Finite Group Databasebeta

← C3×#10592868 #13163209

Subgroups — C3×#10592868 #13163209

Order 1728.

1400 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C2×C33×SD16C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C2×C3×#3755C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63647C211yesmaximal, characteristic, complemented
8#10592868C311yesmaximal, characteristic, direct product factor
9C3×#5169—34nomaximal
10C23×C3×C3⋊D3—21no—
11C2×C33×Q8C2211yescharacteristic, complemented
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638—21no—
14C3×D4×C3⋊D3—22no—
15C3×D4×C3⋊D3—22no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C33×SD16—22no—
19C2×C33×C8C2211yescharacteristic, complemented
20C3×#3755—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
23C3×#3750—21no—
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
25C2×D4×C3⋊D3C2×C311yescharacteristic
26C2×C3×D3×D4—34no—
27C2×C32×SD16—24no—
28C2×C32×SD16D314yescomplemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64155C2×C311yescharacteristic, complemented
31C2×#3755C2×C311yescharacteristic, complemented
32C3×#946—34no—
33C2×C3×#215—34no—
34#63659C2×C311yescharacteristic, complemented
35#63648C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37C3×#832—34no—
38C3×#821—34no—
39C3×(C2×D3)⋊C8—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—42no—
44C22×C3×C3⋊D3—42no—
45C22×C3×C3⋊D3D411yescharacteristic
46C33×Q8—21no—
47C33×Q8—21no—
48C33×D4—21no—
49C33×D4D412yes—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—24no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)D412yes—
57C3×C4×C3⋊D3—21no—
58C3×C4×C3⋊D3—21no—
59C33×C8—21no—
60C3×C3⋊(C3⋊C8)—21no—
61C3×#450—91no—
62#5169—34no—
63C23×C3⋊D3—21no—
64C23×C3×D3—64no—
65C2×C32×Q8—24no—
66C2×C32×Q8C2×D314yescomplemented
67C2×C32×Q8C22×C311yescharacteristic, complemented
68C2×C32×D4—24no—
69C2×C32×D4C2×D314yes—
70C2×C32×D4C22×C311yescharacteristic
71C2×#638—21no—
72D4×C3⋊D3—22no—
73D4×C3⋊D3—22no—
74C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
75C2×C4×C3⋊D3C22×C311yescharacteristic
76C2×C3×#67—64no—
77C3×D3×D4—68no—
78C3×D3×D4—68no—
79C2×C3×D12—34no—
80C2×C3×C4×D3—34no—
81C32×SD16—216no—
82C32×SD16—28no—
83C32×SD16—22no—
84C2×C32×C8—24no—
85C2×C32×C8C2×D314yescomplemented
86C2×C32×C8C22×C311yescharacteristic, complemented
87#3755—22no—
88#3752—21no—
89C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
90#3750—21no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
92C3×#215—68no—
93C3×(C2×D3)⋊C4—64no—
94C3×C4⋊Dic3—34no—
95C3×Dic3⋊C4—64no—
96C2×C3×C3⋊C8—34no—
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⋊D3SD1612yescomplemented
108C33×C4—41no—
109C33×C4C2×D411yescharacteristic
110C33×C4C2×D411yescharacteristic
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—41no—
113C22×C3×D4—91no—
114C2×D3×D4—34no—
115C2×C3×SD16—24no—
116C2×C3×SD16—91no—
117C2×C3×SD16C3×D314yescomplemented
118C2×C3×SD16C3⋊D311yescharacteristic, complemented
119C3×C22⋊Q8—91no—
120#946—34no—
121C2×#215—34no—
122C3×#102—91no—
123C3×#102—91no—
124C3×C22⋊C8—91no—
125#832—34no—
126#821—34no—
127(C2×D3)⋊C8—34no—
128C23×C32—21no—
129C23×C32—24no—
130C23×C32—44no—
131C22×C3⋊D3—24no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—42no—
134C22×C3⋊D3—42no—
135C22×C3⋊D3C3×D411yescharacteristic
136C22×C3×D3—34no—
137C22×C3×D3—616no—
138C22×C3×D3—64no—
139C22×C3×D3—128no—
140C22×C3×D3—128no—
141C32×Q8—28no—
142C32×Q8—24no—
143C32×Q8—21no—
144C32×Q8—21no—
145C32×Q8—24no—
146C32×Q8—44no—
147C32×D4—21no—
148C32×D4—28no—
149C32×D4—24no—
150C32×D4—44no—
151C32×D4#6718yes—
152C32×D4C3×D412yes—
153C2×C32×C4—21no—
154C2×C32×C4—24no—
155C2×C32×C4—24no—
156C2×C32×C4—44no—
157C2×C32×C4C22×D314yes—
158C2×C32×C4C23×C311yescharacteristic
159#638—24no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3C3×D411yescharacteristic
162C4⋊(C3⋊D3)—21no—
163C4⋊(C3⋊D3)C3×D412yes—
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166C3×#67—616no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—64no—
169C3×D12—38no—
170C3×D12—64no—
171C3×C4×D3—64no—
172C3×C4×D3—64no—
173C32×C8—28no—
174C32×C8—24no—
175C32×C8—21no—
176C3⋊(C3⋊C8)—21no—
177C3×C3⋊C8—64no—
178#450—91noSylow
179C2×C33—42no—
180C2×C33#12011yescharacteristic
181C2×C33C2×SD1611yescharacteristic
182C2×C33AGL(1,ℤ/8)11yescharacteristic
183C3×C3⋊D3—22no—
184C3×C3⋊D3—42no—
185C24×C3—181no—
186C23×D3—64no—
187C2×C3×Q8—24no—
188C2×C3×Q8C2×C3×D314yescomplemented
189C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
190C2×C3×D4—24no—
191C2×C3×D4—91no—
192C2×C3×D4—181no—
193C2×C3×D4—182no—
194C2×C3×D4—182no—
195C2×C3×D4C2×C3×D314yes—
196C2×C3×D4C2×C3⋊D311yescharacteristic
197C22×C3×C4—91no—
198C2×#67—64no—
199D3×D4—68no—
200D3×D4—68no—
201C2×D12—34no—
202C2×C4×D3—34no—
203C3×SD16—216no—
204C3×SD16—28no—
205C3×SD16—22no—
206C3×SD16—182no—
207C2×C3×C8—24no—
208C2×C3×C8—91no—
209C2×C3×C8C2×C3×D314yescomplemented
210C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
211C3×C4⋊C4—91no—
212C3×C4⋊C4—181no—
213C3×C22⋊C4—181no—
214#215—68no—
215(C2×D3)⋊C4—64no—
216C4⋊Dic3—34no—
217Dic3⋊C4—64no—
218C2×C3⋊C8—34no—
219C22×C32—22no—
220C22×C32—24no—
221C22×C32—28no—
222C22×C32—48no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—44no—
226C22×C32—44no—
227C22×C32—84no—
228C22×C32—84no—
229C22×C32D3×D414yes—
230C22×C32C2×C3×D411yescharacteristic
231C2×C3⋊D3—21no—
232C2×C3⋊D3—21no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3—48no—
235C2×C3⋊D3—41no—
236C2×C3⋊D3—41no—
237C2×C3⋊D3C3×SD1612yescomplemented
238C2×C3×D3—38no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—68no—
242C2×C3×D3—1232no—
243C2×C3×D3—124no—
244C2×C3×D3—124no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—48no—
248C32×C4—44no—
249C32×C4—41no—
250C32×C4D3×D414yes—
251C32×C4C2×#6714yes—
252C32×C4C2×C3×D411yescharacteristic
253C32×C4C2×C3×D411yescharacteristic
254C3⋊Dic3—21no—
255C3⋊Dic3—41no—
256C3×Dic3—64no—
257C3×Dic3—124no—
258C22×D4—91no—
259C2×SD16—91no—
260C2×SD16C3×C3⋊D311yescharacteristic, complemented
261C22⋊Q8—91no—
262#102—91no—
263#102—91no—
264C22⋊C8—91no—
265C33#45011yescharacteristic, complemented, Sylow
266C23×C3—24no—
267C23×C3—21no—
268C23×C3—44no—
269C23×C3—91no—
270C23×C3—184no—
271C23×C3—181no—
272C23×C3—362no—
273C23×C3—362no—
274C22×D3—34no—
275C22×D3—616no—
276C22×D3—64no—
277C22×D3—128no—
278C22×D3—128no—
279C3×Q8—28no—
280C3×Q8—24no—
281C3×Q8—24no—
282C3×Q8—21no—
283C3×Q8—21no—
284C3×Q8—44no—
285C3×D4—24no—
286C3×D4—21no—
287C3×D4—28no—
288C3×D4—44no—
289C3×D4—92no—
290C3×D4—184no—
291C3×D4—181no—
292C3×D4C3×#6718yes—
293C3×D4#63812yes—
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—44no—
298C2×C3×C4—91no—
299C2×C3×C4—181no—
300C2×C3×C4—181no—
301C2×C3×C4—181no—
302C2×C3×C4C22×C3×D314yes—
303C2×C3×C4C22×C3⋊D311yescharacteristic
304#67—616no—
305C2×Dic3—34no—
306C2×Dic3—64no—
307D12—38no—
308D12—64no—
309C4×D3—64no—
310C4×D3—64no—
311C3×C8—28no—
312C3×C8—24no—
313C3×C8—21no—
314C3×C8—181no—
315C3⋊C8—64no—
316C2×C32—24no—
317C2×C32—24no—
318C2×C32—24no—
319C2×C32—42no—
320C2×C32—48no—
321C2×C32—88no—
322C2×C32D3×SD1614yes—
323C2×C32#91414yes—
324C2×C32#93714yes—
325C2×C32C3×#12011yescharacteristic
326C2×C32C2×C3×SD1611yescharacteristic
327C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
328C3⋊D3—22no—
329C3⋊D3—42no—
330C3×D3—68no—
331C3×D3—128no—
332C24—181no—
333C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
334C2×D4—91no—
335C2×D4—181no—
336C2×D4—182no—
337C2×D4—182no—
338C2×D4C2×C3×C3⋊D311yescharacteristic
339C22×C4—91no—
340SD16—22no—
341SD16—182no—
342C2×C8—91no—
343C2×C8C2×C3×C3⋊D311yescharacteristic, complemented
344C4⋊C4—91no—
345C4⋊C4—181no—
346C22⋊C4—181no—
347C22×C3—28no—
348C22×C3—22no—
349C22×C3—24no—
350C22×C3—48no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—84no—
356C22×C3—84no—
357C22×C3—92no—
358C22×C3—181no—
359C22×C3—181no—
360C22×C3—182no—
361C22×C3—368no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3C3×D3×D414yes—
365C22×C3D4×C3⋊D311yescharacteristic
366C2×D3—38no—
367C2×D3—64no—
368C2×D3—64no—
369C2×D3—68no—
370C2×D3—1232no—
371C2×D3—124no—
372C2×D3—124no—
373C3×C4—24no—
374C3×C4—24no—
375C3×C4—48no—
376C3×C4—44no—
377C3×C4—41no—
378C3×C4—181no—
379C3×C4—361no—
380C3×C4C3×D3×D414yes—
381C3×C4C2×C3×#6714yes—
382C3×C4D4×C3⋊D311yescharacteristic
383C3×C4C2×#63811yescharacteristic
384Dic3—64no—
385Dic3—124no—
386C32—24no—
387C32#516914yescomplemented
388C32C3×#45011yescharacteristic, complemented
389C23—21no—
390C23—91no—
391C23—184no—
392C23—181no—
393C23—362no—
394C23—362no—
395Q8—21no—
396Q8—21no—
397D4—21no—
398D4—92no—
399D4—184no—
400D4—181no—
401D4C3×#63812yes—
402C2×C4—21no—
403C2×C4—91no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4C22×C3×C3⋊D311yescharacteristic
408C8—21no—
409C8—181no—
410C2×C3—24no—
411C2×C3—24no—
412C2×C3—24no—
413C2×C3—48no—
414C2×C3—42no—
415C2×C3—88no—
416C2×C3—182no—
417C2×C3—362no—
418C2×C3C3×D3×SD1614yes—
419C2×C3C3×#91414yes—
420C2×C3C3×#93714yes—
421C2×C3SD16×C3⋊D311yescharacteristic
422C2×C3#6412311yescharacteristic
423C2×C3#6414611yescharacteristic
424D3—68no—
425D3—128no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—92no—
430C22—181no—
431C22—181no—
432C22—182no—
433C22—368no—
434C22—361no—
435C22—361no—
436C22C3×D4×C3⋊D311yescharacteristic
437C4—41no—
438C4—181no—
439C4—361no—
440C4C3×D4×C3⋊D311yescharacteristic
441C4C2×C3×#63811yescharacteristic
442C3—24no—
443C3C3×#516914yescomplemented
444C3#1059286811yescharacteristic, direct product factor
445C2—42no—
446C2—182no—
447C2—362no—
448C2C3×SD16×C3⋊D311yescharacteristic
449C2C3×#6412311yescharacteristic
450C2C3×#6414611yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C3×D4×C3⋊D3C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#450
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C33×SD16C2semidirect
3. C3×#64155C2semidirect
4. C2×C3×#3755C2semidirect
5. C3×#63659C2semidirect
6. C3×#63648C2semidirect
7. C3×#63647C2semidirect
8. #10592868C3direct (N × H)
11. C2×C33×Q8C22semidirect
19. C2×C33×C8C22semidirect
22. C3×C4⋊(C3⋊Dic3)C22semidirect
24. C2×C3×C3⋊(C3⋊C8)C22semidirect
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
30. #64155C2×C3semidirect
31. C2×#3755C2×C3semidirect
34. #63659C2×C3semidirect
35. #63648C2×C3semidirect
36. #63647C2×C3semidirect
66. C2×C32×Q8C2×D3semidirect
67. C2×C32×Q8C22×C3semidirect
85. C2×C32×C8C2×D3semidirect
86. C2×C32×C8C22×C3semidirect
89. C4⋊(C3⋊Dic3)C22×C3semidirect
91. C2×C3⋊(C3⋊C8)C22×C3semidirect
107. C2×C3×C3⋊D3SD16semidirect
117. C2×C3×SD16C3×D3semidirect
118. C2×C3×SD16C3⋊D3semidirect
188. C2×C3×Q8C2×C3×D3semidirect
189. C2×C3×Q8C2×C3⋊D3semidirect
209. C2×C3×C8C2×C3×D3semidirect
210. C2×C3×C8C2×C3⋊D3semidirect
237. C2×C3⋊D3C3×SD16semidirect
260. C2×SD16C3×C3⋊D3semidirect
265. C33#450semidirect
333. C2×Q8C2×C3×C3⋊D3semidirect
343. C2×C8C2×C3×C3⋊D3semidirect
387. C32#5169semidirect
388. C32C3×#450semidirect
443. C3C3×#5169semidirect
444. C3#10592868direct (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×C33×SD162
C3×#641552
C2×C3×#37552
C3×#636592
C3×#636482
C3×#636472
#105928683
C3×#51693