Finite Group Databasebeta

← C3×#24226 #10642833

Subgroups — C3×#24226 #10642833

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 522 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×#2328C211yesmaximal, characteristic, complemented
5C3×#2040C212yesmaximal, complemented
6C3×#2006C212yesmaximal, complemented
7C3×#1963C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9C3×#1792C212yesmaximal, complemented
10#24226C311yesmaximal, 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×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic, complemented
23C3×#463—24no—
24C3×#461—22no—
25C3×#459—22no—
26C3×#453—28no—
27C3×#423C2212yescomplemented
28C2×C3×#104C2214yescomplemented
29C2×C3×#104C2214yescomplemented
30C3×#417C2214yescomplemented
31C2×C3×#100C2212yes—
32C3×#390C2211yescharacteristic
33C3×#390C2211yescharacteristic, complemented
34C3×#386—22no—
35C3×#386—22no—
36C3×#382C2211yescharacteristic, complemented
37C3×#376C2211yescharacteristic, complemented
38C3×#343C2212yescomplemented
39#3496C2×C311yescharacteristic, complemented
40#3036C2×C312yescomplemented
41C2×#453C2×C312yescomplemented
42#2328C2×C311yescharacteristic, complemented
43#2040C2×C312yescomplemented
44#2006C2×C312yescomplemented
45#1963C2×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—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—24no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4D412yes—
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—22no—
98C3×C4⋊C8—22no—
99C3×#104—28no—
100C3×#104—44no—
101C3×#103—24no—
102C3×#102—24no—
103C3×#100—24no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—22no—
107C2×H(5,𝔽2)C22×C312yes—
108C2×AGL(1,ℤ/8)C22×C314yescomplemented
109#547—22no—
110#546—24no—
111D42—22no—
112#537C22×C312yes—
113#534—22no—
114#532C22×C312yes—
115C2×#127C22×C311yescharacteristic
116C2×#121—22no—
117C2×C4×D4C22×C311yescharacteristic
118C2×C4×D4C22×C311yescharacteristic, complemented
119#463—24no—
120#461—22no—
121#459—22no—
122#453—28no—
123#423C22×C312yescomplemented
124C2×#104C22×C314yescomplemented
125C2×#104C22×C314yescomplemented
126#417C22×C314yescomplemented
127C2×#100C22×C312yes—
128#390C22×C311yescharacteristic
129#390C22×C311yescharacteristic, complemented
130#386—22no—
131#386—22no—
132#382C22×C311yescharacteristic, complemented
133#376C22×C311yescharacteristic, complemented
134#343C22×C312yescomplemented
135C24×C3—24no—
136C24×C3—21no—
137C24×C3C2×D412yes—
138C3×#41—24no—
139C3×#41—24no—
140C3×#41—24no—
141C3×#41—24no—
142C3×#41—48no—
143C3×#41—48no—
144C2×C3×Q8C2×D414yes—
145C2×C3×D4—28no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—28no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—48no—
157C2×C3×D4—48no—
158C2×C3×D4—48no—
159C2×C3×D4—48no—
160C2×C3×D4—84no—
161C2×C3×D4—82no—
162C2×C3×D4—82no—
163C2×C3×D4C2×D414yes—
164C2×C3×D4C2×D412yes—
165C2×C3×D4C2×D412yescomplemented
166C2×C3×D4#4112yes—
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—42no—
175C22×C3×C4—82no—
176C22×C3×C4—82no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C3×SD16—48no—
182C3×D8—48no—
183C3×M16—24no—
184C3×M16—24no—
185C2×C3×C8—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—41no—
196C3×C22⋊C4—41no—
197C3×C22⋊C4—82no—
198C3×C22⋊C4—82no—
199C3×C42—21no—
200C3×C42—22no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203C3×C42C2×D412yescomplemented
204H(5,𝔽2)—24no—
205H(5,𝔽2)—24no—
206H(5,𝔽2)—44no—
207C2×#41—24no—
208C2×#41C23×C314yes—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4C23×C311yescharacteristic
214C22×D4C23×C311yescharacteristic
215C23×C4—21no—
216C23×C4C3×D412yes—
217AGL(1,ℤ/8)—28no—
218AGL(1,ℤ/8)—44no—
219C2×SD16—24no—
220C2×D8—24no—
221C2×M16C23×C314yes—
222C4⋊Q8C3×D412yescomplemented
223#127—22no—
224#127C3×D412yescomplemented
225#127C3×D414yescomplemented
226#124C3×D414yescomplemented
227#123—28no—
228C22⋊Q8—24no—
229#121—24no—
230#121—24no—
231#121—24no—
232#121—22no—
233#121—24no—
234#121—22no—
235#120—28no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C23×C311yescharacteristic
244C2×C4⋊C4C23×C311yescharacteristic
245C2×C22⋊C4—24no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4C3×D412yes—
250C2×C42C23×C311yescharacteristic
251C2×C42C23×C311yescharacteristic
252C2×C42C23×C311yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255#104—28no—
256#104—44no—
257#103—24no—
258#102—24no—
259#100—24no—
260#95—21no—
261#95—21no—
262#95—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—22no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—48no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—84no—
275C23×C3—84no—
276C23×C3#12114yes—
277C23×C3C22×D411yescharacteristic
278C3×Q8—24no—
279C3×Q8—24no—
280C3×D4—24no—
281C3×D4—24no—
282C3×D4—24no—
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—44no—
286C3×D4—416no—
287C3×D4—44no—
288C3×D4—44no—
289C3×D4—42no—
290C3×D4—82no—
291C3×D4—88no—
292C3×D4—84no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—44no—
299C2×C3×C4—48no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—44no—
303C2×C3×C4—44no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—82no—
311C2×C3×C4—84no—
312C2×C3×C4#12011yescharacteristic
313C2×C3×C4#12012yes—
314C2×C3×C4#12011yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D412yes—
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4H(5,𝔽2)12yes—
319C3×C8—44no—
320C24—24no—
321C24—21no—
322C24C2×C3×D412yes—
323#41—24no—
324#41—24no—
325#41—24no—
326#41—24no—
327#41—48no—
328#41—48no—
329C2×Q8C2×C3×D414yes—
330C2×D4—28no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—28no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—84no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4C2×C3×D414yes—
349C2×D4C2×C3×D412yes—
350C2×D4C2×C3×D412yescomplemented
351C2×D4C3×#4112yes—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—42no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4C2×C3×D412yes—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C24×C311yescharacteristic
366SD16—48no—
367D8—48no—
368M16—24no—
369M16—24no—
370C2×C8—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C42—21no—
385C42—22no—
386C42C2×C3×D412yes—
387C42C2×C3×D412yes—
388C42C2×C3×D412yescomplemented
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—24no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—168no—
405C22×C3C2×#12011yescharacteristic
406C22×C3C2×#12111yescharacteristic
407C22×C3#53511yescharacteristic
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—44no—
413C3×C4—42no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4D4212yes—
417C3×C4#54612yes—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—48no—
427C23—41no—
428C23—41no—
429C23—84no—
430C23—84no—
431C23C3×#12114yes—
432C23C22×C3×D411yescharacteristic
433Q8—24no—
434Q8—24no—
435D4—24no—
436D4—24no—
437D4—24no—
438D4—24no—
439D4—24no—
440D4—44no—
441D4—416no—
442D4—44no—
443D4—44no—
444D4—42no—
445D4—82no—
446D4—88no—
447D4—84no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—82no—
466C2×C4—84no—
467C2×C4C3×#12011yescharacteristic
468C2×C4C3×#12012yes—
469C2×C4C3×#12011yescharacteristic
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C22×C3×D412yes—
472C2×C4C2×C3×#4111yescharacteristic
473C2×C4C3×H(5,𝔽2)12yes—
474C8—44no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—44no—
478C2×C3—82no—
479C2×C3—84no—
480C2×C3#240811yescharacteristic
481C2×C3#301911yescharacteristic
482C2×C3#302111yescharacteristic
483C22—21no—
484C22—21no—
485C22—24no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—82no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22—168no—
499C22C2×C3×#12011yescharacteristic
500C22C2×C3×#12111yescharacteristic
501C22C3×#53511yescharacteristic
502C4—21no—
503C4—21no—
504C4—22no—
505C4—42no—
506C4—44no—
507C4—42no—
508C4—81no—
509C4—81no—
510C4C3×D4212yes—
511C4C3×#54612yes—
512C3#2422611yescharacteristic, direct product factor, Sylow
513C2—21no—
514C2—21no—
515C2—44no—
516C2—82no—
517C2—84no—
518C2C3×#240811yescharacteristic
519C2C3×#301911yescharacteristic
520C2C3×#302111yescharacteristic
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×#2328C2semidirect
5. C3×#2040C2semidirect
6. C3×#2006C2semidirect
7. C3×#1963C2semidirect
8. C3×#1800C2semidirect
9. C3×#1792C2semidirect
10. #24226C3direct (N × H)
12. C2×C3×AGL(1,ℤ/8)C22semidirect
22. C2×C3×C4×D4C22semidirect
27. C3×#423C22semidirect
28. C2×C3×#104C22semidirect
29. C2×C3×#104C22semidirect
30. C3×#417C22semidirect
33. C3×#390C22semidirect
36. C3×#382C22semidirect
37. C3×#376C22semidirect
38. C3×#343C22semidirect
39. #3496C2×C3semidirect
40. #3036C2×C3semidirect
41. C2×#453C2×C3semidirect
42. #2328C2×C3semidirect
43. #2040C2×C3semidirect
44. #2006C2×C3semidirect
45. #1963C2×C3semidirect
46. #1800C2×C3semidirect
47. #1792C2×C3semidirect
66. C3×C4⋊Q8D4semidirect
68. C3×#127D4semidirect
69. C3×#127D4semidirect
70. C3×#124D4semidirect
108. C2×AGL(1,ℤ/8)C22×C3semidirect
118. C2×C4×D4C22×C3semidirect
123. #423C22×C3semidirect
124. C2×#104C22×C3semidirect
125. C2×#104C22×C3semidirect
126. #417C22×C3semidirect
129. #390C22×C3semidirect
132. #382C22×C3semidirect
133. #376C22×C3semidirect
134. #343C22×C3semidirect
165. C2×C3×D4C2×D4semidirect
203. C3×C42C2×D4semidirect
222. C4⋊Q8C3×D4semidirect
224. #127C3×D4semidirect
225. #127C3×D4semidirect
226. #124C3×D4semidirect
350. C2×D4C2×C3×D4semidirect
388. C42C2×C3×D4semidirect
512. C3#24226direct (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×#23282
C3×#20402
C3×#20062
C3×#19632
C3×#18002
C3×#17922
#242263