Finite Group Databasebeta

← C3×#29595 #10648202

Subgroups — C3×#29595 #10648202

Order 768.

1520 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic, complemented
3C2×C3×#448C211yesmaximal, characteristic, complemented
4C3×#2847C211yesmaximal, characteristic
5C3×#2036C214yesmaximal, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1823C211yesmaximal, characteristic, complemented
8C3×#1787C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic, complemented
10#29595C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×C3×SD16C2212yescomplemented
14C2×C3×C22⋊Q8—24no—
15C2×C3×C22⋊Q8—22no—
16C2×C3×C22⋊Q8—24no—
17C2×C3×C22⋊Q8C2214yes—
18C2×C3×#121C2211yescharacteristic
19C2×C3×#121C2212yes—
20C2×C3×#120—21no—
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4—21no—
24C3×#480—24no—
25C3×#448—24no—
26C2×C3×#107C2212yescomplemented
27C2×C3×#103C2212yescomplemented
28C2×C3×#103C2214yescomplemented
29C2×C3×#102C2212yescomplemented
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2214yes—
33C3×#386—21no—
34C3×#386—22no—
35C3×#380C2212yes—
36C3×#380C2211yescharacteristic
37C3×#340C2214yescomplemented
38C22×C22⋊Q8C2×C311yescharacteristic, complemented
39C2×#480C2×C311yescharacteristic, complemented
40C2×#448C2×C311yescharacteristic, complemented
41#2847C2×C311yescharacteristic
42#2036C2×C314yescomplemented
43#2000C2×C314yescomplemented
44#1823C2×C311yescharacteristic, complemented
45#1787C2×C311yescharacteristic, complemented
46#1779C2×C311yescharacteristic, complemented
47C25×C3D411yescharacteristic
48C22×C3×Q8—22no—
49C22×C3×Q8—21no—
50C22×C3×Q8—44no—
51C22×C3×Q8D414yescomplemented
52C22×C3×Q8C2312yes—
53C22×C3×D4—22no—
54C22×C3×D4—21no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×C3×C4—22no—
59C23×C3×C4—22no—
60C23×C3×C4—44no—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D412yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C2×C3×SD16—24no—
66C2×C3×SD16—44no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×C22⋊Q8—216no—
70C3×C22⋊Q8—28no—
71C3×C22⋊Q8—28no—
72C3×#121—22no—
73C3×#121—22no—
74C3×#121—44no—
75C3×#120—44no—
76C2×C3×C4⋊C4—28no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D414yescomplemented
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—28no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—21no—
90C3×#107—24no—
91C3×#103—24no—
92C3×#103—44no—
93C3×#102—24no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—44no—
97C3×#95—22no—
98C3×#95—21no—
99C23×Q8C22×C311yescharacteristic
100C24×C4C22×C311yescharacteristic
101C22×SD16C22×C312yescomplemented
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8C22×C314yes—
106C2×#121C22×C311yescharacteristic
107C2×#121C22×C312yes—
108C2×#120—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C22⋊C4—21no—
112#480—24no—
113#448—24no—
114C2×#107C22×C312yescomplemented
115C2×#103C22×C312yescomplemented
116C2×#103C22×C314yescomplemented
117C2×#102C22×C312yescomplemented
118C2×C22⋊C8C22×C311yescharacteristic
119C2×C22⋊C8C22×C311yescharacteristic
120C2×C22⋊C8C22×C314yes—
121#386—21no—
122#386—22no—
123#380C22×C311yescharacteristic
124#380C22×C312yes—
125#340C22×C314yescomplemented
126C24×C3—22no—
127C24×C3—24no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—21no—
131C24×C3—42no—
132C24×C3C2×D412yes—
133C24×C3C2×D411yescharacteristic
134C2×C3×Q8—24no—
135C2×C3×Q8—22no—
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—48no—
139C2×C3×Q8—44no—
140C2×C3×Q8—48no—
141C2×C3×Q8C2×D414yes—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—24no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—44no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—48no—
168C22×C3×C4—48no—
169C22×C3×C4—44no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2411yescharacteristic
177C3×SD16—48no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—24no—
181C2×C3×C8—24no—
182C3×C4⋊C4—216no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4C2×D414yes—
189C3×C22⋊C4—216no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C25C3×D411yescharacteristic
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—44no—
198C22×Q8C3×D414yescomplemented
199C22×Q8C23×C312yes—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4C23×C311yescharacteristic
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—44no—
208C23×C4C3×D411yescharacteristic
209C23×C4C3×D412yes—
210C23×C4C23×C312yes—
211C23×C4C23×C311yescharacteristic
212C2×SD16—24no—
213C2×SD16—44no—
214C22×C8C23×C312yes—
215C22×C8C23×C312yes—
216C22⋊Q8—216no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219#121—22no—
220#121—22no—
221#121—44no—
222#120—44no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4C3×D414yescomplemented
228C2×C4⋊C4C23×C312yes—
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C4⋊C4C23×C312yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237#107—24no—
238#103—24no—
239#103—44no—
240#102—24no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—44no—
244#95—22no—
245#95—21no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—28no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—48no—
256C23×C3—48no—
257C23×C3—44no—
258C23×C3—44no—
259C23×C3—41no—
260C23×C3—42no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3#12012yes—
265C23×C3#13512yes—
266C23×C3AGL(1,ℤ/8)12yes—
267C23×C3#13714yes—
268C23×C3C22×D411yescharacteristic
269C3×Q8—24no—
270C3×Q8—24no—
271C3×Q8—48no—
272C3×Q8—44no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—88no—
276C3×D4—84no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—48no—
286C2×C3×C4—44no—
287C2×C3×C4—48no—
288C2×C3×C4—44no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—48no—
292C2×C3×C4—44no—
293C2×C3×C4—44no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4#12012yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4H(5,𝔽2)14yes—
304C3×C8—42no—
305C3×C8—42no—
306C24—22no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24—21no—
311C24—42no—
312C24C2×C3×D412yes—
313C24C2×C3×D411yescharacteristic
314C2×Q8—24no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—48no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8C2×C3×D414yes—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—24no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4C2×C3×D412yes—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D412yes—
355C22×C4C2×C3×D412yes—
356C22×C4C24×C311yescharacteristic
357SD16—48no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—24no—
361C2×C8—24no—
362C4⋊C4—216no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—24no—
368C4⋊C4C2×C3×D414yes—
369C22⋊C4—216no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22×C3—22no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—416no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—81no—
387C22×C3—82no—
388C22×C3—82no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3#44914yes—
392C22×C3C2×#12011yescharacteristic
393C22×C3#53412yes—
394C22×C3C2×#13511yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3C2×#13712yes—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—44no—
400C3×C4—44no—
401C3×C4—82no—
402C3×C4—81no—
403C3×C4#53412yes—
404C3×C4#53512yes—
405C23—22no—
406C23—22no—
407C23—28no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—48no—
415C23—48no—
416C23—44no—
417C23—44no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—42no—
423C23C3×#12012yes—
424C23C3×#13512yes—
425C23C3×AGL(1,ℤ/8)12yes—
426C23C3×#13714yes—
427C23C22×C3×D411yescharacteristic
428Q8—24no—
429Q8—24no—
430Q8—48no—
431Q8—44no—
432D4—42no—
433D4—42no—
434D4—88no—
435D4—84no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4C3×#12012yes—
460C2×C4C22×C3×D411yescharacteristic
461C2×C4C22×C3×D411yescharacteristic
462C2×C4C3×H(5,𝔽2)14yes—
463C8—42no—
464C8—42no—
465C2×C3—24no—
466C2×C3—44no—
467C2×C3—161no—
468C2×C3#261311yescharacteristic
469C2×C3C2×#44911yescharacteristic
470C2×C3#300411yescharacteristic
471C2×C3#319012yes—
472C2×C3#319312yes—
473C22—24no—
474C22—24no—
475C22—22no—
476C22—22no—
477C22—24no—
478C22—24no—
479C22—24no—
480C22—416no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—82no—
486C22—82no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22C3×#44914yes—
491C22C2×C3×#12011yescharacteristic
492C22C3×#53412yes—
493C22C2×C3×#13511yescharacteristic
494C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
495C22C2×C3×#13712yes—
496C4—22no—
497C4—44no—
498C4—42no—
499C4—44no—
500C4—82no—
501C4—81no—
502C4C3×#53412yes—
503C4C3×#53512yes—
504C3#2959511yescharacteristic, direct product factor, Sylow
505C2—24no—
506C2—44no—
507C2—161no—
508C2C3×#261311yescharacteristic
509C2C2×C3×#44911yescharacteristic
510C2C3×#300411yescharacteristic
511C2C3×#319012yes—
512C2C3×#319312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊Q8C2semidirect
2. C2×C3×#480C2semidirect
3. C2×C3×#448C2semidirect
5. C3×#2036C2semidirect
6. C3×#2000C2semidirect
7. C3×#1823C2semidirect
8. C3×#1787C2semidirect
9. C3×#1779C2semidirect
10. #29595C3direct (N × H)
13. C22×C3×SD16C22semidirect
26. C2×C3×#107C22semidirect
27. C2×C3×#103C22semidirect
28. C2×C3×#103C22semidirect
29. C2×C3×#102C22semidirect
37. C3×#340C22semidirect
38. C22×C22⋊Q8C2×C3semidirect
39. C2×#480C2×C3semidirect
40. C2×#448C2×C3semidirect
42. #2036C2×C3semidirect
43. #2000C2×C3semidirect
44. #1823C2×C3semidirect
45. #1787C2×C3semidirect
46. #1779C2×C3semidirect
51. C22×C3×Q8D4semidirect
80. C2×C3×C4⋊C4D4semidirect
101. C22×SD16C22×C3semidirect
114. C2×#107C22×C3semidirect
115. C2×#103C22×C3semidirect
116. C2×#103C22×C3semidirect
117. C2×#102C22×C3semidirect
125. #340C22×C3semidirect
198. C22×Q8C3×D4semidirect
227. C2×C4⋊C4C3×D4semidirect
504. C3#29595direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×C3×#4802
C2×C3×#4482
C3×#28472
C3×#20362
C3×#20002
C3×#18232
C3×#17872
C3×#17792
#295953