Finite Group Databasebeta

← C3×#85310 #11922595

Subgroups — C3×#85310 #11922595

Order 1152.

880 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 92 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×C4⋊S4C212yesmaximal, complemented
2C2×C3×C4⋊S4C211yesmaximal, characteristic, complemented
3C2×C3×C4×S4C211yesmaximal, characteristic, complemented
4C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×(C2×S4)⋊C4C212yesmaximal, complemented
6#85310C311yesmaximal, characteristic, direct product factor
7C3×#2416—31nomaximal
8C22×C3×S4C2212yes—
9C22×C3×S4C2211yescharacteristic
10C2×C3×C4×A4C2212yes—
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C3×C4⋊S4—22no—
14C3×C4⋊S4—24no—
15C3×C4×S4—22no—
16C3×#895—41nomaximal
17C2×C4⋊S4C2×C312yescomplemented
18C2×C4⋊S4C2×C311yescharacteristic, complemented
19C2×C4×S4C2×C311yescharacteristic, complemented
20C2×C3×#121—31no—
21C2×C3×#121—32no—
22C2×C3×#121—31no—
23C2×C3×C4×D4—31no—
24C22×C3×C4⋊C4D311yescharacteristic, complemented
25A4×C4⋊C4—21no—
26A4×C4⋊C4C2×C311yescharacteristic, complemented
27(C2×S4)⋊C4C2×C312yescomplemented
28C3×#396—32no—
29C3×#394—32no—
30C3×#390—31no—
31C3×#389—31no—
32C3×#386—32no—
33C3×#380—31no—
34C22×C3×A4C2311yescharacteristic
35C2×C3×S4—21no—
36C2×C3×S4—21no—
37C2×C3×S4—22no—
38C2×C3×S4—22no—
39C2×C3×S4—22no—
40C2×C3×S4D412yescomplemented
41C2×C3×D12—41no—
42C2×C3×D12—42no—
43C2×C3×C4×D3—41no—
44C3×C4×A4—22no—
45C3×C4×A4D412yescomplemented
46C3×SL(2,ℤ/4)—21no—
47C32×C4⋊C4—41no—
48C3×(C2×D3)⋊C4—42no—
49#2416—31noSylow
50C22×S4C22×C312yes—
51C22×S4C22×C311yescharacteristic
52C22×C3×D4—31no—
53C22×C3×D4—31no—
54C22×C3×D4—32no—
55C22×C3×D4—32no—
56C23×C3×C4—31no—
57C23×C3×C4C2×D312yes—
58C23×C3×C4C2×D311yescharacteristic
59C2×C4×A4—21no—
60C2×C4×A4—22no—
61C2×C4×A4C22×C312yes—
62C2×C4×A4C22×C311yescharacteristic
63C2×SL(2,ℤ/4)C22×C311yescharacteristic
64C4⋊S4—24no—
65C4⋊S4—22no—
66C4×S4—22no—
67C3×#121—61no—
68C3×#121—62no—
69C3×#121—64no—
70C3×#121—62no—
71C3×#121—64no—
72C3×#121—62no—
73C3×#121—61no—
74C3×C4×D4—62no—
75C3×C4×D4—62no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C4⋊C4—31no—
78C2×C3×C4⋊C4—32no—
79C2×C3×C4⋊C4—31no—
80C2×C3×C4⋊C4—32no—
81C2×C3×C4⋊C4—31no—
82C2×C3×C22⋊C4—32no—
83C2×C3×C22⋊C4—32no—
84C2×C3×C22⋊C4—32no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C22⋊C4—32no—
87C2×C3×C22⋊C4—31no—
88C2×C3×C22⋊C4—32no—
89C2×C3×C42—31no—
90#895—41no—
91C3×#95—32no—
92C3×#95—32no—
93C22×C3×D3—41no—
94C22×C3×D3—42no—
95C2×C3×A4C2×D411yescharacteristic
96C2×C3×A4C2×D411yescharacteristic
97C2×C3×A4#4111yescharacteristic
98C3×S4—22no—
99C3×S4—42no—
100C2×C32×C4—41no—
101C2×C32×C4—42no—
102C2×C3×Dic3—41no—
103C3×D12—82no—
104C3×D12—84no—
105C3×C4×D3—82no—
106C2×#121—31no—
107C2×#121—32no—
108C2×#121—31no—
109C2×C4×D4—31no—
110C22×C4⋊C4C3×D311yescharacteristic, complemented
111#396—32no—
112#394—32no—
113#390—31no—
114#389—31no—
115#386—32no—
116#380—31no—
117C24×C3—31no—
118C24×C3—32no—
119C24×C3C22×D311yescharacteristic
120C22×A4—21no—
121C22×A4C23×C311yescharacteristic
122C2×S4—21no—
123C2×S4—21no—
124C2×S4—22no—
125C2×S4—22no—
126C2×S4—22no—
127C2×S4C3×D412yescomplemented
128C2×C3×D4—32no—
129C2×C3×D4—32no—
130C2×C3×D4—61no—
131C2×C3×D4—62no—
132C2×C3×D4—62no—
133C2×C3×D4—62no—
134C2×C3×D4—61no—
135C2×C3×D4—61no—
136C2×C3×D4—61no—
137C2×C3×D4—61no—
138C2×C3×D4—64no—
139C2×C3×D4—62no—
140C2×C3×D4—62no—
141C2×C3×D4—62no—
142C2×C3×D4—62no—
143C2×C3×D4—62no—
144C2×C3×D4—62no—
145C2×C3×D4—62no—
146C2×C3×D4—61no—
147C2×C3×D4—122no—
148C22×C3×C4—22no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—32no—
152C22×C3×C4—31no—
153C22×C3×C4—32no—
154C22×C3×C4—32no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—32no—
158C22×C3×C4—62no—
159C22×C3×C4—61no—
160C22×C3×C4—62no—
161C22×C3×C4—62no—
162C22×C3×C4—61no—
163C22×C3×C4D1212yescomplemented
164C2×D12—41no—
165C2×D12—42no—
166C2×C4×D3—41no—
167C4×A4—22no—
168C4×A4—22no—
169C4×A4—42no—
170C4×A4C3×D412yescomplemented
171SL(2,ℤ/4)—21no—
172C3×C4⋊C4—32no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—62no—
177C3×C4⋊C4—62no—
178C3×C4⋊C4—64no—
179C3×C4⋊C4—81no—
180C3×C4⋊C4S411yescharacteristic, complemented
181C3×C22⋊C4—62no—
182C3×C22⋊C4—64no—
183C3×C22⋊C4—64no—
184C3×C22⋊C4—62no—
185C3×C22⋊C4—61no—
186C3×C22⋊C4—64no—
187C3×C22⋊C4—61no—
188C3×C22⋊C4—62no—
189C3×C22⋊C4—122no—
190C3×C42—62no—
191(C2×D3)⋊C4—42no—
192C22×C32—41no—
193C2×C3×D3—42no—
194C2×C3×D3—81no—
195C2×C3×D3—81no—
196C2×C3×D3—82no—
197C2×C3×D3—82no—
198C2×C3×D3—82no—
199C3×A4#12111yescharacteristic, complemented
200C32×C4—42no—
201C32×C4—82no—
202C3×Dic3—81no—
203C22×D4—31no—
204C22×D4—31no—
205C22×D4—32no—
206C22×D4—32no—
207C23×C4—31no—
208C23×C4C2×C3×D312yes—
209C23×C4C2×C3×D311yescharacteristic
210#121—61no—
211#121—62no—
212#121—64no—
213#121—62no—
214#121—64no—
215#121—62no—
216#121—61no—
217C4×D4—62no—
218C4×D4—62no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—32no—
225C2×C22⋊C4—31no—
226C2×C22⋊C4—32no—
227C2×C22⋊C4—32no—
228C2×C22⋊C4—32no—
229C2×C22⋊C4—32no—
230C2×C22⋊C4—31no—
231C2×C22⋊C4—32no—
232C2×C42—31no—
233#95—32no—
234#95—32no—
235C23×C3—31no—
236C23×C3—31no—
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—32no—
240C23×C3—61no—
241C23×C3—61no—
242C23×C3—61no—
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—62no—
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3—61no—
252C23×C3—62no—
253C23×C3C2×D1211yescharacteristic
254C23×C3D3×D411yescharacteristic
255C23×C3#23911yescharacteristic
256C22×D3—41no—
257C22×D3—42no—
258C2×A4—21no—
259C2×A4—21no—
260C2×A4—21no—
261C2×A4C2×C3×D411yescharacteristic
262C2×A4C2×C3×D411yescharacteristic
263C2×A4C3×#4111yescharacteristic
264S4—22no—
265S4—42no—
266C3×D4—62no—
267C3×D4—62no—
268C3×D4—62no—
269C3×D4—62no—
270C3×D4—124no—
271C3×D4—122no—
272C3×D4—124no—
273C3×D4—122no—
274C3×D4—122no—
275C3×D4—122no—
276C3×D4—122no—
277C3×D4—122no—
278C2×C3×C4—31no—
279C2×C3×C4—32no—
280C2×C3×C4—32no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—62no—
301C2×C3×C4—61no—
302C2×C3×C4—81no—
303C2×C3×C4—82no—
304C2×C3×C4—122no—
305C2×C3×C4—122no—
306C2×C3×C4C2×S412yes—
307C2×C3×C4C2×S411yescharacteristic
308C2×Dic3—41no—
309D12—82no—
310D12—84no—
311C4×D3—82no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—41no—
315C3×D3—82no—
316C3×D3—162no—
317C24—31no—
318C24—32no—
319C24C22×C3×D311yescharacteristic
320C2×D4—32no—
321C2×D4—32no—
322C2×D4—61no—
323C2×D4—62no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—64no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—61no—
339C2×D4—122no—
340C22×C4—22no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—32no—
344C22×C4—31no—
345C22×C4—32no—
346C22×C4—32no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—32no—
350C22×C4—62no—
351C22×C4—61no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4C3×D1212yescomplemented
356C4⋊C4—32no—
357C4⋊C4—31no—
358C4⋊C4—61no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—64no—
362C4⋊C4C3×S411yescharacteristic, complemented
363C22⋊C4—62no—
364C22⋊C4—64no—
365C22⋊C4—64no—
366C22⋊C4—62no—
367C22⋊C4—61no—
368C22⋊C4—64no—
369C22⋊C4—61no—
370C22⋊C4—62no—
371C22⋊C4—122no—
372C42—62no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—31no—
381C22×C3—31no—
382C22×C3—41no—
383C22×C3—61no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—62no—
387C22×C3—81no—
388C22×C3—121no—
389C22×C3—121no—
390C22×C3—122no—
391C22×C3—122no—
392C22×C3—122no—
393C22×C3—122no—
394C22×C3—121no—
395C22×C3—121no—
396C22×C3—122no—
397C22×C3—122no—
398C22×C3—122no—
399C22×C3#89511yescharacteristic, complemented
400C22×C3C22×S411yescharacteristic
401C2×D3—42no—
402C2×D3—81no—
403C2×D3—81no—
404C2×D3—82no—
405C2×D3—82no—
406C2×D3—82no—
407A4—21no—
408A4C3×#12111yescharacteristic, complemented
409C3×C4—22no—
410C3×C4—32no—
411C3×C4—42no—
412C3×C4—62no—
413C3×C4—62no—
414C3×C4—82no—
415C3×C4—82no—
416C3×C4—121no—
417C3×C4—122no—
418C3×C4—121no—
419C3×C4—162no—
420C3×C4C4⋊S412yescomplemented
421Dic3—81no—
422C32—41noSylow
423C23—31no—
424C23—31no—
425C23—31no—
426C23—31no—
427C23—32no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23—62no—
434C23—62no—
435C23—62no—
436C23—62no—
437C23—62no—
438C23—62no—
439C23—61no—
440C23—62no—
441C23C2×C3×D1211yescharacteristic
442C23C3×D3×D411yescharacteristic
443C23C3×#23911yescharacteristic
444D4—62no—
445D4—62no—
446D4—62no—
447D4—62no—
448D4—124no—
449D4—122no—
450D4—124no—
451D4—122no—
452D4—122no—
453D4—122no—
454D4—122no—
455D4—122no—
456C2×C4—31no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—32no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—62no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4C2×C3×S412yes—
481C2×C4C2×C3×S411yescharacteristic
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—41no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—81no—
490C2×C3—81no—
491C2×C3—81no—
492C2×C3—122no—
493C2×C3—242no—
494C2×C3C2×C4⋊S411yescharacteristic
495C2×C3D4×S411yescharacteristic
496C2×C3#591911yescharacteristic
497D3—82no—
498D3—162no—
499C22—31no—
500C22—31no—
501C22—31no—
502C22—31no—
503C22—31no—
504C22—31no—
505C22—31no—
506C22—31no—
507C22—31no—
508C22—61no—
509C22—62no—
510C22—62no—
511C22—62no—
512C22—121no—
513C22—121no—
514C22—122no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22—121no—
519C22—121no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22C3×#89511yescharacteristic, complemented
524C22C22×C3×S411yescharacteristic
525C4—22no—
526C4—32no—
527C4—62no—
528C4—62no—
529C4—122no—
530C4—121no—
531C4—121no—
532C4C3×C4⋊S412yescomplemented
533C3—41no—
534C3—81no—
535C3#8531011yescharacteristic, direct product factor
536C2—31no—
537C2—31no—
538C2—31no—
539C2—31no—
540C2—122no—
541C2—242no—
542C2C2×C3×C4⋊S411yescharacteristic
543C2C3×D4×S411yescharacteristic
544C2C3×#591911yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
hypercenterC3×C4⋊C4S4
hypocenterA4C3×#121
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊S4C2semidirect
2. C2×C3×C4⋊S4C2semidirect
3. C2×C3×C4×S4C2semidirect
4. C3×A4×C4⋊C4C2semidirect
5. C3×(C2×S4)⋊C4C2semidirect
6. #85310C3direct (N × H)
17. C2×C4⋊S4C2×C3semidirect
18. C2×C4⋊S4C2×C3semidirect
19. C2×C4×S4C2×C3semidirect
24. C22×C3×C4⋊C4D3semidirect
26. A4×C4⋊C4C2×C3semidirect
27. (C2×S4)⋊C4C2×C3semidirect
40. C2×C3×S4D4semidirect
45. C3×C4×A4D4semidirect
110. C22×C4⋊C4C3×D3semidirect
127. C2×S4C3×D4semidirect
163. C22×C3×C4D12semidirect
170. C4×A4C3×D4semidirect
180. C3×C4⋊C4S4semidirect
199. C3×A4#121semidirect
355. C22×C4C3×D12semidirect
362. C4⋊C4C3×S4semidirect
399. C22×C3#895semidirect
408. A4C3×#121semidirect
420. C3×C4C4⋊S4semidirect
523. C22C3×#895semidirect
532. C4C3×C4⋊S4semidirect
535. C3#85310direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊S42
C2×C3×C4×S42
C3×A4×C4⋊C42
C3×(C2×S4)⋊C42
#853103
C3×#24163
C3×#8954