Finite Group Databasebeta

← C3×#24642 #10643249

Subgroups — C3×#24642 #10643249

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic, complemented
3C3×#2541C211yesmaximal, characteristic, complemented
4C3×#2357C211yesmaximal, characteristic, complemented
5C3×#2054C212yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1982C212yesmaximal, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24642C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#127C2212yes—
17C2×C3×#121—22no—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C3×#459—24no—
21C3×#442—24no—
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#97C2211yescharacteristic
32C3×#392—21no—
33C3×#392—21no—
34C3×#390C2212yes—
35C3×#390C2212yes—
36C3×#388—22no—
37C3×#382C2211yescharacteristic
38C3×#381—21no—
39C3×C4×C4⋊C4C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#459C2×C311yescharacteristic, complemented
45C2×#442C2×C311yescharacteristic, complemented
46#2541C2×C311yescharacteristic, complemented
47#2357C2×C311yescharacteristic, complemented
48#2054C2×C312yescomplemented
49#2011C2×C312yescomplemented
50#1982C2×C312yescomplemented
51#1961C2×C311yescharacteristic, complemented
52#1940C2×C311yescharacteristic, complemented
53#1926C2×C312yescomplemented
54#1804C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4—21no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C2×C3×D8—24no—
67C2×C3×D8—22no—
68C2×C3×D8—42no—
69C2×C3×D8—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#127—24no—
74C3×#127—22no—
75C3×#127—22no—
76C3×#127—42no—
77C3×#121—48no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4D414yescomplemented
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C42—22no—
93C2×C3×C42C2312yes—
94C2×C3×C42C2311yescharacteristic
95C3×#106—22no—
96C3×C4⋊C8—22no—
97C3×C4⋊C8—42no—
98C3×#102—22no—
99C3×#102—24no—
100C3×#102—22no—
101C3×#102—24no—
102C3×#102—42no—
103C3×#102—42no—
104C3×#97—22no—
105C3×#95—21no—
106C3×#95—22no—
107C3×#95—21no—
108C3×#95D411yescharacteristic
109C3×#95D411yescharacteristic
110C22×D8C22×C311yescharacteristic
111C22×D8C22×C312yes—
112C2×#127C22×C311yescharacteristic
113C2×#127C22×C312yes—
114C2×#121—22no—
115C2×C4×D4C22×C311yescharacteristic
116C2×C4×D4C22×C311yescharacteristic
117#459—24no—
118#442—24no—
119C2×#106C22×C311yescharacteristic
120C2×C4⋊C8C22×C311yescharacteristic
121C2×C4⋊C8C22×C312yes—
122C2×#102C22×C311yescharacteristic
123C2×#102C22×C311yescharacteristic
124C2×#102C22×C312yes—
125C2×#102C22×C312yes—
126C2×#102C22×C312yes—
127C2×#102C22×C312yes—
128C2×#97C22×C311yescharacteristic
129#392—21no—
130#392—21no—
131#390C22×C312yes—
132#390C22×C312yes—
133#388—22no—
134#382C22×C311yescharacteristic
135#381—21no—
136C4×C4⋊C4C22×C312yes—
137C4×C4⋊C4C22×C311yescharacteristic
138C4×C22⋊C4—21no—
139#340C22×C312yes—
140#336C22×C312yes—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—22no—
144C2×C3×D4—21no—
145C2×C3×D4—22no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—44no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—44no—
167C2×C3×D4—82no—
168C2×C3×D4—82no—
169C2×C3×D4—82no—
170C2×C3×D4C2×D412yes—
171C2×C3×D4#4112yes—
172C22×C3×C4—21no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—44no—
177C22×C3×C4—42no—
178C22×C3×C4—82no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2411yescharacteristic
184C3×D8—28no—
185C3×D8—48no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8#4112yes—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—44no—
204C3×C22⋊C4—42no—
205C3×C22⋊C4—44no—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—82no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—82no—
211C3×C42C2×D412yes—
212C3×C42#4112yes—
213C22×D4—21no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4C23×C312yes—
218C22×D4C23×C311yescharacteristic
219C22×D4C23×C311yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C2×D8—22no—
223C2×D8—22no—
224C2×D8—24no—
225C2×D8—22no—
226C2×D8—42no—
227C2×D8—42no—
228C22×C8C23×C312yes—
229C22×C8C23×C311yescharacteristic
230C22×C8C23×C311yescharacteristic
231#127—24no—
232#127—22no—
233#127—22no—
234#127—42no—
235#121—48no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4C3×D414yescomplemented
240C2×C4⋊C4C3×D412yes—
241C2×C4⋊C4C23×C311yescharacteristic
242C2×C4⋊C4C23×C312yes—
243C2×C4⋊C4C23×C311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C42—22no—
251C2×C42C23×C312yes—
252C2×C42C23×C311yescharacteristic
253#106—22no—
254C4⋊C8—22no—
255C4⋊C8—42no—
256#102—22no—
257#102—24no—
258#102—22no—
259#102—24no—
260#102—42no—
261#102—42no—
262#97—22no—
263#95—21no—
264#95—22no—
265#95—21no—
266#95C3×D411yescharacteristic
267#95C3×D411yescharacteristic
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3C22×D411yescharacteristic
286C3×D4—22no—
287C3×D4—22no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—42no—
291C3×D4—41no—
292C3×D4—42no—
293C3×D4—41no—
294C3×D4—42no—
295C3×D4—41no—
296C3×D4—41no—
297C3×D4—42no—
298C3×D4—88no—
299C3×D4—84no—
300C3×D4—84no—
301C3×D4—88no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—84no—
317C2×C3×C4—84no—
318C2×C3×C4#12112yes—
319C2×C3×C4#12112yes—
320C2×C3×C4AGL(1,ℤ/8)14yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4H(5,𝔽2)11yescharacteristic
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C3×C8—22no—
328C3×C8—41no—
329C3×C8—42no—
330C24—21no—
331C24—21no—
332C24—22no—
333C2×D4—21no—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—44no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4C2×C3×D412yes—
360C2×D4C3×#4112yes—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—82no—
368C22×C4C2×C3×D412yes—
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D412yes—
372C22×C4C24×C311yescharacteristic
373D8—28no—
374D8—48no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8C3×#4112yes—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4C2×C3×D412yes—
391C22⋊C4—24no—
392C22⋊C4—44no—
393C22⋊C4—42no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C42—22no—
398C42—22no—
399C42—82no—
400C42C2×C3×D412yes—
401C42C3×#4112yes—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—44no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—82no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3C2×#12111yescharacteristic
420C22×C3#54611yescharacteristic
421C22×C3#55011yescharacteristic
422C22×C3C2×AGL(1,ℤ/8)12yes—
423C22×C3#57612yes—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—24no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—82no—
430C3×C4—81no—
431C3×C4#52511yescharacteristic
432C3×C4#53511yescharacteristic
433C3×C4#55311yescharacteristic
434C3×C4#55511yescharacteristic
435C23—22no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23C22×C3×D411yescharacteristic
453D4—22no—
454D4—22no—
455D4—22no—
456D4—22no—
457D4—42no—
458D4—41no—
459D4—42no—
460D4—41no—
461D4—42no—
462D4—41no—
463D4—41no—
464D4—42no—
465D4—88no—
466D4—84no—
467D4—84no—
468D4—88no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4C3×#12112yes—
486C2×C4C3×#12112yes—
487C2×C4C3×AGL(1,ℤ/8)14yes—
488C2×C4C22×C3×D411yescharacteristic
489C2×C4C2×C3×#4111yescharacteristic
490C2×C4C2×C3×#4111yescharacteristic
491C2×C4C2×C3×#4111yescharacteristic
492C2×C4C3×H(5,𝔽2)11yescharacteristic
493C2×C4C3×H(5,𝔽2)11yescharacteristic
494C8—22no—
495C8—42no—
496C8—41no—
497C2×C3—82no—
498C2×C3—82no—
499C2×C3—162no—
500C2×C3#241411yescharacteristic
501C2×C3#303811yescharacteristic
502C2×C3#304111yescharacteristic
503C2×C3#331812yes—
504C2×C3#339912yes—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—44no—
510C22—82no—
511C22—82no—
512C22—81no—
513C22—81no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22C2×C3×#12111yescharacteristic
523C22C3×#54611yescharacteristic
524C22C3×#55011yescharacteristic
525C22C2×C3×AGL(1,ℤ/8)12yes—
526C22C3×#57612yes—
527C4—22no—
528C4—22no—
529C4—24no—
530C4—42no—
531C4—42no—
532C4—82no—
533C4—81no—
534C4C3×#52511yescharacteristic
535C4C3×#53511yescharacteristic
536C4C3×#55311yescharacteristic
537C4C3×#55511yescharacteristic
538C3#2464211yescharacteristic, direct product factor, Sylow
539C2—82no—
540C2—82no—
541C2—162no—
542C2C3×#241411yescharacteristic
543C2C3×#303811yescharacteristic
544C2C3×#304111yescharacteristic
545C2C3×#331812yes—
546C2C3×#339912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#459C2semidirect
2. C2×C3×#442C2semidirect
3. C3×#2541C2semidirect
4. C3×#2357C2semidirect
5. C3×#2054C2semidirect
6. C3×#2011C2semidirect
7. C3×#1982C2semidirect
8. C3×#1961C2semidirect
9. C3×#1940C2semidirect
10. C3×#1926C2semidirect
11. C3×#1804C2semidirect
12. #24642C3direct (N × H)
44. C2×#459C2×C3semidirect
45. C2×#442C2×C3semidirect
46. #2541C2×C3semidirect
47. #2357C2×C3semidirect
48. #2054C2×C3semidirect
49. #2011C2×C3semidirect
50. #1982C2×C3semidirect
51. #1961C2×C3semidirect
52. #1940C2×C3semidirect
53. #1926C2×C3semidirect
54. #1804C2×C3semidirect
81. C3×C4×D4D4semidirect
239. C4×D4C3×D4semidirect
538. C3#24642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×C3×#4422
C3×#25412
C3×#23572
C3×#20542
C3×#20112
C3×#19822
C3×#19612
C3×#19402
C3×#19262
C3×#18042
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