Finite Group Databasebeta

← C3×#24209 #10642816

Subgroups — C3×#24209 #10642816

Order 768.

1514 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#461C212yesmaximal, complemented
3C2×C3×#448C212yesmaximal
4C3×C4×#121C211yesmaximal, characteristic
5C3×#2043C212yesmaximal, complemented
6C3×#2013C212yesmaximal, complemented
7C3×#1981C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9C3×#1787C212yesmaximal, complemented
10#24209C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2212yes—
12C22×C3×SD16C2214yes—
13C2×C3×C4⋊Q8—28no—
14C2×C3×C4⋊Q8—21no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×#121C2211yescharacteristic
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2211yescharacteristic
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4—21no—
23C22×C3×C42C2211yescharacteristic
24C3×#461—28no—
25C3×#448—28no—
26C2×C3×C4⋊C8C2212yes—
27C2×C3×#103C2214yes—
28C2×C3×#103C2214yes—
29C2×C3×#103C2214yes—
30C2×C3×C22⋊C8C2212yes—
31C3×#390C2211yescharacteristic
32C3×#389C2211yescharacteristic
33C3×#388—21no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C22⋊C4—21no—
38C3×#336C2212yes—
39C22×C4⋊Q8C2×C311yescharacteristic, complemented
40C2×#461C2×C312yescomplemented
41C2×#448C2×C312yes—
42C4×#121C2×C311yescharacteristic
43#2043C2×C312yescomplemented
44#2013C2×C312yescomplemented
45#1981C2×C312yescomplemented
46#1797C2×C311yescharacteristic, complemented
47#1787C2×C312yescomplemented
48C22×C3×Q8—28no—
49C22×C3×Q8—22no—
50C22×C3×Q8—24no—
51C22×C3×Q8—24no—
52C22×C3×Q8—44no—
53C22×C3×Q8C2314yes—
54C22×C3×D4—21no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—22no—
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C2×C3×SD16—28no—
62C2×C3×SD16—28no—
63C2×C3×SD16—44no—
64C22×C3×C8C2314yes—
65C3×C4⋊Q8—28no—
66C3×C4⋊Q8—28no—
67C3×C4⋊Q8—28no—
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8D418yescomplemented
70C3×#121—22no—
71C3×#121D414yes—
72C3×C4×D4—24no—
73C3×C4×D4—44no—
74C2×C3×C4⋊C4—28no—
75C2×C3×C4⋊C4—28no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—24no—
90C2×C3×C42—21no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C3×C4⋊C8—24no—
96C3×#103—28no—
97C3×#103—28no—
98C3×#103—44no—
99C3×C22⋊C8—24no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—21no—
103C23×Q8C22×C312yes—
104C22×SD16C22×C314yes—
105C2×C4⋊Q8—28no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C22×C312yes—
109C2×C4⋊Q8C22×C312yes—
110C2×#121C22×C311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4C22×C311yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C42C22×C311yescharacteristic
116#461—28no—
117#448—28no—
118C2×C4⋊C8C22×C312yes—
119C2×#103C22×C314yes—
120C2×#103C22×C314yes—
121C2×#103C22×C314yes—
122C2×C22⋊C8C22×C312yes—
123#390C22×C311yescharacteristic
124#389C22×C311yescharacteristic
125#388—21no—
126#386—21no—
127#380C22×C311yescharacteristic
128C4×C4⋊C4C22×C311yescharacteristic
129C4×C22⋊C4—21no—
130#336C22×C312yes—
131C24×C3—21no—
132C24×C3C2×D411yescharacteristic
133C2×C3×Q8—28no—
134C2×C3×Q8—24no—
135C2×C3×Q8—24no—
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—416no—
140C2×C3×Q8—416no—
141C2×C3×Q8—44no—
142C2×C3×Q8—44no—
143C2×C3×Q8C2×D418yes—
144C2×C3×D4—24no—
145C2×C3×D4—22no—
146C2×C3×D4—21no—
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4C2×D412yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—28no—
159C22×C3×C4—21no—
160C22×C3×C4—24no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—44no—
164C22×C3×C4—44no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4SD1614yescomplemented
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
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—28no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×D412yes—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—42no—
199C3×C42C2×D414yes—
200C22×Q8—28no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8C23×C314yes—
206C22×D4—21no—
207C22×D4C23×C311yescharacteristic
208C23×C4—22no—
209C23×C4—21no—
210C23×C4C3×D411yescharacteristic
211C23×C4C3×D411yescharacteristic
212C23×C4C23×C311yescharacteristic
213C2×SD16—28no—
214C2×SD16—28no—
215C2×SD16—44no—
216C22×C8C23×C314yes—
217C4⋊Q8—28no—
218C4⋊Q8—28no—
219C4⋊Q8—28no—
220C4⋊Q8—24no—
221C4⋊Q8C3×D418yescomplemented
222#121—22no—
223#121C3×D414yes—
224C4×D4—24no—
225C4×D4—44no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C42—24no—
242C2×C42—21no—
243C2×C42C3×D411yescharacteristic
244C2×C42C3×D411yescharacteristic
245C2×C42C23×C311yescharacteristic
246C2×C42C23×C311yescharacteristic
247C4⋊C8—24no—
248#103—28no—
249#103—28no—
250#103—44no—
251C22⋊C8—24no—
252#95—21no—
253#95—21no—
254#95—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—22no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3#12112yes—
267C23×C3C2×SD1612yes—
268C23×C3#13712yes—
269C23×C3C22×D411yescharacteristic
270C3×Q8—28no—
271C3×Q8—28no—
272C3×Q8—416no—
273C3×Q8—44no—
274C3×Q8—44no—
275C3×D4—24no—
276C3×D4—48no—
277C3×D4—42no—
278C2×C3×C4—28no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—28no—
290C2×C3×C4—28no—
291C2×C3×C4—22no—
292C2×C3×C4—416no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—84no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4#12112yes—
307C2×C3×C4C2×SD1612yes—
308C2×C3×C4C2×SD1612yes—
309C2×C3×C4#13712yes—
310C2×C3×C4#13712yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D412yes—
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)12yes—
315C3×C8—44no—
316C24—21no—
317C24C2×C3×D411yescharacteristic
318C2×Q8—28no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8C2×C3×D418yes—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4C2×C3×D412yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4C3×SD1614yescomplemented
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D412yes—
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C3×#4112yes—
361C22×C4C24×C311yescharacteristic
362SD16—416no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4C2×C3×D412yes—
375C22⋊C4—24no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—42no—
384C42C2×C3×D414yes—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3#46114yescomplemented
399C22×C3C2×#12011yescharacteristic
400C22×C3C2×#12111yescharacteristic
401C22×C3#53511yescharacteristic
402C22×C3C22×SD1611yescharacteristic
403C22×C3C2×#13711yescharacteristic
404C22×C3C2×#13711yescharacteristic
405C22×C3C2×#13711yescharacteristic
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—48no—
411C3×C4—42no—
412C3×C4—81no—
413C3×C4—81no—
414C3×C4#44814yescomplemented
415C3×C4D4212yes—
416C3×C4#54612yes—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23C3×#12112yes—
429C23C2×C3×SD1612yes—
430C23C3×#13712yes—
431C23C22×C3×D411yescharacteristic
432Q8—28no—
433Q8—28no—
434Q8—416no—
435Q8—44no—
436Q8—44no—
437D4—24no—
438D4—48no—
439D4—42no—
440C2×C4—28no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—416no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—84no—
466C2×C4C3×#12012yes—
467C2×C4C3×#12012yes—
468C2×C4C3×#12112yes—
469C2×C4C2×C3×SD1612yes—
470C2×C4C2×C3×SD1612yes—
471C2×C4C3×#13712yes—
472C2×C4C3×#13712yes—
473C2×C4C22×C3×D411yescharacteristic
474C2×C4C22×C3×D412yes—
475C2×C4C2×C3×#4111yescharacteristic
476C2×C4C3×H(5,𝔽2)12yes—
477C8—44no—
478C2×C3—24no—
479C2×C3—82no—
480C2×C3#240811yescharacteristic
481C2×C3C2×#44811yescharacteristic
482C2×C3#300411yescharacteristic
483C2×C3C2×#46111yescharacteristic
484C2×C3#303911yescharacteristic
485C2×C3#320611yescharacteristic
486C2×C3#321811yescharacteristic
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22C3×#46114yescomplemented
501C22C2×C3×#12011yescharacteristic
502C22C2×C3×#12111yescharacteristic
503C22C3×#53511yescharacteristic
504C22C22×C3×SD1611yescharacteristic
505C22C2×C3×#13711yescharacteristic
506C22C2×C3×#13711yescharacteristic
507C22C2×C3×#13711yescharacteristic
508C4—22no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—48no—
513C4—42no—
514C4—81no—
515C4—81no—
516C4C3×#44814yescomplemented
517C4C3×D4212yes—
518C4C3×#54612yes—
519C3#2420911yescharacteristic, direct product factor, Sylow
520C2—24no—
521C2—82no—
522C2C3×#240811yescharacteristic
523C2C2×C3×#44811yescharacteristic
524C2C3×#300411yescharacteristic
525C2C2×C3×#46111yescharacteristic
526C2C3×#303911yescharacteristic
527C2C3×#320611yescharacteristic
528C2C3×#321811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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×#461C2semidirect
5. C3×#2043C2semidirect
6. C3×#2013C2semidirect
7. C3×#1981C2semidirect
8. C3×#1797C2semidirect
9. C3×#1787C2semidirect
10. #24209C3direct (N × H)
39. C22×C4⋊Q8C2×C3semidirect
40. C2×#461C2×C3semidirect
43. #2043C2×C3semidirect
44. #2013C2×C3semidirect
45. #1981C2×C3semidirect
46. #1797C2×C3semidirect
47. #1787C2×C3semidirect
69. C3×C4⋊Q8D4semidirect
168. C22×C3×C4SD16semidirect
221. C4⋊Q8C3×D4semidirect
353. C22×C4C3×SD16semidirect
398. C22×C3#461semidirect
414. C3×C4#448semidirect
500. C22C3×#461semidirect
516. C4C3×#448semidirect
519. C3#24209direct (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×#4612
C2×C3×#4482
C3×C4×#1212
C3×#20432
C3×#20132
C3×#19812
C3×#17972
C3×#17872
#242093