Finite Group Databasebeta

← C3×#25656 #10644263

Subgroups — C3×#25656 #10644263

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#491C211yesmaximal, characteristic, complemented
3C2×C3×#478C211yesmaximal, characteristic
4C3×#2584C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal
6C3×#2042C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1981C211yesmaximal, characteristic, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1863C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic
12#25656C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124C2212yes—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#117—21no—
24C3×#491—24no—
25C3×#478—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C3×#398—21no—
38C3×#395—21no—
39C3×#389—22no—
40C3×#389C2212yes—
41C3×#389C2212yes—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#556C2×C311yescharacteristic
47C2×#491C2×C311yescharacteristic, complemented
48C2×#478C2×C311yescharacteristic
49#2584C2×C311yescharacteristic, complemented
50#2042C2×C312yes—
51#2042C2×C312yes—
52#1985C2×C311yescharacteristic, complemented
53#1981C2×C311yescharacteristic, complemented
54#1865C2×C312yescomplemented
55#1863C2×C312yescomplemented
56#1762C2×C311yescharacteristic
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×C3×C8C2312yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×#126—216no—
67C3×#125—22no—
68C3×#125—22no—
69C3×#124—24no—
70C3×#124—24no—
71C3×C22⋊Q8—28no—
72C3×C4×Q8—24no—
73C3×C4×Q8—24no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—22no—
77C3×C4×D4D414yes—
78C3×#117—28no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4D412yes—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C42—22no—
96C2×C3×C42—21no—
97C2×C3×C42C2312yes—
98C2×C3×C42C2311yescharacteristic
99C3×#107—22no—
100C3×#106—22no—
101C3×C4⋊C8—22no—
102C3×#103—24no—
103C3×#103—24no—
104C3×#103—24no—
105C3×#103—24no—
106C3×#102—22no—
107C3×#102—22no—
108C3×#102—24no—
109C3×#97—22no—
110C3×#95—21no—
111C3×#95—22no—
112C3×#95—22no—
113#556—24no—
114#556—24no—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×#126—22no—
117C2×#125C22×C311yescharacteristic
118C2×#124C22×C312yes—
119C2×C22⋊Q8—21no—
120C2×C4×Q8C22×C312yes—
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×D4C22×C311yescharacteristic
123C2×#117—21no—
124#491—24no—
125#478—24no—
126C2×#107C22×C311yescharacteristic
127C2×#106C22×C311yescharacteristic
128C2×C4⋊C8C22×C311yescharacteristic
129C2×#103C22×C312yes—
130C2×#103C22×C312yes—
131C2×#103C22×C312yes—
132C2×#103C22×C312yes—
133C2×#102C22×C311yescharacteristic
134C2×#102C22×C311yescharacteristic
135C2×#102C22×C312yes—
136C2×#97C22×C311yescharacteristic
137#398—21no—
138#395—21no—
139#389—22no—
140#389C22×C312yes—
141#389C22×C312yes—
142C4×C4⋊C4C22×C311yescharacteristic
143#340C22×C312yes—
144#336C22×C312yes—
145#336C22×C312yes—
146C24×C3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C2×C3×Q8C2×D414yes—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—44no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C2×C3×C8—22no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C3×C4⋊C4—28no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—82no—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4#4114yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—24no—
198C3×C42—28no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—81no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204C22×Q8C3×D411yescharacteristic
205C22×Q8C23×C312yes—
206C22×D4C23×C311yescharacteristic
207C23×C4—21no—
208C22×C8C23×C312yes—
209C22×C8C23×C311yescharacteristic
210C22×C8C23×C311yescharacteristic
211C4⋊Q8—22no—
212C4⋊Q8C3×D414yes—
213#126—216no—
214#125—22no—
215#125—22no—
216#124—24no—
217#124—24no—
218C22⋊Q8—28no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×D4—22no—
224C4×D4C3×D414yes—
225#117—28no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4C3×D411yescharacteristic
234C2×C4⋊C4C3×D412yes—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C312yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42C23×C312yes—
245C2×C42C23×C311yescharacteristic
246#107—22no—
247#106—22no—
248C4⋊C8—22no—
249#103—24no—
250#103—24no—
251#103—24no—
252#103—24no—
253#102—22no—
254#102—22no—
255#102—24no—
256#97—22no—
257#95—21no—
258#95—22no—
259#95—22no—
260C23×C3—22no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3C22×D411yescharacteristic
266C3×Q8—24no—
267C3×Q8—24no—
268C3×Q8—44no—
269C3×Q8—42no—
270C3×Q8—42no—
271C3×D4—22no—
272C3×D4—22no—
273C3×D4—42no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—48no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C42−1+412yes—
303C3×C8—41no—
304C3×C8—41no—
305C3×C8—42no—
306C24—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—42no—
314C2×Q8C2×C3×D414yes—
315C2×D4—21no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4C2×C3×D412yes—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—44no—
326C22×C4C2×C3×D412yes—
327C22×C4C2×C3×D412yes—
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C24×C311yescharacteristic
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—82no—
353C4⋊C4C2×C3×D412yes—
354C4⋊C4C3×#4114yes—
355C22⋊C4—28no—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C42—28no—
359C42—22no—
360C42—22no—
361C42—81no—
362C42C2×C3×D412yes—
363C42C2×C3×D412yes—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3C2×#12011yescharacteristic
370C22×C3#54812yes—
371C22×C3#57512yes—
372C22×C3#57711yescharacteristic
373C22×C3#57811yescharacteristic
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—44no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—81no—
382C3×C4—81no—
383C3×C4#53211yescharacteristic
384C3×C4#53611yescharacteristic
385C3×C4#54712yes—
386C23—22no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23C22×C3×D411yescharacteristic
392Q8—24no—
393Q8—24no—
394Q8—44no—
395Q8—42no—
396Q8—42no—
397D4—22no—
398D4—22no—
399D4—42no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4C3×#12012yes—
424C2×C4C3×#12012yes—
425C2×C4C22×C3×D411yescharacteristic
426C2×C4C22×C3×D411yescharacteristic
427C2×C4C2×C3×#4112yes—
428C2×C4C3×2−1+412yes—
429C8—41no—
430C8—41no—
431C8—42no—
432C2×C3—82no—
433C2×C3#243211yescharacteristic
434C2×C3#300511yescharacteristic
435C2×C3#301311yescharacteristic
436C2×C3#337612yes—
437C2×C3#337712yes—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22C2×C3×#12011yescharacteristic
444C22C3×#54812yes—
445C22C3×#57512yes—
446C22C3×#57711yescharacteristic
447C22C3×#57811yescharacteristic
448C4—22no—
449C4—22no—
450C4—44no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—44no—
455C4—81no—
456C4—81no—
457C4C3×#53211yescharacteristic
458C4C3×#53611yescharacteristic
459C4C3×#54712yes—
460C3#2565611yescharacteristic, direct product factor, Sylow
461C2—82no—
462C2C3×#243211yescharacteristic
463C2C3×#300511yescharacteristic
464C2C3×#301311yescharacteristic
465C2C3×#337612yes—
466C2C3×#337712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
2. C2×C3×#491C2semidirect
4. C3×#2584C2semidirect
7. C3×#1985C2semidirect
8. C3×#1981C2semidirect
9. C3×#1865C2semidirect
10. C3×#1863C2semidirect
12. #25656C3direct (N × H)
47. C2×#491C2×C3semidirect
49. #2584C2×C3semidirect
52. #1985C2×C3semidirect
53. #1981C2×C3semidirect
54. #1865C2×C3semidirect
55. #1863C2×C3semidirect
460. C3#25656direct (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×#4912
C2×C3×#4782
C3×#25842
C3×#20422
C3×#19852
C3×#19812
C3×#18652
C3×#18632
C3×#17622
#256563