Finite Group Databasebeta

← C3×#85489 #11922773

Subgroups — C3×#85489 #11922773

Order 1152.

766 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×A4C211yesmaximal, characteristic, complemented
2C3×#5912C211yesmaximal, characteristic, complemented
3C3×#5415C212yesmaximal, complemented
4C3×#5414C212yesmaximal, complemented
5C3×#5409C211yesmaximal, characteristic, complemented
6#85489C311yesmaximal, characteristic, direct product factor
7C3×#3003—31nomaximal
8C23×C3×A4—21no—
9C3×D4×A4—21no—
10C3×D4×A4C2212yes—
11C2×C3×C4×A4C2211yescharacteristic
12C3×GL(2,ℤ/4)—21no—
13C3×C4⋊S4C2211yescharacteristic, complemented
14C3×C4×S4—21no—
15C3×A4⋊Q8C2211yescharacteristic, complemented
16C3×#932—41nomaximal
17C3×A4⋊C8C2212yescomplemented
18C23×C3×D4D311yescharacteristic, complemented
19C2×D4×A4—21no—
20C2×D4×A4C2×C311yescharacteristic, complemented
21#5912C2×C311yescharacteristic, complemented
22C2×C3×AGL(1,ℤ/8)—31no—
23C2×C3×AGL(1,ℤ/8)—31no—
24C3×#525—31no—
25#5415C2×C312yescomplemented
26#5414C2×C312yescomplemented
27#5409C2×C311yescharacteristic, complemented
28C3×#450—32no—
29C3×#450—32no—
30C3×#447—32no—
31C3×#447—32no—
32C3×#418—31no—
33C3×#418—31no—
34C3×#407—31no—
35C22×C3×A4—22no—
36C22×C3×A4—42no—
37C22×C3×A4D411yescharacteristic
38C2×C3×S4—21no—
39C2×C32×D4—41no—
40C3×#235—41no—
41C3×C4×A4D411yescharacteristic
42C3×C4×A4C2311yescharacteristic
43C3×SL(2,ℤ/4)—21no—
44C3×#214—42no—
45C3×#213—42no—
46C3×#208—41no—
47#3003—31noSylow
48C25×C3—21no—
49C23×A4—22no—
50C23×A4—21no—
51C2×C3×#41—31no—
52C22×C3×D4—21no—
53C22×C3×D4—31no—
54C22×C3×D4—31no—
55C22×C3×D4—32no—
56C22×C3×D4—61no—
57C22×C3×D4—61no—
58C22×C3×D4C2×D312yes—
59C23×C3×C4C2×D311yescharacteristic
60D4×A4—22no—
61D4×A4—21no—
62D4×A4—22no—
63D4×A4C22×C312yes—
64C2×C4×A4—21no—
65C2×C4×A4C22×C311yescharacteristic
66GL(2,ℤ/4)—21no—
67C4⋊S4C22×C311yescharacteristic, complemented
68C4×S4—21no—
69A4⋊Q8C22×C311yescharacteristic, complemented
70C3×AGL(1,ℤ/8)—62no—
71C3×AGL(1,ℤ/8)—62no—
72C3×AGL(1,ℤ/8)—62no—
73C3×AGL(1,ℤ/8)—61no—
74C3×AGL(1,ℤ/8)—61no—
75C2×C3×SD16—32no—
76C2×C3×SD16—32no—
77C2×C3×D8—32no—
78C2×C3×D8—32no—
79C2×C3×M16—31no—
80C2×C3×M16—31no—
81C3×#123—61no—
82C3×C22⋊Q8—31no—
83C3×C22⋊Q8—31no—
84C3×#121—31no—
85C3×#121—31no—
86C3×#120—61no—
87C3×C4×D4—61no—
88C3×C4×D4—61no—
89C3×#117—31no—
90#932—41no—
91A4⋊C8C22×C312yescomplemented
92C3×#102—32no—
93C3×#102—32no—
94C3×#102—32no—
95C3×#102—32no—
96C3×C22⋊C8—32no—
97C3×C22⋊C8—32no—
98C23×C32—81no—
99C2×C3×A4—21no—
100C2×C3×A4—42no—
101C2×C3×A4C2×D411yescharacteristic
102C3×S4—41no—
103C32×D4—42no—
104C32×D4—81no—
105C2×C32×C4—41no—
106C3×#67—81no—
107C3×D12—41no—
108C3×C4×D3—81no—
109C3×Dic6—41no—
110C3×C3⋊C8—42no—
111C23×D4C3×D311yescharacteristic, complemented
112C2×AGL(1,ℤ/8)—31no—
113C2×AGL(1,ℤ/8)—31no—
114#525—31no—
115#450—32no—
116#450—32no—
117#447—32no—
118#447—32no—
119#418—31no—
120#418—31no—
121#407—31no—
122C24×C3—22no—
123C24×C3—42no—
124C24×C3—61no—
125C24×C3—62no—
126C24×C3—61no—
127C24×C3—122no—
128C24×C3#6711yescharacteristic
129C22×A4—24no—
130C22×A4—22no—
131C22×A4—21no—
132C22×A4—44no—
133C22×A4—42no—
134C22×A4C3×D411yescharacteristic
135C2×S4—21no—
136C3×#41—61no—
137C3×#41—61no—
138C3×#41—61no—
139C3×#41—61no—
140C2×C3×Q8—31no—
141C2×C3×D4—32no—
142C2×C3×D4—32no—
143C2×C3×D4—32no—
144C2×C3×D4—32no—
145C2×C3×D4—32no—
146C2×C3×D4—31no—
147C2×C3×D4—31no—
148C2×C3×D4—41no—
149C2×C3×D4—61no—
150C2×C3×D4—61no—
151C2×C3×D4—61no—
152C2×C3×D4—62no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4—62no—
156C2×C3×D4—64no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C2×C3×D4—61no—
160C2×C3×D4—81no—
161C2×C3×D4S411yescharacteristic, complemented
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—61no—
167C22×C3×C4#6711yescharacteristic
168C22×C3×C4C22×D311yescharacteristic
169#235—41no—
170C4×A4—21no—
171C4×A4—21no—
172C4×A4C3×D411yescharacteristic
173C4×A4C23×C311yescharacteristic
174SL(2,ℤ/4)—21no—
175C3×SD16—64no—
176C3×SD16—62no—
177C3×SD16—62no—
178C3×D8—64no—
179C3×D8—62no—
180C3×D8—62no—
181C3×M16—61no—
182C3×M16—61no—
183C3×M16—61no—
184C3×M16—61no—
185C2×C3×C8—32no—
186C2×C3×C8—32no—
187C3×C4⋊C4—31no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—61no—
190C3×C4⋊C4—61no—
191C3×C22⋊C4—61no—
192C3×C22⋊C4—61no—
193C3×C22⋊C4—61no—
194C3×C22⋊C4—61no—
195C3×C22⋊C4—61no—
196C3×C42—61no—
197#214—42no—
198#213—42no—
199#208—41no—
200C22×C32—41no—
201C22×C32—82no—
202C22×C32—162no—
203C2×C3×D3—81no—
204C3×A4AGL(1,ℤ/8)11yescharacteristic, complemented
205C32×C4—41no—
206C32×C4—41no—
207C3×Dic3—81no—
208C25—21no—
209C2×#41—31no—
210C22×D4—21no—
211C22×D4—31no—
212C22×D4—31no—
213C22×D4—32no—
214C22×D4—61no—
215C22×D4—61no—
216C22×D4C2×C3×D312yes—
217C23×C4C2×C3×D311yescharacteristic
218AGL(1,ℤ/8)—62no—
219AGL(1,ℤ/8)—62no—
220AGL(1,ℤ/8)—62no—
221AGL(1,ℤ/8)—61no—
222AGL(1,ℤ/8)—61no—
223C2×SD16—32no—
224C2×SD16—32no—
225C2×D8—32no—
226C2×D8—32no—
227C2×M16—31no—
228C2×M16—31no—
229#123—61no—
230C22⋊Q8—31no—
231C22⋊Q8—31no—
232#121—31no—
233#121—31no—
234#120—61no—
235C4×D4—61no—
236C4×D4—61no—
237#117—31no—
238#102—32no—
239#102—32no—
240#102—32no—
241#102—32no—
242C22⋊C8—32no—
243C22⋊C8—32no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—31no—
247C23×C3—31no—
248C23×C3—42no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3—61no—
252C23×C3—62no—
253C23×C3—62no—
254C23×C3—61no—
255C23×C3—61no—
256C23×C3—81no—
257C23×C3—82no—
258C23×C3—122no—
259C23×C3—124no—
260C23×C3—122no—
261C23×C3—122no—
262C23×C3—122no—
263C23×C3—124no—
264C23×C3—122no—
265C23×C3C2×#6711yescharacteristic
266C2×A4—22no—
267C2×A4—21no—
268C2×A4—21no—
269C2×A4—44no—
270C2×A4—42no—
271C2×A4C2×C3×D411yescharacteristic
272S4—41no—
273C3×Q8—61no—
274C3×Q8—61no—
275C3×D4—21no—
276C3×D4—34no—
277C3×D4—32no—
278C3×D4—42no—
279C3×D4—62no—
280C3×D4—62no—
281C3×D4—62no—
282C3×D4—61no—
283C3×D4—61no—
284C3×D4—61no—
285C3×D4—81no—
286C3×D4—82no—
287C3×D4—82no—
288C3×D4—121no—
289C3×D4—122no—
290C3×D4—121no—
291C3×D4—121no—
292C3×D4—121no—
293C3×D4C2×S412yes—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—41no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—81no—
307C2×C3×C4—121no—
308C2×C3×C4—121no—
309C2×C3×C4C2×S411yescharacteristic
310#67—81no—
311D12—41no—
312C4×D3—81no—
313Dic6—41no—
314C3×C8—62no—
315C3×C8—62no—
316C3⋊C8—42no—
317C2×C32—41no—
318C2×C32—81no—
319C2×C32—162no—
320C3×D3—161no—
321C24—22no—
322C24—42no—
323C24—61no—
324C24—62no—
325C24—61no—
326C24—122no—
327C24C3×#6711yescharacteristic
328#41—61no—
329#41—61no—
330#41—61no—
331#41—61no—
332C2×Q8—31no—
333C2×D4—32no—
334C2×D4—32no—
335C2×D4—32no—
336C2×D4—32no—
337C2×D4—32no—
338C2×D4—31no—
339C2×D4—31no—
340C2×D4—61no—
341C2×D4—61no—
342C2×D4—61no—
343C2×D4—62no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—62no—
347C2×D4—64no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4C3×S411yescharacteristic, complemented
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—61no—
357C22×C4C3×#6711yescharacteristic
358C22×C4C22×C3×D311yescharacteristic
359SD16—64no—
360SD16—62no—
361SD16—62no—
362D8—64no—
363D8—62no—
364D8—62no—
365M16—61no—
366M16—61no—
367M16—61no—
368M16—61no—
369C2×C8—32no—
370C2×C8—32no—
371C4⋊C4—31no—
372C4⋊C4—31no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4—61no—
378C22⋊C4—61no—
379C22⋊C4—61no—
380C42—61no—
381C22×C3—22no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—32no—
385C22×C3—31no—
386C22×C3—42no—
387C22×C3—41no—
388C22×C3—61no—
389C22×C3—61no—
390C22×C3—61no—
391C22×C3—62no—
392C22×C3—82no—
393C22×C3—84no—
394C22×C3—81no—
395C22×C3—122no—
396C22×C3—122no—
397C22×C3—122no—
398C22×C3—122no—
399C22×C3—122no—
400C22×C3—121no—
401C22×C3—121no—
402C22×C3—124no—
403C22×C3—124no—
404C22×C3—121no—
405C22×C3—162no—
406C22×C3—164no—
407C22×C3#93211yescharacteristic, complemented
408C22×C3GL(2,ℤ/4)11yescharacteristic
409C2×D3—81no—
410A4—21no—
411A4C3×AGL(1,ℤ/8)11yescharacteristic, complemented
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—81no—
417C3×C4—81no—
418C3×C4—121no—
419C3×C4—121no—
420C3×C4—121no—
421C3×C4GL(2,ℤ/4)11yescharacteristic
422C3×C4C22×S411yescharacteristic
423Dic3—81no—
424C32—41noSylow
425C23—21no—
426C23—21no—
427C23—31no—
428C23—31no—
429C23—42no—
430C23—62no—
431C23—62no—
432C23—61no—
433C23—62no—
434C23—62no—
435C23—61no—
436C23—61no—
437C23—122no—
438C23—124no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23—124no—
443C23—122no—
444C23C2×C3×#6711yescharacteristic
445Q8—61no—
446Q8—61no—
447D4—21no—
448D4—34no—
449D4—32no—
450D4—62no—
451D4—62no—
452D4—62no—
453D4—61no—
454D4—61no—
455D4—61no—
456D4—121no—
457D4—121no—
458D4—122no—
459D4—121no—
460D4—121no—
461D4C2×C3×S412yes—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—31no—
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—31no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4C2×C3×S411yescharacteristic
476C8—62no—
477C8—62no—
478C2×C3—21no—
479C2×C3—31no—
480C2×C3—31no—
481C2×C3—42no—
482C2×C3—41no—
483C2×C3—61no—
484C2×C3—81no—
485C2×C3—82no—
486C2×C3—81no—
487C2×C3—122no—
488C2×C3—162no—
489C2×C3—164no—
490C2×C3—241no—
491C2×C3C2×GL(2,ℤ/4)11yescharacteristic
492D3—161no—
493C22—22no—
494C22—31no—
495C22—31no—
496C22—32no—
497C22—31no—
498C22—42no—
499C22—61no—
500C22—61no—
501C22—61no—
502C22—62no—
503C22—122no—
504C22—122no—
505C22—122no—
506C22—122no—
507C22—122no—
508C22—121no—
509C22—121no—
510C22—124no—
511C22—124no—
512C22—121no—
513C22C3×#93211yescharacteristic, complemented
514C22C3×GL(2,ℤ/4)11yescharacteristic
515C4—31no—
516C4—31no—
517C4—121no—
518C4—121no—
519C4—121no—
520C4C3×GL(2,ℤ/4)11yescharacteristic
521C4C22×C3×S411yescharacteristic
522C3—41no—
523C3—81no—
524C3#8548911yescharacteristic, direct product factor
525C2—21no—
526C2—31no—
527C2—31no—
528C2—42no—
529C2—61no—
530C2—122no—
531C2—241no—
532C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C3C2×GL(2,ℤ/4)
commutator, cosocleC4×A4C23×C3
FrattiniC4C22×C3×S4
FittingC23×C3×D4D3
socleC23×C3C2×#67
hypercenterC2×C3×D4S4
hypocenterA4C3×AGL(1,ℤ/8)
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×A4C2semidirect
2. C3×#5912C2semidirect
3. C3×#5415C2semidirect
4. C3×#5414C2semidirect
5. C3×#5409C2semidirect
6. #85489C3direct (N × H)
13. C3×C4⋊S4C22semidirect
15. C3×A4⋊Q8C22semidirect
17. C3×A4⋊C8C22semidirect
18. C23×C3×D4D3semidirect
20. C2×D4×A4C2×C3semidirect
21. #5912C2×C3semidirect
25. #5415C2×C3semidirect
26. #5414C2×C3semidirect
27. #5409C2×C3semidirect
67. C4⋊S4C22×C3semidirect
69. A4⋊Q8C22×C3semidirect
91. A4⋊C8C22×C3semidirect
111. C23×D4C3×D3semidirect
161. C2×C3×D4S4semidirect
204. C3×A4AGL(1,ℤ/8)semidirect
351. C2×D4C3×S4semidirect
407. C22×C3#932semidirect
411. A4C3×AGL(1,ℤ/8)semidirect
513. C22C3×#932semidirect
524. C3#85489direct (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×A42
C3×#59122
C3×#54152
C3×#54142
C3×#54092
#854893
C3×#30033
C3×#9324