Finite Group Databasebeta

← C4×C4⋊(C3⋊S4) #11922837

Subgroups — C4×C4⋊(C3⋊S4) #11922837

Order 1152.

1054 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 123 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⋊S4)C211yesmaximal, characteristic
2C2×C4×C3⋊S4C212yesmaximal, complemented
3C3×C42×A4C211yesmaximal, characteristic, complemented
4#10593647C212yesmaximal, complemented
5C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
6C4×C4⋊S4—33nomaximal
7C4×#930—31nomaximal
8C22×C3⋊S4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3⋊SL(2,ℤ/4)C2212yes—
13C4⋊(C3⋊S4)C414yesdirect product factor
14C4×C3⋊S4—24no—
15C4×C4⋊(C3⋊D3)—41nomaximal
16C2×C4⋊S4—33no—
17C2×C4×S4—36no—
18C22×C3×C42D311yescharacteristic, complemented
19C2×#930—31no—
20C2×C4×#67—32no—
21C2×C4×D12—31no—
22C42×A4D313yescomplemented
23(C2×S4)⋊C4—36no—
24C4⋊SL(2,ℤ/4)—33no—
25#5213—32no—
26#5212—31no—
27#4940—32no—
28#4939—31no—
29C4×(C2×D3)⋊C4—32no—
30C4×C4⋊Dic3—31no—
31#4928—31no—
32C22×C3×A4C2311yescharacteristic
33C2×C3⋊S4—22no—
34C2×C3⋊S4—22no—
35C2×C3⋊S4C2×C414yescomplemented
36C2×C4⋊(C3⋊D3)—41no—
37C2×C4×C3⋊D3—42no—
38C3×C4×A4—21no—
39C3×C4×A4C2×C412yescomplemented
40C3×C4×A4D412yescomplemented
41C3⋊SL(2,ℤ/4)—22no—
42C32×C42—41no—
43#3752—42no—
44C4⋊(C3⋊Dic3)—41no—
45C4×#121—91noSylow
46C22×S4—36no—
47C23×C3×C4C2×D311yescharacteristic
48C23×C3×C4C2×D311yescharacteristic
49C23×C3×C4C2×D311yescharacteristic
50C22×#67—32no—
51C22×D12—31no—
52C22×C4×D3—32no—
53C2×C4×A4C2×D313yes—
54C2×C4×A4C2×D313yes—
55C2×C4×A4C2×D313yes—
56C2×SL(2,ℤ/4)—36no—
57C4⋊S4—312no—
58C4×S4—612no—
59C2×C3×C42—31no—
60C2×C3×C42—31no—
61C2×C3×C42—31no—
62C2×C3×C42—31no—
63C2×#217—32no—
64#930—34no—
65#930—34no—
66C4×#67—64no—
67C4×#67—64no—
68C2×(C2×D3)⋊C4—32no—
69C2×(C2×D3)⋊C4—32no—
70C2×(C2×D3)⋊C4—32no—
71C2×(C2×D3)⋊C4—32no—
72C2×C4⋊Dic3—31no—
73C2×C4⋊Dic3—31no—
74C2×C4⋊Dic3—31no—
75C2×Dic3⋊C4—32no—
76C2×C4×Dic3—32no—
77C4×D12—61no—
78C4×D12—61no—
79C4×D12—61no—
80C4×D12—61no—
81C4×D12—123no—
82C3⋊#95—32no—
83C3⋊#95—32no—
84C3⋊#95—32no—
85C22×C3⋊D3—42no—
86C2×C3×A4C22×C411yescharacteristic
87C2×C3×A4C2×D411yescharacteristic
88C2×C3×A4#4111yescharacteristic
89C3⋊S4—24no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4—41no—
93C2×C3⋊Dic3—42no—
94C4⋊(C3⋊D3)—44no—
95C4×C3⋊D3—84no—
96C2×#121—91no—
97C2×C4×D4—91no—
98C2×C4×D4—92no—
99C22×C42C3⋊D311yescharacteristic, complemented
100#390—91no—
101#389—91no—
102#388—92no—
103#386—92no—
104#380—91no—
105C4×C4⋊C4—91no—
106C4×C22⋊C4—92no—
107C24×C3C22×D311yescharacteristic
108C23×D3—32no—
109C22×A4C22×D313yes—
110C2×S4—312no—
111C2×S4—66no—
112C2×S4—66no—
113C22×C3×C4—21no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—32no—
120C22×C3×C4—32no—
121C22×C3×C4—31no—
122C22×C3×C4—61no—
123C22×C3×C4C4×D312yescomplemented
124C22×C3×C4D1212yescomplemented
125C2×#67—34no—
126C2×#67—34no—
127C2×#67—62no—
128C2×#67—62no—
129C2×#67—62no—
130C2×#67—62no—
131C22×Dic3—32no—
132C22×Dic3—32no—
133C22×Dic3—32no—
134C2×D12—34no—
135C2×D12—61no—
136C2×D12—62no—
137C2×D12—61no—
138C2×D12—123no—
139C2×C4×D3—62no—
140C2×C4×D3—64no—
141C2×C4×D3—64no—
142C2×C4×D3—62no—
143C2×C4×D3—126no—
144C4×A4—23no—
145C4×A4C4×D316yescomplemented
146C4×A4D1216yescomplemented
147SL(2,ℤ/4)—66no—
148C3×C42—31no—
149C3×C42—31no—
150C3×C42—31no—
151C3×C42—43no—
152C3×C42—61no—
153C3×C42S411yescharacteristic, complemented
154#217—62no—
155#217—62no—
156(C2×D3)⋊C4—38no—
157(C2×D3)⋊C4—62no—
158(C2×D3)⋊C4—64no—
159(C2×D3)⋊C4—64no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—126no—
162C4⋊Dic3—34no—
163C4⋊Dic3—61no—
164C4⋊Dic3—62no—
165C4⋊Dic3—61no—
166C4⋊Dic3—123no—
167Dic3⋊C4—64no—
168C4×Dic3—64no—
169C22×C32—41no—
170C2×C3⋊D3—44no—
171C2×C3⋊D3—82no—
172C2×C3⋊D3—82no—
173C3×A4C4×D411yescharacteristic, complemented
174C32×C4—42no—
175C32×C4—42no—
176C32×C4—81no—
177C3⋊Dic3—82no—
178C22×D4—91no—
179C22×D4—92no—
180C23×C4—92no—
181C23×C4C2×C3⋊D311yescharacteristic
182C23×C4C2×C3⋊D311yescharacteristic
183C23×C4C2×C3⋊D311yescharacteristic
184#121—94no—
185#121—94no—
186C4×D4—181no—
187C4×D4—181no—
188C4×D4—181no—
189C4×D4—184no—
190C4×D4—184no—
191C4×D4—181no—
192C2×C4⋊C4—91no—
193C2×C4⋊C4—91no—
194C2×C4⋊C4—92no—
195C2×C4⋊C4—91no—
196C2×C22⋊C4—92no—
197C2×C22⋊C4—92no—
198C2×C22⋊C4—92no—
199C2×C22⋊C4—92no—
200C2×C22⋊C4—92no—
201C2×C42—31no—
202C2×C42—31no—
203C2×C42—31no—
204C2×C42—31no—
205C2×C42—92no—
206#95—92no—
207#95—92no—
208#95—92no—
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—31no—
212C23×C3—31no—
213C23×C3C2×C4×D311yescharacteristic
214C23×C3C2×D1211yescharacteristic
215C23×C3#23511yescharacteristic
216C22×D3—34no—
217C22×D3—62no—
218C22×D3—62no—
219C22×D3—62no—
220C22×D3—62no—
221C22×D3—62no—
222C22×D3—62no—
223C22×D3—126no—
224C2×A4C2×C4×D313yes—
225C2×A4C2×D1213yes—
226C2×A4#23513yes—
227S4—612no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—32no—
236C2×C3×C4—32no—
237C2×C3×C4—31no—
238C2×C3×C4—43no—
239C2×C3×C4—43no—
240C2×C3×C4—43no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4C2×S411yescharacteristic
247C2×C3×C4C2×S411yescharacteristic
248C2×C3×C4C2×S411yescharacteristic
249#67—64no—
250#67—64no—
251#67—64no—
252#67—64no—
253C2×Dic3—34no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—126no—
263D12—64no—
264D12—64no—
265D12—1212no—
266C4×D3—124no—
267C4×D3—124no—
268C4×D3—2412no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C3⋊D3—84no—
273C24—92no—
274C24C22×C3⋊D311yescharacteristic
275C2×D4—94no—
276C2×D4—94no—
277C2×D4—94no—
278C2×D4—181no—
279C2×D4—182no—
280C2×D4—182no—
281C2×D4—182no—
282C2×D4—182no—
283C2×D4—182no—
284C2×D4—181no—
285C22×C4—21no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—31no—
291C22×C4—32no—
292C22×C4—32no—
293C22×C4—31no—
294C22×C4—61no—
295C22×C4—92no—
296C22×C4—92no—
297C22×C4—92no—
298C22×C4—182no—
299C22×C4—184no—
300C22×C4—184no—
301C22×C4—182no—
302C22×C4C4×C3⋊D312yescomplemented
303C22×C4C4⋊(C3⋊D3)12yescomplemented
304C4⋊C4—94no—
305C4⋊C4—184no—
306C4⋊C4—181no—
307C4⋊C4—182no—
308C4⋊C4—181no—
309C22⋊C4—98no—
310C22⋊C4—184no—
311C22⋊C4—182no—
312C22⋊C4—184no—
313C22⋊C4—182no—
314C22⋊C4—182no—
315C22⋊C4—182no—
316C42—31no—
317C42—31no—
318C42—31no—
319C42—61no—
320C42—184no—
321C42C3⋊S411yescharacteristic, complemented
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—43no—
332C22×C3—61no—
333C22×C3C4×D1211yescharacteristic, complemented
334C22×C3C22×S411yescharacteristic
335C2×D3—64no—
336C2×D3—64no—
337C2×D3—64no—
338C2×D3—1212no—
339C2×D3—122no—
340C2×D3—122no—
341C2×D3—122no—
342C2×D3—122no—
343C2×D3—246no—
344C2×D3—246no—
345A4C4×D1213yescomplemented
346C3×C4—21no—
347C3×C4—32no—
348C3×C4—32no—
349C3×C4—46no—
350C3×C4—46no—
351C3×C4—61no—
352C3×C4—83no—
353C3×C4C4×S412yescomplemented
354C3×C4C4⋊S412yescomplemented
355Dic3—64no—
356Dic3—122no—
357Dic3—122no—
358Dic3—246no—
359C32—41noSylow
360C23—31no—
361C23—31no—
362C23—31no—
363C23—31no—
364C23—94no—
365C23—182no—
366C23—182no—
367C23—182no—
368C23—182no—
369C23—182no—
370C23—182no—
371C23C2×C4×C3⋊D311yescharacteristic
372C23C2×C4⋊(C3⋊D3)11yescharacteristic
373C23#382811yescharacteristic
374D4—184no—
375D4—184no—
376D4—184no—
377D4—184no—
378D4—184no—
379D4—184no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—31no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—94no—
396C2×C4—182no—
397C2×C4—182no—
398C2×C4—182no—
399C2×C4—182no—
400C2×C4—182no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4—364no—
405C2×C4—364no—
406C2×C4C2×C3⋊S411yescharacteristic
407C2×C4C2×C3⋊S411yescharacteristic
408C2×C4C2×C3⋊S411yescharacteristic
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—43no—
414C2×C3—43no—
415C2×C3—43no—
416C2×C3C2×C4×S411yescharacteristic
417C2×C3C2×C4⋊S411yescharacteristic
418C2×C3#591211yescharacteristic
419D3—124no—
420D3—2412no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—31no—
428C22—31no—
429C22—31no—
430C22—61no—
431C22—184no—
432C22—184no—
433C22—184no—
434C22—362no—
435C22—362no—
436C22—362no—
437C22—362no—
438C22C4×C4⋊(C3⋊D3)11yescharacteristic, complemented
439C22C22×C3⋊S411yescharacteristic
440C4—21no—
441C4—32no—
442C4—32no—
443C4—61no—
444C4—184no—
445C4—362no—
446C4—362no—
447C4C4×C3⋊S412yescomplemented
448C4C4⋊(C3⋊S4)12yesdirect product factor
449C3—43no—
450C3C4×C4⋊S411yescharacteristic, complemented
451C2—31no—
452C2—31no—
453C2—31no—
454C2—31no—
455C2—364no—
456C2C2×C4×C3⋊S411yescharacteristic
457C2C2×C4⋊(C3⋊S4)11yescharacteristic
458C2#1059657511yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutatorC2×C3×A4C22×C4
FrattiniC22C22×C3⋊S4
FittingC22×C3×C42D3
socleC24×C3C22×D3
prime radicalC2×C4⋊(C3⋊S4)C2
cosocleC22×C3×A4C23
hypercenterC42C3⋊S4
hypocenterC3×A4C4×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊S4C2semidirect
3. C3×C42×A4C2semidirect
4. #10593647C2semidirect
5. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
13. C4⋊(C3⋊S4)C4direct (N × H)
18. C22×C3×C42D3semidirect
22. C42×A4D3semidirect
35. C2×C3⋊S4C2×C4semidirect
39. C3×C4×A4C2×C4semidirect
40. C3×C4×A4D4semidirect
99. C22×C42C3⋊D3semidirect
123. C22×C3×C4C4×D3semidirect
124. C22×C3×C4D12semidirect
145. C4×A4C4×D3semidirect
146. C4×A4D12semidirect
153. C3×C42S4semidirect
173. C3×A4C4×D4semidirect
302. C22×C4C4×C3⋊D3semidirect
303. C22×C4C4⋊(C3⋊D3)semidirect
321. C42C3⋊S4semidirect
333. C22×C3C4×D12semidirect
345. A4C4×D12semidirect
353. C3×C4C4×S4semidirect
354. C3×C4C4⋊S4semidirect
438. C22C4×C4⋊(C3⋊D3)semidirect
447. C4C4×C3⋊S4semidirect
448. C4C4⋊(C3⋊S4)direct (N × H)
450. 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⋊S4)2
C2×C4×C3⋊S42
C3×C42×A42
#105936472
C4⋊(C3⋊SL(2,ℤ/4))2
C4×C4⋊S43
C4×#9303
C4×C4⋊(C3⋊D3)4