Finite Group Databasebeta

← C3×A4×#124 #11919700

Subgroups — C3×A4×#124 #11919700

Order 1152.

1032 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 176 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×Q8×A4C211yesmaximal, characteristic, complemented
2C2×C3×D4×A4C211yesmaximal, characteristic
3C3×A4×C22⋊C4C214yesmaximal, complemented
4C3×C42×A4C211yesmaximal, characteristic, complemented
5C22×C3×#124C311yesmaximal, characteristic, complemented
6A4×#124C313yesmaximal, direct product factor
7C23×C3×A4C2212yes—
8C3×Q8×A4—22no—
9C3×D4×A4—22no—
10C2×C3×C4×A4C2212yes—
11C2×C3×C4×A4C2212yes—
12C2×C3×C4×A4C2211yescharacteristic
13C32×#124—41nomaximal
14C23×C3×Q8C2×C311yescharacteristic, complemented
15C23×C3×D4C2×C311yescharacteristic
16C2×Q8×A4C2×C313yescomplemented
17C2×D4×A4C2×C313yes—
18C2×C3×#124—31no—
19C2×C3×#124—31no—
20C2×C3×#124—34no—
21C2×C3×#124—31no—
22C2×C3×#124—31no—
23C22×C3×C22⋊C4C2×C314yescomplemented
24C22×C3×C42C2×C311yescharacteristic, complemented
25A4×C22⋊C4C2×C3112yescomplemented
26C42×A4C2×C313yescomplemented
27C22×C3×A4—22no—
28C22×C3×A4—24no—
29C22×C3×A4C2311yescharacteristic
30C2×C32×Q8—41no—
31C2×C32×D4—41no—
32C3×C4×A4—22no—
33C3×C4×A4—22no—
34C3×C4×A4D412yes—
35C32×C22⋊C4—44no—
36C32×C42—41no—
37C22×#124C3211yescharacteristic, complemented, Sylow
38C25×C3C22×C312yes—
39C23×A4C22×C316yes—
40C22×C3×Q8—22no—
41C22×C3×Q8—31no—
42C22×C3×Q8—32no—
43C22×C3×Q8—31no—
44C22×C3×Q8—62no—
45C22×C3×D4—22no—
46C22×C3×D4—31no—
47C22×C3×D4—31no—
48C22×C3×D4—32no—
49C22×C3×D4—62no—
50C23×C3×C4C22×C312yes—
51C23×C3×C4C22×C312yes—
52C23×C3×C4C22×C311yescharacteristic
53Q8×A4—26no—
54D4×A4—26no—
55C2×C4×A4C22×C316yes—
56C2×C4×A4C22×C316yes—
57C2×C4×A4C22×C313yes—
58C3×#124—34no—
59C3×#124—34no—
60C3×#124—32no—
61C3×#124—34no—
62C3×#124—31no—
63C3×#124—31no—
64C3×#124—34no—
65C3×#124—31no—
66C3×#124—43no—
67C3×#124A411yescharacteristic, direct product factor
68C2×C3×C22⋊C4—34no—
69C2×C3×C22⋊C4—34no—
70C2×C3×C22⋊C4—34no—
71C2×C3×C22⋊C4—34no—
72C2×C3×C42—31no—
73C2×C3×C42—31no—
74C2×C3×C42—32no—
75C23×C32—42no—
76C2×C3×A4—42no—
77C2×C3×A4C2×D411yescharacteristic
78C2×C3×A4#4112yes—
79C32×Q8—82no—
80C32×D4—82no—
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C32×C4—42no—
84C23×Q8C2×C3211yescharacteristic, complemented
85C23×D4C2×C3211yescharacteristic
86C2×#124—31no—
87C2×#124—31no—
88C2×#124—34no—
89C2×#124—31no—
90C2×#124—31no—
91C22×C22⋊C4C2×C3214yescomplemented
92C22×C42C2×C3211yescharacteristic, complemented
93C24×C3—22no—
94C24×C3—24no—
95C24×C3—32no—
96C24×C3—32no—
97C24×C3—64no—
98C24×C3—62no—
99C24×C3C23×C311yescharacteristic
100C22×A4—26no—
101C22×A4—212no—
102C22×A4C23×C313yes—
103C2×C3×Q8—32no—
104C2×C3×Q8—32no—
105C2×C3×Q8—31no—
106C2×C3×Q8—43no—
107C2×C3×Q8—62no—
108C2×C3×Q8—62no—
109C2×C3×Q8—62no—
110C2×C3×Q8—64no—
111C2×C3×Q8—62no—
112C2×C3×Q8—64no—
113C2×C3×Q8C2×A411yescharacteristic, complemented
114C2×C3×D4—32no—
115C2×C3×D4—32no—
116C2×C3×D4—31no—
117C2×C3×D4—43no—
118C2×C3×D4—62no—
119C2×C3×D4—62no—
120C2×C3×D4—64no—
121C2×C3×D4—62no—
122C2×C3×D4—64no—
123C2×C3×D4—62no—
124C2×C3×D4C2×A411yescharacteristic
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—32no—
130C22×C3×C4—32no—
131C22×C3×C4—32no—
132C22×C3×C4—32no—
133C22×C3×C4—32no—
134C22×C3×C4—62no—
135C22×C3×C4—62no—
136C22×C3×C4C3×D412yes—
137C4×A4—26no—
138C4×A4—26no—
139C4×A4C3×D416yes—
140C3×C22⋊C4—38no—
141C3×C22⋊C4—34no—
142C3×C22⋊C4—34no—
143C3×C22⋊C4—34no—
144C3×C22⋊C4—412no—
145C3×C22⋊C4C2×A414yescomplemented
146C3×C42—32no—
147C3×C42—32no—
148C3×C42—31no—
149C3×C42—43no—
150C3×C42C2×A411yescharacteristic, complemented
151C22×C32—41no—
152C22×C32—82no—
153C22×C32—84no—
154C3×A4#12411yescharacteristic, direct product factor
155C32×C4—42no—
156C32×C4—82no—
157C32×C4—82no—
158C25C22×C3212yes—
159C22×Q8—22no—
160C22×Q8—31no—
161C22×Q8—32no—
162C22×Q8—31no—
163C22×Q8—62no—
164C22×D4—22no—
165C22×D4—31no—
166C22×D4—31no—
167C22×D4—32no—
168C22×D4—62no—
169C23×C4C22×C3212yes—
170C23×C4C22×C3212yes—
171C23×C4C22×C3211yescharacteristic
172#124—34no—
173#124—34no—
174#124—32no—
175#124—34no—
176#124—31no—
177#124—31no—
178#124—34no—
179#124—31no—
180#124C3×A411yescharacteristic, direct product factor
181C2×C22⋊C4—34no—
182C2×C22⋊C4—34no—
183C2×C22⋊C4—34no—
184C2×C22⋊C4—34no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—32no—
188C23×C3—31no—
189C23×C3—32no—
190C23×C3—31no—
191C23×C3—32no—
192C23×C3—42no—
193C23×C3—46no—
194C23×C3—62no—
195C23×C3—62no—
196C23×C3—64no—
197C23×C3—64no—
198C23×C3—62no—
199C23×C3—62no—
200C23×C3—64no—
201C23×C3—64no—
202C23×C3—122no—
203C23×C3—124no—
204C23×C3—124no—
205C23×C3C2×C3×D411yescharacteristic
206C23×C3C3×#4112yes—
207C23×C3C22×A412yes—
208C2×A4—46no—
209C2×A4C2×C3×D413yes—
210C2×A4C3×#4116yes—
211C3×Q8—22no—
212C3×Q8—64no—
213C3×Q8—64no—
214C3×Q8—62no—
215C3×Q8—86no—
216C3×D4—22no—
217C3×D4—64no—
218C3×D4—64no—
219C3×D4—62no—
220C3×D4—86no—
221C2×C3×C4—32no—
222C2×C3×C4—31no—
223C2×C3×C4—32no—
224C2×C3×C4—32no—
225C2×C3×C4—32no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—43no—
229C2×C3×C4—46no—
230C2×C3×C4—46no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4C22×A412yes—
240C2×C3×C4C22×A412yes—
241C2×C3×C4C22×A411yescharacteristic
242C2×C32—41no—
243C2×C32—42no—
244C2×C32—162no—
245C24—22no—
246C24—24no—
247C24—32no—
248C24—32no—
249C24—62no—
250C24—64no—
251C24C23×C3211yescharacteristic
252C2×Q8—32no—
253C2×Q8—32no—
254C2×Q8—31no—
255C2×Q8—62no—
256C2×Q8—62no—
257C2×Q8—62no—
258C2×Q8—64no—
259C2×Q8—62no—
260C2×Q8—64no—
261C2×Q8C2×C3×A411yescharacteristic, complemented
262C2×D4—32no—
263C2×D4—32no—
264C2×D4—31no—
265C2×D4—62no—
266C2×D4—62no—
267C2×D4—64no—
268C2×D4—62no—
269C2×D4—64no—
270C2×D4—62no—
271C2×D4C2×C3×A411yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—32no—
279C22×C4—32no—
280C22×C4—32no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4C32×D412yes—
284C22⋊C4—38no—
285C22⋊C4—34no—
286C22⋊C4—34no—
287C22⋊C4—34no—
288C22⋊C4C2×C3×A414yescomplemented
289C42—32no—
290C42—32no—
291C42—31no—
292C42C2×C3×A411yescharacteristic, complemented
293C22×C3—22no—
294C22×C3—24no—
295C22×C3—31no—
296C22×C3—32no—
297C22×C3—32no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—43no—
303C22×C3—62no—
304C22×C3—64no—
305C22×C3—86no—
306C22×C3—812no—
307C22×C3—122no—
308C22×C3—122no—
309C22×C3—124no—
310C22×C3—122no—
311C22×C3—122no—
312C22×C3—124no—
313C22×C3C3×#12411yescharacteristic, complemented
314C22×C3C23×A411yescharacteristic
315A4C3×#12413yesdirect product factor
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—32no—
319C3×C4—46no—
320C3×C4—62no—
321C3×C4—62no—
322C3×C4—86no—
323C3×C4—86no—
324C3×C4D4×A412yes—
325C32—41noSylow
326C23—31no—
327C23—32no—
328C23—31no—
329C23—32no—
330C23—42no—
331C23—62no—
332C23—62no—
333C23—64no—
334C23—64no—
335C23—62no—
336C23—62no—
337C23—64no—
338C23—64no—
339C23—122no—
340C23—124no—
341C23—124no—
342C23C2×C32×D411yescharacteristic
343C23C32×#4112yes—
344C23C22×C3×A412yes—
345Q8—22no—
346Q8—64no—
347Q8—64no—
348Q8—62no—
349D4—22no—
350D4—64no—
351D4—64no—
352D4—62no—
353C2×C4—32no—
354C2×C4—31no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—62no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4C22×C3×A412yes—
369C2×C4C22×C3×A412yes—
370C2×C4C22×C3×A411yescharacteristic
371C2×C3—31no—
372C2×C3—32no—
373C2×C3—31no—
374C2×C3—42no—
375C2×C3—43no—
376C2×C3—46no—
377C2×C3—122no—
378C2×C3—166no—
379C2×C3C2×D4×A411yescharacteristic
380C2×C3A4×#4112yes—
381C22—22no—
382C22—24no—
383C22—31no—
384C22—32no—
385C22—32no—
386C22—31no—
387C22—31no—
388C22—32no—
389C22—32no—
390C22—62no—
391C22—64no—
392C22—122no—
393C22—122no—
394C22—124no—
395C22—122no—
396C22—122no—
397C22—124no—
398C22C32×#12411yescharacteristic, complemented
399C22C23×C3×A411yescharacteristic
400C4—22no—
401C4—22no—
402C4—32no—
403C4—62no—
404C4—62no—
405C4C3×D4×A412yes—
406C3—43no—
407C3A4×#12411yescharacteristic, direct product factor
408C2—31no—
409C2—31no—
410C2—32no—
411C2—42no—
412C2—122no—
413C2C2×C3×D4×A411yescharacteristic
414C2C3×A4×#4112yes—
subgroupG/H
centerC22×C3C23×A4
commutator, cosocleC24C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×#124C3
socleC24×C3C23×C3
prime radicalC2×C3×D4×A4C2
hypercenterC3×#124A4
hypocenterC22C32×#124
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
3. C3×A4×C22⋊C4C2semidirect
4. C3×C42×A4C2semidirect
5. C22×C3×#124C3semidirect
6. A4×#124C3direct (N × H)
14. C23×C3×Q8C2×C3semidirect
16. C2×Q8×A4C2×C3semidirect
23. C22×C3×C22⋊C4C2×C3semidirect
24. C22×C3×C42C2×C3semidirect
25. A4×C22⋊C4C2×C3semidirect
26. C42×A4C2×C3semidirect
37. C22×#124C32semidirect
67. C3×#124A4direct (N × H)
84. C23×Q8C2×C32semidirect
91. C22×C22⋊C4C2×C32semidirect
92. C22×C42C2×C32semidirect
113. C2×C3×Q8C2×A4semidirect
145. C3×C22⋊C4C2×A4semidirect
150. C3×C42C2×A4semidirect
154. C3×A4#124direct (N × H)
180. #124C3×A4direct (N × H)
261. C2×Q8C2×C3×A4semidirect
288. C22⋊C4C2×C3×A4semidirect
292. C42C2×C3×A4semidirect
313. C22×C3C3×#124semidirect
315. A4C3×#124direct (N × H)
398. C22C32×#124semidirect
407. C3A4×#124direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C2×C3×D4×A42
C3×A4×C22⋊C42
C3×C42×A42
C22×C3×#1243
A4×#1243
C32×#1244