Finite Group Databasebeta

← C3×#24327 #10642934

Subgroups — C3×#24327 #10642934

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic, complemented
3C2×C3×C4×D8C211yesmaximal, characteristic
4C3×#2538C211yesmaximal, characteristic, complemented
5C3×#2059C212yesmaximal, complemented
6C3×#2025C212yesmaximal, complemented
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1984C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal
10C3×#1864C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24327C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C3×#554—24no—
15C3×#554—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—22no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C22×C3×C4⋊C4—21no—
26C3×#476—24no—
27C3×C4×D8—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×#106C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×C4×C8C2211yescharacteristic
36C3×#396—22no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#384C2211yescharacteristic
40C3×#380—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#554C2×C311yescharacteristic, complemented
47C2×#476C2×C311yescharacteristic, complemented
48C2×C4×D8C2×C311yescharacteristic
49#2538C2×C311yescharacteristic, complemented
50#2059C2×C312yescomplemented
51#2025C2×C312yescomplemented
52#1990C2×C311yescharacteristic, complemented
53#1984C2×C311yescharacteristic, complemented
54#1879C2×C312yes—
55#1864C2×C312yescomplemented
56#1802C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—22no—
63C23×C3×C4—21no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C2×C3×D8Q814yescomplemented
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yes—
72C3×#125—24no—
73C3×#125—24no—
74C3×C22⋊Q8—28no—
75C3×C22⋊Q8—216no—
76C3×C22⋊Q8—44no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4D414yes—
85C2×C3×C4⋊C4—28no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D412yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#107—22no—
106C3×#106—24no—
107C3×C4⋊C8—22no—
108C3×#102—22no—
109C3×#102—24no—
110C3×#102—22no—
111C3×#102—24no—
112C3×C4×C8—22no—
113C3×#95—22no—
114C22×D8C22×C311yescharacteristic
115#554—24no—
116#554—24no—
117C2×C4⋊Q8C22×C311yescharacteristic
118C2×#125C22×C312yes—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—22no—
122C2×C4×Q8C22×C311yescharacteristic
123C2×C4×D4C22×C312yes—
124C2×C4×D4C22×C311yescharacteristic
125C2×C4×D4C22×C311yescharacteristic
126C22×C4⋊C4—21no—
127#476—24no—
128C4×D8—24no—
129C2×#107C22×C311yescharacteristic
130C2×#106C22×C312yes—
131C2×C4⋊C8C22×C311yescharacteristic
132C2×#102C22×C312yes—
133C2×#102C22×C311yescharacteristic
134C2×#102C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×C4×C8C22×C311yescharacteristic
137#396—22no—
138#390C22×C312yes—
139#389C22×C312yes—
140#384C22×C311yescharacteristic
141#380—21no—
142#340C22×C312yes—
143#340C22×C312yes—
144#340C22×C312yes—
145#336C22×C312yes—
146#336C22×C312yes—
147C24×C3—21no—
148C24×C3—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—81no—
154C2×C3×Q8—81no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
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×D4C2×D412yes—
164C2×C3×D4C2×Q812yes—
165C2×C3×D4C2×Q812yes—
166C2×C3×D4#4112yes—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—28no—
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—42no—
177C22×C3×C4—42no—
178C22×C3×C4—82no—
179C22×C3×C4—82no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
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×Q812yes—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—44no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—81no—
209C3×C4⋊C4C2×D414yes—
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—82no—
216C3×C42—22no—
217C3×C42—22no—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220C22×Q8—21no—
221C22×Q8C3×D411yescharacteristic
222C22×D4C23×C311yescharacteristic
223C22×D4C23×C311yescharacteristic
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—21no—
227C2×D8—22no—
228C2×D8—22no—
229C2×D8C3×Q814yescomplemented
230C22×C8C23×C311yescharacteristic
231C22×C8C23×C312yes—
232C22×C8C23×C311yescharacteristic
233C4⋊Q8—22no—
234C4⋊Q8C3×D414yes—
235#125—24no—
236#125—24no—
237C22⋊Q8—28no—
238C22⋊Q8—216no—
239C22⋊Q8—44no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4C3×D414yes—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4C3×D411yescharacteristic
256C2×C4⋊C4C3×D412yes—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C312yes—
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C4⋊C4C23×C312yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C42C23×C312yes—
267C2×C42C23×C311yescharacteristic
268#107—22no—
269#106—24no—
270C4⋊C8—22no—
271#102—22no—
272#102—24no—
273#102—22no—
274#102—24no—
275C4×C8—22no—
276#95—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—44no—
293C3×Q8—84no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4C22⋊Q814yescomplemented
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—48no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—88no—
324C2×C3×C4—84no—
325C2×C3×C4#12012yes—
326C2×C3×C4#12012yes—
327C2×C3×C4#13512yes—
328C2×C3×C4AGL(1,ℤ/8)12yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×Q811yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C42−1+411yescharacteristic
334C2×C3×C42−1+411yescharacteristic
335C3×C8—22no—
336C3×C8—41no—
337C3×C8—42no—
338C24—21no—
339C24—21no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—81no—
345C2×Q8—81no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4C2×C3×D412yes—
355C2×D4C2×C3×Q812yes—
356C2×D4C2×C3×Q812yes—
357C2×D4C3×#4112yes—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4C2×C3×D412yes—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C24×C311yescharacteristic
376D8—28no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—22no—
385C2×C8C2×C3×Q812yes—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—81no—
400C4⋊C4C2×C3×D414yes—
401C4⋊C4C2×C3×D412yes—
402C22⋊C4—28no—
403C22⋊C4—24no—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C22⋊C4—82no—
407C42—22no—
408C42—22no—
409C42C2×C3×D412yes—
410C42C2×C3×D412yes—
411C22×C3—24no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54811yescharacteristic
427C22×C3D4×Q811yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
430C22×C3#57611yescharacteristic
431C22×C3#57711yescharacteristic
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—44no—
437C3×C4—82no—
438C3×C4—81no—
439C3×C4#44914yes—
440C3×C4C2×C22⋊Q811yescharacteristic
441C3×C4#53611yescharacteristic
442C3×C4#54811yescharacteristic
443C3×C4D4×Q811yescharacteristic
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23C22×C3×D411yescharacteristic
459Q8—44no—
460Q8—84no—
461D4—22no—
462D4—22no—
463D4—22no—
464D4—22no—
465D4—22no—
466D4—41no—
467D4—41no—
468D4C3×C22⋊Q814yescomplemented
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—88no—
491C2×C4—84no—
492C2×C4C3×#12012yes—
493C2×C4C3×#12012yes—
494C2×C4C3×#13512yes—
495C2×C4C3×AGL(1,ℤ/8)12yes—
496C2×C4C22×C3×D411yescharacteristic
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C22×C3×Q811yescharacteristic
499C2×C4C2×C3×#4111yescharacteristic
500C2×C4C3×2−1+411yescharacteristic
501C2×C4C3×2−1+411yescharacteristic
502C8—22no—
503C8—41no—
504C8—42no—
505C2×C3—44no—
506C2×C3—82no—
507C2×C3#240911yescharacteristic
508C2×C3C2×#44911yescharacteristic
509C2×C3#301211yescharacteristic
510C2×C3#336011yescharacteristic
511C2×C3#336611yescharacteristic
512C2×C3#338011yescharacteristic
513C2×C3#338411yescharacteristic
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22C2×C3×#12011yescharacteristic
529C22C3×#54811yescharacteristic
530C22C3×D4×Q811yescharacteristic
531C22C2×C3×#13511yescharacteristic
532C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
533C22C3×#57611yescharacteristic
534C22C3×#57711yescharacteristic
535C4—22no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—81no—
541C4—82no—
542C4C3×#44914yes—
543C4C2×C3×C22⋊Q811yescharacteristic
544C4C3×#53611yescharacteristic
545C4C3×#54811yescharacteristic
546C4C3×D4×Q811yescharacteristic
547C3#2432711yescharacteristic, direct product factor, Sylow
548C2—44no—
549C2—82no—
550C2C3×#240911yescharacteristic
551C2C2×C3×#44911yescharacteristic
552C2C3×#301211yescharacteristic
553C2C3×#336011yescharacteristic
554C2C3×#336611yescharacteristic
555C2C3×#338011yescharacteristic
556C2C3×#338411yescharacteristic
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×#554C2semidirect
2. C2×C3×#476C2semidirect
4. C3×#2538C2semidirect
5. C3×#2059C2semidirect
6. C3×#2025C2semidirect
7. C3×#1990C2semidirect
8. C3×#1984C2semidirect
10. C3×#1864C2semidirect
11. C3×#1802C2semidirect
12. #24327C3direct (N × H)
46. C2×#554C2×C3semidirect
47. C2×#476C2×C3semidirect
49. #2538C2×C3semidirect
50. #2059C2×C3semidirect
51. #2025C2×C3semidirect
52. #1990C2×C3semidirect
53. #1984C2×C3semidirect
55. #1864C2×C3semidirect
56. #1802C2×C3semidirect
66. C2×C3×D8Q8semidirect
229. C2×D8C3×Q8semidirect
301. C3×D4C22⋊Q8semidirect
468. D4C3×C22⋊Q8semidirect
547. C3#24327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#4762
C2×C3×C4×D82
C3×#25382
C3×#20592
C3×#20252
C3×#19902
C3×#19842
C3×#18792
C3×#18642
C3×#18022
#243273