Finite Group Databasebeta

← C3×#85485 #11922769

Subgroups — C3×#85485 #11922769

Order 1152.

894 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C3×C4×A4C211yesmaximal, characteristic, complemented
2C2×C3×GL(2,ℤ/4)C211yesmaximal, characteristic
3C3×#5432C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C212yesmaximal, complemented
5C3×SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6#85485C311yesmaximal, characteristic, direct product factor
7C3×#2849—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C22×C3×S4C2211yescharacteristic
10C2×C3×C4×A4—22no—
11C2×C3×C4×A4C2212yes—
12C3×GL(2,ℤ/4)—22no—
13C2×C3×SL(2,ℤ/4)C2211yescharacteristic
14C2×C3×SL(2,ℤ/4)C2212yes—
15C3×#929—41nomaximal
16C24×C3×C4D311yescharacteristic, complemented
17C22×C4×A4—21no—
18C22×C4×A4C2×C311yescharacteristic, complemented
19C2×GL(2,ℤ/4)C2×C311yescharacteristic
20C2×C3×#123—31no—
21C2×C3×#123—31no—
22C2×C3×#120—31no—
23#5432C2×C311yescharacteristic, complemented
24(C2×S4)⋊C4C2×C312yescomplemented
25SL(2,ℤ/4)⋊C4C2×C312yescomplemented
26C3×#386—32no—
27C3×#386—32no—
28C3×#385—32no—
29C3×#385—32no—
30C3×#381—31no—
31C3×#381—31no—
32C3×#379—31no—
33C22×C3×A4—22no—
34C22×C3×A4D412yes—
35C22×C3×A4C2311yescharacteristic
36C2×C3×S4—21no—
37C2×C3×S4—22no—
38C22×C32×C4—41no—
39C2×C3×#67—41no—
40C3×C4×A4—22no—
41C3×SL(2,ℤ/4)—21no—
42C3×SL(2,ℤ/4)—22no—
43C3×#217—41no—
44C3×(C2×D3)⋊C4—42no—
45C3×Dic3⋊C4—42no—
46#2849—31noSylow
47C25×C3C2×D311yescharacteristic
48C23×A4—21no—
49C23×A4C22×C311yescharacteristic
50C22×S4C22×C311yescharacteristic
51C22×C3×D4—31no—
52C22×C3×D4—31no—
53C22×C3×D4—31no—
54C23×C3×C4—22no—
55C23×C3×C4—31no—
56C23×C3×C4—31no—
57C23×C3×C4—32no—
58C23×C3×C4—62no—
59C23×C3×C4C2×D312yes—
60C2×C4×A4—24no—
61C2×C4×A4—22no—
62C2×C4×A4—22no—
63C2×C4×A4C22×C312yes—
64GL(2,ℤ/4)—22no—
65C2×SL(2,ℤ/4)C22×C311yescharacteristic
66C2×SL(2,ℤ/4)C22×C312yes—
67C3×#123—64no—
68C3×#123—61no—
69C3×#123—62no—
70C3×#123—61no—
71C3×#120—62no—
72C3×#120—62no—
73C2×C3×C4⋊C4—32no—
74C2×C3×C4⋊C4—32no—
75C2×C3×C22⋊C4—31no—
76C2×C3×C22⋊C4—31no—
77C2×C3×C22⋊C4—32no—
78C2×C3×C22⋊C4—31no—
79C2×C3×C22⋊C4—31no—
80C2×C3×C22⋊C4—32no—
81C2×C3×C22⋊C4—32no—
82C2×C3×C22⋊C4—32no—
83C2×C3×C22⋊C4—31no—
84#929—41no—
85C3×#95—32no—
86C3×#95—32no—
87C3×#95—32no—
88C3×#95—32no—
89C23×C32—41no—
90C22×C3×D3—41no—
91C2×C3×A4—22no—
92C2×C3×A4C2×D411yescharacteristic
93C2×C3×A4#4112yes—
94C3×S4—41no—
95C2×C32×C4—42no—
96C2×C32×C4—82no—
97C3×#67—82no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—42no—
100C24×C4C3×D311yescharacteristic, complemented
101C2×#123—31no—
102C2×#123—31no—
103C2×#120—31no—
104#386—32no—
105#386—32no—
106#385—32no—
107#385—32no—
108#381—31no—
109#381—31no—
110#379—31no—
111C24×C3—22no—
112C24×C3—31no—
113C24×C3—31no—
114C24×C3—32no—
115C24×C3—31no—
116C24×C3—62no—
117C24×C3#6712yes—
118C24×C3C22×D311yescharacteristic
119C22×A4—24no—
120C22×A4—22no—
121C22×A4—22no—
122C22×A4—21no—
123C22×A4C3×D412yes—
124C22×A4C23×C311yescharacteristic
125C2×S4—21no—
126C2×S4—22no—
127C2×C3×D4—62no—
128C2×C3×D4—61no—
129C2×C3×D4—62no—
130C2×C3×D4—62no—
131C2×C3×D4—61no—
132C2×C3×D4—62no—
133C2×C3×D4—64no—
134C2×C3×D4—62no—
135C2×C3×D4—61no—
136C2×C3×D4—61no—
137C22×C3×C4—22no—
138C22×C3×C4—32no—
139C22×C3×C4—34no—
140C22×C3×C4—31no—
141C22×C3×C4—32no—
142C22×C3×C4—32no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—32no—
146C22×C3×C4—32no—
147C22×C3×C4—31no—
148C22×C3×C4—41no—
149C22×C3×C4—62no—
150C22×C3×C4—62no—
151C22×C3×C4—61no—
152C22×C3×C4—62no—
153C22×C3×C4—64no—
154C22×C3×C4—64no—
155C22×C3×C4—62no—
156C22×C3×C4—81no—
157C22×C3×C4S411yescharacteristic, complemented
158C2×#67—41no—
159C4×A4—24no—
160C4×A4—22no—
161SL(2,ℤ/4)—21no—
162SL(2,ℤ/4)—22no—
163C3×C4⋊C4—64no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—62no—
166C3×C22⋊C4—64no—
167C3×C22⋊C4—62no—
168C3×C22⋊C4—62no—
169C3×C22⋊C4—64no—
170C3×C22⋊C4—62no—
171C3×C22⋊C4—61no—
172C3×C22⋊C4—62no—
173C3×C22⋊C4—62no—
174C3×C22⋊C4—61no—
175C3×C22⋊C4—62no—
176C3×C22⋊C4—61no—
177C3×C22⋊C4—61no—
178C3×C22⋊C4—62no—
179#217—41no—
180(C2×D3)⋊C4—42no—
181Dic3⋊C4—42no—
182C22×C32—42no—
183C22×C32—41no—
184C22×C32—82no—
185C2×C3×D3—81no—
186C2×C3×D3—82no—
187C3×A4#12311yescharacteristic, complemented
188C32×C4—82no—
189C3×Dic3—81no—
190C3×Dic3—82no—
191C25C2×C3×D311yescharacteristic
192C22×D4—31no—
193C22×D4—31no—
194C22×D4—31no—
195C23×C4—22no—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4—32no—
199C23×C4—62no—
200C23×C4C2×C3×D312yes—
201#123—64no—
202#123—61no—
203#123—62no—
204#123—61no—
205#120—62no—
206#120—62no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—32no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—31no—
211C2×C22⋊C4—31no—
212C2×C22⋊C4—31no—
213C2×C22⋊C4—32no—
214C2×C22⋊C4—32no—
215C2×C22⋊C4—32no—
216C2×C22⋊C4—31no—
217C2×C22⋊C4—32no—
218#95—32no—
219#95—32no—
220#95—32no—
221#95—32no—
222C23×C3—22no—
223C23×C3—31no—
224C23×C3—32no—
225C23×C3—31no—
226C23×C3—32no—
227C23×C3—34no—
228C23×C3—32no—
229C23×C3—32no—
230C23×C3—31no—
231C23×C3—41no—
232C23×C3—62no—
233C23×C3—62no—
234C23×C3—62no—
235C23×C3—61no—
236C23×C3—64no—
237C23×C3—62no—
238C23×C3—64no—
239C23×C3—61no—
240C23×C3—61no—
241C23×C3—62no—
242C23×C3—62no—
243C23×C3—61no—
244C23×C3—81no—
245C23×C3#23512yes—
246C23×C3C2×#6711yescharacteristic
247C23×C3C2×S411yescharacteristic
248C22×D3—41no—
249C2×A4—24no—
250C2×A4—22no—
251C2×A4—22no—
252C2×A4—21no—
253C2×A4C2×C3×D411yescharacteristic
254C2×A4C3×#4112yes—
255S4—41no—
256C3×D4—122no—
257C3×D4—121no—
258C3×D4—124no—
259C3×D4—122no—
260C3×D4—121no—
261C3×D4—122no—
262C2×C3×C4—22no—
263C2×C3×C4—34no—
264C2×C3×C4—32no—
265C2×C3×C4—42no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—64no—
269C2×C3×C4—64no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4—62no—
280C2×C3×C4—61no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—61no—
285C2×C3×C4—82no—
286C2×C3×C4—84no—
287C2×C3×C4—82no—
288C2×C3×C4C2×S412yes—
289#67—82no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×C32—42no—
293C2×C32—41no—
294C2×C32—82no—
295C3×D3—161no—
296C24—22no—
297C24—31no—
298C24—31no—
299C24—32no—
300C24—31no—
301C24—62no—
302C24C3×#6712yes—
303C24C22×C3×D311yescharacteristic
304C2×D4—62no—
305C2×D4—61no—
306C2×D4—62no—
307C2×D4—62no—
308C2×D4—61no—
309C2×D4—62no—
310C2×D4—64no—
311C2×D4—61no—
312C2×D4—62no—
313C2×D4—61no—
314C22×C4—22no—
315C22×C4—32no—
316C22×C4—34no—
317C22×C4—31no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—32no—
323C22×C4—32no—
324C22×C4—31no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4—64no—
330C22×C4—64no—
331C22×C4—62no—
332C22×C4C3×S411yescharacteristic, complemented
333C4⋊C4—64no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C22⋊C4—64no—
337C22⋊C4—62no—
338C22⋊C4—62no—
339C22⋊C4—61no—
340C22⋊C4—62no—
341C22⋊C4—62no—
342C22⋊C4—61no—
343C22⋊C4—62no—
344C22⋊C4—64no—
345C22⋊C4—61no—
346C22⋊C4—62no—
347C22⋊C4—61no—
348C22⋊C4—62no—
349C22×C3—22no—
350C22×C3—32no—
351C22×C3—31no—
352C22×C3—32no—
353C22×C3—32no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—34no—
357C22×C3—32no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—62no—
361C22×C3—62no—
362C22×C3—64no—
363C22×C3—61no—
364C22×C3—62no—
365C22×C3—64no—
366C22×C3—62no—
367C22×C3—82no—
368C22×C3—84no—
369C22×C3—82no—
370C22×C3—81no—
371C22×C3—121no—
372C22×C3—121no—
373C22×C3—122no—
374C22×C3—122no—
375C22×C3—121no—
376C22×C3#92911yescharacteristic, complemented
377C22×C3GL(2,ℤ/4)12yes—
378C22×C3C22×S411yescharacteristic
379C2×D3—81no—
380C2×D3—82no—
381A4—21no—
382A4C3×#12311yescharacteristic, complemented
383C3×C4—22no—
384C3×C4—62no—
385C3×C4—82no—
386C3×C4—84no—
387C3×C4—122no—
388C3×C4—121no—
389C3×C4—121no—
390C3×C4—122no—
391C3×C4—121no—
392Dic3—81no—
393Dic3—82no—
394C32—41noSylow
395C23—22no—
396C23—31no—
397C23—32no—
398C23—31no—
399C23—32no—
400C23—34no—
401C23—32no—
402C23—32no—
403C23—31no—
404C23—62no—
405C23—62no—
406C23—62no—
407C23—61no—
408C23—64no—
409C23—62no—
410C23—64no—
411C23—61no—
412C23—61no—
413C23—62no—
414C23—62no—
415C23—61no—
416C23C3×#23512yes—
417C23C2×C3×#6711yescharacteristic
418C23C2×C3×S411yescharacteristic
419D4—122no—
420D4—121no—
421D4—124no—
422D4—122no—
423D4—122no—
424D4—121no—
425C2×C4—22no—
426C2×C4—34no—
427C2×C4—32no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—64no—
431C2×C4—64no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—61no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—61no—
447C2×C4C2×C3×S412yes—
448C2×C3—22no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—32no—
452C2×C3—42no—
453C2×C3—41no—
454C2×C3—62no—
455C2×C3—82no—
456C2×C3—84no—
457C2×C3—82no—
458C2×C3—81no—
459C2×C3—241no—
460C2×C3#591212yes—
461C2×C3C2×GL(2,ℤ/4)11yescharacteristic
462D3—161no—
463C22—22no—
464C22—32no—
465C22—31no—
466C22—32no—
467C22—32no—
468C22—31no—
469C22—31no—
470C22—34no—
471C22—32no—
472C22—62no—
473C22—62no—
474C22—64no—
475C22—61no—
476C22—62no—
477C22—64no—
478C22—62no—
479C22—121no—
480C22—121no—
481C22—122no—
482C22—122no—
483C22—121no—
484C22C3×#92911yescharacteristic, complemented
485C22C3×GL(2,ℤ/4)12yes—
486C22C22×C3×S411yescharacteristic
487C4—22no—
488C4—62no—
489C4—121no—
490C4—121no—
491C4—122no—
492C4—122no—
493C4—121no—
494C3—41no—
495C3—81no—
496C3#8548511yescharacteristic, direct product factor
497C2—22no—
498C2—31no—
499C2—31no—
500C2—32no—
501C2—62no—
502C2—241no—
503C2C3×#591212yes—
504C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC24×C3×C4D3
socleC24×C3C22×D3
prime radicalC2×C3×GL(2,ℤ/4)C2
hypercenterC22×C3×C4S4
hypocenterA4C3×#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×A4C2semidirect
3. C3×#5432C2semidirect
4. C3×(C2×S4)⋊C4C2semidirect
5. C3×SL(2,ℤ/4)⋊C4C2semidirect
6. #85485C3direct (N × H)
16. C24×C3×C4D3semidirect
18. C22×C4×A4C2×C3semidirect
23. #5432C2×C3semidirect
24. (C2×S4)⋊C4C2×C3semidirect
25. SL(2,ℤ/4)⋊C4C2×C3semidirect
100. C24×C4C3×D3semidirect
157. C22×C3×C4S4semidirect
187. C3×A4#123semidirect
332. C22×C4C3×S4semidirect
376. C22×C3#929semidirect
382. A4C3×#123semidirect
484. C22C3×#929semidirect
496. C3#85485direct (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×A42
C2×C3×GL(2,ℤ/4)2
C3×#54322
C3×(C2×S4)⋊C42
C3×SL(2,ℤ/4)⋊C42
#854853
C3×#28493
C3×#9294