Finite Group Databasebeta

← C3×#85513 #11922797

Subgroups — C3×#85513 #11922797

Order 1152.

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

normal subgroups only

This group has 712 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
1C3×A4×#41C211yesmaximal, characteristic, complemented
2C2×C3×C4⋊S4C211yesmaximal, characteristic, complemented
3C3×#5417C212yesmaximal, complemented
4C3×#5415C212yesmaximal, complemented
5C3×#5409C211yesmaximal, characteristic, complemented
6#85513C311yesmaximal, characteristic, direct product factor
7C3×#3006—31nomaximal
8C22×C3×S4—21no—
9C3×Q8×A4C2211yescharacteristic, complemented
10C3×D4×A4—21no—
11C3×D4×A4C2211yescharacteristic, complemented
12C2×C3×C4×A4—21no—
13C2×C3×C4×A4C2211yescharacteristic
14C3×C4⋊S4—21no—
15C3×C4⋊S4C2212yes—
16C3×#949—41nomaximal
17C3×A4⋊C8C2212yescomplemented
18C22×C3×#41D311yescharacteristic, complemented
19A4×#41—21no—
20A4×#41C2×C311yescharacteristic, complemented
21C2×C4⋊S4C2×C311yescharacteristic, complemented
22C2×C3×AGL(1,ℤ/8)—31no—
23C2×C3×AGL(1,ℤ/8)—31no—
24C2×C3×#121—31no—
25#5417C2×C312yescomplemented
26#5415C2×C312yescomplemented
27#5409C2×C311yescharacteristic, complemented
28C3×#449—32no—
29C3×#449—32no—
30C3×#448—32no—
31C3×#447—32no—
32C3×#417—31no—
33C3×#417—31no—
34C3×#407—31no—
35C22×C3×A4—21no—
36C22×C3×A4D411yescharacteristic
37C2×C3×S4—22no—
38C2×C3×S4—42no—
39C32×#41—41no—
40C2×C3×D12—41no—
41C3×C4×A4—21no—
42C3×C4×A4D411yescharacteristic
43C3×C4×A4C2311yescharacteristic
44C3×#215—42no—
45C3×#213—42no—
46C3×#208—41no—
47#3006—31noSylow
48C22×S4—21no—
49C2×C3×#41—31no—
50C2×C3×#41—31no—
51C2×C3×#41—31no—
52C2×C3×#41—31no—
53C2×C3×#41—61no—
54C2×C3×#41—61no—
55C22×C3×Q8C2×D311yescharacteristic, complemented
56C22×C3×D4—21no—
57C22×C3×D4—31no—
58C22×C3×D4—61no—
59C22×C3×D4C2×D311yescharacteristic, complemented
60C23×C3×C4—21no—
61C23×C3×C4C2×D311yescharacteristic
62Q8×A4—21no—
63Q8×A4C22×C311yescharacteristic, complemented
64D4×A4—22no—
65D4×A4—21no—
66D4×A4—21no—
67D4×A4C22×C311yescharacteristic, complemented
68C2×C4×A4—22no—
69C2×C4×A4—21no—
70C2×C4×A4—21no—
71C2×C4×A4C22×C311yescharacteristic
72C4⋊S4—21no—
73C4⋊S4C22×C312yes—
74C3×AGL(1,ℤ/8)—62no—
75C3×AGL(1,ℤ/8)—62no—
76C3×AGL(1,ℤ/8)—61no—
77C3×AGL(1,ℤ/8)—61no—
78C3×AGL(1,ℤ/8)—61no—
79C3×AGL(1,ℤ/8)—61no—
80C2×C3×SD16—32no—
81C2×C3×SD16—32no—
82C2×C3×D8—32no—
83C2×C3×D8—32no—
84C2×C3×M16—31no—
85C2×C3×M16—31no—
86C3×#121—32no—
87C3×#121—32no—
88C3×#121—61no—
89C3×#121—61no—
90C2×C3×C4⋊C4—31no—
91C2×C3×C22⋊C4—61no—
92#949—41no—
93A4⋊C8C22×C312yescomplemented
94C3×#103—32no—
95C3×#103—32no—
96C3×#102—32no—
97C3×#102—32no—
98C3×C22⋊C8—32no—
99C3×C22⋊C8—32no—
100C22×C3×D3—81no—
101C2×C3×A4—21no—
102C2×C3×A4—41no—
103C2×C3×A4C2×D411yescharacteristic
104C3×S4—42no—
105C32×Q8—41no—
106C32×D4—41no—
107C32×D4—81no—
108C2×C32×C4—41no—
109C2×C32×C4—81no—
110C3×D12—42no—
111C3×D12—81no—
112C3×C3⋊C8—42no—
113C22×#41C3×D311yescharacteristic, complemented
114C2×AGL(1,ℤ/8)—31no—
115C2×AGL(1,ℤ/8)—31no—
116C2×#121—31no—
117#449—32no—
118#449—32no—
119#448—32no—
120#447—32no—
121#417—31no—
122#417—31no—
123#407—31no—
124C24×C3—21no—
125C24×C3—61no—
126C24×C3#6711yescharacteristic
127C22×A4—22no—
128C22×A4—21no—
129C22×A4—21no—
130C22×A4C3×D411yescharacteristic
131C2×S4—22no—
132C2×S4—42no—
133C3×#41—32no—
134C3×#41—32no—
135C3×#41—31no—
136C3×#41—31no—
137C3×#41—31no—
138C3×#41—41no—
139C3×#41—61no—
140C3×#41—61no—
141C3×#41—61no—
142C3×#41—62no—
143C3×#41—62no—
144C3×#41—81no—
145C3×#41S411yescharacteristic, complemented
146C2×C3×Q8—31no—
147C2×C3×Q8—31no—
148C2×C3×Q8—61no—
149C2×C3×D4—31no—
150C2×C3×D4—31no—
151C2×C3×D4—32no—
152C2×C3×D4—61no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4—61no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C2×C3×D4—61no—
160C2×C3×D4—61no—
161C2×C3×D4—62no—
162C2×C3×D4—62no—
163C2×C3×D4—61no—
164C2×C3×D4—122no—
165C2×C3×D4—122no—
166C22×C3×C4—21no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4—61no—
172C22×C3×C4—61no—
173C22×C3×C4—61no—
174C22×C3×C4—61no—
175C22×C3×C4—61no—
176C22×C3×C4#6711yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C2×D12—41no—
179C4×A4—22no—
180C4×A4—21no—
181C4×A4—21no—
182C4×A4—21no—
183C4×A4C3×D411yescharacteristic
184C4×A4C23×C311yescharacteristic
185C3×SD16—64no—
186C3×SD16—62no—
187C3×SD16—62no—
188C3×D8—64no—
189C3×D8—62no—
190C3×D8—62no—
191C3×M16—61no—
192C3×M16—61no—
193C3×M16—61no—
194C3×M16—61no—
195C2×C3×C8—32no—
196C2×C3×C8—32no—
197C3×C4⋊C4—32no—
198C3×C4⋊C4—61no—
199C3×C22⋊C4—62no—
200C3×C22⋊C4—62no—
201#215—42no—
202#213—42no—
203#208—41no—
204C22×C32—41no—
205C22×C32—81no—
206C2×C3×D3—82no—
207C2×C3×D3—162no—
208C3×A4AGL(1,ℤ/8)11yescharacteristic, complemented
209C32×C4—41no—
210C32×C4—41no—
211C32×C4—81no—
212C2×#41—31no—
213C2×#41—31no—
214C2×#41—31no—
215C2×#41—31no—
216C2×#41—61no—
217C2×#41—61no—
218C22×Q8C2×C3×D311yescharacteristic, complemented
219C22×D4—21no—
220C22×D4—31no—
221C22×D4—61no—
222C22×D4C2×C3×D311yescharacteristic, complemented
223C23×C4—21no—
224C23×C4C2×C3×D311yescharacteristic
225AGL(1,ℤ/8)—62no—
226AGL(1,ℤ/8)—62no—
227AGL(1,ℤ/8)—61no—
228AGL(1,ℤ/8)—61no—
229AGL(1,ℤ/8)—61no—
230AGL(1,ℤ/8)—61no—
231C2×SD16—32no—
232C2×SD16—32no—
233C2×D8—32no—
234C2×D8—32no—
235C2×M16—31no—
236C2×M16—31no—
237#121—32no—
238#121—32no—
239#121—61no—
240#121—61no—
241C2×C4⋊C4—31no—
242C2×C22⋊C4—61no—
243#103—32no—
244#103—32no—
245#102—32no—
246#102—32no—
247C22⋊C8—32no—
248C22⋊C8—32no—
249C23×C3—21no—
250C23×C3—31no—
251C23×C3—31no—
252C23×C3—41no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—61no—
256C23×C3—62no—
257C23×C3—121no—
258C23×C3—121no—
259C23×C3—122no—
260C23×C3—122no—
261C23×C3—121no—
262C23×C3C2×#6711yescharacteristic
263C22×D3—81no—
264C2×A4—21no—
265C2×A4—21no—
266C2×A4—41no—
267C2×A4—42no—
268C2×A4—41no—
269C2×A4C2×C3×D411yescharacteristic
270S4—42no—
271C3×Q8—32no—
272C3×Q8—31no—
273C3×Q8—41no—
274C3×Q8—61no—
275C3×Q8—81no—
276C3×Q8C2×S411yescharacteristic, complemented
277C3×D4—21no—
278C3×D4—32no—
279C3×D4—31no—
280C3×D4—41no—
281C3×D4—61no—
282C3×D4—61no—
283C3×D4—62no—
284C3×D4—61no—
285C3×D4—62no—
286C3×D4—61no—
287C3×D4—62no—
288C3×D4—81no—
289C3×D4—82no—
290C3×D4—81no—
291C3×D4—121no—
292C3×D4—122no—
293C3×D4—124no—
294C3×D4—122no—
295C3×D4—121no—
296C3×D4C2×S411yescharacteristic, complemented
297C2×C3×C4—21no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—41no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—62no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—62no—
314C2×C3×C4—81no—
315C2×C3×C4—82no—
316C2×C3×C4—81no—
317C2×C3×C4—122no—
318C2×C3×C4C2×S411yescharacteristic
319D12—42no—
320D12—81no—
321C3×C8—62no—
322C3×C8—62no—
323C3⋊C8—42no—
324C2×C32—41no—
325C2×C32—81no—
326C2×C32—161no—
327C3×D3—162no—
328C24—21no—
329C24—61no—
330C24C3×#6711yescharacteristic
331#41—32no—
332#41—32no—
333#41—31no—
334#41—31no—
335#41—31no—
336#41—61no—
337#41—61no—
338#41—61no—
339#41—62no—
340#41—62no—
341#41C3×S411yescharacteristic, complemented
342C2×Q8—31no—
343C2×Q8—31no—
344C2×Q8—61no—
345C2×D4—31no—
346C2×D4—31no—
347C2×D4—32no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—61no—
360C2×D4—122no—
361C2×D4—122no—
362C22×C4—21no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—31no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4C3×#6711yescharacteristic
373C22×C4C22×C3×D311yescharacteristic
374SD16—64no—
375SD16—62no—
376SD16—62no—
377D8—64no—
378D8—62no—
379D8—62no—
380M16—61no—
381M16—61no—
382M16—61no—
383M16—61no—
384C2×C8—32no—
385C2×C8—32no—
386C4⋊C4—32no—
387C4⋊C4—61no—
388C22⋊C4—62no—
389C22⋊C4—62no—
390C22×C3—21no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—41no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—81no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3—121no—
406C22×C3—121no—
407C22×C3—122no—
408C22×C3—122no—
409C22×C3—122no—
410C22×C3—242no—
411C22×C3—242no—
412C22×C3#94911yescharacteristic, complemented
413C22×C3GL(2,ℤ/4)11yescharacteristic
414C2×D3—82no—
415C2×D3—162no—
416A4—21no—
417A4C3×AGL(1,ℤ/8)11yescharacteristic, complemented
418C3×C4—21no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—61no—
424C3×C4—81no—
425C3×C4—82no—
426C3×C4—81no—
427C3×C4—81no—
428C3×C4—122no—
429C3×C4GL(2,ℤ/4)11yescharacteristic
430C3×C4C22×S411yescharacteristic
431C32—41noSylow
432C23—21no—
433C23—31no—
434C23—31no—
435C23—41no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—62no—
440C23—121no—
441C23—121no—
442C23—122no—
443C23—122no—
444C23—121no—
445C23C2×C3×#6711yescharacteristic
446Q8—32no—
447Q8—31no—
448Q8—61no—
449Q8C2×C3×S411yescharacteristic, complemented
450D4—21no—
451D4—32no—
452D4—31no—
453D4—61no—
454D4—61no—
455D4—62no—
456D4—61no—
457D4—62no—
458D4—61no—
459D4—62no—
460D4—121no—
461D4—122no—
462D4—124no—
463D4—122no—
464D4—121no—
465D4C2×C3×S411yescharacteristic, complemented
466C2×C4—21no—
467C2×C4—31no—
468C2×C4—31no—
469C2×C4—31no—
470C2×C4—31no—
471C2×C4—31no—
472C2×C4—31no—
473C2×C4—31no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—62no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—62no—
482C2×C4—122no—
483C2×C4C2×C3×S411yescharacteristic
484C8—62no—
485C8—62no—
486C2×C3—21no—
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—41no—
490C2×C3—41no—
491C2×C3—61no—
492C2×C3—81no—
493C2×C3—81no—
494C2×C3—121no—
495C2×C3—161no—
496C2×C3—161no—
497C2×C3—162no—
498C2×C3—242no—
499C2×C3C2×GL(2,ℤ/4)11yescharacteristic
500D3—162no—
501C22—21no—
502C22—31no—
503C22—31no—
504C22—31no—
505C22—61no—
506C22—61no—
507C22—61no—
508C22—61no—
509C22—61no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22—121no—
514C22—122no—
515C22—122no—
516C22—122no—
517C22—242no—
518C22—242no—
519C22C3×#94911yescharacteristic, complemented
520C22C3×GL(2,ℤ/4)11yescharacteristic
521C4—21no—
522C4—31no—
523C4—31no—
524C4—61no—
525C4—122no—
526C4C3×GL(2,ℤ/4)11yescharacteristic
527C4C22×C3×S411yescharacteristic
528C3—41no—
529C3—81no—
530C3#8551311yescharacteristic, direct product factor
531C2—21no—
532C2—31no—
533C2—31no—
534C2—41no—
535C2—61no—
536C2—121no—
537C2—242no—
538C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C3C2×GL(2,ℤ/4)
commutator, cosocleC4×A4C23×C3
FrattiniC4C22×C3×S4
FittingC22×C3×#41D3
socleC23×C3C2×#67
hypercenterC3×#41S4
hypocenterA4C3×AGL(1,ℤ/8)
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
1. C3×A4×#41C2semidirect
2. C2×C3×C4⋊S4C2semidirect
3. C3×#5417C2semidirect
4. C3×#5415C2semidirect
5. C3×#5409C2semidirect
6. #85513C3direct (N × H)
9. C3×Q8×A4C22semidirect
11. C3×D4×A4C22semidirect
17. C3×A4⋊C8C22semidirect
18. C22×C3×#41D3semidirect
20. A4×#41C2×C3semidirect
21. C2×C4⋊S4C2×C3semidirect
25. #5417C2×C3semidirect
26. #5415C2×C3semidirect
27. #5409C2×C3semidirect
55. C22×C3×Q8C2×D3semidirect
59. C22×C3×D4C2×D3semidirect
63. Q8×A4C22×C3semidirect
67. D4×A4C22×C3semidirect
93. A4⋊C8C22×C3semidirect
113. C22×#41C3×D3semidirect
145. C3×#41S4semidirect
208. C3×A4AGL(1,ℤ/8)semidirect
218. C22×Q8C2×C3×D3semidirect
222. C22×D4C2×C3×D3semidirect
276. C3×Q8C2×S4semidirect
296. C3×D4C2×S4semidirect
341. #41C3×S4semidirect
412. C22×C3#949semidirect
417. A4C3×AGL(1,ℤ/8)semidirect
449. Q8C2×C3×S4semidirect
465. D4C2×C3×S4semidirect
519. C22C3×#949semidirect
530. C3#85513direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4×#412
C2×C3×C4⋊S42
C3×#54172
C3×#54152
C3×#54092
#855133
C3×#30063
C3×#9494