Finite Group Databasebeta

← C3×#24338 #10642945

Subgroups — C3×#24338 #10642945

Order 768.

1296 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
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic, complemented
3C2×C3×#442C211yesmaximal, characteristic
4C3×#2580C211yesmaximal, characteristic, complemented
5C3×#2058C212yesmaximal, complemented
6C3×#2023C212yesmaximal, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal
10C3×#1864C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24338C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8—21no—
14C22×C3×D8C2211yescharacteristic
15C3×D4×Q8—24no—
16C3×D4×Q8—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8—22no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C3×#476—24no—
26C3×#442—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#97C2211yescharacteristic
35C3×#399—21no—
36C3×#393—21no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#385—22no—
40C3×#382C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×D4×Q8C2×C311yescharacteristic, complemented
47C2×#476C2×C311yescharacteristic, complemented
48C2×#442C2×C311yescharacteristic
49#2580C2×C311yescharacteristic, complemented
50#2058C2×C312yescomplemented
51#2023C2×C312yescomplemented
52#1993C2×C311yescharacteristic, complemented
53#1985C2×C311yescharacteristic, complemented
54#1879C2×C312yes—
55#1864C2×C312yescomplemented
56#1804C2×C311yescharacteristic, complemented
57C22×C3×Q8—28no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—48no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—22no—
65C23×C3×C4—21no—
66C23×C3×C4—21no—
67C2×C3×D8—22no—
68C2×C3×D8—22no—
69C2×C3×D8Q814yescomplemented
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C3×C4⋊Q8—24no—
74C3×C4⋊Q8—24no—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8D414yes—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—216no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4D414yes—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4D412yes—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C42C2312yes—
102C2×C3×C42C2311yescharacteristic
103C3×#106—22no—
104C3×#106—24no—
105C3×C4⋊C8—22no—
106C3×#102—22no—
107C3×#102—24no—
108C3×#102—22no—
109C3×#102—24no—
110C3×#97—22no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—22no—
114C23×Q8—21no—
115C22×D8C22×C311yescharacteristic
116D4×Q8—24no—
117D4×Q8—24no—
118C2×C4⋊Q8C22×C312yes—
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—21no—
122C2×C4×Q8C22×C311yescharacteristic
123C2×C4×D4C22×C311yescharacteristic
124C2×C4×D4C22×C312yes—
125C2×C4×D4C22×C311yescharacteristic
126#476—24no—
127#442—24no—
128C2×#106C22×C311yescharacteristic
129C2×#106C22×C312yes—
130C2×C4⋊C8C22×C311yescharacteristic
131C2×#102C22×C312yes—
132C2×#102C22×C311yescharacteristic
133C2×#102C22×C312yes—
134C2×#102C22×C311yescharacteristic
135C2×#97C22×C311yescharacteristic
136#399—21no—
137#393—21no—
138#390C22×C312yes—
139#389C22×C312yes—
140#385—22no—
141#382C22×C311yescharacteristic
142#340C22×C312yes—
143#340C22×C312yes—
144#340C22×C312yes—
145#336C22×C312yes—
146#336C22×C312yes—
147C24×C3—21no—
148C24×C3—21no—
149C2×C3×Q8—28no—
150C2×C3×Q8—22no—
151C2×C3×Q8—24no—
152C2×C3×Q8—24no—
153C2×C3×Q8—44no—
154C2×C3×Q8—44no—
155C2×C3×Q8—432no—
156C2×C3×Q8—44no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4C2×D412yes—
166C2×C3×D4C2×Q812yes—
167C2×C3×D4C2×Q812yescomplemented
168C2×C3×D4#4112yes—
169C22×C3×C4—28no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—44no—
176C22×C3×C4—44no—
177C22×C3×C4—44no—
178C22×C3×C4—42no—
179C22×C3×C4—82no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2411yescharacteristic
185C3×D8—28no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8C2×Q812yescomplemented
195C3×C4⋊C4—28no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—82no—
206C3×C4⋊C4C2×D414yes—
207C3×C4⋊C4C2×D412yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—44no—
212C3×C22⋊C4—82no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42C2×D412yes—
216C3×C42C2×D412yes—
217C22×Q8—28no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—48no—
221C22×Q8C3×D411yescharacteristic
222C22×D4C23×C311yescharacteristic
223C22×D4C23×C311yescharacteristic
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—21no—
227C2×D8—22no—
228C2×D8—22no—
229C2×D8C3×Q814yescomplemented
230C22×C8C23×C311yescharacteristic
231C22×C8C23×C312yes—
232C22×C8C23×C311yescharacteristic
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—22no—
236C4⋊Q8C3×D414yes—
237C22⋊Q8—28no—
238C22⋊Q8—216no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4C3×D414yes—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C3×D412yes—
251C2×C4⋊C4C3×D411yescharacteristic
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C312yes—
254C2×C4⋊C4C23×C312yes—
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C42C23×C312yes—
262C2×C42C23×C311yescharacteristic
263#106—22no—
264#106—24no—
265C4⋊C8—22no—
266#102—22no—
267#102—24no—
268#102—22no—
269#102—24no—
270#97—22no—
271#95—21no—
272#95—21no—
273#95—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C3×Q8—28no—
286C3×Q8—48no—
287C3×Q8—416no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—22no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4C22⋊Q814yes—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—48no—
306C2×C3×C4—416no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—84no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4AGL(1,ℤ/8)14yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×Q811yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×C3×C42−1+411yescharacteristic
325C3×C8—22no—
326C3×C8—41no—
327C3×C8—42no—
328C24—21no—
329C24—21no—
330C2×Q8—28no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—432no—
337C2×Q8—44no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×C3×D412yes—
347C2×D4C2×C3×Q812yes—
348C2×D4C2×C3×Q812yescomplemented
349C2×D4C3×#4112yes—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—82no—
361C22×C4C2×C3×D412yes—
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D412yes—
365C22×C4C24×C311yescharacteristic
366D8—28no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8C2×C3×Q812yescomplemented
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—82no—
387C4⋊C4C2×C3×D414yes—
388C4⋊C4C2×C3×D412yes—
389C22⋊C4—28no—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—82no—
394C42—22no—
395C42—22no—
396C42C2×C3×D412yes—
397C42C2×C3×D412yes—
398C22×C3—24no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3D4×Q811yescharacteristic
411C22×C3C2×AGL(1,ℤ/8)12yes—
412C22×C3#57712yes—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—22no—
416C3×C4—44no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—82no—
420C3×C4—81no—
421C3×C4C2×C22⋊Q811yescharacteristic
422C3×C4#53611yescharacteristic
423C3×C4#54811yescharacteristic
424C3×C4D4×Q811yescharacteristic
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23C22×C3×D411yescharacteristic
436Q8—28no—
437Q8—48no—
438Q8—416no—
439D4—22no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—22no—
444D4—41no—
445D4—41no—
446D4C3×C22⋊Q814yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—48no—
457C2×C4—416no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—84no—
467C2×C4C3×#12012yes—
468C2×C4C3×#12012yes—
469C2×C4C3×AGL(1,ℤ/8)14yes—
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C22×C3×Q811yescharacteristic
473C2×C4C2×C3×#4111yescharacteristic
474C2×C4C3×2−1+411yescharacteristic
475C2×C4C3×2−1+411yescharacteristic
476C8—22no—
477C8—41no—
478C8—42no—
479C2×C3—44no—
480C2×C3—82no—
481C2×C3#240911yescharacteristic
482C2×C3#300311yescharacteristic
483C2×C3#301011yescharacteristic
484C2×C3#337312yes—
485C2×C3#338412yes—
486C22—24no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—81no—
494C22—81no—
495C22—82no—
496C22C2×C3×#12011yescharacteristic
497C22C3×#54811yescharacteristic
498C22C3×D4×Q811yescharacteristic
499C22C2×C3×AGL(1,ℤ/8)12yes—
500C22C3×#57712yes—
501C4—22no—
502C4—24no—
503C4—22no—
504C4—44no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—82no—
509C4C2×C3×C22⋊Q811yescharacteristic
510C4C3×#53611yescharacteristic
511C4C3×#54811yescharacteristic
512C4C3×D4×Q811yescharacteristic
513C3#2433811yescharacteristic, direct product factor, Sylow
514C2—44no—
515C2—82no—
516C2C3×#240911yescharacteristic
517C2C3×#300311yescharacteristic
518C2C3×#301011yescharacteristic
519C2C3×#337312yes—
520C2C3×#338412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×C3×#476C2semidirect
4. C3×#2580C2semidirect
5. C3×#2058C2semidirect
6. C3×#2023C2semidirect
7. C3×#1993C2semidirect
8. C3×#1985C2semidirect
10. C3×#1864C2semidirect
11. C3×#1804C2semidirect
12. #24338C3direct (N × H)
46. C2×D4×Q8C2×C3semidirect
47. C2×#476C2×C3semidirect
49. #2580C2×C3semidirect
50. #2058C2×C3semidirect
51. #2023C2×C3semidirect
52. #1993C2×C3semidirect
53. #1985C2×C3semidirect
55. #1864C2×C3semidirect
56. #1804C2×C3semidirect
69. C2×C3×D8Q8semidirect
167. C2×C3×D4C2×Q8semidirect
194. C2×C3×C8C2×Q8semidirect
229. C2×D8C3×Q8semidirect
348. C2×D4C2×C3×Q8semidirect
375. C2×C8C2×C3×Q8semidirect
513. C3#24338direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#4762
C2×C3×#4422
C3×#25802
C3×#20582
C3×#20232
C3×#19932
C3×#19852
C3×#18792
C3×#18642
C3×#18042
#243383