Finite Group Databasebeta

← C3×#24222 #10642829

Subgroups — C3×#24222 #10642829

Order 768.

1586 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3496C211yesmaximal, characteristic, complemented
2C3×#3036C212yesmaximal, complemented
3C2×C3×#453C212yesmaximal, complemented
4C3×C4×#127C211yesmaximal, characteristic, complemented
5C3×#2039C212yesmaximal, complemented
6C3×#2006C212yesmaximal, complemented
7C3×#1962C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9C3×#1792C212yesmaximal, complemented
10#24222C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×H(5,𝔽2)C2212yes—
12C2×C3×AGL(1,ℤ/8)C2214yescomplemented
13C3×#547—22no—
14C3×#546—24no—
15C3×D42—22no—
16C3×#537C2212yes—
17C3×#534—22no—
18C3×#532C2212yes—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#121—22no—
21C2×C3×C4×D4—22no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic, complemented
24C3×#463—24no—
25C3×#461—22no—
26C3×#459—22no—
27C3×#453—28no—
28C3×#423C2212yescomplemented
29C2×C3×#104C2214yescomplemented
30C2×C3×#104C2214yescomplemented
31C3×#417C2214yescomplemented
32C2×C3×#100C2212yes—
33C3×#390—22no—
34C3×#390C2211yescharacteristic
35C3×#390C2211yescharacteristic, complemented
36C3×#384C2211yescharacteristic, complemented
37C3×C43C2211yescharacteristic, complemented
38C3×#343C2212yescomplemented
39#3496C2×C311yescharacteristic, complemented
40#3036C2×C312yescomplemented
41C2×#453C2×C312yescomplemented
42C4×#127C2×C311yescharacteristic, complemented
43#2039C2×C312yescomplemented
44#2006C2×C312yescomplemented
45#1962C2×C312yescomplemented
46#1800C2×C311yescharacteristic, complemented
47#1792C2×C312yescomplemented
48C3×H(5,𝔽2)—24no—
49C3×H(5,𝔽2)—24no—
50C3×H(5,𝔽2)—44no—
51C2×C3×#41—24no—
52C2×C3×#41C2314yes—
53C22×C3×D4—24no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—24no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4D412yes—
61C3×AGL(1,ℤ/8)—28no—
62C3×AGL(1,ℤ/8)—44no—
63C2×C3×SD16—24no—
64C2×C3×D8—24no—
65C2×C3×M16C2314yes—
66C3×C4⋊Q8D412yescomplemented
67C3×#127—22no—
68C3×#127D412yescomplemented
69C3×#127D414yescomplemented
70C3×#124D414yescomplemented
71C3×#123—28no—
72C3×C22⋊Q8—24no—
73C3×#121—24no—
74C3×#121—24no—
75C3×#121—24no—
76C3×#121—22no—
77C3×#121—24no—
78C3×#121—22no—
79C3×#120—28no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—48no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—24no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4D412yes—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C3×C4⋊C8—22no—
102C3×C4⋊C8—22no—
103C3×#104—28no—
104C3×#104—44no—
105C3×#103—24no—
106C3×#102—24no—
107C3×#100—24no—
108C2×H(5,𝔽2)C22×C312yes—
109C2×AGL(1,ℤ/8)C22×C314yescomplemented
110#547—22no—
111#546—24no—
112D42—22no—
113#537C22×C312yes—
114#534—22no—
115#532C22×C312yes—
116C2×#127C22×C311yescharacteristic
117C2×#121—22no—
118C2×C4×D4—22no—
119C2×C4×D4C22×C311yescharacteristic
120C2×C4×D4C22×C311yescharacteristic, complemented
121#463—24no—
122#461—22no—
123#459—22no—
124#453—28no—
125#423C22×C312yescomplemented
126C2×#104C22×C314yescomplemented
127C2×#104C22×C314yescomplemented
128#417C22×C314yescomplemented
129C2×#100C22×C312yes—
130#390—22no—
131#390C22×C311yescharacteristic
132#390C22×C311yescharacteristic, complemented
133#384C22×C311yescharacteristic, complemented
134C43C22×C311yescharacteristic, complemented
135#343C22×C312yescomplemented
136C24×C3—24no—
137C24×C3—21no—
138C24×C3C2×D412yes—
139C3×#41—24no—
140C3×#41—24no—
141C3×#41—24no—
142C3×#41—24no—
143C3×#41—48no—
144C3×#41—48no—
145C2×C3×Q8C2×D414yes—
146C2×C3×D4—28no—
147C2×C3×D4—28no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—44no—
158C2×C3×D4—44no—
159C2×C3×D4—48no—
160C2×C3×D4—48no—
161C2×C3×D4—48no—
162C2×C3×D4—48no—
163C2×C3×D4—84no—
164C2×C3×D4C2×D414yes—
165C2×C3×D4C2×D412yes—
166C2×C3×D4C2×D412yescomplemented
167C2×C3×D4#4112yes—
168C22×C3×C4—21no—
169C22×C3×C4—24no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—44no—
176C22×C3×C4—44no—
177C22×C3×C4—41no—
178C22×C3×C4—41no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C3×SD16—48no—
184C3×D8—48no—
185C3×M16—24no—
186C3×M16—24no—
187C2×C3×C8—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—44no—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—42no—
201C3×C42—24no—
202C3×C42—21no—
203C3×C42—22no—
204C3×C42—42no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207C3×C42C2×D412yescomplemented
208H(5,𝔽2)—24no—
209H(5,𝔽2)—24no—
210H(5,𝔽2)—44no—
211C2×#41—24no—
212C2×#41C23×C314yes—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4C23×C311yescharacteristic
218C22×D4C23×C311yescharacteristic
219C23×C4—21no—
220C23×C4C3×D412yes—
221AGL(1,ℤ/8)—28no—
222AGL(1,ℤ/8)—44no—
223C2×SD16—24no—
224C2×D8—24no—
225C2×M16C23×C314yes—
226C4⋊Q8C3×D412yescomplemented
227#127—22no—
228#127C3×D412yescomplemented
229#127C3×D414yescomplemented
230#124C3×D414yescomplemented
231#123—28no—
232C22⋊Q8—24no—
233#121—24no—
234#121—24no—
235#121—24no—
236#121—22no—
237#121—24no—
238#121—22no—
239#120—28no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—48no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C311yescharacteristic
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4C3×D412yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C23×C311yescharacteristic
259C2×C42C23×C311yescharacteristic
260C2×C42C23×C311yescharacteristic
261C4⋊C8—22no—
262C4⋊C8—22no—
263#104—28no—
264#104—44no—
265#103—24no—
266#102—24no—
267#100—24no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—24no—
271C23×C3—22no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—48no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—84no—
280C23×C3—84no—
281C23×C3#12114yes—
282C23×C3C22×D411yescharacteristic
283C3×Q8—24no—
284C3×Q8—24no—
285C3×D4—24no—
286C3×D4—24no—
287C3×D4—24no—
288C3×D4—24no—
289C3×D4—24no—
290C3×D4—416no—
291C3×D4—416no—
292C3×D4—44no—
293C3×D4—44no—
294C3×D4—42no—
295C3×D4—44no—
296C3×D4—82no—
297C3×D4—84no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—48no—
308C2×C3×C4—41no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—82no—
316C2×C3×C4—82no—
317C2×C3×C4—82no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4#12011yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D412yes—
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4H(5,𝔽2)12yes—
325C3×C8—44no—
326C24—24no—
327C24—21no—
328C24C2×C3×D412yes—
329#41—24no—
330#41—24no—
331#41—24no—
332#41—24no—
333#41—48no—
334#41—48no—
335C2×Q8C2×C3×D414yes—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—84no—
354C2×D4C2×C3×D414yes—
355C2×D4C2×C3×D412yes—
356C2×D4C2×C3×D412yescomplemented
357C2×D4C3×#4112yes—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C24×C311yescharacteristic
373SD16—48no—
374D8—48no—
375M16—24no—
376M16—24no—
377C2×C8—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C22⋊C4—28no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—42no—
391C42—24no—
392C42—21no—
393C42—22no—
394C42—42no—
395C42C2×C3×D412yes—
396C42C2×C3×D412yes—
397C42C2×C3×D412yescomplemented
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—24no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—42no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—168no—
414C22×C3C2×#12011yescharacteristic
415C22×C3C2×#12111yescharacteristic
416C22×C3#53511yescharacteristic
417C3×C4—24no—
418C3×C4—21no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—44no—
422C3×C4—42no—
423C3×C4—81no—
424C3×C4#45312yescomplemented
425C3×C4D4212yes—
426C3×C4#54612yes—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—48no—
436C23—41no—
437C23—41no—
438C23—84no—
439C23—84no—
440C23C3×#12114yes—
441C23C22×C3×D411yescharacteristic
442Q8—24no—
443Q8—24no—
444D4—24no—
445D4—24no—
446D4—24no—
447D4—24no—
448D4—24no—
449D4—416no—
450D4—416no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—42no—
455D4—82no—
456D4—84no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—48no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4C3×#12012yes—
478C2×C4C3×#12011yescharacteristic
479C2×C4C3×#12011yescharacteristic
480C2×C4C22×C3×D411yescharacteristic
481C2×C4C22×C3×D412yes—
482C2×C4C2×C3×#4111yescharacteristic
483C2×C4C3×H(5,𝔽2)12yes—
484C8—44no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—44no—
488C2×C3—82no—
489C2×C3—84no—
490C2×C3#240811yescharacteristic
491C2×C3C2×#45311yescharacteristic
492C2×C3#301811yescharacteristic
493C22—21no—
494C22—21no—
495C22—24no—
496C22—44no—
497C22—44no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—42no—
502C22—44no—
503C22—44no—
504C22—82no—
505C22—82no—
506C22—81no—
507C22—81no—
508C22—168no—
509C22C2×C3×#12011yescharacteristic
510C22C2×C3×#12111yescharacteristic
511C22C3×#53511yescharacteristic
512C4—24no—
513C4—21no—
514C4—22no—
515C4—42no—
516C4—44no—
517C4—42no—
518C4—81no—
519C4C3×#45312yescomplemented
520C4C3×D4212yes—
521C4C3×#54612yes—
522C3#2422211yescharacteristic, direct product factor, Sylow
523C2—21no—
524C2—21no—
525C2—44no—
526C2—82no—
527C2—84no—
528C2C3×#240811yescharacteristic
529C2C2×C3×#45311yescharacteristic
530C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3496C2semidirect
2. C3×#3036C2semidirect
3. C2×C3×#453C2semidirect
4. C3×C4×#127C2semidirect
5. C3×#2039C2semidirect
6. C3×#2006C2semidirect
7. C3×#1962C2semidirect
8. C3×#1800C2semidirect
9. C3×#1792C2semidirect
10. #24222C3direct (N × H)
12. C2×C3×AGL(1,ℤ/8)C22semidirect
23. C2×C3×C4×D4C22semidirect
28. C3×#423C22semidirect
29. C2×C3×#104C22semidirect
30. C2×C3×#104C22semidirect
31. C3×#417C22semidirect
35. C3×#390C22semidirect
36. C3×#384C22semidirect
37. C3×C43C22semidirect
38. C3×#343C22semidirect
39. #3496C2×C3semidirect
40. #3036C2×C3semidirect
41. C2×#453C2×C3semidirect
42. C4×#127C2×C3semidirect
43. #2039C2×C3semidirect
44. #2006C2×C3semidirect
45. #1962C2×C3semidirect
46. #1800C2×C3semidirect
47. #1792C2×C3semidirect
66. C3×C4⋊Q8D4semidirect
68. C3×#127D4semidirect
69. C3×#127D4semidirect
70. C3×#124D4semidirect
109. C2×AGL(1,ℤ/8)C22×C3semidirect
120. C2×C4×D4C22×C3semidirect
125. #423C22×C3semidirect
126. C2×#104C22×C3semidirect
127. C2×#104C22×C3semidirect
128. #417C22×C3semidirect
132. #390C22×C3semidirect
133. #384C22×C3semidirect
134. C43C22×C3semidirect
135. #343C22×C3semidirect
166. C2×C3×D4C2×D4semidirect
207. C3×C42C2×D4semidirect
226. C4⋊Q8C3×D4semidirect
228. #127C3×D4semidirect
229. #127C3×D4semidirect
230. #124C3×D4semidirect
356. C2×D4C2×C3×D4semidirect
397. C42C2×C3×D4semidirect
424. C3×C4#453semidirect
519. C4C3×#453semidirect
522. C3#24222direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34962
C3×#30362
C2×C3×#4532
C3×C4×#1272
C3×#20392
C3×#20062
C3×#19622
C3×#18002
C3×#17922
#242223