Finite Group Databasebeta

← C3×#32903 #10651510

Subgroups — C3×#32903 #10651510

Order 768.

1576 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#460C212yesmaximal, complemented
3C2×C3×#448C212yesmaximal, complemented
4C3×#2856C211yesmaximal, characteristic
5C3×#1880C214yesmaximal, complemented
6C3×#1874C212yesmaximal, complemented
7C3×#1803C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#32903C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×SD16C2214yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#121—21no—
14C2×C3×#121—21no—
15C2×C3×#121C2212yes—
16C2×C3×C4×Q8—28no—
17C2×C3×C4×Q8—21no—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2212yes—
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C42—21no—
24C22×C3×C42C2211yescharacteristic
25C3×#460—28no—
26C3×#448—28no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2214yes—
29C2×C3×#102C2214yes—
30C2×C3×C22⋊C8C2212yes—
31C3×#390—21no—
32C3×#390—21no—
33C3×#390C2212yes—
34C3×#384C2211yescharacteristic
35C3×#340C2214yes—
36C3×#336C2212yes—
37C22×C4×Q8C2×C311yescharacteristic, complemented
38C2×#460C2×C312yescomplemented
39C2×#448C2×C312yescomplemented
40#2856C2×C311yescharacteristic
41#1880C2×C314yescomplemented
42#1874C2×C312yescomplemented
43#1803C2×C312yescomplemented
44#1797C2×C311yescharacteristic, complemented
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—21no—
48C22×C3×Q8—22no—
49C22×C3×Q8D414yescomplemented
50C22×C3×Q8C2312yes—
51C22×C3×D4—22no—
52C22×C3×D4—21no—
53C22×C3×D4—21no—
54C22×C3×D4C2312yes—
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×C3×SD16—28no—
61C2×C3×SD16—28no—
62C2×C3×SD16—28no—
63C22×C3×C8C2314yes—
64C3×#127—24no—
65C3×#121—24no—
66C3×#121—24no—
67C3×#121—44no—
68C3×#121—44no—
69C3×C4×Q8—28no—
70C3×C4×Q8—28no—
71C3×C4×Q8—24no—
72C3×C4×Q8—24no—
73C3×C4×Q8—24no—
74C3×C4×Q8D418yescomplemented
75C2×C3×C4⋊C4—28no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—28no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42—24no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C3×C4⋊C8—24no—
96C3×#103—28no—
97C3×#102—28no—
98C3×C22⋊C8—24no—
99C23×Q8C22×C311yescharacteristic
100C22×SD16C22×C314yes—
101C2×#127C22×C311yescharacteristic
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C22×C312yes—
105C2×C4×Q8—28no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C22×C312yes—
109C2×C4×Q8C22×C312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C42—21no—
113C22×C42C22×C311yescharacteristic
114#460—28no—
115#448—28no—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C314yes—
118C2×#102C22×C314yes—
119C2×C22⋊C8C22×C312yes—
120#390—21no—
121#390—21no—
122#390C22×C312yes—
123#384C22×C311yescharacteristic
124#340C22×C314yes—
125#336C22×C312yes—
126C24×C3—22no—
127C24×C3C2×D411yescharacteristic
128C2×C3×Q8—24no—
129C2×C3×Q8—216no—
130C2×C3×Q8—24no—
131C2×C3×Q8—216no—
132C2×C3×Q8—24no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8C2×D414yes—
137C2×C3×Q8C2×D414yes—
138C2×C3×Q8#4114yes—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—24no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—41no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4SD1614yescomplemented
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4#4112yes—
176C22×C3×C4C2411yescharacteristic
177C3×SD16—416no—
178C2×C3×C8—24no—
179C2×C3×C8—24no—
180C2×C3×C8—24no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×D414yes—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—44no—
193C3×C42—28no—
194C3×C42—28no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D414yes—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—21no—
202C22×Q8—22no—
203C22×Q8C3×D414yescomplemented
204C22×Q8C23×C312yes—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C23×C312yes—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C3×D411yescharacteristic
212C23×C4C3×D411yescharacteristic
213C23×C4C23×C311yescharacteristic
214C2×SD16—28no—
215C2×SD16—28no—
216C2×SD16—28no—
217C22×C8C23×C314yes—
218#127—24no—
219#121—24no—
220#121—24no—
221#121—44no—
222#121—44no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8C3×D418yescomplemented
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C23×C312yes—
237C2×C4⋊C4C23×C312yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C42—28no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42C3×D411yescharacteristic
246C2×C42C3×D411yescharacteristic
247C2×C42C23×C311yescharacteristic
248C2×C42C23×C311yescharacteristic
249C4⋊C8—24no—
250#103—28no—
251#102—28no—
252C22⋊C8—24no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3#12112yes—
265C23×C3C2×SD1612yes—
266C23×C3AGL(1,ℤ/8)12yes—
267C23×C3C22×D411yescharacteristic
268C3×Q8—216no—
269C3×Q8—24no—
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8#12118yescomplemented
273C3×D4—44no—
274C3×D4—44no—
275C3×D4—84no—
276C3×D4—88no—
277C3×D4—84no—
278C2×C3×C4—216no—
279C2×C3×C4—21no—
280C2×C3×C4—24no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—28no—
291C2×C3×C4—21no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—44no—
296C2×C3×C4—48no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12012yes—
304C2×C3×C4#12112yes—
305C2×C3×C4C2×SD1612yes—
306C2×C3×C4C2×SD1612yes—
307C2×C3×C4#13712yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D412yes—
310C2×C3×C4C2×#4112yes—
311C2×C3×C4C2×#4111yescharacteristic
312C3×C8—44no—
313C24—22no—
314C24C2×C3×D411yescharacteristic
315C2×Q8—24no—
316C2×Q8—216no—
317C2×Q8—24no—
318C2×Q8—216no—
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8C2×C3×D414yes—
324C2×Q8C2×C3×D414yes—
325C2×Q8C3×#4114yes—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C3×SD1614yescomplemented
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C3×#4112yes—
363C22×C4C24×C311yescharacteristic
364SD16—416no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C4⋊C4—28no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4C2×C3×D414yes—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C42—28no—
381C42—28no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42C2×C3×D414yes—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3#46014yescomplemented
400C22×C3C2×#12011yescharacteristic
401C22×C3C2×#12111yescharacteristic
402C22×C3#52511yescharacteristic
403C22×C3C22×SD1611yescharacteristic
404C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C3C2×#13711yescharacteristic
406C22×C3#57511yescharacteristic
407C3×C4—28no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—41no—
412C3×C4—44no—
413C3×C4—82no—
414C3×C4#44814yescomplemented
415C3×C4C2×#12112yes—
416C3×C4#52512yes—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23C3×#12112yes—
429C23C2×C3×SD1612yes—
430C23C3×AGL(1,ℤ/8)12yes—
431C23C22×C3×D411yescharacteristic
432Q8—216no—
433Q8—24no—
434Q8—24no—
435Q8—24no—
436Q8C3×#12118yescomplemented
437D4—44no—
438D4—44no—
439D4—84no—
440D4—88no—
441D4—84no—
442C2×C4—216no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—44no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4C3×#12012yes—
467C2×C4C3×#12012yes—
468C2×C4C3×#12112yes—
469C2×C4C2×C3×SD1612yes—
470C2×C4C2×C3×SD1612yes—
471C2×C4C3×#13712yes—
472C2×C4C22×C3×D411yescharacteristic
473C2×C4C22×C3×D412yes—
474C2×C4C2×C3×#4112yes—
475C2×C4C2×C3×#4111yescharacteristic
476C8—44no—
477C2×C3—24no—
478C2×C3—162no—
479C2×C3#284711yescharacteristic
480C2×C3C2×#44811yescharacteristic
481C2×C3#300611yescharacteristic
482C2×C3C2×#46011yescharacteristic
483C2×C3#303711yescharacteristic
484C2×C3#310311yescharacteristic
485C2×C3#311211yescharacteristic
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22C3×#46014yescomplemented
500C22C2×C3×#12011yescharacteristic
501C22C2×C3×#12111yescharacteristic
502C22C3×#52511yescharacteristic
503C22C22×C3×SD1611yescharacteristic
504C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
505C22C2×C3×#13711yescharacteristic
506C22C3×#57511yescharacteristic
507C4—22no—
508C4—22no—
509C4—28no—
510C4—22no—
511C4—41no—
512C4—44no—
513C4—82no—
514C4C3×#44814yescomplemented
515C4C2×C3×#12112yes—
516C4C3×#52512yes—
517C3#3290311yescharacteristic, direct product factor, Sylow
518C2—24no—
519C2—162no—
520C2C3×#284711yescharacteristic
521C2C2×C3×#44811yescharacteristic
522C2C3×#300611yescharacteristic
523C2C2×C3×#46011yescharacteristic
524C2C3×#303711yescharacteristic
525C2C3×#310311yescharacteristic
526C2C3×#311211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×Q8C2semidirect
2. C2×C3×#460C2semidirect
3. C2×C3×#448C2semidirect
5. C3×#1880C2semidirect
6. C3×#1874C2semidirect
7. C3×#1803C2semidirect
8. C3×#1797C2semidirect
9. #32903C3direct (N × H)
37. C22×C4×Q8C2×C3semidirect
38. C2×#460C2×C3semidirect
39. C2×#448C2×C3semidirect
41. #1880C2×C3semidirect
42. #1874C2×C3semidirect
43. #1803C2×C3semidirect
44. #1797C2×C3semidirect
49. C22×C3×Q8D4semidirect
74. C3×C4×Q8D4semidirect
168. C22×C3×C4SD16semidirect
203. C22×Q8C3×D4semidirect
228. C4×Q8C3×D4semidirect
272. C3×Q8#121semidirect
355. C22×C4C3×SD16semidirect
399. C22×C3#460semidirect
414. C3×C4#448semidirect
436. Q8C3×#121semidirect
499. C22C3×#460semidirect
514. C4C3×#448semidirect
517. C3#32903direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×C3×#4602
C2×C3×#4482
C3×#28562
C3×#18802
C3×#18742
C3×#18032
C3×#17972
#329033