Finite Group Databasebeta

← C3×#25628 #10644235

Subgroups — C3×#25628 #10644235

Order 768.

1218 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic, complemented
2C2×C3×#492C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic
4C3×#2548C211yesmaximal, characteristic, complemented
5C3×#2012C214yesmaximal, complemented
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1880C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25628C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2214yes—
12C22×C3×D8C2212yes—
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#126—22no—
18C2×C3×#124—21no—
19C2×C3×#124C2212yes—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C3×#492—24no—
26C3×#459—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2214yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#394—22no—
33C3×#390—21no—
34C3×#390C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#556C2×C311yescharacteristic, complemented
39C2×#492C2×C311yescharacteristic, complemented
40C2×#459C2×C311yescharacteristic
41#2548C2×C311yescharacteristic, complemented
42#2012C2×C314yescomplemented
43#1984C2×C311yescharacteristic, complemented
44#1981C2×C311yescharacteristic, complemented
45#1880C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C22×C3×Q8D411yescharacteristic
48C22×C3×Q8C2312yes—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4C2312yes—
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4—21no—
54C2×C3×SD16—28no—
55C2×C3×SD16—28no—
56C2×C3×SD16—44no—
57C2×C3×D8—24no—
58C2×C3×D8—44no—
59C22×C3×C8C2312yes—
60C22×C3×C8C2312yes—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8D414yes—
63C3×#127—22no—
64C3×#127—22no—
65C3×#126—216no—
66C3×#124—24no—
67C3×#124—24no—
68C3×#124—44no—
69C3×C22⋊Q8—28no—
70C3×C4×Q8—24no—
71C3×C4×Q8—24no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C3×#117—28no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C42—22no—
90C2×C3×C42—21no—
91C2×C3×C42C2312yes—
92C2×C3×C42C2311yescharacteristic
93C3×C4⋊C8—22no—
94C3×#103—44no—
95C3×#102—24no—
96C3×#102—24no—
97C3×C4×C8—22no—
98C3×#95—21no—
99C22×SD16C22×C314yes—
100C22×D8C22×C312yes—
101#556—24no—
102#556—24no—
103C2×C4⋊Q8C22×C311yescharacteristic
104C2×#127C22×C311yescharacteristic
105C2×#126—22no—
106C2×#124—21no—
107C2×#124C22×C312yes—
108C2×C22⋊Q8—21no—
109C2×C4×Q8C22×C312yes—
110C2×C4×Q8C22×C311yescharacteristic
111C2×C4×D4C22×C311yescharacteristic
112C2×#117—21no—
113#492—24no—
114#459—24no—
115C2×C4⋊C8C22×C311yescharacteristic
116C2×#103C22×C314yes—
117C2×#102C22×C312yes—
118C2×#102C22×C312yes—
119C2×C4×C8C22×C311yescharacteristic
120#394—22no—
121#390—21no—
122#390C22×C314yes—
123C4×C4⋊C4C22×C311yescharacteristic
124#340C22×C312yes—
125#336C22×C314yes—
126C24×C3—21no—
127C24×C3—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—24no—
133C2×C3×Q8—24no—
134C2×C3×Q8C2×D414yes—
135C2×C3×D4—21no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—24no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4—82no—
147C2×C3×D4C2×D412yes—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—22no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C3×SD16—416no—
160C3×D8—48no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—24no—
165C2×C3×C8#4114yes—
166C3×C4⋊C4—28no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4C2×D412yes—
175C3×C22⋊C4—28no—
176C3×C22⋊C4—28no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—48no—
179C3×C22⋊C4—84no—
180C3×C42—28no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—42no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186C22×Q8C3×D411yescharacteristic
187C22×Q8C23×C312yes—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4C23×C312yes—
191C22×D4C23×C311yescharacteristic
192C23×C4—21no—
193C2×SD16—28no—
194C2×SD16—28no—
195C2×SD16—44no—
196C2×D8—24no—
197C2×D8—44no—
198C22×C8C23×C312yes—
199C22×C8C23×C312yes—
200C4⋊Q8—22no—
201C4⋊Q8C3×D414yes—
202#127—22no—
203#127—22no—
204#126—216no—
205#124—24no—
206#124—24no—
207#124—44no—
208C22⋊Q8—28no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×D4—22no—
214C4×D4C3×D414yes—
215#117—28no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C3×D411yescharacteristic
220C2×C4⋊C4C3×D412yes—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42C23×C312yes—
231C2×C42C23×C311yescharacteristic
232C4⋊C8—22no—
233#103—44no—
234#102—24no—
235#102—24no—
236C4×C8—22no—
237#95—21no—
238C23×C3—22no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—44no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—44no—
248C23×C3C22×D411yescharacteristic
249C3×Q8—28no—
250C3×Q8—42no—
251C3×Q8—42no—
252C3×Q8—44no—
253C3×D4—22no—
254C3×D4—22no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—42no—
258C3×D4—44no—
259C3×D4—84no—
260C3×D4—84no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—44no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—48no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4#12012yes—
277C2×C3×C4#12012yes—
278C2×C3×C4#13512yes—
279C2×C3×C4AGL(1,ℤ/8)12yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C4H(5,𝔽2)12yes—
284C3×C8—24no—
285C3×C8—42no—
286C24—21no—
287C24—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8C2×C3×D414yes—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—82no—
307C2×D4C2×C3×D412yes—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C24×C311yescharacteristic
319SD16—416no—
320D8—48no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8C3×#4114yes—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4C2×C3×D412yes—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—24no—
338C22⋊C4—48no—
339C22⋊C4—84no—
340C42—28no—
341C42—22no—
342C42—22no—
343C42—42no—
344C42C2×C3×D412yes—
345C42C2×C3×D412yes—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—84no—
352C22×C3—84no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3C2×#12011yescharacteristic
357C22×C3#55012yes—
358C22×C3C2×#13511yescharacteristic
359C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C3#57712yes—
361C3×C4—22no—
362C3×C4—44no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—82no—
367C3×C4#44914yes—
368C3×C4#53211yescharacteristic
369C3×C4#53511yescharacteristic
370C3×C4#54612yes—
371C23—22no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—44no—
379C23—42no—
380C23—44no—
381C23C22×C3×D411yescharacteristic
382Q8—28no—
383Q8—44no—
384Q8—42no—
385Q8—42no—
386D4—22no—
387D4—22no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—44no—
392D4—84no—
393D4—84no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4C3×#12012yes—
410C2×C4C3×#12012yes—
411C2×C4C3×#13512yes—
412C2×C4C3×AGL(1,ℤ/8)12yes—
413C2×C4C22×C3×D411yescharacteristic
414C2×C4C22×C3×D411yescharacteristic
415C2×C4C2×C3×#4112yes—
416C2×C4C3×H(5,𝔽2)12yes—
417C8—24no—
418C8—42no—
419C2×C3—82no—
420C2×C3—162no—
421C2×C3#243111yescharacteristic
422C2×C3C2×#44911yescharacteristic
423C2×C3#301011yescharacteristic
424C2×C3#339412yes—
425C2×C3#339712yes—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—82no—
430C22—82no—
431C22—84no—
432C22—84no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22C2×C3×#12011yescharacteristic
437C22C3×#55012yes—
438C22C2×C3×#13511yescharacteristic
439C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
440C22C3×#57712yes—
441C4—22no—
442C4—44no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—82no—
447C4C3×#44914yes—
448C4C3×#53211yescharacteristic
449C4C3×#53511yescharacteristic
450C4C3×#54612yes—
451C3#2562811yescharacteristic, direct product factor, Sylow
452C2—82no—
453C2—162no—
454C2C3×#243111yescharacteristic
455C2C2×C3×#44911yescharacteristic
456C2C3×#301011yescharacteristic
457C2C3×#339412yes—
458C2C3×#339712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#556C2semidirect
2. C2×C3×#492C2semidirect
4. C3×#2548C2semidirect
5. C3×#2012C2semidirect
6. C3×#1984C2semidirect
7. C3×#1981C2semidirect
8. C3×#1880C2semidirect
9. C3×C4×#102C2semidirect
10. #25628C3direct (N × H)
38. C2×#556C2×C3semidirect
39. C2×#492C2×C3semidirect
41. #2548C2×C3semidirect
42. #2012C2×C3semidirect
43. #1984C2×C3semidirect
44. #1981C2×C3semidirect
45. #1880C2×C3semidirect
46. C4×#102C2×C3semidirect
451. C3#25628direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4922
C2×C3×#4592
C3×#25482
C3×#20122
C3×#19842
C3×#19812
C3×#18802
C3×C4×#1022
#256283