Finite Group Databasebeta

← C3×#29594 #10648201

Subgroups — C3×#29594 #10648201

Order 768.

1864 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic, complemented
2C2×C3×#481C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic, complemented
4C3×#2847C211yesmaximal, characteristic, complemented
5C3×#2034C214yesmaximal, complemented
6C3×#1999C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1786C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic, complemented
10#29594C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4—21no—
12C23×C3×D4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C22×C3×D8C2212yescomplemented
15C2×C3×#121—24no—
16C2×C3×#121—22no—
17C2×C3×#121—24no—
18C2×C3×#121C2214yes—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#121C2212yescomplemented
21C2×C3×#120—21no—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4—21no—
24C3×#481—24no—
25C3×#447—24no—
26C2×C3×#106C2212yescomplemented
27C2×C3×#102C2212yescomplemented
28C2×C3×#102C2212yescomplemented
29C2×C3×#102C2214yescomplemented
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2214yescomplemented
33C3×#386—22no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×#380C2212yescomplemented
37C3×#340C2214yescomplemented
38C22×#121C2×C311yescharacteristic, complemented
39C2×#481C2×C311yescharacteristic, complemented
40C2×#447C2×C311yescharacteristic, complemented
41#2847C2×C311yescharacteristic, complemented
42#2034C2×C314yescomplemented
43#1999C2×C314yescomplemented
44#1824C2×C311yescharacteristic, complemented
45#1786C2×C311yescharacteristic, complemented
46#1779C2×C311yescharacteristic, complemented
47C25×C3—21no—
48C25×C3D411yescharacteristic
49C22×C3×D4—28no—
50C22×C3×D4—24no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4—21no—
54C22×C3×D4—22no—
55C22×C3×D4—44no—
56C22×C3×D4—48no—
57C22×C3×D4D414yescomplemented
58C22×C3×D4C2311yescharacteristic
59C22×C3×D4C2312yes—
60C23×C3×C4—21no—
61C23×C3×C4—22no—
62C23×C3×C4—22no—
63C23×C3×C4—44no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D412yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C2×C3×D8—24no—
69C2×C3×D8—44no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×#121—216no—
73C3×#121—28no—
74C3×#121—28no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#121—44no—
78C3×#120—44no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D414yescomplemented
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—28no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—22no—
91C3×#106—24no—
92C3×#102—24no—
93C3×#102—24no—
94C3×#102—44no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×C22⋊C8—44no—
98C3×#95—21no—
99C3×#95—22no—
100C23×D4—21no—
101C23×D4C22×C311yescharacteristic
102C24×C4C22×C311yescharacteristic
103C22×D8C22×C312yescomplemented
104C2×#121—24no—
105C2×#121—22no—
106C2×#121—24no—
107C2×#121C22×C311yescharacteristic
108C2×#121C22×C314yes—
109C2×#121C22×C312yescomplemented
110C2×#120—21no—
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C22⋊C4—21no—
113#481—24no—
114#447—24no—
115C2×#106C22×C312yescomplemented
116C2×#102C22×C312yescomplemented
117C2×#102C22×C312yescomplemented
118C2×#102C22×C314yescomplemented
119C2×C22⋊C8C22×C311yescharacteristic
120C2×C22⋊C8C22×C311yescharacteristic
121C2×C22⋊C8C22×C314yescomplemented
122#386—22no—
123#386—21no—
124#380C22×C311yescharacteristic
125#380C22×C312yescomplemented
126#340C22×C314yescomplemented
127C24×C3—22no—
128C24×C3—24no—
129C24×C3—24no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—21no—
134C24×C3—42no—
135C24×C3—44no—
136C24×C3—44no—
137C24×C3—42no—
138C24×C3—42no—
139C24×C3C2×D412yes—
140C24×C3C2×D411yescharacteristic
141C2×C3×D4—216no—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—48no—
150C2×C3×D4—44no—
151C2×C3×D4—48no—
152C2×C3×D4—432no—
153C2×C3×D4—48no—
154C2×C3×D4—48no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—44no—
161C2×C3×D4C2×D414yes—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—48no—
177C22×C3×C4—48no—
178C22×C3×C4—44no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2411yescharacteristic
184C3×D8—48no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—24no—
188C2×C3×C8—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—216no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—42no—
200C25—21no—
201C25C3×D411yescharacteristic
202C22×D4—28no—
203C22×D4—24no—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—44no—
209C22×D4—48no—
210C22×D4C3×D414yescomplemented
211C22×D4C23×C311yescharacteristic
212C22×D4C23×C312yes—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—44no—
217C23×C4C3×D411yescharacteristic
218C23×C4C3×D412yes—
219C23×C4C23×C312yes—
220C23×C4C23×C311yescharacteristic
221C2×D8—24no—
222C2×D8—44no—
223C22×C8C23×C312yes—
224C22×C8C23×C312yes—
225#121—216no—
226#121—28no—
227#121—28no—
228#121—22no—
229#121—22no—
230#121—44no—
231#120—44no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C3×D414yescomplemented
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C4⋊C4C23×C312yes—
237C2×C4⋊C4C23×C312yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244#106—24no—
245#102—24no—
246#102—24no—
247#102—44no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—44no—
251#95—21no—
252#95—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—28no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—24no—
263C23×C3—48no—
264C23×C3—48no—
265C23×C3—44no—
266C23×C3—48no—
267C23×C3—48no—
268C23×C3—44no—
269C23×C3—44no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—42no—
273C23×C3—48no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3—88no—
280C23×C3—88no—
281C23×C3#12012yes—
282C23×C3#13512yes—
283C23×C3AGL(1,ℤ/8)14yes—
284C23×C3AGL(1,ℤ/8)12yes—
285C23×C3C22×D411yescharacteristic
286C3×D4—24no—
287C3×D4—24no—
288C3×D4—48no—
289C3×D4—432no—
290C3×D4—42no—
291C3×D4—44no—
292C3×D4—42no—
293C3×D4—88no—
294C3×D4—84no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—48no—
303C2×C3×C4—44no—
304C2×C3×C4—48no—
305C2×C3×C4—44no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4#12012yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4H(5,𝔽2)14yes—
318C3×C8—42no—
319C3×C8—42no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—42no—
328C24—44no—
329C24—44no—
330C24—42no—
331C24—42no—
332C24C2×C3×D412yes—
333C24C2×C3×D411yescharacteristic
334C2×D4—216no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—432no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—44no—
354C2×D4C2×C3×D414yes—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D412yes—
375C22×C4C2×C3×D412yes—
376C22×C4C24×C311yescharacteristic
377D8—48no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—24no—
387C4⋊C4C2×C3×D414yes—
388C22⋊C4—216no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—22no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—416no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—816no—
409C22×C3—84no—
410C22×C3—84no—
411C22×C3—81no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3#44914yes—
417C22×C3C2×#12011yescharacteristic
418C22×C3#53412yes—
419C22×C3C2×#13511yescharacteristic
420C22×C3C2×AGL(1,ℤ/8)12yes—
421C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—44no—
425C3×C4—82no—
426C3×C4—81no—
427C3×C4#53412yes—
428C3×C4#53512yes—
429C23—22no—
430C23—22no—
431C23—28no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—48no—
440C23—48no—
441C23—44no—
442C23—48no—
443C23—48no—
444C23—44no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—48no—
449C23—44no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—41no—
454C23—42no—
455C23—88no—
456C23—88no—
457C23C3×#12012yes—
458C23C3×#13512yes—
459C23C3×AGL(1,ℤ/8)14yes—
460C23C3×AGL(1,ℤ/8)12yes—
461C23C22×C3×D411yescharacteristic
462D4—24no—
463D4—24no—
464D4—48no—
465D4—432no—
466D4—42no—
467D4—44no—
468D4—42no—
469D4—88no—
470D4—84no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—48no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4C3×#12012yes—
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C3×H(5,𝔽2)14yes—
494C8—42no—
495C8—42no—
496C2×C3—24no—
497C2×C3—44no—
498C2×C3—84no—
499C2×C3—161no—
500C2×C3#261311yescharacteristic
501C2×C3C2×#44911yescharacteristic
502C2×C3#300311yescharacteristic
503C2×C3#318812yes—
504C2×C3#319212yes—
505C22—24no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—416no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—816no—
521C22—84no—
522C22—84no—
523C22—82no—
524C22—82no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22C3×#44914yes—
529C22C2×C3×#12011yescharacteristic
530C22C3×#53412yes—
531C22C2×C3×#13511yescharacteristic
532C22C2×C3×AGL(1,ℤ/8)12yes—
533C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
534C4—22no—
535C4—44no—
536C4—42no—
537C4—82no—
538C4—81no—
539C4C3×#53412yes—
540C4C3×#53512yes—
541C3#2959411yescharacteristic, direct product factor, Sylow
542C2—24no—
543C2—44no—
544C2—84no—
545C2—161no—
546C2C3×#261311yescharacteristic
547C2C2×C3×#44911yescharacteristic
548C2C3×#300311yescharacteristic
549C2C3×#318812yes—
550C2C3×#319212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×#121C2semidirect
2. C2×C3×#481C2semidirect
3. C2×C3×#447C2semidirect
4. C3×#2847C2semidirect
5. C3×#2034C2semidirect
6. C3×#1999C2semidirect
7. C3×#1824C2semidirect
8. C3×#1786C2semidirect
9. C3×#1779C2semidirect
10. #29594C3direct (N × H)
14. C22×C3×D8C22semidirect
20. C2×C3×#121C22semidirect
26. C2×C3×#106C22semidirect
27. C2×C3×#102C22semidirect
28. C2×C3×#102C22semidirect
29. C2×C3×#102C22semidirect
32. C2×C3×C22⋊C8C22semidirect
36. C3×#380C22semidirect
37. C3×#340C22semidirect
38. C22×#121C2×C3semidirect
39. C2×#481C2×C3semidirect
40. C2×#447C2×C3semidirect
41. #2847C2×C3semidirect
42. #2034C2×C3semidirect
43. #1999C2×C3semidirect
44. #1824C2×C3semidirect
45. #1786C2×C3semidirect
46. #1779C2×C3semidirect
57. C22×C3×D4D4semidirect
81. C2×C3×C4⋊C4D4semidirect
103. C22×D8C22×C3semidirect
109. C2×#121C22×C3semidirect
115. C2×#106C22×C3semidirect
116. C2×#102C22×C3semidirect
117. C2×#102C22×C3semidirect
118. C2×#102C22×C3semidirect
121. C2×C22⋊C8C22×C3semidirect
125. #380C22×C3semidirect
126. #340C22×C3semidirect
210. C22×D4C3×D4semidirect
234. C2×C4⋊C4C3×D4semidirect
541. C3#29594direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×C3×#4812
C2×C3×#4472
C3×#28472
C3×#20342
C3×#19992
C3×#18242
C3×#17862
C3×#17792
#295943