Finite Group Databasebeta

← C3×#10592737 #13163074

Subgroups — C3×#10592737 #13163074

Order 1728.

1180 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
1C33×#121C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3C3×#64139C211yesmaximal, characteristic, complemented
4C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7C3×#63634C211yesmaximal, characteristic, complemented
8#10592737C311yesmaximal, characteristic, direct product factor
9C3×#5038—34nomaximal
10C2×C33×D4—21no—
11C2×C33×D4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C2×C3×#638—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C33×C22⋊C4—21no—
17C3×#3753—22no—
18C3×#3752—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#121—24no—
24C32×#121D314yescomplemented
25C32×#121C2×C311yescharacteristic, complemented
26C2×#3753C2×C311yescharacteristic, complemented
27#64139C2×C311yescharacteristic, complemented
28C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
29C2×C3×#213—34no—
30C3×#930—34no—
31C22×C3×C3⋊C8—34no—
32#63659C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic, complemented
34#63634C2×C311yescharacteristic, complemented
35C3×#832—34no—
36C3×#809—34no—
37C3×#807—34no—
38C23×C33—21no—
39C23×C33D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C33×D4—42no—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C3×#638—42no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×C3⋊(C3⋊C8)D412yescomplemented
55C3×#466—91no—
56#5038—34no—
57C2×C32×D4—28no—
58C2×C32×D4—24no—
59C2×C32×D4—24no—
60C2×C32×D4—21no—
61C2×C32×D4C2×D314yes—
62C2×C32×D4C22×C311yescharacteristic
63C22×C32×C4—24no—
64C22×C32×C4C2×D314yes—
65C22×C32×C4C22×C311yescharacteristic
66C2×#638—21no—
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
68C2×C3×#67—64no—
69C2×C3×D12—34no—
70C32×C4⋊C4—24no—
71C32×C4⋊C4C2×D314yes—
72C32×C4⋊C4C22×C311yescharacteristic
73C32×C22⋊C4—28no—
74C32×C22⋊C4—24no—
75C32×C22⋊C4—21no—
76#3753—22no—
77#3752—21no—
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)—22no—
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×#213—68no—
83C3×(C2×D3)⋊C4—64no—
84C3×C4⋊Dic3—34no—
85C2×C3×C3⋊C8—34no—
86C2×C3×C3⋊C8—34no—
87C2×C3×C3⋊C8—68no—
88C22×C33—21no—
89C22×C33—21no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33—41no—
93C22×C33D812yescomplemented
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C33×C4—21no—
99C33×C4—41no—
100C33×C4C2×D411yescharacteristic
101C33×C4#4111yescharacteristic
102C3×C3⋊Dic3—41no—
103C2×C3×D8—91no—
104C22×C3×C8—91no—
105C3×#121—24no—
106C3×#121—91no—
107C3×#121C3×D314yescomplemented
108C3×#121C3⋊D311yescharacteristic, complemented
109C2×#213—34no—
110#930—34no—
111C22×C3⋊C8—34no—
112C3×#107—91no—
113C3×#102—91no—
114C3×#102—91no—
115#832—34no—
116#809—34no—
117#807—34no—
118C23×C32—28no—
119C23×C32—24no—
120C23×C32—21no—
121C23×C32—24no—
122C23×C32#6714yes—
123C23×C32C3×D411yescharacteristic
124C22×C3⋊D3—21no—
125C22×C3×D3—64no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—24no—
129C32×D4—24no—
130C32×D4—44no—
131C32×D4—44no—
132C32×D4—416no—
133C32×D4—48no—
134C32×D4—42no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—28no—
138C2×C32×C4—24no—
139C2×C32×C4—21no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—44no—
145C2×C32×C4—44no—
146C2×C32×C4#6714yes—
147C2×C32×C4C3×D411yescharacteristic
148C2×C32×C4C22×D314yes—
149C2×C32×C4C23×C311yescharacteristic
150#638—42no—
151C2×C3⋊Dic3—21no—
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)—21no—
154C3×#67—128no—
155C2×C3×Dic3—64no—
156C3×D12—64no—
157C3×D12—64no—
158C3⋊(C3⋊C8)—21no—
159C3⋊(C3⋊C8)C3×D412yescomplemented
160C3×C3⋊C8—38no—
161C3×C3⋊C8—64no—
162#466—91noSylow
163C2×C33—22no—
164C2×C33—81no—
165C2×C33#12111yescharacteristic
166C2×C33C2×D811yescharacteristic
167C2×C33#13511yescharacteristic
168C3×C3⋊D3—81no—
169C2×C3×D4—28no—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—24no—
173C2×C3×D4—91no—
174C2×C3×D4—181no—
175C2×C3×D4C2×C3×D314yes—
176C2×C3×D4C2×C3⋊D311yescharacteristic
177C22×C3×C4—24no—
178C22×C3×C4C2×C3×D314yes—
179C22×C3×C4C2×C3⋊D311yescharacteristic
180C2×#67—64no—
181C2×D12—34no—
182C3×D8—182no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C2×C3×C8—182no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—91no—
188C3×C4⋊C4C2×C3×D314yes—
189C3×C4⋊C4C2×C3⋊D311yescharacteristic
190C3×C22⋊C4—28no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—181no—
194#213—68no—
195(C2×D3)⋊C4—64no—
196C4⋊Dic3—34no—
197C2×C3⋊C8—34no—
198C2×C3⋊C8—34no—
199C2×C3⋊C8—68no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—28no—
203C22×C32—24no—
204C22×C32—24no—
205C22×C32—24no—
206C22×C32—44no—
207C22×C32—44no—
208C22×C32—48no—
209C22×C32—48no—
210C22×C32—48no—
211C22×C32—44no—
212C22×C32—44no—
213C22×C32—44no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32#21318yescomplemented
218C22×C32C3×D812yescomplemented
219C22×C32C2×#6714yes—
220C22×C32C2×C3×D411yescharacteristic
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3×D3—124no—
225C2×C3×D3—124no—
226C2×C3×D3—124no—
227C32×C4—21no—
228C32×C4—24no—
229C32×C4—24no—
230C32×C4—24no—
231C32×C4—44no—
232C32×C4—48no—
233C32×C4—44no—
234C32×C4—41no—
235C32×C4D3×D414yes—
236C32×C4#23714yes—
237C32×C4C2×C3×D411yescharacteristic
238C32×C4C3×#4111yescharacteristic
239C3⋊Dic3—41no—
240C3×Dic3—124no—
241C2×D8—91no—
242C22×C8—91no—
243#121—91no—
244#121C3×C3⋊D311yescharacteristic, complemented
245#107—91no—
246#102—91no—
247#102—91no—
248C33#46611yescharacteristic, complemented, Sylow
249C23×C3—28no—
250C23×C3—24no—
251C23×C3—21no—
252C23×C3—24no—
253C23×C3—181no—
254C23×C3C3×#6714yes—
255C23×C3#63811yescharacteristic
256C22×D3—64no—
257C3×D4—24no—
258C3×D4—24no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—44no—
262C3×D4—44no—
263C3×D4—416no—
264C3×D4—48no—
265C3×D4—42no—
266C3×D4—181no—
267C3×D4—181no—
268C3×D4—362no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—28no—
274C2×C3×C4—24no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—181no—
281C2×C3×C4C3×#6714yes—
282C2×C3×C4#63811yescharacteristic
283C2×C3×C4C22×C3×D314yes—
284C2×C3×C4C22×C3⋊D311yescharacteristic
285#67—128no—
286C2×Dic3—64no—
287D12—64no—
288D12—64no—
289C3×C8—92no—
290C3×C8—181no—
291C3⋊C8—38no—
292C3⋊C8—64no—
293C2×C32—22no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—28no—
298C2×C32—48no—
299C2×C32—88no—
300C2×C32—84no—
301C2×C32—81no—
302C2×C32C2×#21314yes—
303C2×C32#93914yes—
304C2×C32#95014yes—
305C2×C32C3×#12111yescharacteristic
306C2×C32C2×C3×D811yescharacteristic
307C2×C32C3×#13511yescharacteristic
308C3⋊D3—81no—
309C3×D3—244no—
310C2×D4—21no—
311C2×D4—91no—
312C2×D4—181no—
313C2×D4C2×C3×C3⋊D311yescharacteristic
314C22×C4C2×C3×C3⋊D311yescharacteristic
315D8—182no—
316C2×C8—91no—
317C2×C8—91no—
318C2×C8—182no—
319C4⋊C4—91no—
320C4⋊C4C2×C3×C3⋊D311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—181no—
323C22×C3—24no—
324C22×C3—24no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—28no—
328C22×C3—24no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—48no—
332C22×C3—48no—
333C22×C3—48no—
334C22×C3—44no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3C3×#21318yescomplemented
344C22×C3#375312yescomplemented
345C22×C3C2×C3×#6714yes—
346C22×C3C2×#63811yescharacteristic
347C2×D3—124no—
348C2×D3—124no—
349C2×D3—124no—
350C3×C4—24no—
351C3×C4—21no—
352C3×C4—24no—
353C3×C4—24no—
354C3×C4—44no—
355C3×C4—48no—
356C3×C4—44no—
357C3×C4—41no—
358C3×C4—361no—
359C3×C4C3×D3×D414yes—
360C3×C4C3×#23714yes—
361C3×C4D4×C3⋊D311yescharacteristic
362C3×C4#383011yescharacteristic
363Dic3—124no—
364C32—24no—
365C32#503814yescomplemented
366C32C3×#46611yescharacteristic, complemented
367C23—21no—
368C23—181no—
369C23C3×#63811yescharacteristic
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×#63811yescharacteristic
381C2×C4C22×C3×C3⋊D311yescharacteristic
382C8—92no—
383C8—181no—
384C2×C3—28no—
385C2×C3—22no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—48no—
390C2×C3—88no—
391C2×C3—84no—
392C2×C3—81no—
393C2×C3—721no—
394C2×C3C2×C3×#21314yes—
395C2×C3C3×#93914yes—
396C2×C3C3×#95014yes—
397C2×C3C2×#375311yescharacteristic
398C2×C3#6414811yescharacteristic
399C2×C3#6415911yescharacteristic
400D3—244no—
401C22—21no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—361no—
407C22—361no—
408C22—361no—
409C22C3×#375312yescomplemented
410C22C2×C3×#63811yescharacteristic
411C4—21no—
412C4—41no—
413C4—361no—
414C4C3×D4×C3⋊D311yescharacteristic
415C4C3×#383011yescharacteristic
416C3—24no—
417C3C3×#503814yescomplemented
418C3#1059273711yescharacteristic, direct product factor
419C2—22no—
420C2—81no—
421C2—721no—
422C2C2×C3×#375311yescharacteristic
423C2C3×#6414811yescharacteristic
424C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#121C2
socleC22×C33C2×D4
hypercenterC3×#121C3⋊D3
hypocenterC32C3×#466
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C33×#121C2semidirect
2. C2×C3×#3753C2semidirect
3. C3×#64139C2semidirect
4. C22×C3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63659C2semidirect
6. C3×#63636C2semidirect
7. C3×#63634C2semidirect
8. #10592737C3direct (N × H)
24. C32×#121D3semidirect
25. C32×#121C2×C3semidirect
26. C2×#3753C2×C3semidirect
27. #64139C2×C3semidirect
28. C22×C3⋊(C3⋊C8)C2×C3semidirect
32. #63659C2×C3semidirect
33. #63636C2×C3semidirect
34. #63634C2×C3semidirect
54. C3×C3⋊(C3⋊C8)D4semidirect
93. C22×C33D8semidirect
107. C3×#121C3×D3semidirect
108. C3×#121C3⋊D3semidirect
159. C3⋊(C3⋊C8)C3×D4semidirect
217. C22×C32#213semidirect
218. C22×C32C3×D8semidirect
244. #121C3×C3⋊D3semidirect
248. C33#466semidirect
343. C22×C3C3×#213semidirect
344. C22×C3#3753semidirect
365. C32#5038semidirect
366. C32C3×#466semidirect
409. C22C3×#3753semidirect
417. C3C3×#5038semidirect
418. C3#10592737direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1212
C2×C3×#37532
C3×#641392
C22×C3×C3⋊(C3⋊C8)2
C3×#636592
C3×#636362
C3×#636342
#105927373
C3×#50383