Finite Group Databasebeta

← C4⋊C4×C3⋊S4 #11922856

Subgroups — C4⋊C4×C3⋊S4 #11922856

Order 1152.

1022 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 115 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×C4×C3⋊S4C212yesmaximal
2C2×C4×C3⋊S4C211yesmaximal, characteristic
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
5#10593645C212yesmaximal, complemented
6C4⋊C4×S4—33nomaximal
7C4⋊C4×#67—31nomaximal
8C22×C3⋊S4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
13C4×C3⋊S4—24no—
14C4×C3⋊S4C414yescomplemented
15C4⋊C4×C3⋊D3—41nomaximal
16C2×C4×S4—33no—
17C2×C4×S4—36no—
18C22×C3×C4⋊C4D311yescharacteristic, complemented
19C2×C4×#67—31no—
20C2×C4×#67—32no—
21C2×D3×C4⋊C4—31no—
22A4×C4⋊C4D313yescomplemented
23C4⋊SL(2,ℤ/4)—33no—
24SL(2,ℤ/4)⋊C4—36no—
25#5212—31no—
26#5210—32no—
27#4989—32no—
28#4987—31no—
29#4976—32no—
30#4975—31no—
31Dic3×C4⋊C4—31no—
32C22×C3×A4C2311yescharacteristic
33C2×C3⋊S4C2×C412yes—
34C2×C3⋊S4D412yes—
35C2×C3⋊S4Q812yes—
36C2×C4×C3⋊D3—41no—
37C2×C4×C3⋊D3—42no—
38C3×C4×A4—22no—
39C3×C4×A4C2×C412yescomplemented
40C3⋊SL(2,ℤ/4)—22no—
41C3⋊SL(2,ℤ/4)C2×C412yescomplemented
42C32×C4⋊C4—41no—
43C4⋊(C3⋊Dic3)—41no—
44#3750—42no—
45D4×C4⋊C4—91noSylow
46C22×S4—33no—
47C23×C3×C4C2×D312yes—
48C23×C3×C4C2×D311yescharacteristic
49C22×#67—31no—
50C22×C4×D3—31no—
51C22×C4×D3—32no—
52C2×C4×A4C2×D316yes—
53C2×C4×A4C2×D313yes—
54C2×SL(2,ℤ/4)—33no—
55C2×SL(2,ℤ/4)—36no—
56C4×S4—312no—
57C4×S4—612no—
58C2×C3×C4⋊C4—31no—
59C2×C3×C4⋊C4—31no—
60C2×C3×C4⋊C4—32no—
61C2×#217—31no—
62C2×#217—32no—
63C4×#67—34no—
64C4×#67—34no—
65C4×#67—64no—
66C4×#67—64no—
67C2×(C2×D3)⋊C4—31no—
68C2×(C2×D3)⋊C4—32no—
69C2×C4⋊Dic3—31no—
70C2×C4⋊Dic3—31no—
71C2×Dic3⋊C4—31no—
72C2×Dic3⋊C4—32no—
73C2×Dic3⋊C4—32no—
74C2×Dic3⋊C4—32no—
75C2×C4×Dic3—31no—
76C2×C4×Dic3—32no—
77D3×C4⋊C4—61no—
78D3×C4⋊C4—62no—
79D3×C4⋊C4—61no—
80D3×C4⋊C4—123no—
81C3⋊#95—32no—
82C3⋊#95—32no—
83C3⋊#95—32no—
84C22×C3⋊D3—41no—
85C2×C3×A4C22×C411yescharacteristic
86C2×C3×A4C2×D411yescharacteristic
87C2×C3×A4C2×Q811yescharacteristic
88C3⋊S4C4⋊C414yesdirect product factor
89C2×C32×C4—41no—
90C2×C32×C4—42no—
91C2×C3⋊Dic3—41no—
92C2×C3⋊Dic3—42no—
93C4×C3⋊D3—44no—
94C4×C3⋊D3—84no—
95C2×C4×D4—91no—
96C2×C4×D4—92no—
97C22×C4⋊C4—91no—
98C22×C4⋊C4C3⋊D311yescharacteristic, complemented
99#389—92no—
100#385—92no—
101#385—92no—
102#384—91no—
103#380—91no—
104#380—91no—
105C4×C4⋊C4—91no—
106C24×C3C22×D311yescharacteristic
107C23×D3—31no—
108C22×A4C22×D313yes—
109C2×S4—36no—
110C2×S4—36no—
111C2×S4—36no—
112C22×C3×C4—22no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C22×C3×C4—32no—
118C22×C3×C4—62no—
119C22×C3×C4C4×D312yescomplemented
120C2×#67—32no—
121C2×#67—32no—
122C2×#67—32no—
123C2×#67—32no—
124C2×#67—32no—
125C2×#67—32no—
126C22×Dic3—31no—
127C22×Dic3—31no—
128C22×Dic3—31no—
129C22×Dic3—32no—
130C22×Dic3—32no—
131C2×C4×D3—34no—
132C2×C4×D3—61no—
133C2×C4×D3—62no—
134C2×C4×D3—61no—
135C2×C4×D3—62no—
136C2×C4×D3—64no—
137C2×C4×D3—62no—
138C2×C4×D3—123no—
139C2×C4×D3—126no—
140C2×C4×D3—122no—
141C4×A4—26no—
142C4×A4C4×D316yescomplemented
143SL(2,ℤ/4)—36no—
144SL(2,ℤ/4)—66no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—43no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4S411yescharacteristic, complemented
150#217—32no—
151#217—32no—
152#217—62no—
153#217—62no—
154(C2×D3)⋊C4—34no—
155(C2×D3)⋊C4—64no—
156C4⋊Dic3—61no—
157C4⋊Dic3—62no—
158C4⋊Dic3—61no—
159C4⋊Dic3—123no—
160Dic3⋊C4—34no—
161Dic3⋊C4—64no—
162Dic3⋊C4—62no—
163Dic3⋊C4—62no—
164Dic3⋊C4—126no—
165Dic3⋊C4—122no—
166C4×Dic3—34no—
167C4×Dic3—64no—
168C22×C32—41no—
169C2×C3⋊D3—42no—
170C2×C3⋊D3—42no—
171C2×C3⋊D3—42no—
172C3×A4C2×C4⋊C411yescharacteristic, complemented
173C32×C4—42no—
174C32×C4—82no—
175C3⋊Dic3—42no—
176C3⋊Dic3—82no—
177C22×D4—91no—
178C23×C4—91no—
179C23×C4—92no—
180C23×C4C2×C3⋊D312yes—
181C23×C4C2×C3⋊D311yescharacteristic
182C4×D4—94no—
183C4×D4—94no—
184C4×D4—184no—
185C4×D4—184no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—91no—
190C2×C4⋊C4—91no—
191C2×C4⋊C4—92no—
192C2×C4⋊C4—92no—
193C2×C4⋊C4—91no—
194C2×C4⋊C4—92no—
195C2×C4⋊C4—181no—
196C2×C4⋊C4—182no—
197C2×C4⋊C4—181no—
198C2×C22⋊C4—91no—
199C2×C22⋊C4—91no—
200C2×C22⋊C4—92no—
201C2×C22⋊C4—92no—
202C2×C42—91no—
203C2×C42—92no—
204#95—92no—
205#95—92no—
206#95—92no—
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—31no—
210C23×C3—31no—
211C23×C3C2×C4×D311yescharacteristic
212C23×C3D3×D411yescharacteristic
213C23×C3D3×Q811yescharacteristic
214C22×D3—32no—
215C22×D3—32no—
216C22×D3—32no—
217C22×D3—61no—
218C22×D3—61no—
219C22×D3—61no—
220C22×D3—61no—
221C22×D3—123no—
222C2×A4C2×C4×D313yes—
223C2×A4D3×D413yes—
224C2×A4D3×Q813yes—
225S4—312no—
226C2×C3×C4—31no—
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—43no—
232C2×C3×C4—46no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—61no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4C2×S412yes—
239C2×C3×C4C2×S411yescharacteristic
240#67—34no—
241#67—34no—
242#67—34no—
243#67—34no—
244C2×Dic3—32no—
245C2×Dic3—32no—
246C2×Dic3—32no—
247C2×Dic3—32no—
248C2×Dic3—32no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—123no—
260C2×Dic3—126no—
261C4×D3—64no—
262C4×D3—64no—
263C4×D3—1212no—
264C4×D3—124no—
265C4×D3—124no—
266C4×D3—2412no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C3⋊D3—44no—
271C24—91no—
272C24C22×C3⋊D311yescharacteristic
273C2×D4—92no—
274C2×D4—92no—
275C2×D4—92no—
276C2×D4—92no—
277C2×D4—92no—
278C2×D4—92no—
279C22×C4—22no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—32no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—62no—
286C22×C4—94no—
287C22×C4—91no—
288C22×C4—91no—
289C22×C4—91no—
290C22×C4—92no—
291C22×C4—92no—
292C22×C4—181no—
293C22×C4—182no—
294C22×C4—182no—
295C22×C4—184no—
296C22×C4—181no—
297C22×C4—182no—
298C22×C4—362no—
299C22×C4C4×C3⋊D312yescomplemented
300C4⋊C4—32no—
301C4⋊C4—31no—
302C4⋊C4—61no—
303C4⋊C4—94no—
304C4⋊C4—182no—
305C4⋊C4—181no—
306C4⋊C4—182no—
307C4⋊C4—184no—
308C4⋊C4—182no—
309C4⋊C4—181no—
310C4⋊C4—362no—
311C4⋊C4C3⋊S411yescharacteristic, direct product factor
312C22⋊C4—94no—
313C22⋊C4—92no—
314C22⋊C4—92no—
315C22⋊C4—184no—
316C22⋊C4—182no—
317C22⋊C4—182no—
318C42—94no—
319C42—184no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—43no—
330C22×C3—61no—
331C22×C3D3×C4⋊C411yescharacteristic, complemented
332C22×C3C22×S411yescharacteristic
333C2×D3—34no—
334C2×D3—62no—
335C2×D3—62no—
336C2×D3—62no—
337C2×D3—62no—
338C2×D3—62no—
339C2×D3—62no—
340C2×D3—126no—
341C2×D3—126no—
342C2×D3—126no—
343A4D3×C4⋊C413yescomplemented
344C3×C4—22no—
345C3×C4—32no—
346C3×C4—46no—
347C3×C4—62no—
348C3×C4—86no—
349C3×C4C4×S412yescomplemented
350Dic3—34no—
351Dic3—62no—
352Dic3—62no—
353Dic3—126no—
354Dic3—122no—
355Dic3—122no—
356Dic3—246no—
357C32—41noSylow
358C23—31no—
359C23—31no—
360C23—31no—
361C23—31no—
362C23—92no—
363C23—92no—
364C23—92no—
365C23—181no—
366C23—181no—
367C23—181no—
368C23—181no—
369C23C2×C4×C3⋊D311yescharacteristic
370C23D4×C3⋊D311yescharacteristic
371C23Q8×C3⋊D311yescharacteristic
372D4—94no—
373D4—94no—
374D4—94no—
375D4—94no—
376C2×C4—31no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—61no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—92no—
390C2×C4—92no—
391C2×C4—184no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—184no—
397C2×C4—182no—
398C2×C4—182no—
399C2×C4—182no—
400C2×C4—182no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—364no—
404C2×C4—364no—
405C2×C4C2×C3⋊S412yes—
406C2×C4C2×C3⋊S411yescharacteristic
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—43no—
412C2×C3—43no—
413C2×C3—43no—
414C2×C3C2×C4×S411yescharacteristic
415C2×C3D4×S411yescharacteristic
416C2×C3Q8×S411yescharacteristic
417D3—64no—
418D3—1212no—
419C22—31no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—31no—
428C22—61no—
429C22—94no—
430C22—182no—
431C22—182no—
432C22—182no—
433C22—182no—
434C22—182no—
435C22—182no—
436C22C4⋊C4×C3⋊D311yescharacteristic, complemented
437C22C22×C3⋊S411yescharacteristic
438C4—22no—
439C4—32no—
440C4—62no—
441C4—94no—
442C4—182no—
443C4—182no—
444C4—362no—
445C4—362no—
446C4C4×C3⋊S412yescomplemented
447C3—43no—
448C3C4⋊C4×S411yescharacteristic, complemented
449C2—31no—
450C2—31no—
451C2—31no—
452C2—31no—
453C2—184no—
454C2C2×C4×C3⋊S411yescharacteristic
455C2D4×C3⋊S411yescharacteristic
456C2Q8×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutatorC2×C3×A4C22×C4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3⋊S4C22
cosocleC22×C3×A4C23
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4C2×C4⋊C4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×A4×C4⋊C4C2semidirect
4. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
5. #10593645C2semidirect
14. C4×C3⋊S4C4semidirect
18. C22×C3×C4⋊C4D3semidirect
22. A4×C4⋊C4D3semidirect
39. C3×C4×A4C2×C4semidirect
41. C3⋊SL(2,ℤ/4)C2×C4semidirect
88. C3⋊S4C4⋊C4direct (N × H)
98. C22×C4⋊C4C3⋊D3semidirect
119. C22×C3×C4C4×D3semidirect
142. C4×A4C4×D3semidirect
149. C3×C4⋊C4S4semidirect
172. C3×A4C2×C4⋊C4semidirect
299. C22×C4C4×C3⋊D3semidirect
311. C4⋊C4C3⋊S4direct (N × H)
331. C22×C3D3×C4⋊C4semidirect
343. A4D3×C4⋊C4semidirect
349. C3×C4C4×S4semidirect
436. C22C4⋊C4×C3⋊D3semidirect
446. C4C4×C3⋊S4semidirect
448. C3C4⋊C4×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊S42
C3×A4×C4⋊C42
C4⋊(C3⋊SL(2,ℤ/4))2
#105936452
C4⋊C4×S43
C4⋊C4×#673
C4⋊C4×C3⋊D34