Finite Group Databasebeta

← C3×#25612 #10644219

Subgroups — C3×#25612 #10644219

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#488C211yesmaximal, characteristic, complemented
4C3×#2620C211yesmaximal, characteristic
5C3×#2001C212yesmaximal, complemented
6C3×#2000C212yesmaximal
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1878C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25612C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C3×#541—24no—
16C3×#541—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#126—22no—
19C2×C3×#125—22no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×Q8C2212yes—
26C2×C3×#117C2211yescharacteristic
27C3×#497—24no—
28C3×#488—24no—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×#97C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C3×#396—22no—
37C3×#394—22no—
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#384C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#541C2×C311yescharacteristic, complemented
45C2×#497C2×C311yescharacteristic, complemented
46C2×#488C2×C311yescharacteristic, complemented
47#2620C2×C311yescharacteristic
48#2001C2×C312yescomplemented
49#2000C2×C312yes—
50#1992C2×C311yescharacteristic, complemented
51#1956C2×C311yescharacteristic, complemented
52#1880C2×C312yescomplemented
53#1878C2×C312yescomplemented
54#1750C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2312yes—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×C3×Q16—24no—
61C2×C3×Q16—24no—
62C2×C3×Q16—42no—
63C2×C3×SD16—24no—
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yescomplemented
70C3×#126—216no—
71C3×#125—216no—
72C3×#125—22no—
73C3×#125D414yescomplemented
74C3×#124—22no—
75C3×#124—22no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—42no—
79C3×#121—42no—
80C3×C4×Q8—24no—
81C3×C4×Q8—24no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4D412yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4D412yes—
100C2×C3×C42—22no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C3×#103—24no—
104C3×#103—42no—
105C3×#103—42no—
106C3×#102—24no—
107C3×C22⋊C8—24no—
108C3×#97—22no—
109C3×C4×C8—22no—
110C3×#95—22no—
111C22×Q16C22×C312yes—
112C22×SD16C22×C312yes—
113#541—24no—
114#541—24no—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×#126—22no—
117C2×#125—22no—
118C2×#125C22×C311yescharacteristic
119C2×#124C22×C311yescharacteristic
120C2×C22⋊Q8C22×C312yes—
121C2×C22⋊Q8C22×C311yescharacteristic
122C2×#121C22×C311yescharacteristic
123C2×C4×Q8C22×C312yes—
124C2×#117C22×C311yescharacteristic
125#497—24no—
126#488—24no—
127C2×#103C22×C312yes—
128C2×#103C22×C312yes—
129C2×#103C22×C312yes—
130C2×#102C22×C312yes—
131C2×C22⋊C8C22×C312yes—
132C2×#97C22×C311yescharacteristic
133C2×C4×C8C22×C311yescharacteristic
134#396—22no—
135#394—22no—
136#391C22×C311yescharacteristic
137#390C22×C311yescharacteristic
138#384C22×C311yescharacteristic
139#340C22×C312yes—
140#340C22×C312yes—
141#336C22×C312yes—
142C24×C3—21no—
143C24×C3C2×D411yescharacteristic
144C2×C3×Q8—22no—
145C2×C3×Q8—21no—
146C2×C3×Q8—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×Q8C2×D414yes—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—42no—
159C2×C3×D4—82no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C3×Q16—48no—
174C3×SD16—48no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8#4114yes—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—82no—
191C3×C4⋊C4C2×D414yes—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—82no—
198C3×C42—28no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203C22×Q8C23×C311yescharacteristic
204C22×Q8C23×C312yes—
205C22×D4—21no—
206C22×D4C23×C311yescharacteristic
207C23×C4C23×C311yescharacteristic
208C2×Q16—24no—
209C2×Q16—24no—
210C2×Q16—42no—
211C2×SD16—24no—
212C2×SD16—24no—
213C2×SD16—42no—
214C22×C8C23×C312yes—
215C22×C8C23×C312yes—
216C4⋊Q8—22no—
217C4⋊Q8C3×D414yescomplemented
218#126—216no—
219#125—216no—
220#125—22no—
221#125C3×D414yescomplemented
222#124—22no—
223#124—22no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—42no—
227#121—42no—
228C4×Q8—24no—
229C4×Q8—24no—
230#117—22no—
231#117—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4C3×D412yes—
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C311yescharacteristic
241C2×C4⋊C4C23×C312yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C3×D412yes—
248C2×C42—22no—
249C2×C42C23×C311yescharacteristic
250C2×C42C23×C311yescharacteristic
251#103—24no—
252#103—42no—
253#103—42no—
254#102—24no—
255C22⋊C8—24no—
256#97—22no—
257C4×C8—22no—
258#95—22no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3C22×D411yescharacteristic
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8—42no—
273C3×Q8—42no—
274C3×Q8—42no—
275C3×Q8—42no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—84no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12014yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C2×#4112yes—
306C2×C3×C4H(5,𝔽2)12yes—
307C3×C8—24no—
308C3×C8—42no—
309C24—21no—
310C24C2×C3×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8C2×C3×D414yes—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—42no—
326C2×D4—82no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D412yes—
339C22×C4C24×C311yescharacteristic
340Q16—48no—
341SD16—48no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8C3×#4114yes—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—42no—
357C4⋊C4—82no—
358C4⋊C4C2×C3×D414yes—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C42—28no—
366C42—21no—
367C42—21no—
368C42C2×C3×D412yes—
369C42C2×C3×D412yes—
370C22×C3—42no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3—82no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—81no—
379C22×C3—82no—
380C22×C3C2×#12011yescharacteristic
381C22×C3#53211yescharacteristic
382C22×C3#53511yescharacteristic
383C22×C3#57712yes—
384C22×C3#57812yes—
385C3×C4—24no—
386C3×C4—41no—
387C3×C4—44no—
388C3×C4—42no—
389C3×C4—44no—
390C3×C4—81no—
391C3×C4—81no—
392C3×C4—81no—
393C3×C4#54612yes—
394C3×C4#55012yes—
395C23—22no—
396C23—22no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C22×C3×D411yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412D4—42no—
413D4—42no—
414D4—84no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4C3×#12014yes—
439C2×C4C22×C3×D411yescharacteristic
440C2×C4C22×C3×D411yescharacteristic
441C2×C4C2×C3×#4112yes—
442C2×C4C3×H(5,𝔽2)12yes—
443C8—24no—
444C8—42no—
445C2×C3—81no—
446C2×C3—161no—
447C2×C3#243111yescharacteristic
448C2×C3#301312yes—
449C2×C3#317812yes—
450C2×C3#322211yescharacteristic
451C2×C3#322311yescharacteristic
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—42no—
457C22—82no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—82no—
462C22C2×C3×#12011yescharacteristic
463C22C3×#53211yescharacteristic
464C22C3×#53511yescharacteristic
465C22C3×#57712yes—
466C22C3×#57812yes—
467C4—24no—
468C4—41no—
469C4—44no—
470C4—42no—
471C4—44no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4C3×#54612yes—
476C4C3×#55012yes—
477C3#2561211yescharacteristic, direct product factor, Sylow
478C2—81no—
479C2—161no—
480C2C3×#243111yescharacteristic
481C2C3×#301312yes—
482C2C3×#317812yes—
483C2C3×#322211yescharacteristic
484C2C3×#322311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
1. C2×C3×#541C2semidirect
2. C2×C3×#497C2semidirect
3. C2×C3×#488C2semidirect
5. C3×#2001C2semidirect
7. C3×#1992C2semidirect
8. C3×#1956C2semidirect
9. C3×#1880C2semidirect
10. C3×#1878C2semidirect
11. C3×#1750C2semidirect
12. #25612C3direct (N × H)
44. C2×#541C2×C3semidirect
45. C2×#497C2×C3semidirect
46. C2×#488C2×C3semidirect
48. #2001C2×C3semidirect
50. #1992C2×C3semidirect
51. #1956C2×C3semidirect
52. #1880C2×C3semidirect
53. #1878C2×C3semidirect
54. #1750C2×C3semidirect
69. C3×C4⋊Q8D4semidirect
73. C3×#125D4semidirect
217. C4⋊Q8C3×D4semidirect
221. #125C3×D4semidirect
477. C3#25612direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#4972
C2×C3×#4882
C3×#26202
C3×#20012
C3×#20002
C3×#19922
C3×#19562
C3×#18802
C3×#18782
C3×#17502
#256123