Finite Group Databasebeta

← C3×#10592472 #13162810

Subgroups — C3×#10592472 #13162810

Order 1728.

1388 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3C2×C3×#3745C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C33×#102C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592472C311yesmaximal, characteristic, direct product factor
9C3×#4773—34nomaximal
10C23×C3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3—22no—
15C2×C3×C4⋊(C3⋊D3)—21no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C33×C8C2211yescharacteristic, complemented
20C33×C4⋊C4C2211yescharacteristic, complemented
21C3×#3753—22no—
22C3×#3752—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
24C3×#3745—22no—
25C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D3×D4—34no—
27C2×#3753C2×C311yescharacteristic, complemented
28C2×#3745C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic, complemented
30C2×C3×#213—34no—
31C2×C3×D24—34no—
32C3×#895—34no—
33C32×#102—24no—
34C32×#102D314yescomplemented
35C32×#102C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63636C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39C3×#809—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊D3—42no—
45C22×C3×C3⋊D3—42no—
46C22×C3×C3⋊D3D411yescharacteristic
47C33×D4—21no—
48C33×D4D412yescomplemented
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C3×#638—24no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—42no—
57C3×C4⋊(C3⋊D3)D412yescomplemented
58C3×C4×C3⋊D3—21no—
59C3×C4×C3⋊D3—21no—
60C33×C8—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×#447—91no—
63#4773—34no—
64C23×C3⋊D3—21no—
65C23×C3×D3—64no—
66C2×C32×D4—24no—
67C2×C32×D4C2×D314yes—
68C2×C32×D4C22×C311yescharacteristic
69C2×#638—21no—
70D4×C3⋊D3—22no—
71D4×C3⋊D3—22no—
72C2×C4⋊(C3⋊D3)—21no—
73C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
74C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
75C2×C4×C3⋊D3C22×C311yescharacteristic
76C2×C3×#67—64no—
77C3×D3×D4—68no—
78C3×D3×D4—68no—
79C2×C3×D12—34no—
80C2×C3×D12—34no—
81C2×C3×D12—64no—
82C2×C3×C4×D3—34no—
83C2×C32×C8—24no—
84C2×C32×C8C2×D314yescomplemented
85C2×C32×C8C22×C311yescharacteristic, complemented
86C32×C4⋊C4—24no—
87C32×C4⋊C4C2×D314yescomplemented
88C32×C4⋊C4C22×C311yescharacteristic, complemented
89#3753—22no—
90#3752—21no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
92#3745—22no—
93C3×#213—68no—
94C3×(C2×D3)⋊C4—64no—
95C2×C3×C3⋊C8—34no—
96C3×D24—68no—
97C22×C33—22no—
98C22×C33—41no—
99C22×C33—41no—
100C22×C33C2×D411yescharacteristic
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—22no—
104C2×C3×C3⋊D3—48no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3D812yescomplemented
111C33×C4—41no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3×C3⋊Dic3—21no—
115C22×C3×D4—91no—
116C2×D3×D4—34no—
117C2×C3×D8—91no—
118C2×C3×D8—91no—
119C3×#121—91no—
120C2×#213—34no—
121C2×D24—34no—
122#895—34no—
123C3×#102—24no—
124C3×#102—91no—
125C3×#102C3×D314yescomplemented
126C3×#102C3⋊D311yescharacteristic, complemented
127C3×C22⋊C8—91no—
128(C2×D3)⋊C8—34no—
129#809—34no—
130C23×C32—21no—
131C23×C32—24no—
132C23×C32—44no—
133C22×C3⋊D3—24no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3—42no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3C3×D411yescharacteristic
139C22×C3×D3—34no—
140C22×C3×D3—616no—
141C22×C3×D3—64no—
142C22×C3×D3—64no—
143C22×C3×D3—128no—
144C22×C3×D3—128no—
145C32×D4—21no—
146C32×D4—28no—
147C32×D4—24no—
148C32×D4—44no—
149C32×D4D1218yescomplemented
150C32×D4C3×D412yescomplemented
151C2×C32×C4—21no—
152C2×C32×C4—24no—
153C2×C32×C4—24no—
154C2×C32×C4—44no—
155C2×C32×C4C22×D314yes—
156C2×C32×C4C23×C311yescharacteristic
157#638—24no—
158C2×C3⋊Dic3C3×D411yescharacteristic
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)—21no—
161C4⋊(C3⋊D3)—21no—
162C4⋊(C3⋊D3)—42no—
163C4⋊(C3⋊D3)C3×D412yescomplemented
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166C3×#67—616no—
167C2×C3×Dic3—34no—
168C3×D12—38no—
169C3×D12—64no—
170C3×D12—64no—
171C3×D12—64no—
172C3×D12—128no—
173C3×C4×D3—64no—
174C3×C4×D3—64no—
175C32×C8—21no—
176C32×C8—24no—
177C32×C8—44no—
178C3⋊(C3⋊C8)—21no—
179C3×C3⋊C8—64no—
180#447—91noSylow
181C2×C33—42no—
182C2×C33#12011yescharacteristic
183C2×C33C2×D811yescharacteristic
184C2×C33AGL(1,ℤ/8)11yescharacteristic
185C3×C3⋊D3—22no—
186C3×C3⋊D3—42no—
187C3×C3⋊D3—81no—
188C24×C3—181no—
189C23×D3—64no—
190C2×C3×D4—24no—
191C2×C3×D4—91no—
192C2×C3×D4—91no—
193C2×C3×D4—181no—
194C2×C3×D4—181no—
195C2×C3×D4—182no—
196C2×C3×D4—182no—
197C2×C3×D4C2×C3×D314yes—
198C2×C3×D4C2×C3⋊D311yescharacteristic
199C22×C3×C4—91no—
200C2×#67—64no—
201D3×D4—68no—
202D3×D4—68no—
203C2×D12—34no—
204C2×D12—34no—
205C2×D12—64no—
206C2×C4×D3—34no—
207C3×D8—182no—
208C3×D8—182no—
209C2×C3×C8—24no—
210C2×C3×C8—91no—
211C2×C3×C8C2×C3×D314yescomplemented
212C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
213C3×C4⋊C4—24no—
214C3×C4⋊C4C2×C3×D314yescomplemented
215C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
216C3×C22⋊C4—181no—
217#213—68no—
218(C2×D3)⋊C4—64no—
219C2×C3⋊C8—34no—
220D24—68no—
221C22×C32—22no—
222C22×C32—24no—
223C22×C32—28no—
224C22×C32—48no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—44no—
228C22×C32—44no—
229C22×C32—84no—
230C22×C32—84no—
231C22×C32D3×D414yes—
232C22×C32C2×C3×D411yescharacteristic
233C2×C3⋊D3—21no—
234C2×C3⋊D3—21no—
235C2×C3⋊D3—22no—
236C2×C3⋊D3—48no—
237C2×C3⋊D3—41no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—41no—
240C2×C3⋊D3—41no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3C3×D812yescomplemented
243C2×C3×D3—38no—
244C2×C3×D3—64no—
245C2×C3×D3—64no—
246C2×C3×D3—68no—
247C2×C3×D3—1232no—
248C2×C3×D3—124no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252C2×C3×D3—124no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4—41no—
256C32×C4—44no—
257C32×C4—84no—
258C32×C4C2×D1214yes—
259C32×C4D3×D414yes—
260C32×C4C2×C3×D411yescharacteristic
261C32×C4C2×C3×D411yescharacteristic
262C3⋊Dic3—21no—
263C3×Dic3—64no—
264C22×D4—91no—
265C2×D8—91no—
266C2×D8—91no—
267#121—91no—
268#102—91no—
269#102C3×C3⋊D311yescharacteristic, complemented
270C22⋊C8—91no—
271C33#44711yescharacteristic, complemented, Sylow
272C23×C3—24no—
273C23×C3—21no—
274C23×C3—44no—
275C23×C3—91no—
276C23×C3—184no—
277C23×C3—181no—
278C23×C3—181no—
279C23×C3—362no—
280C23×C3—362no—
281C22×D3—34no—
282C22×D3—616no—
283C22×D3—64no—
284C22×D3—64no—
285C22×D3—128no—
286C22×D3—128no—
287C3×D4—24no—
288C3×D4—21no—
289C3×D4—28no—
290C3×D4—44no—
291C3×D4—92no—
292C3×D4—184no—
293C3×D4—181no—
294C3×D4—181no—
295C3×D4—181no—
296C3×D4—362no—
297C3×D4C3×D1218yescomplemented
298C3×D4C4⋊(C3⋊D3)12yescomplemented
299C2×C3×C4—24no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—44no—
303C2×C3×C4—91no—
304C2×C3×C4—181no—
305C2×C3×C4—181no—
306C2×C3×C4C22×C3×D314yes—
307C2×C3×C4C22×C3⋊D311yescharacteristic
308#67—616no—
309C2×Dic3—34no—
310D12—38no—
311D12—64no—
312D12—64no—
313D12—64no—
314D12—128no—
315C4×D3—64no—
316C4×D3—64no—
317C3×C8—24no—
318C3×C8—21no—
319C3×C8—44no—
320C3×C8—181no—
321C3⋊C8—64no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—42no—
326C2×C32—48no—
327C2×C32—88no—
328C2×C32#88214yes—
329C2×C32D3×D814yes—
330C2×C32#91414yes—
331C2×C32C3×#12011yescharacteristic
332C2×C32C2×C3×D811yescharacteristic
333C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
334C3⋊D3—22no—
335C3⋊D3—42no—
336C3⋊D3—81no—
337C3×D3—68no—
338C3×D3—128no—
339C3×D3—244no—
340C24—181no—
341C2×D4—91no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—181no—
345C2×D4—182no—
346C2×D4—182no—
347C2×D4C2×C3×C3⋊D311yescharacteristic
348C22×C4—91no—
349D8—182no—
350D8—182no—
351C2×C8—91no—
352C2×C8C2×C3×C3⋊D311yescharacteristic, complemented
353C4⋊C4C2×C3×C3⋊D311yescharacteristic, complemented
354C22⋊C4—181no—
355C22×C3—28no—
356C22×C3—22no—
357C22×C3—24no—
358C22×C3—48no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—84no—
364C22×C3—84no—
365C22×C3—92no—
366C22×C3—181no—
367C22×C3—181no—
368C22×C3—182no—
369C22×C3—368no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3—361no—
374C22×C3—361no—
375C22×C3C3×D3×D414yes—
376C22×C3D4×C3⋊D311yescharacteristic
377C2×D3—38no—
378C2×D3—64no—
379C2×D3—64no—
380C2×D3—68no—
381C2×D3—1232no—
382C2×D3—124no—
383C2×D3—124no—
384C2×D3—124no—
385C2×D3—124no—
386C2×D3—124no—
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—44no—
390C3×C4—41no—
391C3×C4—84no—
392C3×C4—181no—
393C3×C4C2×C3×D1214yes—
394C3×C4C3×D3×D414yes—
395C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
396C3×C4D4×C3⋊D311yescharacteristic
397Dic3—64no—
398C32—24no—
399C32#477314yescomplemented
400C32C3×#44711yescharacteristic, complemented
401C23—21no—
402C23—91no—
403C23—184no—
404C23—181no—
405C23—181no—
406C23—362no—
407C23—362no—
408D4—21no—
409D4—92no—
410D4—184no—
411D4—181no—
412D4—181no—
413D4—181no—
414D4—362no—
415D4C3×C4⋊(C3⋊D3)12yescomplemented
416C2×C4—21no—
417C2×C4—91no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4C22×C3×C3⋊D311yescharacteristic
421C8—21no—
422C8—181no—
423C2×C3—24no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—48no—
427C2×C3—42no—
428C2×C3—88no—
429C2×C3—182no—
430C2×C3—362no—
431C2×C3—721no—
432C2×C3C3×#88214yes—
433C2×C3C3×D3×D814yes—
434C2×C3C3×#91414yes—
435C2×C3#6409111yescharacteristic
436C2×C3D8×C3⋊D311yescharacteristic
437C2×C3#6412311yescharacteristic
438D3—68no—
439D3—128no—
440D3—244no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—92no—
445C22—181no—
446C22—181no—
447C22—182no—
448C22—368no—
449C22—361no—
450C22—361no—
451C22—361no—
452C22—361no—
453C22—361no—
454C22C3×D4×C3⋊D311yescharacteristic
455C4—41no—
456C4—181no—
457C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
458C4C3×D4×C3⋊D311yescharacteristic
459C3—24no—
460C3C3×#477314yescomplemented
461C3#1059247211yescharacteristic, direct product factor
462C2—42no—
463C2—182no—
464C2—362no—
465C2—721no—
466C2C3×#6409111yescharacteristic
467C2C3×D8×C3⋊D311yescharacteristic
468C2C3×#6412311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#447
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C2×C3×#3753C2semidirect
3. C2×C3×#3745C2semidirect
4. C3×#64104C2semidirect
5. C33×#102C2semidirect
6. C3×#63647C2semidirect
7. C3×#63636C2semidirect
8. #10592472C3direct (N × H)
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C33×C8C22semidirect
20. C33×C4⋊C4C22semidirect
23. C2×C3×C3⋊(C3⋊C8)C22semidirect
25. C2×D4×C3⋊D3C2×C3semidirect
27. C2×#3753C2×C3semidirect
28. C2×#3745C2×C3semidirect
29. #64104C2×C3semidirect
34. C32×#102D3semidirect
35. C32×#102C2×C3semidirect
36. #63647C2×C3semidirect
37. #63636C2×C3semidirect
48. C33×D4D4semidirect
57. C3×C4⋊(C3⋊D3)D4semidirect
74. C2×C4⋊(C3⋊D3)C22×C3semidirect
84. C2×C32×C8C2×D3semidirect
85. C2×C32×C8C22×C3semidirect
87. C32×C4⋊C4C2×D3semidirect
88. C32×C4⋊C4C22×C3semidirect
91. C2×C3⋊(C3⋊C8)C22×C3semidirect
110. C2×C3×C3⋊D3D8semidirect
125. C3×#102C3×D3semidirect
126. C3×#102C3⋊D3semidirect
149. C32×D4D12semidirect
150. C32×D4C3×D4semidirect
163. C4⋊(C3⋊D3)C3×D4semidirect
211. C2×C3×C8C2×C3×D3semidirect
212. C2×C3×C8C2×C3⋊D3semidirect
214. C3×C4⋊C4C2×C3×D3semidirect
215. C3×C4⋊C4C2×C3⋊D3semidirect
242. C2×C3⋊D3C3×D8semidirect
269. #102C3×C3⋊D3semidirect
271. C33#447semidirect
297. C3×D4C3×D12semidirect
298. C3×D4C4⋊(C3⋊D3)semidirect
352. C2×C8C2×C3×C3⋊D3semidirect
353. C4⋊C4C2×C3×C3⋊D3semidirect
399. C32#4773semidirect
400. C32C3×#447semidirect
415. D4C3×C4⋊(C3⋊D3)semidirect
460. C3C3×#4773semidirect
461. C3#10592472direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×C3×#37532
C2×C3×#37452
C3×#641042
C33×#1022
C3×#636472
C3×#636362
#105924723
C3×#47733