×C3⋊C8—32no—
139C3×C3⋊C8—61no—
140D4×C9—21no—
141D4×C9—21no—
142D4×C9—41no—
143D4×C9—41no—
144D4×C9—44no—
145D4×C9—42no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—22no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—41no—
153C2×C4×C9—41no—
154C2×C4×C9C3×D411yescharacteristic
155C2×C4×C9C23×C311yescharacteristic
156#611—42no—
157C2×Dic9—21no—
158D36—21no—
159D36—21no—
160C9⋊C8—21no—
161C9⋊C8C3×D412yescomplemented
162#466—91noSylow
163C2×C3×C9—22no—
164C2×C3×C9—81no—
165C2×C3×C9#12111yescharacteristic
166C2×C3×C9C2×D811yescharacteristic
167C2×C3×C9#13511yescharacteristic
168C3×D9—81no—
169C2×C3×D4—22no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—91no—
174C2×C3×D4—181no—
175C2×C3×D4C2×D911yescharacteristic
176C2×C3×D4C2×C3×D311yescharacteristic
177C22×C3×C4—21no—
178C22×C3×C4C2×D911yescharacteristic
179C22×C3×C4C2×C3×D311yescharacteristic
180C2×#67—61no—
181C2×D12—31no—
182C3×D8—182no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C2×C3×C8—182no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—91no—
188C3×C4⋊C4C2×D911yescharacteristic
189C3×C4⋊C4C2×C3×D311yescharacteristic
190C3×C22⋊C4—22no—
191C3×C22⋊C4—21no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—181no—
194#213—62no—
195(C2×D3)⋊C4—61no—
196C4⋊Dic3—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—62no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32#21312yes—
206C22×C32C2×#6711yescharacteristic
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—21no—
211C32×C4—41no—
212C32×C4D3×D411yescharacteristic
213C32×C4#23711yescharacteristic
214C3×Dic3—121no—
215C22×C9—21no—
216C22×C9—21no—
217C22×C9—22no—
218C22×C9—21no—
219C22×C9—41no—
220C22×C9—41no—
221C22×C9—42no—
222C22×C9—42no—
223C22×C9—42no—
224C22×C9—41no—
225C22×C9—41no—
226C22×C9—41no—
227C22×C9C3×D812yescomplemented
228C22×C9C2×C3×D411yescharacteristic
229C2×D9—41no—
230C2×D9—41no—
231C2×D9—41no—
232C4×C9—21no—
233C4×C9—21no—
234C4×C9—21no—
235C4×C9—41no—
236C4×C9—42no—
237C4×C9—41no—
238C4×C9C2×C3×D411yescharacteristic
239C4×C9C3×#4111yescharacteristic
240Dic9—41no—
241C2×D8—91no—
242C22×C8—91no—
243#121—91no—
244#121C3×D911yescharacteristic, complemented
245#107—91no—
246#102—91no—
247#102—91no—
248C3×C9#46611yescharacteristic, complemented, Sylow
249C23×C3—22no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—181no—
254C23×C3#61111yescharacteristic
255C23×C3C3×#6711yescharacteristic
256C22×D3—61no—
257C3×D4—21no—
258C3×D4—21no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—41no—
262C3×D4—41no—
263C3×D4—44no—
264C3×D4—42no—
265C3×D4—42no—
266C3×D4—181no—
267C3×D4—181no—
268C3×D4—362no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—181no—
281C2×C3×C4#61111yescharacteristic
282C2×C3×C4C22×D911yescharacteristic
283C2×C3×C4C3×#6711yescharacteristic
284C2×C3×C4C22×C3×D311yescharacteristic
285#67—122no—
286C2×Dic3—61no—
287D12—61no—
288D12—61no—
289C3×C8—92no—
290C3×C8—181no—
291C3⋊C8—32no—
292C3⋊C8—61no—
293C2×C32—22no—
294C2×C32—81no—
295C2×C32C2×#21311yescharacteristic
296C2×C32#93911yescharacteristic
297C2×C32#95011yescharacteristic
298C3×D3—241no—
299C2×C9—22no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—42no—
304C2×C9—82no—
305C2×C9—81no—
306C2×C9C3×#12111yescharacteristic
307C2×C9C2×C3×D811yescharacteristic
308C2×C9C3×#13511yescharacteristic
309D9—81no—
310C2×D4—21no—
311C2×D4—91no—
312C2×D4—181no—
313C2×D4C2×C3×D911yescharacteristic
314C22×C4C2×C3×D911yescharacteristic
315D8—182no—
316C2×C8—91no—
317C2×C8—91no—
318C2×C8—182no—
319C4⋊C4—91no—
320C4⋊C4C2×C3×D911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—181no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3#367312yescomplemented
344C22×C3C2×#61111yescharacteristic
345C22×C3C3×#21312yes—
346C22×C3C2×C3×#6711yescharacteristic
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—121no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—41no—
355C3×C4—42no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—361no—
359C3×C4D4×D911yescharacteristic
360C3×C4#369911yescharacteristic
361C3×C4C3×D3×D411yescharacteristic
362C3×C4C3×#23711yescharacteristic
363Dic3—121no—
364C32#503811yescharacteristic
365C9—21no—
366C9C3×#46611yescharacteristic, complemented
367C23—21no—
368C23—181no—
369C23C3×#61111yescharacteristic
370D4—21no—
371D4—21no—
372D4—42no—
373D4—181no—
374D4—181no—
375D4—362no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—181no—
380C2×C4C3×#61111yescharacteristic
381C2×C4C22×C3×D911yescharacteristic
382C8—92no—
383C8—181no—
384C2×C3—22no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—42no—
390C2×C3—82no—
391C2×C3—81no—
392C2×C3—81no—
393C2×C3—721no—
394C2×C3C2×#367311yescharacteristic
395C2×C3#6350111yescharacteristic
396C2×C3#6351311yescharacteristic
397C2×C3C2×C3×#21311yescharacteristic
398C2×C3C3×#93911yescharacteristic
399C2×C3C3×#95011yescharacteristic
400D3—241no—
401C22—21no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—361no—
407C22—361no—
408C22—361no—
409C22C3×#367312yescomplemented
410C22C2×C3×#61111yescharacteristic
411C4—21no—
412C4—41no—
413C4—361no—
414C4C3×D4×D911yescharacteristic
415C4C3×#369911yescharacteristic
416C3—21no—
417C3#1058872111yescharacteristic, direct product factor
418C3C3×#503811yescharacteristic
419C2—22no—
420C2—81no—
421C2—721no—
422C2C2×C3×#367311yescharacteristic
423C2C3×#6350111yescharacteristic
424C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#121C2
socleC22×C32C2×#67
hypercenterC3×#121D9
hypocenterC9C3×#466
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C3×C9×#121C2semidirect
2. C2×C3×#3673C2semidirect
3. C3×#63492C2semidirect
4. C22×C3×C9⋊C8C2semidirect
5. C3×#63392C2semidirect
6. C3×#63369C2semidirect
7. C3×#63366C2semidirect
9. #10588721C3direct (N × H)
31. C9×#121C2×C3semidirect
32. C2×#3673C2×C3semidirect
33. #63492C2×C3semidirect
34. C22×C9⋊C8C2×C3semidirect
35. #63392C2×C3semidirect
36. #63369C2×C3semidirect
37. #63366C2×C3semidirect
54. C3×C9⋊C8D4semidirect
93. C22×C3×C9D8semidirect
107. C3×#121D9semidirect
161. C9⋊C8C3×D4semidirect
227. C22×C9C3×D8semidirect
244. #121C3×D9semidirect
248. C3×C9#466semidirect
343. C22×C3#3673semidirect
366. C9C3×#466semidirect
409. C22C3×#3673semidirect
417. C3#10588721direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1212
C2×C3×#36732
C3×#634922
C22×C3×C9⋊C82
C3×#633922
C3×#633692
C3×#633662
C3×#50383
#105887213