Finite Group Databasebeta

← #11921983

Subgroups — #11921983

Order 1152.

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

normal subgroups only

This group has 908 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
1C2×C4⋊(C3⋊S4)C211yesmaximal, characteristic
2C2×(C3⋊S4)⋊C4C212yesmaximal, complemented
3A4×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5#10593242C212yesmaximal, complemented
6#85308—31nomaximal
7#73467—31nomaximal
8C22×C3⋊S4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11C4⋊(C3⋊S4)C414yescomplemented
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C2×C3×SL(2,ℤ/4)C2211yescharacteristic
14(C3⋊S4)⋊C4—24no—
15#63894—41nomaximal
16C2×C4⋊S4—31no—
17C2×C4⋊S4—61no—
18C2×C4×S4—32no—
19C2×#930—31no—
20C22×C4⋊Dic3D311yescharacteristic, complemented
21C2×#67⋊C4—32no—
22C2×C4×D12—31no—
23A4×C4⋊C4—31no—
24(C2×S4)⋊C4—32no—
25C4×SL(2,ℤ/4)D311yescharacteristic, complemented
26C3×C4×C22⋊C4—31no—
27#4958—32no—
28#4939—31no—
29C4×C4⋊Dic3—31no—
30#4928—31no—
31#4670—32no—
32#4668—32no—
33C22×C3×A4C2311yescharacteristic
34C2×C3⋊S4—22no—
35C2×C3⋊S4—22no—
36C2×C3⋊S4C2×C414yescomplemented
37C2×C4⋊(C3⋊D3)—41no—
38C3×C4×A4C2×C412yescomplemented
39C2×#624—42no—
40Dic3×A4—22no—
41C3×SL(2,ℤ/4)—21no—
42C3×SL(2,ℤ/4)D412yescomplemented
43C3×C4⋊Dic3—41no—
44C3×C4×Dic3—41no—
45#3722—42no—
46#2361—91noSylow
47C22×S4—32no—
48C22×S4—62no—
49C23×C3×C4C2×D311yescharacteristic
50C22×#67—32no—
51C23×Dic3C2×D312yes—
52C22×D12—31no—
53C22×C4×D3—32no—
54C2×C4×A4—21no—
55C2×C4×A4—32no—
56C2×C4×A4C2×D311yescharacteristic
57C2×SL(2,ℤ/4)C2×D311yescharacteristic
58C2×SL(2,ℤ/4)C2×D311yescharacteristic
59C4⋊S4—34no—
60C4⋊S4—64no—
61C4×S4—64no—
62C2×C3×C22⋊C4—31no—
63C2×C3×C22⋊C4—31no—
64C2×C3×C42—31no—
65C2×C3×C42—31no—
66#930—34no—
67#930—34no—
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—32no—
75C2×C4⋊Dic3—31no—
76C2×Dic3⋊C4—32no—
77C2×C4×Dic3—32no—
78#893—121no—
79#67⋊C4—64no—
80#67⋊C4—64no—
81C4×D12—61no—
82C4×D12—62no—
83C4×D12—61no—
84C3×#95—31no—
85C3×#95—31no—
86C3⋊#95—32no—
87C3⋊#95—32no—
88C22×C3⋊D3—42no—
89C2×C3×A4C22×C411yescharacteristic
90C2×C3×A4C2×D411yescharacteristic
91C2×C3×A4#4111yescharacteristic
92C3⋊S4—24no—
93C2×C32×C4—41no—
94C4⋊(C3⋊D3)—44no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—42no—
98#624—84no—
99C2×#121—91no—
100C2×C4×D4—91no—
101C2×C4×D4—92no—
102C22×C4⋊C4—31no—
103#390—91no—
104#389—91no—
105#388—92no—
106#386—92no—
107#385—92no—
108C4×C4⋊C4—91no—
109C4×C22⋊C4—31no—
110C24×C3C22×D311yescharacteristic
111C23×D3—32no—
112C22×A4—21no—
113C22×A4C22×D311yescharacteristic
114C2×S4—34no—
115C2×S4—64no—
116C2×S4—62no—
117C2×S4—62no—
118C2×S4—122no—
119C2×S4—122no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—32no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4C4×D312yescomplemented
128C2×#67—34no—
129C2×#67—34no—
130C2×#67—62no—
131C2×#67—62no—
132C2×#67—62no—
133C2×#67—62no—
134C22×Dic3—22no—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—62no—
139C2×D12—34no—
140C2×D12—61no—
141C2×D12—62no—
142C2×D12—61no—
143C2×D12—121no—
144C2×D12—241no—
145C2×C4×D3—62no—
146C2×C4×D3—64no—
147C2×C4×D3—62no—
148C2×C4×D3—122no—
149C2×C4×D3—122no—
150C4×A4—22no—
151C4×A4—62no—
152C4×A4C4×D312yescomplemented
153SL(2,ℤ/4)—21no—
154SL(2,ℤ/4)D1212yescomplemented
155C3×C4⋊C4—121no—
156C3×C22⋊C4—32no—
157C3×C22⋊C4—32no—
158C3×C22⋊C4—61no—
159C3×C22⋊C4—61no—
160C3×C42—61no—
161C3×C42—62no—
162C3×C42—61no—
163(C2×D3)⋊C4—38no—
164(C2×D3)⋊C4—62no—
165(C2×D3)⋊C4—64no—
166(C2×D3)⋊C4—62no—
167(C2×D3)⋊C4—122no—
168(C2×D3)⋊C4—122no—
169C4⋊Dic3—34no—
170C4⋊Dic3—31no—
171C4⋊Dic3—32no—
172C4⋊Dic3—61no—
173C4⋊Dic3S411yescharacteristic, complemented
174Dic3⋊C4—64no—
175C4×Dic3—41no—
176C4×Dic3—64no—
177C22×C32—41no—
178C2×C3⋊D3—44no—
179C2×C3⋊D3—82no—
180C2×C3⋊D3—82no—
181C3×A4C4×D411yescharacteristic, complemented
182C32×C4—42no—
183C3×Dic3—42no—
184C3×Dic3—82no—
185C3×Dic3—81no—
186C22×D4—91no—
187C22×D4—92no—
188C23×C4—32no—
189C23×C4—92no—
190C23×C4D3211yescharacteristic
191#121—94no—
192#121—94no—
193C4×D4—181no—
194C4×D4—182no—
195C4×D4—181no—
196C4×D4—184no—
197C4×D4—184no—
198C2×C4⋊C4—91no—
199C2×C4⋊C4—91no—
200C2×C4⋊C4—92no—
201C2×C4⋊C4—91no—
202C2×C4⋊C4—92no—
203C2×C22⋊C4—31no—
204C2×C22⋊C4—31no—
205C2×C22⋊C4—92no—
206C2×C22⋊C4—92no—
207C2×C22⋊C4—92no—
208C2×C22⋊C4—92no—
209C2×C42—31no—
210C2×C42—31no—
211C2×C42—92no—
212#95—31no—
213#95—31no—
214#95—92no—
215#95—92no—
216C23×C3—31no—
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—31no—
220C23×C3C2×C4×D311yescharacteristic
221C23×C3D3×D411yescharacteristic
222C23×C3#23911yescharacteristic
223C22×D3—34no—
224C22×D3—62no—
225C22×D3—62no—
226C22×D3—62no—
227C22×D3—62no—
228C22×D3—62no—
229C22×D3—62no—
230C22×D3—122no—
231C22×D3—242no—
232C2×A4—21no—
233C2×A4—21no—
234C2×A4—21no—
235C2×A4C2×C4×D311yescharacteristic
236C2×A4C2×D1211yescharacteristic
237C2×A4#23511yescharacteristic
238S4—64no—
239S4—124no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—41no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—81no—
259C2×C3×C4—122no—
260C2×C3×C4C2×S411yescharacteristic
261#67—64no—
262#67—64no—
263#67—64no—
264#67—64no—
265C2×Dic3—34no—
266C2×Dic3—32no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3C2×S412yes—
276D12—64no—
277D12—64no—
278D12—124no—
279D12—244no—
280C4×D3—124no—
281C4×D3—124no—
282C4×D3—244no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C3⋊D3—84no—
287C24—92no—
288C24C2×D3211yescharacteristic
289C2×D4—94no—
290C2×D4—94no—
291C2×D4—94no—
292C2×D4—181no—
293C2×D4—182no—
294C2×D4—182no—
295C2×D4—182no—
296C2×D4—182no—
297C2×D4—182no—
298C2×D4—181no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—32no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—62no—
307C22×C4—92no—
308C22×C4—92no—
309C22×C4—92no—
310C22×C4—182no—
311C22×C4—184no—
312C22×C4—182no—
313C22×C4—182no—
314C22×C4—362no—
315C22×C4#62412yescomplemented
316C4⋊C4—31no—
317C4⋊C4—92no—
318C4⋊C4—94no—
319C4⋊C4—91no—
320C4⋊C4—181no—
321C4⋊C4—184no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—98no—
327C22⋊C4—182no—
328C22⋊C4—184no—
329C22⋊C4—182no—
330C22⋊C4—362no—
331C42—61no—
332C42—62no—
333C42—61no—
334C42—184no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—31no—
344C22×C3—41no—
345C22×C3—61no—
346C22×C3—81no—
347C22×C3#89311yescharacteristic, complemented
348C22×C3C22×S411yescharacteristic
349C2×D3—64no—
350C2×D3—64no—
351C2×D3—64no—
352C2×D3—124no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—244no—
358C2×D3—242no—
359C2×D3—242no—
360C2×D3—482no—
361C2×D3—482no—
362A4—21no—
363A4C4×D1211yescharacteristic, complemented
364C3×C4—32no—
365C3×C4—42no—
366C3×C4—62no—
367C3×C4—62no—
368C3×C4—82no—
369C3×C4—121no—
370C3×C4—121no—
371C3×C4—242no—
372C3×C4C4×S412yescomplemented
373Dic3—22no—
374Dic3—42no—
375Dic3—64no—
376Dic3—62no—
377Dic3—81no—
378C32—41noSylow
379C23—31no—
380C23—31no—
381C23—31no—
382C23—31no—
383C23—94no—
384C23—182no—
385C23—182no—
386C23—182no—
387C23—182no—
388C23—182no—
389C23—182no—
390C23#379911yescharacteristic
391C23D3×D1211yescharacteristic
392C23C2×#62411yescharacteristic
393D4—184no—
394D4—184no—
395D4—184no—
396D4—184no—
397D4—184no—
398D4—184no—
399C2×C4—31no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—94no—
418C2×C4—92no—
419C2×C4—182no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4—182no—
425C2×C4—364no—
426C2×C4—364no—
427C2×C4D3×S411yescharacteristic
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—81no—
436C2×C3—81no—
437C2×C3—81no—
438C2×C3C2×C4×S411yescharacteristic
439C2×C3D4×S411yescharacteristic
440C2×C3#591911yescharacteristic
441D3—124no—
442D3—244no—
443D3—484no—
444C22—31no—
445C22—31no—
446C22—31no—
447C22—31no—
448C22—31no—
449C22—31no—
450C22—31no—
451C22—31no—
452C22—31no—
453C22—61no—
454C22—184no—
455C22—184no—
456C22—184no—
457C22—362no—
458C22—362no—
459C22—362no—
460C22—362no—
461C22#6389411yescharacteristic, complemented
462C22C2×D3×S411yescharacteristic
463C4—32no—
464C4—62no—
465C4—62no—
466C4—62no—
467C4—121no—
468C4—121no—
469C4—184no—
470C4—182no—
471C4(C3⋊S4)⋊C412yescomplemented
472C3—41no—
473C3—81no—
474C3#8530811yescharacteristic, complemented
475C2—31no—
476C2—31no—
477C2—31no—
478C2—31no—
479C2—364no—
480C2#1059642111yescharacteristic
481C2D12×S411yescharacteristic
482C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC2×C4⋊(C3⋊S4)C2
cosocleC22×C3×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4C4×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊S4)⋊C4C2semidirect
3. A4×C4⋊Dic3C2semidirect
4. C3×C4×SL(2,ℤ/4)C2semidirect
5. #10593242C2semidirect
11. C4⋊(C3⋊S4)C4semidirect
20. C22×C4⋊Dic3D3semidirect
25. C4×SL(2,ℤ/4)D3semidirect
36. C2×C3⋊S4C2×C4semidirect
38. C3×C4×A4C2×C4semidirect
42. C3×SL(2,ℤ/4)D4semidirect
127. C22×C3×C4C4×D3semidirect
152. C4×A4C4×D3semidirect
154. SL(2,ℤ/4)D12semidirect
173. C4⋊Dic3S4semidirect
181. C3×A4C4×D4semidirect
315. C22×C4#624semidirect
347. C22×C3#893semidirect
363. A4C4×D12semidirect
372. C3×C4C4×S4semidirect
461. C22#63894semidirect
471. C4(C3⋊S4)⋊C4semidirect
474. C3#85308semidirect

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×(C3⋊S4)⋊C42
A4×C4⋊Dic32
C3×C4×SL(2,ℤ/4)2
#105932422
#853083
#734673
#638944