Finite Group Databasebeta

← C3×#25658 #10644265

Subgroups — C3×#25658 #10644265

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×C3×#490C211yesmaximal, characteristic
3C2×C3×#479C211yesmaximal, characteristic, complemented
4C3×#2584C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2042C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1982C211yesmaximal, characteristic
9C3×#1864C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25658C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#124C2212yes—
19C2×C3×#123—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2212yes—
24C3×#490—24no—
25C3×#479—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
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×#559C2×C311yescharacteristic
47C2×#490C2×C311yescharacteristic
48C2×#479C2×C311yescharacteristic, complemented
49#2584C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2042C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1982C2×C311yescharacteristic
54#1864C2×C312yescomplemented
55#1862C2×C312yescomplemented
56#1763C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4—22no—
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×#127—22no—
67C3×#127D414yes—
68C3×#125—22no—
69C3×#125—22no—
70C3×#124—28no—
71C3×#124—24no—
72C3×#124—24no—
73C3×#123—216no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8D414yes—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
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—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C42—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2312yes—
98C2×C3×C42C2311yescharacteristic
99C3×#107—22no—
100C3×#106—22no—
101C3×C4⋊C8—22no—
102C3×#103—22no—
103C3×#103—22no—
104C3×#103—24no—
105C3×#102—24no—
106C3×#102—24no—
107C3×#102—24no—
108C3×#102—24no—
109C3×#97—22no—
110C3×#95—21no—
111C3×#95—22no—
112C3×#95—22no—
113#559—24no—
114#559—24no—
115C2×#127C22×C311yescharacteristic
116C2×#125C22×C311yescharacteristic
117C2×#124—21no—
118C2×#124C22×C312yes—
119C2×#123—22no—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4C22×C312yes—
124#490—24no—
125#479—24no—
126C2×#107C22×C311yescharacteristic
127C2×#106C22×C311yescharacteristic
128C2×C4⋊C8C22×C311yescharacteristic
129C2×#103C22×C311yescharacteristic
130C2×#103C22×C311yescharacteristic
131C2×#103C22×C312yes—
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yes—
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—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—42no—
152C2×C3×Q8C2×D412yes—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—44no—
161C2×C3×D4—81no—
162C2×C3×D4—81no—
163C2×C3×D4C2×D414yes—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—28no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—82no—
171C22×C3×C4—82no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×C3×C8—22no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—44no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—82no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4#4114yes—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—28no—
202C3×C42—24no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—81no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208C22×Q8C3×D411yescharacteristic
209C22×Q8C23×C311yescharacteristic
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C23×C312yes—
213C23×C4—22no—
214C22×C8C23×C312yes—
215C22×C8C23×C311yescharacteristic
216C22×C8C23×C311yescharacteristic
217#127—22no—
218#127C3×D414yes—
219#125—22no—
220#125—22no—
221#124—28no—
222#124—24no—
223#124—24no—
224#123—216no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8C3×D414yes—
229C4×D4—28no—
230C4×D4—24no—
231C4×D4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4C3×D411yescharacteristic
238C2×C4⋊C4C3×D412yes—
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C311yescharacteristic
241C2×C4⋊C4C23×C311yescharacteristic
242C2×C4⋊C4C23×C312yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C23×C312yes—
249C2×C42C23×C311yescharacteristic
250#107—22no—
251#106—22no—
252C4⋊C8—22no—
253#103—22no—
254#103—22no—
255#103—24no—
256#102—24no—
257#102—24no—
258#102—24no—
259#102—24no—
260#97—22no—
261#95—21no—
262#95—22no—
263#95—22no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3C22×D411yescharacteristic
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—44no—
275C3×Q8—42no—
276C3×D4—24no—
277C3×D4—24no—
278C3×D4—48no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—84no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—88no—
305C2×C3×C4#12012yes—
306C2×C3×C4#12012yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4112yes—
310C2×C3×C42−1+412yes—
311C3×C8—41no—
312C3×C8—41no—
313C3×C8—42no—
314C24—22no—
315C2×Q8—22no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—42no—
320C2×Q8C2×C3×D412yes—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—44no—
329C2×D4—81no—
330C2×D4—81no—
331C2×D4C2×C3×D414yes—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—28no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C24×C311yescharacteristic
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—82no—
365C4⋊C4C2×C3×D412yes—
366C4⋊C4C3×#4114yes—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—28no—
370C42—24no—
371C42—22no—
372C42—22no—
373C42—81no—
374C42C2×C3×D412yes—
375C42C2×C3×D412yes—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3C2×#12011yescharacteristic
384C22×C3#54812yes—
385C22×C3#57512yes—
386C22×C3#57611yescharacteristic
387C22×C3#57711yescharacteristic
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—44no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4#53211yescharacteristic
398C3×C4#53611yescharacteristic
399C3×C4#54712yes—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23C22×C3×D411yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—44no—
411Q8—42no—
412D4—24no—
413D4—24no—
414D4—48no—
415D4—42no—
416D4—42no—
417D4—84no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—88no—
441C2×C4C3×#12012yes—
442C2×C4C3×#12012yes—
443C2×C4C22×C3×D411yescharacteristic
444C2×C4C22×C3×D411yescharacteristic
445C2×C4C2×C3×#4112yes—
446C2×C4C3×2−1+412yes—
447C8—41no—
448C8—41no—
449C8—42no—
450C2×C3—84no—
451C2×C3#243211yescharacteristic
452C2×C3#300511yescharacteristic
453C2×C3#301211yescharacteristic
454C2×C3#337012yes—
455C2×C3#337112yes—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22C2×C3×#12011yescharacteristic
464C22C3×#54812yes—
465C22C3×#57512yes—
466C22C3×#57611yescharacteristic
467C22C3×#57711yescharacteristic
468C4—22no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—44no—
475C4—81no—
476C4—81no—
477C4C3×#53211yescharacteristic
478C4C3×#53611yescharacteristic
479C4C3×#54712yes—
480C3#2565811yescharacteristic, direct product factor, Sylow
481C2—84no—
482C2C3×#243211yescharacteristic
483C2C3×#300511yescharacteristic
484C2C3×#301211yescharacteristic
485C2C3×#337012yes—
486C2C3×#337112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
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
3. C2×C3×#479C2semidirect
4. C3×#2584C2semidirect
5. C3×#2042C2semidirect
6. C3×#2042C2semidirect
7. C3×#1985C2semidirect
9. C3×#1864C2semidirect
10. C3×#1862C2semidirect
11. C3×#1763C2semidirect
12. #25658C3direct (N × H)
48. C2×#479C2×C3semidirect
49. #2584C2×C3semidirect
50. #2042C2×C3semidirect
51. #2042C2×C3semidirect
52. #1985C2×C3semidirect
54. #1864C2×C3semidirect
55. #1862C2×C3semidirect
56. #1763C2×C3semidirect
480. C3#25658direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4902
C2×C3×#4792
C3×#25842
C3×#20422
C3×#19852
C3×#19822
C3×#18642
C3×#18622
C3×#17632
#256583