Finite Group Databasebeta

← C4×#64204 #11921950

Subgroups — C4×#64204 #11921950

Order 1152.

680 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 87 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×#64204C211yesmaximal, characteristic
2C4×Dic3×A4C211yesmaximal, characteristic
3C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic
5Dic3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
6SL(2,ℤ/4)⋊Dic3C211yesmaximal, characteristic
7#10593238C211yesmaximal, characteristic
8C4×A4⋊Q8—31nomaximal
9C4×#878—31nomaximal
10C2×C3×C4×A4C2211yescharacteristic
11C2×Dic3×A4C2211yescharacteristic
12C2×Dic3×A4C2211yescharacteristic
13C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3×SL(2,ℤ/4)C2211yescharacteristic
16C2×C3×SL(2,ℤ/4)C2211yescharacteristic
17#64204C414yesdirect product factor
18C4×#627—41nomaximal
19C2×A4⋊Q8—31no—
20C22×C4×Dic3D311yescharacteristic
21C2×#878—31no—
22C2×C4×Dic6—31no—
23C42×A4—31no—
24C4⋊SL(2,ℤ/4)—31no—
25SL(2,ℤ/4)⋊C4—31no—
26SL(2,ℤ/4)⋊C4—31no—
27C4×SL(2,ℤ/4)—31no—
28C4×SL(2,ℤ/4)—61no—
29C4×SL(2,ℤ/4)D311yescharacteristic
30C3×C4×C22⋊C4—31no—
31C4×#217—31no—
32#4950—31no—
33#4946—31no—
34#4942—31no—
35C4×C4⋊Dic3—31no—
36C4×Dic3⋊C4—31no—
37C4×Dic3⋊C4—31no—
38#4655—31no—
39#4654—31no—
40#4647—31no—
41#4646—31no—
42C22×C3×A4C2311yescharacteristic
43C3×C4×A4Q812yescomplemented
44C2×#627—41no—
45Dic3×A4—21no—
46Dic3×A4C2×C412yes—
47C3⋊SL(2,ℤ/4)—21no—
48C3⋊SL(2,ℤ/4)C2×C412yes—
49C3×SL(2,ℤ/4)—21no—
50C3×SL(2,ℤ/4)C2×C412yes—
51C4×C3⋊Dic3—41no—
52C3×C4×Dic3—41no—
53C3×C4×Dic3—41no—
54#3724—41no—
55Dic3⋊Dic3—41no—
56Dic3⋊Dic3—41no—
57C4×C22⋊Q8—91noSylow
58C23×C3×C4C2×D311yescharacteristic
59C23×Dic3C2×D311yescharacteristic
60C23×Dic3C2×D311yescharacteristic
61C22×Dic6—31no—
62C2×C4×A4—21no—
63C2×C4×A4—31no—
64C2×C4×A4—31no—
65C2×C4×A4C2×D311yescharacteristic
66C2×SL(2,ℤ/4)—31no—
67C2×SL(2,ℤ/4)—31no—
68C2×SL(2,ℤ/4)—61no—
69C2×SL(2,ℤ/4)—61no—
70C2×SL(2,ℤ/4)C2×D311yescharacteristic
71C2×SL(2,ℤ/4)C2×D311yescharacteristic
72A4⋊Q8—34no—
73C2×C3×C22⋊C4—31no—
74C2×C3×C22⋊C4—31no—
75C2×C3×C42—31no—
76C2×C3×C42—31no—
77C2×#217—31no—
78C2×#217—31no—
79C2×C4⋊Dic3—31no—
80C2×C4⋊Dic3—31no—
81C2×C4⋊Dic3—31no—
82C2×Dic3⋊C4—31no—
83C2×Dic3⋊C4—31no—
84C2×Dic3⋊C4—31no—
85C2×Dic3⋊C4—31no—
86C2×Dic3⋊C4—31no—
87C2×Dic3⋊C4—31no—
88C2×C4×Dic3—31no—
89C2×C4×Dic3—31no—
90C2×C4×Dic3—31no—
91C2×C4×Dic3—31no—
92C2×C4×Dic3—31no—
93C2×C4×Dic3—31no—
94#878—34no—
95#878—34no—
96C4×Dic6—61no—
97C4×Dic6—61no—
98C4×Dic6—61no—
99C4×Dic6—61no—
100C4×Dic6—121no—
101C3×#95—31no—
102C3×#95—31no—
103C3⋊#95—31no—
104C3⋊#95—31no—
105C3⋊#95—31no—
106C3⋊#95—31no—
107C3⋊#95—31no—
108C3⋊#95—31no—
109C3⋊#95—31no—
110C3⋊#95—31no—
111C2×C3×A4C22×C411yescharacteristic
112C2×C3×A4C2×Q811yescharacteristic
113C2×C3×A4#4111yescharacteristic
114C2×C32×C4—41no—
115C2×C3⋊Dic3—41no—
116C2×C3⋊Dic3—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121#627—44no—
122C2×C22⋊Q8—91no—
123C2×C4×Q8—91no—
124C22×C42—31no—
125#391—91no—
126#389—91no—
127#387—91no—
128#387—91no—
129#385—91no—
130#385—91no—
131#380—91no—
132C4×C4⋊C4—91no—
133C4×C4⋊C4—91no—
134C4×C4⋊C4—91no—
135C4×C22⋊C4—31no—
136C4×C22⋊C4—91no—
137C24×C3C22×D311yescharacteristic
138C22×A4—21no—
139C22×A4C22×D311yescharacteristic
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—32no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4Dic612yescomplemented
148C22×Dic3—21no—
149C22×Dic3—31no—
150C22×Dic3—31no—
151C22×Dic3—31no—
152C22×Dic3—32no—
153C22×Dic3—31no—
154C22×Dic3—31no—
155C22×Dic3—31no—
156C22×Dic3—31no—
157C22×Dic3—31no—
158C22×Dic3—61no—
159C22×Dic3C4×D312yes—
160C2×Dic6—34no—
161C2×Dic6—61no—
162C2×Dic6—62no—
163C2×Dic6—61no—
164C2×Dic6—121no—
165C4×A4—22no—
166C4×A4—32no—
167C4×A4—61no—
168C4×A4Dic612yescomplemented
169SL(2,ℤ/4)—21no—
170SL(2,ℤ/4)—32no—
171SL(2,ℤ/4)—62no—
172SL(2,ℤ/4)—62no—
173SL(2,ℤ/4)—61no—
174SL(2,ℤ/4)C4×D312yes—
175C3×C22⋊C4—32no—
176C3×C22⋊C4—32no—
177C3×C22⋊C4—61no—
178C3×C22⋊C4—61no—
179C3×C42—61no—
180C3×C42—62no—
181C3×C42—61no—
182C3×C42—121no—
183#217—32no—
184#217—32no—
185#217—61no—
186#217—61no—
187C4⋊Dic3—34no—
188C4⋊Dic3—61no—
189C4⋊Dic3—62no—
190C4⋊Dic3—61no—
191C4⋊Dic3—121no—
192Dic3⋊C4—34no—
193Dic3⋊C4—34no—
194Dic3⋊C4—61no—
195Dic3⋊C4—61no—
196Dic3⋊C4—62no—
197Dic3⋊C4—62no—
198Dic3⋊C4—61no—
199Dic3⋊C4—61no—
200Dic3⋊C4—121no—
201Dic3⋊C4—121no—
202C4×Dic3—31no—
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—41no—
206C4×Dic3—61no—
207C4×Dic3—61no—
208C4×Dic3—62no—
209C4×Dic3—61no—
210C4×Dic3—121no—
211C4×Dic3—241no—
212C4×Dic3S411yescharacteristic
213C22×C32—41no—
214C3×A4C4×Q811yescharacteristic, complemented
215C32×C4—42no—
216C3⋊Dic3—42no—
217C3⋊Dic3—81no—
218C3×Dic3—42no—
219C3×Dic3—42no—
220C3×Dic3—81no—
221C3×Dic3—81no—
222C22×Q8—91no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4D3211yescharacteristic
226C22⋊Q8—94no—
227C22⋊Q8—94no—
228C4×Q8—181no—
229C4×Q8—181no—
230C4×Q8—181no—
231C4×Q8—181no—
232C2×C4⋊C4—91no—
233C2×C4⋊C4—91no—
234C2×C4⋊C4—91no—
235C2×C4⋊C4—91no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—91no—
238C2×C4⋊C4—91no—
239C2×C4⋊C4—91no—
240C2×C4⋊C4—91no—
241C2×C22⋊C4—31no—
242C2×C22⋊C4—31no—
243C2×C22⋊C4—91no—
244C2×C22⋊C4—91no—
245C2×C42—31no—
246C2×C42—31no—
247C2×C42—91no—
248C2×C42—91no—
249C2×C42—91no—
250C2×C42—91no—
251C2×C42—91no—
252C2×C42—91no—
253#95—31no—
254#95—31no—
255#95—91no—
256#95—91no—
257#95—91no—
258#95—91no—
259#95—91no—
260#95—91no—
261#95—91no—
262#95—91no—
263C23×C3—31no—
264C23×C3—31no—
265C23×C3—31no—
266C23×C3—31no—
267C23×C3C2×Dic611yescharacteristic
268C23×C3C2×C4×D311yescharacteristic
269C23×C3#23511yescharacteristic
270C2×A4—21no—
271C2×A4—21no—
272C2×A4—21no—
273C2×A4C2×Dic611yescharacteristic
274C2×A4C2×C4×D311yescharacteristic
275C2×A4#23511yescharacteristic
276C2×C3×C4—31no—
277C2×C3×C4—32no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—32no—
281C2×C3×C4—32no—
282C2×C3×C4—41no—
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×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—81no—
295C2×C3×C4—121no—
296C2×C3×C4—121no—
297C2×C3×C4C2×S411yescharacteristic
298C2×Dic3—31no—
299C2×Dic3—32no—
300C2×Dic3—32no—
301C2×Dic3—32no—
302C2×Dic3—32no—
303C2×Dic3—32no—
304C2×Dic3—32no—
305C2×Dic3—31no—
306C2×Dic3—41no—
307C2×Dic3—41no—
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—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—121no—
323C2×Dic3—121no—
324C2×Dic3—241no—
325C2×Dic3—241no—
326C2×Dic3C2×S411yescharacteristic
327C2×Dic3C2×S411yescharacteristic
328Dic6—64no—
329Dic6—64no—
330Dic6—124no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—41no—
334C24C2×D3211yescharacteristic
335C2×Q8—94no—
336C2×Q8—181no—
337C2×Q8—181no—
338C2×Q8—182no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—32no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—32no—
347C22×C4—61no—
348C22×C4—91no—
349C22×C4—91no—
350C22×C4—91no—
351C22×C4—91no—
352C22×C4—92no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—181no—
358C22×C4#62712yescomplemented
359C4⋊C4—94no—
360C4⋊C4—94no—
361C4⋊C4—94no—
362C4⋊C4—181no—
363C4⋊C4—181no—
364C4⋊C4—181no—
365C4⋊C4—181no—
366C4⋊C4—181no—
367C4⋊C4—181no—
368C4⋊C4—182no—
369C4⋊C4—182no—
370C4⋊C4—182no—
371C22⋊C4—32no—
372C22⋊C4—32no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C22⋊C4—92no—
376C22⋊C4—92no—
377C22⋊C4—181no—
378C22⋊C4—181no—
379C42—31no—
380C42—61no—
381C42—62no—
382C42—61no—
383C42—91no—
384C42—91no—
385C42—91no—
386C42—181no—
387C42—181no—
388C42—182no—
389C42—181no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—31no—
396C22×C3—31no—
397C22×C3—31no—
398C22×C3—31no—
399C22×C3—41no—
400C22×C3—61no—
401C22×C3—81no—
402C22×C3C4×Dic611yescharacteristic, complemented
403C22×C3C22×S411yescharacteristic
404A4—21no—
405A4C4×Dic611yescharacteristic, complemented
406C3×C4—32no—
407C3×C4—42no—
408C3×C4—62no—
409C3×C4—62no—
410C3×C4—82no—
411C3×C4—122no—
412C3×C4—121no—
413C3×C4—121no—
414C3×C4—241no—
415C3×C4A4⋊Q812yescomplemented
416Dic3—21no—
417Dic3—32no—
418Dic3—42no—
419Dic3—62no—
420Dic3—62no—
421Dic3—61no—
422Dic3—81no—
423Dic3—122no—
424Dic3—121no—
425Dic3—121no—
426Dic3—242no—
427Dic3—241no—
428Dic3—242no—
429Dic3C4×S412yes—
430C32—41noSylow
431C23—31no—
432C23—31no—
433C23—31no—
434C23—31no—
435C23#379811yescharacteristic
436C23C4×D3211yescharacteristic
437C23C2×#62711yescharacteristic
438Q8—184no—
439Q8—184no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—32no—
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—92no—
460C2×C4—92no—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—92no—
465C2×C4—92no—
466C2×C4—91no—
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×C4—181no—
481C2×C4D3×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×C3C2×A4⋊Q811yescharacteristic
493C2×C3C2×C4×S411yescharacteristic
494C2×C3#591211yescharacteristic
495C22—31no—
496C22—31no—
497C22—31no—
498C22—31no—
499C22—31no—
500C22—31no—
501C22—31no—
502C22—31no—
503C22—31no—
504C22—61no—
505C22C4×#62711yescharacteristic, complemented
506C22C2×D3×S411yescharacteristic
507C4—32no—
508C4—32no—
509C4—62no—
510C4—61no—
511C4—62no—
512C4—92no—
513C4—121no—
514C4—121no—
515C4—182no—
516C4—181no—
517C4—182no—
518C4—361no—
519C4—361no—
520C4#6420412yesdirect product factor
521C3—41no—
522C3—81no—
523C3C4×A4⋊Q811yescharacteristic, complemented
524C2—31no—
525C2—31no—
526C2—31no—
527C2—31no—
528C2#1059642611yescharacteristic
529C2C4×D3×S411yescharacteristic
530C2C2×#6420411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×S4
commutatorC2×C3×A4C22×C4
FrattiniC22C2×D3×S4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radical, cosocleC22×C3×A4C23
hypocenterC3×A4C4×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
17. #64204C4direct (N × H)
43. C3×C4×A4Q8semidirect
147. C22×C3×C4Dic6semidirect
168. C4×A4Dic6semidirect
214. C3×A4C4×Q8semidirect
358. C22×C4#627semidirect
402. C22×C3C4×Dic6semidirect
405. A4C4×Dic6semidirect
415. C3×C4A4⋊Q8semidirect
505. C22C4×#627semidirect
520. C4#64204direct (N × H)
523. C3C4×A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#642042
C4×Dic3×A42
C4×C3⋊SL(2,ℤ/4)2
C3×C4×SL(2,ℤ/4)2
Dic3⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊Dic32
#105932382
C4×A4⋊Q83
C4×#8783
C4×#6274