Finite Group Databasebeta

← #11922862

Subgroups — #11922862

Order 1152.

664 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4⋊C4C211yesmaximal, characteristic, complemented
2#10593647C211yesmaximal, characteristic
3#10593647C211yesmaximal, characteristic
4#10593647C211yesmaximal, characteristic
5C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
6#10593645C211yesmaximal, characteristic, complemented
7C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8#85313—33nomaximal
9#82175—31nomaximal
10C22×C3⋊S4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3×C4×A4C2211yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
16C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
17#64107—41nomaximal
18C22×C3×C4⋊C4D311yescharacteristic, complemented
19C2×#898—31no—
20A4×C4⋊C4D313yescomplemented
21(C2×S4)⋊C4—33no—
22(C2×S4)⋊C4—33no—
23(C2×S4)⋊C4—33no—
24C4⋊SL(2,ℤ/4)—33no—
25SL(2,ℤ/4)⋊C4—33no—
26C4×SL(2,ℤ/4)—33no—
27#5213—31no—
28#5213—31no—
29#5213—31no—
30#5212—31no—
31#5210—31no—
32C4×#217—31no—
33#4994—31no—
34#4992—31no—
35#4990—31no—
36#4986—31no—
37#4985—31no—
38#4980—31no—
39#4978—31no—
40C22×C3×A4C2311yescharacteristic
41C2×C3⋊S4—21no—
42C2×C3⋊S4—21no—
43C2×C3⋊S4—21no—
44C3×C4×A4—21no—
45C3×C4×A4—21no—
46C3×C4×A4—21no—
47C3⋊SL(2,ℤ/4)—21no—
48C3⋊SL(2,ℤ/4)—21no—
49C3⋊SL(2,ℤ/4)—21no—
50C32×C4⋊C4—41no—
51#3752—41no—
52#3752—41no—
53#3752—41no—
54C4⋊(C3⋊Dic3)—41no—
55#3750—41no—
56C4×C3⋊Dic3—41no—
57#2481—91noSylow
58C22×S4—33no—
59C23×C3×C4C2×D311yescharacteristic
60C23×C3×C4C2×D311yescharacteristic
61C23×C3×C4C2×D311yescharacteristic
62C22×#67—31no—
63C2×C4×A4C2×D313yes—
64C2×C4×A4C2×D313yes—
65C2×C4×A4C2×D313yes—
66C2×SL(2,ℤ/4)—33no—
67C2×SL(2,ℤ/4)—33no—
68C2×SL(2,ℤ/4)—33no—
69C2×C3×C4⋊C4—31no—
70C2×C3×C4⋊C4—31no—
71C2×C3×C4⋊C4—31no—
72C2×C3×C4⋊C4—31no—
73C2×#217—31no—
74C2×#217—31no—
75C2×#217—31no—
76C2×(C2×D3)⋊C4—31no—
77C2×(C2×D3)⋊C4—31no—
78C2×(C2×D3)⋊C4—31no—
79C2×(C2×D3)⋊C4—31no—
80C2×(C2×D3)⋊C4—31no—
81C2×(C2×D3)⋊C4—31no—
82C2×C4⋊Dic3—31no—
83C2×C4⋊Dic3—31no—
84C2×Dic3⋊C4—31no—
85C2×Dic3⋊C4—31no—
86C2×C4×Dic3—31no—
87C2×C4×Dic3—31no—
88#898—61no—
89#898—61no—
90#898—61no—
91#898—61no—
92#898—123no—
93C3⋊#95—31no—
94C3⋊#95—31no—
95C3⋊#95—31no—
96C3⋊#95—31no—
97C3⋊#95—31no—
98C3⋊#95—31no—
99C3⋊#95—31no—
100C3⋊#95—31no—
101C3⋊#95—31no—
102C3⋊#95—31no—
103C3⋊#95—31no—
104C3⋊#95—31no—
105C22×C3⋊D3—41no—
106C2×C3×A4#4111yescharacteristic
107C2×C3×A4#4111yescharacteristic
108C2×C3×A4#4111yescharacteristic
109C3⋊S4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4—41no—
113C2×C3⋊Dic3—41no—
114C2×C3⋊Dic3—41no—
115C2×C3⋊Dic3—41no—
116C2×#126—91no—
117C22×C4⋊C4C3⋊D311yescharacteristic, complemented
118#400—91no—
119#400—91no—
120#397—91no—
121#396—91no—
122#388—91no—
123#387—91no—
124#387—91no—
125#386—91no—
126#386—91no—
127#386—91no—
128#385—91no—
129#380—91no—
130C4×C22⋊C4—91no—
131C24×C3C22×D311yescharacteristic
132C23×D3—31no—
133C22×A4C22×D313yes—
134C2×S4—63no—
135C2×S4—63no—
136C2×S4—63no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—61no—
147C22×C3×C4—61no—
148C22×C3×C4—61no—
149C2×#67—61no—
150C2×#67—61no—
151C2×#67—61no—
152C2×#67—61no—
153C2×#67—61no—
154C2×#67—61no—
155C22×Dic3—31no—
156C22×Dic3—31no—
157C22×Dic3—31no—
158C22×Dic3—31no—
159C22×Dic3—31no—
160C22×Dic3—31no—
161C22×Dic3—31no—
162C4×A4—23no—
163C4×A4—23no—
164C4×A4—23no—
165SL(2,ℤ/4)—63no—
166SL(2,ℤ/4)—63no—
167SL(2,ℤ/4)—63no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—43no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4S411yescharacteristic, complemented
174#217—61no—
175#217—61no—
176#217—61no—
177#217—61no—
178#217—61no—
179#217—61no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—62no—
184(C2×D3)⋊C4—61no—
185(C2×D3)⋊C4—61no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—123no—
188(C2×D3)⋊C4—123no—
189(C2×D3)⋊C4—123no—
190(C2×D3)⋊C4—121no—
191(C2×D3)⋊C4—121no—
192C4⋊Dic3—61no—
193C4⋊Dic3—61no—
194C4⋊Dic3—123no—
195C4⋊Dic3—121no—
196Dic3⋊C4—61no—
197Dic3⋊C4—62no—
198Dic3⋊C4—61no—
199Dic3⋊C4—123no—
200C4×Dic3—61no—
201C4×Dic3—61no—
202C4×Dic3—123no—
203C4×Dic3—121no—
204C22×C32—41no—
205C2×C3⋊D3—81no—
206C2×C3⋊D3—81no—
207C2×C3⋊D3—81no—
208C3×A4#12611yescharacteristic, complemented
209C32×C4—81no—
210C32×C4—81no—
211C32×C4—81no—
212C3⋊Dic3—81no—
213C3⋊Dic3—81no—
214C3⋊Dic3—81no—
215C22×D4—91no—
216C23×C4C2×C3⋊D311yescharacteristic
217C23×C4C2×C3⋊D311yescharacteristic
218C23×C4C2×C3⋊D311yescharacteristic
219#126—181no—
220#126—181no—
221#126—181no—
222#126—181no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—91no—
228C2×C4⋊C4—91no—
229C2×C4⋊C4—91no—
230C2×C4⋊C4—91no—
231C2×C22⋊C4—91no—
232C2×C22⋊C4—91no—
233C2×C22⋊C4—91no—
234C2×C22⋊C4—91no—
235C2×C22⋊C4—91no—
236C2×C22⋊C4—91no—
237C2×C22⋊C4—91no—
238C2×C22⋊C4—91no—
239C2×C22⋊C4—91no—
240C2×C42—91no—
241C2×C42—91no—
242#95—91no—
243#95—91no—
244#95—91no—
245#95—91no—
246#95—91no—
247#95—91no—
248#95—91no—
249#95—91no—
250#95—91no—
251#95—91no—
252#95—91no—
253#95—91no—
254C23×C3—31no—
255C23×C3—31no—
256C23×C3—31no—
257C23×C3—31no—
258C23×C3#23511yescharacteristic
259C23×C3#23711yescharacteristic
260C23×C3#23911yescharacteristic
261C22×D3—61no—
262C22×D3—61no—
263C22×D3—61no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—61no—
268C22×D3—123no—
269C2×A4#23513yes—
270C2×A4#23713yes—
271C2×A4#23913yes—
272S4—123no—
273C2×C3×C4—32no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—43no—
278C2×C3×C4—43no—
279C2×C3×C4—43no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4C2×S411yescharacteristic
292C2×C3×C4C2×S411yescharacteristic
293C2×C3×C4C2×S411yescharacteristic
294#67—121no—
295#67—121no—
296#67—121no—
297#67—121no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—123no—
320C2×Dic3—123no—
321C2×Dic3—123no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—41no—
325C3⋊D3—161no—
326C24—91no—
327C24C22×C3⋊D311yescharacteristic
328C2×D4—181no—
329C2×D4—181no—
330C2×D4—181no—
331C2×D4—181no—
332C2×D4—181no—
333C2×D4—181no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—91no—
347C22×C4—91no—
348C22×C4—91no—
349C22×C4—91no—
350C22×C4—91no—
351C22×C4—91no—
352C22×C4—91no—
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—31no—
356C4⋊C4—61no—
357C4⋊C4—181no—
358C4⋊C4—181no—
359C4⋊C4—182no—
360C4⋊C4—181no—
361C4⋊C4—181no—
362C4⋊C4—361no—
363C4⋊C4C3⋊S411yescharacteristic, complemented
364C22⋊C4—181no—
365C22⋊C4—181no—
366C22⋊C4—181no—
367C22⋊C4—182no—
368C22⋊C4—181no—
369C22⋊C4—181no—
370C22⋊C4—181no—
371C22⋊C4—181no—
372C22⋊C4—181no—
373C22⋊C4—181no—
374C22⋊C4—181no—
375C22⋊C4—181no—
376C22⋊C4—181no—
377C22⋊C4—361no—
378C22⋊C4—361no—
379C42—181no—
380C42—181no—
381C42—361no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—43no—
392C22×C3—61no—
393C22×C3#89811yescharacteristic, complemented
394C22×C3C22×S411yescharacteristic
395C2×D3—121no—
396C2×D3—121no—
397C2×D3—121no—
398C2×D3—121no—
399C2×D3—121no—
400C2×D3—121no—
401C2×D3—121no—
402C2×D3—243no—
403C2×D3—243no—
404C2×D3—243no—
405A4#89813yescomplemented
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—61no—
410C3×C4—61no—
411C3×C4—61no—
412C3×C4—83no—
413C3×C4—83no—
414C3×C4—83no—
415Dic3—121no—
416Dic3—121no—
417Dic3—121no—
418Dic3—121no—
419Dic3—121no—
420Dic3—121no—
421Dic3—121no—
422Dic3—243no—
423Dic3—243no—
424Dic3—243no—
425C32—41noSylow
426C23—31no—
427C23—31no—
428C23—31no—
429C23—31no—
430C23—181no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23—181no—
435C23—181no—
436C23—181no—
437C23#382811yescharacteristic
438C23#383011yescharacteristic
439C23#383211yescharacteristic
440D4—361no—
441D4—361no—
442D4—361no—
443D4—361no—
444C2×C4—32no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4—181no—
472C2×C4—181no—
473C2×C4—181no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4C2×C3⋊S411yescharacteristic
481C2×C4C2×C3⋊S411yescharacteristic
482C2×C4C2×C3⋊S411yescharacteristic
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—43no—
488C2×C3—43no—
489C2×C3—43no—
490C2×C3#591211yescharacteristic
491C2×C3#591411yescharacteristic
492C2×C3#591911yescharacteristic
493D3—241no—
494D3—483no—
495C22—31no—
496C22—31no—
497C22—31no—
498C22—31no—
499C22—31no—
500C22—31no—
501C22—31no—
502C22—31no—
503C22—31no—
504C22—61no—
505C22—361no—
506C22—361no—
507C22—361no—
508C22—361no—
509C22—361no—
510C22—361no—
511C22—361no—
512C22#6410711yescharacteristic, complemented
513C22C22×C3⋊S411yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—61no—
518C4—61no—
519C4—61no—
520C4—361no—
521C4—361no—
522C4—361no—
523C4—361no—
524C4—361no—
525C4—361no—
526C4—361no—
527C3—43no—
528C3#8531311yescharacteristic, complemented
529C2—31no—
530C2—31no—
531C2—31no—
532C2—31no—
533C2—721no—
534C2#1059657511yescharacteristic
535C2#1059657711yescharacteristic
536C2#1059658211yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3⋊S4C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4#126
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×A4×C4⋊C4C2semidirect
5. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
6. #10593645C2semidirect
7. C4×C3⋊SL(2,ℤ/4)C2semidirect
18. C22×C3×C4⋊C4D3semidirect
20. A4×C4⋊C4D3semidirect
117. C22×C4⋊C4C3⋊D3semidirect
173. C3×C4⋊C4S4semidirect
208. C3×A4#126semidirect
363. C4⋊C4C3⋊S4semidirect
393. C22×C3#898semidirect
405. A4#898semidirect
512. C22#64107semidirect
528. C3#85313semidirect

Maximal subgroups

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

subgroupindex
C3×A4×C4⋊C42
#105936472
C4⋊(C3⋊SL(2,ℤ/4))2
#105936452
C4×C3⋊SL(2,ℤ/4)2
#853133
#821753
#641074