Finite Group Databasebeta

← C3×#25682 #10644289

Subgroups — C3×#25682 #10644289

Order 768.

1088 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic
2C2×C3×#499C211yesmaximal, characteristic, complemented
3C2×C3×#490C211yesmaximal, characteristic, complemented
4C3×#2629C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25682C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#540—24no—
14C3×#540—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—22no—
19C2×C3×#121—22no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117C2211yescharacteristic
23C3×#499—24no—
24C3×#490—24no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×#97C2211yescharacteristic
34C2×C3×C4×C8C2211yescharacteristic
35C3×#398—22no—
36C3×#396—22no—
37C3×#389C2212yes—
38C3×#384C2211yescharacteristic
39C3×#380C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44C2×#540C2×C311yescharacteristic
45C2×#499C2×C311yescharacteristic, complemented
46C2×#490C2×C311yescharacteristic, complemented
47#2629C2×C311yescharacteristic, complemented
48#2034C2×C312yescomplemented
49#2033C2×C312yescomplemented
50#1992C2×C311yescharacteristic, complemented
51#1957C2×C311yescharacteristic, complemented
52#1864C2×C312yescomplemented
53#1862C2×C312yescomplemented
54#1750C2×C311yescharacteristic, complemented
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4C2312yes—
58C23×C3×C4—22no—
59C23×C3×C4C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×#127—22no—
65C3×#127D414yes—
66C3×#125—22no—
67C3×#125—22no—
68C3×#125—22no—
69C3×#125D414yes—
70C3×#123—216no—
71C3×#121—216no—
72C3×#121—24no—
73C3×#121—24no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C3×#117—22no—
77C3×#117—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—24no—
93C3×#106—24no—
94C3×#102—24no—
95C3×#102—24no—
96C3×#102—24no—
97C3×#102—24no—
98C3×C22⋊C8—22no—
99C3×C22⋊C8—22no—
100C3×#97—22no—
101C3×C4×C8—22no—
102C3×#95—22no—
103C3×#95—22no—
104#540—24no—
105#540—24no—
106C2×#127C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×#125C22×C311yescharacteristic
109C2×#123—22no—
110C2×#121—22no—
111C2×#121C22×C312yes—
112C2×C4×D4C22×C312yes—
113C2×#117C22×C311yescharacteristic
114#499—24no—
115#490—24no—
116C2×#107C22×C312yes—
117C2×#106C22×C312yes—
118C2×#102C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×#102C22×C312yes—
122C2×C22⋊C8C22×C311yescharacteristic
123C2×C22⋊C8C22×C311yescharacteristic
124C2×#97C22×C311yescharacteristic
125C2×C4×C8C22×C311yescharacteristic
126#398—22no—
127#396—22no—
128#389C22×C312yes—
129#384C22×C311yescharacteristic
130#380C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C311yescharacteristic
134#336C22×C311yescharacteristic
135C24×C3—22no—
136C24×C3C2×D411yescharacteristic
137C2×C3×D4—28no—
138C2×C3×D4—28no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—82no—
145C2×C3×D4—82no—
146C2×C3×D4—82no—
147C2×C3×D4—82no—
148C2×C3×D4C2×D414yes—
149C22×C3×C4—22no—
150C22×C3×C4—28no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—82no—
160C22×C3×C4—82no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2411yescharacteristic
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8#4112yes—
176C2×C3×C8#4112yes—
177C3×C4⋊C4—28no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—44no—
185C3×C4⋊C4—82no—
186C3×C4⋊C4—82no—
187C3×C4⋊C4C2×D414yes—
188C3×C22⋊C4—28no—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—82no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4C23×C312yes—
199C23×C4—22no—
200C23×C4C23×C311yescharacteristic
201C22×C8C23×C311yescharacteristic
202C22×C8C23×C311yescharacteristic
203C22×C8C23×C311yescharacteristic
204C22×C8C23×C311yescharacteristic
205#127—22no—
206#127C3×D414yes—
207#125—22no—
208#125—22no—
209#125—22no—
210#125C3×D414yes—
211#123—216no—
212#121—216no—
213#121—24no—
214#121—24no—
215C4×D4—24no—
216C4×D4—24no—
217#117—22no—
218#117—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C3×D412yes—
224C2×C4⋊C4C23×C312yes—
225C2×C4⋊C4C23×C312yes—
226C2×C4⋊C4C23×C312yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C3×D412yes—
231C2×C42C23×C311yescharacteristic
232C2×C42C23×C311yescharacteristic
233#107—24no—
234#106—24no—
235#102—24no—
236#102—24no—
237#102—24no—
238#102—24no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241#97—22no—
242C4×C8—22no—
243#95—22no—
244#95—22no—
245C23×C3—21no—
246C23×C3—24no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3C22×D411yescharacteristic
258C3×D4—24no—
259C3×D4—24no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—88no—
263C3×D4—88no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—24no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—88no—
286C2×C3×C4#12014yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C42−1+411yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C3×C8—22no—
294C3×C8—22no—
295C3×C8—41no—
296C3×C8—41no—
297C24—22no—
298C24C2×C3×D411yescharacteristic
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4C2×C3×D414yes—
311C22×C4—22no—
312C22×C4—28no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D412yes—
325C22×C4C2×C3×D412yes—
326C22×C4C24×C311yescharacteristic
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8C3×#4112yes—
338C2×C8C3×#4112yes—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—44no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4C2×C3×D414yes—
350C22⋊C4—28no—
351C22⋊C4—28no—
352C22⋊C4—24no—
353C22⋊C4—82no—
354C42—21no—
355C42—21no—
356C42C2×C3×D412yes—
357C42C2×C3×D412yes—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—44no—
362C22×C3—42no—
363C22×C3—44no—
364C22×C3—42no—
365C22×C3—44no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3C2×#12011yescharacteristic
371C22×C3#53211yescharacteristic
372C22×C3#53611yescharacteristic
373C22×C3#57612yes—
374C22×C3#57712yes—
375C3×C4—24no—
376C3×C4—41no—
377C3×C4—42no—
378C3×C4—44no—
379C3×C4—82no—
380C3×C4—82no—
381C3×C4#54711yescharacteristic
382C3×C4#54711yescharacteristic
383C3×C4#54811yescharacteristic
384C3×C4#54811yescharacteristic
385C23—22no—
386C23—22no—
387C23—21no—
388C23—24no—
389C23—21no—
390C23—21no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23C22×C3×D411yescharacteristic
398D4—24no—
399D4—24no—
400D4—42no—
401D4—42no—
402D4—88no—
403D4—88no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—88no—
426C2×C4C3×#12014yes—
427C2×C4C22×C3×D411yescharacteristic
428C2×C4C22×C3×D411yescharacteristic
429C2×C4C2×C3×#4111yescharacteristic
430C2×C4C2×C3×#4111yescharacteristic
431C2×C4C3×2−1+411yescharacteristic
432C2×C4C3×2−1+411yescharacteristic
433C8—22no—
434C8—22no—
435C8—41no—
436C8—41no—
437C2×C3—81no—
438C2×C3—84no—
439C2×C3#243211yescharacteristic
440C2×C3#301212yes—
441C2×C3#317411yescharacteristic
442C2×C3#317611yescharacteristic
443C2×C3#324112yes—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—44no—
448C22—42no—
449C22—44no—
450C22—42no—
451C22—44no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22C2×C3×#12011yescharacteristic
457C22C3×#53211yescharacteristic
458C22C3×#53611yescharacteristic
459C22C3×#57612yes—
460C22C3×#57712yes—
461C4—24no—
462C4—41no—
463C4—42no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4C3×#54711yescharacteristic
468C4C3×#54711yescharacteristic
469C4C3×#54811yescharacteristic
470C4C3×#54811yescharacteristic
471C3#2568211yescharacteristic, direct product factor, Sylow
472C2—81no—
473C2—84no—
474C2C3×#243211yescharacteristic
475C2C3×#301212yes—
476C2C3×#317411yescharacteristic
477C2C3×#317611yescharacteristic
478C2C3×#324112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#499C2semidirect
3. C2×C3×#490C2semidirect
4. C3×#2629C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1992C2semidirect
8. C3×#1957C2semidirect
9. C3×#1864C2semidirect
10. C3×#1862C2semidirect
11. C3×#1750C2semidirect
12. #25682C3direct (N × H)
45. C2×#499C2×C3semidirect
46. C2×#490C2×C3semidirect
47. #2629C2×C3semidirect
48. #2034C2×C3semidirect
49. #2033C2×C3semidirect
50. #1992C2×C3semidirect
51. #1957C2×C3semidirect
52. #1864C2×C3semidirect
53. #1862C2×C3semidirect
54. #1750C2×C3semidirect
471. C3#25682direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#4992
C2×C3×#4902
C3×#26292
C3×#20342
C3×#20332
C3×#19922
C3×#19572
C3×#18642
C3×#18622
C3×#17502
#256823