Finite Group Databasebeta

← C3×#32902 #10651509

Subgroups — C3×#32902 #10651509

Order 768.

2008 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×C3×#459C212yesmaximal, complemented
3C2×C3×#447C212yesmaximal, complemented
4C3×#2856C211yesmaximal, characteristic, complemented
5C3×#1879C214yesmaximal, complemented
6C3×#1874C212yesmaximal, complemented
7C3×#1802C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#32902C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4—21no—
12C22×C3×D8C2214yes—
13C2×C3×#127C2211yescharacteristic, complemented
14C2×C3×#121—21no—
15C2×C3×#121—21no—
16C2×C3×#121C2212yes—
17C2×C3×C4×D4—28no—
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2212yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4—21no—
24C22×C3×C42C2211yescharacteristic
25C3×#459—28no—
26C3×#447—28no—
27C2×C3×C4⋊C8C2212yescomplemented
28C2×C3×#102C2214yes—
29C2×C3×#102C2214yescomplemented
30C2×C3×C22⋊C8C2212yes—
31C3×#390—21no—
32C3×#390—21no—
33C3×#390C2212yescomplemented
34C3×#384C2211yescharacteristic, complemented
35C3×#340C2214yescomplemented
36C3×#336C2212yescomplemented
37C22×C4×D4C2×C311yescharacteristic, complemented
38C2×#459C2×C312yescomplemented
39C2×#447C2×C312yescomplemented
40#2856C2×C311yescharacteristic, complemented
41#1879C2×C314yescomplemented
42#1874C2×C312yescomplemented
43#1802C2×C312yescomplemented
44#1797C2×C311yescharacteristic, complemented
45C25×C3—21no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4D414yescomplemented
54C22×C3×D4C2312yes—
55C22×C3×D4C2312yes—
56C23×C3×C4—28no—
57C23×C3×C4—21no—
58C23×C3×C4—22no—
59C23×C3×C4—22no—
60C23×C3×C4—44no—
61C23×C3×C4—44no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×D8—28no—
66C2×C3×D8—28no—
67C2×C3×D8—28no—
68C22×C3×C8C2314yes—
69C3×#127—24no—
70C3×#121—24no—
71C3×#121—24no—
72C3×#121—44no—
73C3×#121—44no—
74C3×C4×D4—28no—
75C3×C4×D4—28no—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C3×C4×D4—24no—
79C3×C4×D4D418yescomplemented
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—28no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C42—24no—
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D411yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—24no—
98C3×#102—28no—
99C3×#102—28no—
100C3×C22⋊C8—24no—
101C23×D4C22×C311yescharacteristic
102C24×C4—21no—
103C22×D8C22×C314yes—
104C2×#127C22×C311yescharacteristic, complemented
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C22×C312yes—
108C2×C4×D4—28no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C22×C312yes—
112C2×C4×D4C22×C312yes—
113C22×C4⋊C4C22×C311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C42C22×C311yescharacteristic
116#459—28no—
117#447—28no—
118C2×C4⋊C8C22×C312yescomplemented
119C2×#102C22×C314yes—
120C2×#102C22×C314yescomplemented
121C2×C22⋊C8C22×C312yes—
122#390—21no—
123#390—21no—
124#390C22×C312yescomplemented
125#384C22×C311yescharacteristic, complemented
126#340C22×C314yescomplemented
127#336C22×C312yescomplemented
128C24×C3—24no—
129C24×C3—24no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—42no—
134C24×C3—42no—
135C24×C3—42no—
136C24×C3—42no—
137C24×C3C2×D411yescharacteristic
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—216no—
141C2×C3×D4—216no—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—24no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—44no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4C2×D414yes—
160C2×C3×D4C2×D414yes—
161C2×C3×D4#4114yes—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—28no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—28no—
172C22×C3×C4—22no—
173C22×C3×C4—41no—
174C22×C3×C4—44no—
175C22×C3×C4—432no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—44no—
179C22×C3×C4—44no—
180C22×C3×C4—44no—
181C22×C3×C4D814yescomplemented
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4#4112yes—
189C22×C3×C4C2411yescharacteristic
190C3×D8—416no—
191C2×C3×C8—24no—
192C2×C3×C8—24no—
193C2×C3×C8—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4C2×D414yes—
201C3×C22⋊C4—28no—
202C3×C22⋊C4—28no—
203C3×C22⋊C4—44no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—44no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42C2×D414yes—
210C25—21no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4C3×D414yescomplemented
219C22×D4C23×C312yes—
220C22×D4C23×C312yes—
221C23×C4—28no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C3×D411yescharacteristic
228C23×C4C3×D411yescharacteristic
229C23×C4C23×C311yescharacteristic
230C2×D8—28no—
231C2×D8—28no—
232C2×D8—28no—
233C22×C8C23×C314yes—
234#127—24no—
235#121—24no—
236#121—24no—
237#121—44no—
238#121—44no—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4C3×D418yescomplemented
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C23×C312yes—
250C2×C4⋊C4C23×C312yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C42—24no—
258C2×C42C3×D411yescharacteristic
259C2×C42C3×D411yescharacteristic
260C2×C42C23×C311yescharacteristic
261C2×C42C23×C311yescharacteristic
262C4⋊C8—24no—
263#102—28no—
264#102—28no—
265C22⋊C8—24no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—216no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—48no—
275C23×C3—48no—
276C23×C3—48no—
277C23×C3—48no—
278C23×C3—48no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—48no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3#12112yes—
292C23×C3C2×D812yes—
293C23×C3AGL(1,ℤ/8)12yes—
294C23×C3C22×D411yescharacteristic
295C3×D4—216no—
296C3×D4—24no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—44no—
300C3×D4—44no—
301C3×D4—84no—
302C3×D4—88no—
303C3×D4—84no—
304C3×D4#12118yescomplemented
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—28no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—44no—
319C2×C3×C4—416no—
320C2×C3×C4—48no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—416no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#12012yes—
328C2×C3×C4#12012yes—
329C2×C3×C4#12112yes—
330C2×C3×C4C2×D812yes—
331C2×C3×C4C2×D812yes—
332C2×C3×C4AGL(1,ℤ/8)12yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D412yes—
335C2×C3×C4C2×#4112yes—
336C2×C3×C4C2×#4111yescharacteristic
337C3×C8—44no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24C2×C3×D411yescharacteristic
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—216no—
351C2×D4—216no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4C2×C3×D414yes—
370C2×D4C2×C3×D414yes—
371C2×D4C3×#4114yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—44no—
385C22×C4—432no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4C3×D814yescomplemented
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C3×#4112yes—
399C22×C4C24×C311yescharacteristic
400D8—416no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4C2×C3×D414yes—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42C2×C3×D414yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—28no—
427C22×C3—432no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3#45914yescomplemented
442C22×C3C2×#12011yescharacteristic
443C22×C3C2×#12111yescharacteristic
444C22×C3#52511yescharacteristic
445C22×C3C22×D811yescharacteristic
446C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
447C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
448C22×C3#57511yescharacteristic
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—44no—
453C3×C4—41no—
454C3×C4—82no—
455C3×C4#44714yescomplemented
456C3×C4C2×#12112yes—
457C3×C4#52512yes—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—216no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—48no—
467C23—48no—
468C23—48no—
469C23—48no—
470C23—48no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—48no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C3×#12112yes—
484C23C2×C3×D812yes—
485C23C3×AGL(1,ℤ/8)12yes—
486C23C22×C3×D411yescharacteristic
487D4—216no—
488D4—24no—
489D4—24no—
490D4—24no—
491D4—44no—
492D4—44no—
493D4—84no—
494D4—88no—
495D4—84no—
496D4C3×#12118yescomplemented
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—416no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—416no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4C3×#12012yes—
520C2×C4C3×#12012yes—
521C2×C4C3×#12112yes—
522C2×C4C2×C3×D812yes—
523C2×C4C2×C3×D812yes—
524C2×C4C3×AGL(1,ℤ/8)12yes—
525C2×C4C22×C3×D411yescharacteristic
526C2×C4C22×C3×D412yes—
527C2×C4C2×C3×#4112yes—
528C2×C4C2×C3×#4111yescharacteristic
529C8—44no—
530C2×C3—24no—
531C2×C3—48no—
532C2×C3—162no—
533C2×C3#284711yescharacteristic
534C2×C3C2×#44711yescharacteristic
535C2×C3#300611yescharacteristic
536C2×C3C2×#45911yescharacteristic
537C2×C3#303611yescharacteristic
538C2×C3#310011yescharacteristic
539C2×C3#310911yescharacteristic
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—28no—
547C22—432no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22C3×#45914yescomplemented
562C22C2×C3×#12011yescharacteristic
563C22C2×C3×#12111yescharacteristic
564C22C3×#52511yescharacteristic
565C22C22×C3×D811yescharacteristic
566C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
567C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
568C22C3×#57511yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—44no—
573C4—41no—
574C4—82no—
575C4C3×#44714yescomplemented
576C4C2×C3×#12112yes—
577C4C3×#52512yes—
578C3#3290211yescharacteristic, direct product factor, Sylow
579C2—24no—
580C2—48no—
581C2—162no—
582C2C3×#284711yescharacteristic
583C2C2×C3×#44711yescharacteristic
584C2C3×#300611yescharacteristic
585C2C2×C3×#45911yescharacteristic
586C2C3×#303611yescharacteristic
587C2C3×#310011yescharacteristic
588C2C3×#310911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×D4C2semidirect
2. C2×C3×#459C2semidirect
3. C2×C3×#447C2semidirect
4. C3×#2856C2semidirect
5. C3×#1879C2semidirect
6. C3×#1874C2semidirect
7. C3×#1802C2semidirect
8. C3×#1797C2semidirect
9. #32902C3direct (N × H)
13. C2×C3×#127C22semidirect
27. C2×C3×C4⋊C8C22semidirect
29. C2×C3×#102C22semidirect
33. C3×#390C22semidirect
34. C3×#384C22semidirect
35. C3×#340C22semidirect
36. C3×#336C22semidirect
37. C22×C4×D4C2×C3semidirect
38. C2×#459C2×C3semidirect
39. C2×#447C2×C3semidirect
40. #2856C2×C3semidirect
41. #1879C2×C3semidirect
42. #1874C2×C3semidirect
43. #1802C2×C3semidirect
44. #1797C2×C3semidirect
53. C22×C3×D4D4semidirect
79. C3×C4×D4D4semidirect
104. C2×#127C22×C3semidirect
118. C2×C4⋊C8C22×C3semidirect
120. C2×#102C22×C3semidirect
124. #390C22×C3semidirect
125. #384C22×C3semidirect
126. #340C22×C3semidirect
127. #336C22×C3semidirect
181. C22×C3×C4D8semidirect
218. C22×D4C3×D4semidirect
244. C4×D4C3×D4semidirect
304. C3×D4#121semidirect
391. C22×C4C3×D8semidirect
441. C22×C3#459semidirect
455. C3×C4#447semidirect
496. D4C3×#121semidirect
561. C22C3×#459semidirect
575. C4C3×#447semidirect
578. C3#32902direct (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×D42
C2×C3×#4592
C2×C3×#4472
C3×#28562
C3×#18792
C3×#18742
C3×#18022
C3×#17972
#329023