Finite Group Databasebeta

← A4×#888 #11923162

Subgroups — A4×#888 #11923162

Order 1152.

794 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 111 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×A4×Dic6C212yesmaximal
2C3×A4×C4⋊C4C211yesmaximal, characteristic
3A4×C4⋊Dic3C211yesmaximal, characteristic
4A4×Dic3⋊C4C212yesmaximal
5C4×Dic3×A4C211yesmaximal, characteristic
6C22×#888C311yesmaximal, characteristic, complemented
7A4×C4⋊Q8—31nomaximal
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11C2×Dic3×A4C2211yescharacteristic
12C2×Dic3×A4C2211yescharacteristic
13A4×Dic6—24no—
14C3×#888—41nomaximal
15C23×Dic6C2×C312yes—
16C2×Q8×A4—32no—
17C22×C3×C4⋊C4C2×C311yescharacteristic
18C22×C4⋊Dic3C2×C311yescharacteristic
19C22×Dic3⋊C4C2×C312yes—
20C22×C4×Dic3C2×C311yescharacteristic
21C2×#888—31no—
22C2×#888—32no—
23C2×#888—31no—
24C2×#888—31no—
25C2×#888—32no—
26C2×#888—31no—
27A4×C4⋊C4—21no—
28A4×C4⋊C4—32no—
29A4×C4⋊C4—31no—
30A4×C4⋊C4D311yescharacteristic
31C42×A4—31no—
32C22×C3×A4C2311yescharacteristic
33C2×C3×Dic6—42no—
34C3×C4×A4—22no—
35C3×C4×A4Q812yescomplemented
36Dic3×A4—22no—
37Dic3×A4D412yes—
38Dic3×A4Q812yescomplemented
39C32×C4⋊C4—41no—
40C3×C4⋊Dic3—41no—
41C3×Dic3⋊C4—42no—
42C3×C4×Dic3—41no—
43C22×C4⋊Q8—31noSylow
44C23×C3×C4C22×C312yes—
45C23×C3×C4C22×C311yescharacteristic
46C23×Dic3C22×C312yes—
47C23×Dic3C22×C311yescharacteristic
48C23×Dic3C22×C311yescharacteristic
49C22×Dic6—24no—
50C22×Dic6—32no—
51C22×Dic6—32no—
52C22×Dic6—32no—
53C22×Dic6—32no—
54C22×Dic6—64no—
55Q8×A4—64no—
56C2×C4×A4—21no—
57C2×C4×A4—22no—
58C2×C4×A4—31no—
59C2×C4×A4—31no—
60C2×C4×A4—32no—
61C2×C4×A4C2×D312yes—
62C2×C4×A4C2×D311yescharacteristic
63C3×C4⋊Q8—121no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—32no—
67C2×C4⋊Dic3—31no—
68C2×C4⋊Dic3—31no—
69C2×C4⋊Dic3—32no—
70C2×Dic3⋊C4—32no—
71C2×Dic3⋊C4—32no—
72C2×Dic3⋊C4—32no—
73C2×Dic3⋊C4—32no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—31no—
76C2×C4×Dic3—31no—
77C2×C4×Dic3—31no—
78#888—34no—
79#888—34no—
80#888—32no—
81#888—32no—
82#888—32no—
83#888—32no—
84#888—31no—
85#888—31no—
86#888—32no—
87#888—31no—
88#888A411yescharacteristic, direct product factor
89C2×C3×A4C2×D411yescharacteristic
90C2×C3×A4C2×Q811yescharacteristic
91C2×C3×A4C2×Q811yescharacteristic
92C2×C32×C4—41no—
93C2×C32×C4—42no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—42no—
97C3×Dic6—84no—
98C23×Q8—32no—
99C2×C4⋊Q8—91no—
100C2×C4⋊Q8—91no—
101C2×C4⋊Q8—91no—
102C2×C4⋊Q8—92no—
103C2×C4⋊Q8—91no—
104C2×C4⋊Q8—92no—
105C22×C4⋊C4—32no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4C3×D311yescharacteristic
108C22×C42—31no—
109C24×C3C23×C311yescharacteristic
110C22×A4—21no—
111C22×A4C22×D311yescharacteristic
112C2×C3×Q8—122no—
113C22×C3×C4—22no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—32no—
117C22×C3×C4—32no—
118C22×C3×C4—32no—
119C22×C3×C4—62no—
120C22×C3×C4C3×Q812yescomplemented
121C22×Dic3—22no—
122C22×Dic3—31no—
123C22×Dic3—31no—
124C22×Dic3—31no—
125C22×Dic3—31no—
126C22×Dic3—32no—
127C22×Dic3—32no—
128C22×Dic3—32no—
129C22×Dic3—32no—
130C22×Dic3—62no—
131C22×Dic3C3×D412yes—
132C22×Dic3C3×Q812yescomplemented
133C2×Dic6—34no—
134C2×Dic6—32no—
135C2×Dic6—32no—
136C2×Dic6—32no—
137C2×Dic6—64no—
138C2×Dic6—64no—
139C2×Dic6—64no—
140C2×Dic6—64no—
141C2×Dic6—64no—
142C2×Dic6—64no—
143C2×Dic6—64no—
144C2×Dic6—64no—
145C2×Dic6C2×A412yes—
146C4×A4—22no—
147C4×A4—22no—
148C4×A4—32no—
149C4×A4—32no—
150C4×A4—42no—
151C4×A4—62no—
152C4×A4Dic612yescomplemented
153C3×C4⋊C4—32no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—81no—
158C3×C4⋊C4—122no—
159C3×C4⋊C4—121no—
160C3×C4⋊C4C2×A411yescharacteristic
161C3×C42—121no—
162C4⋊Dic3—32no—
163C4⋊Dic3—31no—
164C4⋊Dic3—32no—
165C4⋊Dic3C2×A411yescharacteristic
166Dic3⋊C4—34no—
167Dic3⋊C4—32no—
168Dic3⋊C4—32no—
169Dic3⋊C4—32no—
170Dic3⋊C4C2×A412yes—
171C4×Dic3—32no—
172C4×Dic3—31no—
173C4×Dic3—31no—
174C4×Dic3—31no—
175C4×Dic3C2×A411yescharacteristic
176C22×C32—41no—
177C3×A4C4⋊Q811yescharacteristic, complemented
178C32×C4—42no—
179C32×C4—82no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—82no—
183C22×Q8—64no—
184C22×Q8—92no—
185C22×Q8—92no—
186C22×Q8—92no—
187C22×Q8—92no—
188C22×Q8—184no—
189C23×C4—31no—
190C23×C4—31no—
191C23×C4—32no—
192C23×C4C2×C3×D312yes—
193C23×C4C2×C3×D311yescharacteristic
194C4⋊Q8—31no—
195C4⋊Q8—91no—
196C4⋊Q8—92no—
197C4⋊Q8—91no—
198C4⋊Q8—92no—
199C4⋊Q8—92no—
200C4⋊Q8—92no—
201C4⋊Q8—94no—
202C4⋊Q8—94no—
203C4⋊Q8—92no—
204C4⋊Q8—91no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—92no—
209C2×C4⋊C4—91no—
210C2×C4⋊C4—92no—
211C2×C4⋊C4—92no—
212C2×C4⋊C4—92no—
213C2×C4⋊C4—92no—
214C2×C4⋊C4—91no—
215C2×C42—91no—
216C2×C42—91no—
217C2×C42—91no—
218C2×C42—91no—
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—31no—
222C23×C3—31no—
223C23×C3C2×C3×D411yescharacteristic
224C23×C3C2×C3×Q811yescharacteristic
225C23×C3C2×C3×Q811yescharacteristic
226C2×A4—21no—
227C2×A4—21no—
228C2×A4—21no—
229C2×A4C2×Dic611yescharacteristic
230C2×A4D3×D411yescharacteristic
231C2×A4D3×Q811yescharacteristic
232C3×Q8—244no—
233C2×C3×C4—32no—
234C2×C3×C4—31no—
235C2×C3×C4—32no—
236C2×C3×C4—32no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—81no—
246C2×C3×C4—82no—
247C2×C3×C4—121no—
248C2×C3×C4—121no—
249C2×C3×C4—122no—
250C2×C3×C4C22×A412yes—
251C2×C3×C4C22×A411yescharacteristic
252C2×Dic3—32no—
253C2×Dic3—32no—
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—32no—
257C2×Dic3—32no—
258C2×Dic3—32no—
259C2×Dic3—32no—
260C2×Dic3—32no—
261C2×Dic3—32no—
262C2×Dic3—32no—
263C2×Dic3—62no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267C2×Dic3C22×A412yes—
268C2×Dic3C22×A411yescharacteristic
269C2×Dic3C22×A411yescharacteristic
270Dic6—24no—
271Dic6—64no—
272Dic6—64no—
273Dic6—68no—
274Dic6—64no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C24C22×C3×D311yescharacteristic
279C2×Q8—32no—
280C2×Q8—92no—
281C2×Q8—92no—
282C2×Q8—94no—
283C2×Q8—92no—
284C2×Q8—184no—
285C2×Q8—184no—
286C2×Q8—184no—
287C2×Q8—184no—
288C2×Q8—184no—
289C2×Q8—184no—
290C2×Q8—184no—
291C2×Q8—184no—
292C22×C4—22no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—91no—
303C22×C4—91no—
304C22×C4—92no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4—92no—
308C22×C4—92no—
309C22×C4—92no—
310C22×C4—182no—
311C22×C4C3×Dic612yescomplemented
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—31no—
315C4⋊C4—32no—
316C4⋊C4—31no—
317C4⋊C4—92no—
318C4⋊C4—92no—
319C4⋊C4—92no—
320C4⋊C4—94no—
321C4⋊C4—92no—
322C4⋊C4—92no—
323C4⋊C4—91no—
324C4⋊C4D3×A411yescharacteristic
325C42—31no—
326C42—91no—
327C42—91no—
328C42—92no—
329C42—91no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—32no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—41no—
341C22×C3—81no—
342C22×C3C3×C4⋊Q811yescharacteristic, complemented
343C22×C3C23×A411yescharacteristic
344A4—21no—
345A4#88811yescharacteristic, direct product factor
346C3×C4—22no—
347C3×C4—32no—
348C3×C4—42no—
349C3×C4—62no—
350C3×C4—82no—
351C3×C4—82no—
352C3×C4—122no—
353C3×C4—122no—
354C3×C4—162no—
355C3×C4—242no—
356C3×C4Q8×A412yescomplemented
357Dic3—22no—
358Dic3—32no—
359Dic3—32no—
360Dic3—62no—
361Dic3D4×A412yes—
362Dic3Q8×A412yescomplemented
363C32—41noSylow
364C23—31no—
365C23—31no—
366C23—31no—
367C23—31no—
368C23C2×C3×Dic611yescharacteristic
369C23C3×D3×D411yescharacteristic
370C23C3×D3×Q811yescharacteristic
371Q8—64no—
372Q8—184no—
373Q8—184no—
374Q8—184no—
375Q8—188no—
376C2×C4—32no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—92no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—92no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—92no—
398C2×C4—92no—
399C2×C4—92no—
400C2×C4—182no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4C2×D3×A412yes—
405C2×C4C2×D3×A411yescharacteristic
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—81no—
414C2×C3—81no—
415C2×C3—81no—
416C2×C3C2×D4×A411yescharacteristic
417C2×C3C2×Q8×A411yescharacteristic
418C2×C3C2×Q8×A411yescharacteristic
419C22—31no—
420C22—31no—
421C22—31no—
422C22—32no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—31no—
428C22—31no—
429C22C3×#88811yescharacteristic, complemented
430C22C22×D3×A411yescharacteristic
431C4—22no—
432C4—32no—
433C4—32no—
434C4—32no—
435C4—62no—
436C4—62no—
437C4—92no—
438C4—92no—
439C4—182no—
440C4A4×Dic612yescomplemented
441C3—41no—
442C3—81no—
443C3A4×C4⋊Q811yescharacteristic, complemented
444C2—31no—
445C2—31no—
446C2—31no—
447C2—31no—
448C2C2×A4×Dic611yescharacteristic
449C2D3×D4×A411yescharacteristic
450C2D3×Q8×A411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, socle, cosocleC24×C3C23×C3
FittingC22×C3×C4⋊C4C2×C3
prime radicalC22×C3×A4C23
hypercenterC4⋊C4D3×A4
hypocenterC22×C3C3×C4⋊Q8
LCSG ▷ C24×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C22×#888C3semidirect
35. C3×C4×A4Q8semidirect
38. Dic3×A4Q8semidirect
88. #888A4direct (N × H)
120. C22×C3×C4C3×Q8semidirect
132. C22×Dic3C3×Q8semidirect
152. C4×A4Dic6semidirect
177. C3×A4C4⋊Q8semidirect
311. C22×C4C3×Dic6semidirect
342. C22×C3C3×C4⋊Q8semidirect
345. A4#888direct (N × H)
356. C3×C4Q8×A4semidirect
362. Dic3Q8×A4semidirect
429. C22C3×#888semidirect
440. C4A4×Dic6semidirect
443. C3A4×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×A4×Dic62
C3×A4×C4⋊C42
A4×C4⋊Dic32
A4×Dic3⋊C42
C4×Dic3×A42
C22×#8883
A4×C4⋊Q83
C3×#8884