Finite Group Databasebeta

← A4×#834 #11921358

Subgroups — A4×#834 #11921358

Order 1152.

598 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 63 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×C3×D4×A4C211yesmaximal, characteristic
2A4×#217C212yesmaximal, complemented
3C22×#834C311yesmaximal, characteristic, complemented
4A4×#99—31nomaximal
5C23×C3×A4C411yescharacteristic, complemented
6C23×C3×A4C2211yescharacteristic
7C3×D4×A4—22no—
8C2×C3×C4×A4C411yescharacteristic, complemented
9C2×Dic3×A4—22no—
10C3×#834—41nomaximal
11C23×C3×D4C2×C311yescharacteristic
12C2×D4×A4—21no—
13C2×D4×A4D311yescharacteristic
14C22×#217C2×C312yescomplemented
15A4×C22⋊C4—32no—
16C2×#834—31no—
17C2×#834—31no—
18C2×#834—32no—
19C22×C3×A4—21no—
20C22×C3×A4—41no—
21C22×C3×A4—41no—
22C22×C3×A4C2×C411yescharacteristic
23C22×C3×A4D412yes—
24C2×C32×D4—41no—
25C3×C4×A4—21no—
26Dic3×A4—42no—
27C3×#217—42no—
28C22×#99—31noSylow
29C25×C3C3×C411yescharacteristic, complemented
30C25×C3C22×C311yescharacteristic
31C23×A4—21no—
32C23×A4—21no—
33C23×A4Dic311yescharacteristic, complemented
34C23×A4C2×D311yescharacteristic
35C22×C3×D4—22no—
36C22×C3×D4—31no—
37C22×C3×D4—31no—
38C22×C3×D4—31no—
39C22×C3×D4—31no—
40C22×C3×D4—62no—
41C23×C3×C4C3×C411yescharacteristic, complemented
42C23×Dic3—22no—
43D4×A4—24no—
44D4×A4—22no—
45C2×C4×A4—21no—
46C2×C4×A4—62no—
47C2×C4×A4Dic311yescharacteristic, complemented
48C2×#217—32no—
49C2×#217—32no—
50C2×#217—62no—
51C3×#99—121no—
52#834—32no—
53#834—31no—
54#834—32no—
55#834A411yescharacteristic, direct product factor
56C23×C32—41no—
57C23×C32—41no—
58C2×C3×A4—21no—
59C2×C3×A4—22no—
60C2×C3×A4—41no—
61C2×C3×A4C22⋊C411yescharacteristic
62C32×D4—82no—
63C2×C32×C4—41no—
64C2×C3×Dic3—82no—
65C23×D4C3×D311yescharacteristic
66C22×C22⋊C4—32no—
67C2×#99—91no—
68C2×#99—92no—
69C2×#99—91no—
70C24×C3—21no—
71C24×C3—32no—
72C24×C3—31no—
73C24×C3—31no—
74C24×C3—31no—
75C24×C3—31no—
76C24×C3—41no—
77C24×C3—41no—
78C24×C3—61no—
79C24×C3—121no—
80C24×C3—121no—
81C24×C3C2×C3×C411yescharacteristic
82C24×C3C3×D412yes—
83C22×A4—22no—
84C22×A4—21no—
85C22×A4—21no—
86C22×A4—22no—
87C22×A4—41no—
88C22×A4—41no—
89C22×A4—42no—
90C22×A4—42no—
91C22×A4C2×Dic311yescharacteristic
92C22×A4#6712yes—
93C2×C3×D4—32no—
94C2×C3×D4—31no—
95C2×C3×D4—31no—
96C2×C3×D4—31no—
97C2×C3×D4—41no—
98C2×C3×D4—62no—
99C2×C3×D4—62no—
100C2×C3×D4—62no—
101C2×C3×D4—62no—
102C2×C3×D4—62no—
103C2×C3×D4—62no—
104C2×C3×D4—62no—
105C2×C3×D4—62no—
106C2×C3×D4—81no—
107C2×C3×D4C2×A411yescharacteristic
108C22×C3×C4—21no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—61no—
112C22×Dic3—42no—
113C22×Dic3—62no—
114C22×Dic3—62no—
115C22×Dic3—122no—
116C4×A4—22no—
117C4×A4—21no—
118C4×A4—122no—
119C3×C22⋊C4—122no—
120#217—32no—
121#217—62no—
122#217—62no—
123#217C2×A412yescomplemented
124C22×C32—41no—
125C22×C32—42no—
126C22×C32—81no—
127C22×C32—161no—
128C22×C32—161no—
129C3×A4#9911yescharacteristic, complemented
130C32×C4—81no—
131C3×Dic3—162no—
132C25C3×Dic311yescharacteristic, complemented
133C25C2×C3×D311yescharacteristic
134C22×D4—22no—
135C22×D4—31no—
136C22×D4—31no—
137C22×D4—31no—
138C22×D4—31no—
139C22×D4—62no—
140C23×C4—62no—
141C23×C4C3×Dic311yescharacteristic, complemented
142C2×C22⋊C4—92no—
143C2×C22⋊C4—92no—
144C2×C22⋊C4—182no—
145#99—31no—
146#99—92no—
147#99—92no—
148#99—91no—
149C23×C3—22no—
150C23×C3—21no—
151C23×C3—32no—
152C23×C3—31no—
153C23×C3—32no—
154C23×C3—31no—
155C23×C3—32no—
156C23×C3—31no—
157C23×C3—32no—
158C23×C3—31no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—61no—
163C23×C3—61no—
164C23×C3—62no—
165C23×C3—61no—
166C23×C3—61no—
167C23×C3—61no—
168C23×C3—81no—
169C23×C3—81no—
170C23×C3—121no—
171C23×C3—121no—
172C23×C3—121no—
173C23×C3—121no—
174C23×C3—121no—
175C23×C3—122no—
176C23×C3—121no—
177C23×C3—122no—
178C23×C3—121no—
179C23×C3—121no—
180C23×C3—121no—
181C23×C3—122no—
182C23×C3—122no—
183C23×C3C3×C22⋊C411yescharacteristic
184C23×C3C4×A411yescharacteristic, complemented
185C23×C3C22×A411yescharacteristic
186C2×A4—22no—
187C2×A4—22no—
188C2×A4—21no—
189C2×A4—21no—
190C2×A4—42no—
191C2×A4—42no—
192C2×A4—41no—
193C2×A4#21711yescharacteristic
194C3×D4—22no—
195C3×D4—62no—
196C3×D4—62no—
197C3×D4—64no—
198C3×D4—62no—
199C3×D4—82no—
200C3×D4—84no—
201C2×C3×C4—31no—
202C2×C3×C4—41no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—81no—
208C2×C3×C4—242no—
209C2×C3×C4C4×A411yescharacteristic, complemented
210C2×Dic3—22no—
211C2×Dic3—62no—
212C2×Dic3—122no—
213C2×Dic3—122no—
214C2×Dic3—122no—
215C2×Dic3—122no—
216C2×C32—41no—
217C2×C32—81no—
218C2×C32—82no—
219C2×C32—161no—
220C24—21no—
221C24—32no—
222C24—31no—
223C24—31no—
224C24—31no—
225C24—31no—
226C24—41no—
227C24—41no—
228C24—61no—
229C24—121no—
230C24—121no—
231C24C2×C3×Dic311yescharacteristic
232C24C3×#6712yes—
233C2×D4—32no—
234C2×D4—31no—
235C2×D4—31no—
236C2×D4—31no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—62no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4D3×A411yescharacteristic
246C22×C4—21no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—61no—
250C22×C4—122no—
251C22×C4—182no—
252C22×C4—182no—
253C22×C4—362no—
254C22⋊C4—32no—
255C22⋊C4—92no—
256C22⋊C4—182no—
257C22⋊C4—182no—
258C22×C3—21no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—32no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—42no—
267C22×C3—62no—
268C22×C3—62no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3—62no—
272C22×C3—61no—
273C22×C3—62no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—81no—
277C22×C3—82no—
278C22×C3—81no—
279C22×C3—82no—
280C22×C3—122no—
281C22×C3—121no—
282C22×C3—121no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3—121no—
286C22×C3—121no—
287C22×C3—122no—
288C22×C3—122no—
289C22×C3—122no—
290C22×C3—161no—
291C22×C3—161no—
292C22×C3—162no—
293C22×C3—162no—
294C22×C3C3×#9911yescharacteristic, complemented
295C22×C3C2×C4×A411yescharacteristic
296C22×C3D4×A412yes—
297A4—21no—
298A4#83411yescharacteristic, direct product factor
299C3×C4—21no—
300C3×C4—61no—
301C3×C4—81no—
302C3×C4—82no—
303C3×C4—482no—
304Dic3—42no—
305Dic3—122no—
306C32—41noSylow
307C23—22no—
308C23—21no—
309C23—32no—
310C23—31no—
311C23—32no—
312C23—31no—
313C23—32no—
314C23—31no—
315C23—32no—
316C23—31no—
317C23—41no—
318C23—61no—
319C23—61no—
320C23—62no—
321C23—61no—
322C23—61no—
323C23—61no—
324C23—121no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23—121no—
329C23—122no—
330C23—121no—
331C23—122no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—122no—
336C23—122no—
337C23C3×#21711yescharacteristic
338C23Dic3×A411yescharacteristic, complemented
339C23C2×D3×A411yescharacteristic
340D4—22no—
341D4—62no—
342D4—62no—
343D4—64no—
344D4—62no—
345C2×C4—31no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—62no—
351C2×C4—182no—
352C2×C4—362no—
353C2×C4—362no—
354C2×C4—362no—
355C2×C4—362no—
356C2×C4Dic3×A411yescharacteristic, complemented
357C2×C3—22no—
358C2×C3—21no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—62no—
364C2×C3—61no—
365C2×C3—82no—
366C2×C3—81no—
367C2×C3—82no—
368C2×C3—81no—
369C2×C3—121no—
370C2×C3—162no—
371C2×C3—161no—
372C2×C3—162no—
373C2×C3A4×C22⋊C411yescharacteristic
374C22—21no—
375C22—31no—
376C22—31no—
377C22—31no—
378C22—32no—
379C22—41no—
380C22—41no—
381C22—62no—
382C22—62no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22—62no—
387C22—61no—
388C22—62no—
389C22—61no—
390C22—122no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—121no—
396C22—121no—
397C22—122no—
398C22—122no—
399C22—122no—
400C22C3×#83411yescharacteristic, complemented
401C22C2×Dic3×A411yescharacteristic
402C22A4×#6712yes—
403C4—21no—
404C4—61no—
405C4—122no—
406C4—362no—
407C3—41no—
408C3—81no—
409C3A4×#9911yescharacteristic, complemented
410C2—22no—
411C2—21no—
412C2—31no—
413C2—31no—
414C2—41no—
415C2—61no—
416C2—62no—
417C2—121no—
418C2A4×#21711yescharacteristic
subgroupG/H
centerC2A4×#217
commutatorC24×C3C2×C3×C4
FrattiniC23C2×D3×A4
FittingC23×C3×D4C2×C3
socleC23×C3C3×C22⋊C4
prime radicalC2×C3×D4×A4C2
cosocleC25×C3C22×C3
hypercenterC2×D4D3×A4
hypocenterC22×C3C3×#99
LCSG ▷ C24×C3 ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. A4×#217C2semidirect
3. C22×#834C3semidirect
5. C23×C3×A4C4semidirect
8. C2×C3×C4×A4C4semidirect
14. C22×#217C2×C3semidirect
29. C25×C3C3×C4semidirect
33. C23×A4Dic3semidirect
41. C23×C3×C4C3×C4semidirect
47. C2×C4×A4Dic3semidirect
55. #834A4direct (N × H)
123. #217C2×A4semidirect
129. C3×A4#99semidirect
132. C25C3×Dic3semidirect
141. C23×C4C3×Dic3semidirect
184. C23×C3C4×A4semidirect
209. C2×C3×C4C4×A4semidirect
294. C22×C3C3×#99semidirect
298. A4#834direct (N × H)
338. C23Dic3×A4semidirect
356. C2×C4Dic3×A4semidirect
400. C22C3×#834semidirect
409. C3A4×#99semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
A4×#2172
C22×#8343
A4×#993
C3×#8344