Finite Group Databasebeta

← C2×C3×#5409 #11922761

Subgroups — C2×C3×#5409 #11922761

Order 1152.

780 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22×C3×C4×A4C211yesmaximal, characteristic
2C3×#5409C214yesmaximal, direct product factor
3C2×C3×A4⋊C8C212yesmaximal, complemented
4C2×#5409C311yesmaximal, characteristic, direct product factor
5C2×C3×#407—31nomaximal
6C23×C3×A4C411yescharacteristic
7C2×C3×C4×A4C412yes—
8C2×C3×C4×A4C411yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×C4×A4C2212yes—
11C2×C3×#208—41nomaximal
12C3×A4⋊C8C2214yescomplemented
13C24×C3×C4D311yescharacteristic
14C22×C4×A4—21no—
15C22×C4×A4C2×C311yescharacteristic
16C22×C3×M16—31no—
17C22×C3×M16—31no—
18#5409C2×C314yesdirect product factor
19C2×A4⋊C8C2×C312yescomplemented
20C3×#407—34no—
21C3×#407—34no—
22C2×C3×C22⋊C8—32no—
23C2×C3×C22⋊C8—32no—
24C22×C3×A4—22no—
25C22×C3×A4C2×C412yes—
26C22×C3×A4C2×C411yescharacteristic
27C22×C32×C4—41no—
28C3×C4×A4C2×C411yescharacteristic
29C3×C4×A4C2×C412yes—
30C3×C4×A4C2311yescharacteristic
31C3×#208—44no—
32C2×C3×C3⋊C8—42no—
33C2×#407—31noSylow
34C25×C3Dic311yescharacteristic
35C23×A4—21no—
36C23×A4C3×C411yescharacteristic
37C23×C3×C4—31no—
38C23×C3×C4—31no—
39C23×C3×C4—31no—
40C23×C3×C4—32no—
41C23×C3×C4—31no—
42C23×C3×C4—32no—
43C23×C3×C4Dic312yes—
44C23×C3×C4Dic311yescharacteristic
45C23×C3×C4C2×D311yescharacteristic
46C23×C3×C4C2×D312yes—
47C2×C4×A4—22no—
48C2×C4×A4—21no—
49C2×C4×A4—21no—
50C2×C4×A4—22no—
51C2×C4×A4C3×C412yes—
52C2×C4×A4C3×C411yescharacteristic
53C2×C4×A4C22×C311yescharacteristic
54C2×C4×A4C22×C312yes—
55C2×C3×M16—34no—
56C2×C3×M16—34no—
57C2×C3×M16—61no—
58C2×C3×M16—61no—
59C2×C3×M16—62no—
60C2×C3×M16—62no—
61C2×C3×M16—61no—
62C2×C3×M16—61no—
63C22×C3×C8—32no—
64C22×C3×C8—32no—
65C2×#208—41no—
66A4⋊C8C22×C314yescomplemented
67C3×C22⋊C8—38no—
68C3×C22⋊C8—34no—
69C3×C22⋊C8—34no—
70C23×C32—41no—
71C2×C3×A4—22no—
72C2×C3×A4M1612yescomplemented
73C2×C3×A4C22×C411yescharacteristic
74C2×C32×C4—41no—
75C2×C32×C4—42no—
76C2×C32×C4—42no—
77C2×C32×C4—41no—
78C3×C3⋊C8—44no—
79C24×C4C3×D311yescharacteristic
80C22×M16—31no—
81C22×M16—31no—
82#407—34no—
83#407—34no—
84C2×C22⋊C8—32no—
85C2×C22⋊C8—32no—
86C24×C3—22no—
87C24×C3—31no—
88C24×C3—31no—
89C24×C3—32no—
90C24×C3—62no—
91C24×C3C2×Dic312yes—
92C24×C3C2×Dic311yescharacteristic
93C22×A4—24no—
94C22×A4—22no—
95C22×A4—22no—
96C22×A4—21no—
97C22×A4C2×C3×C412yes—
98C22×A4C2×C3×C411yescharacteristic
99C22×C3×C4—31no—
100C22×C3×C4—31no—
101C22×C3×C4—32no—
102C22×C3×C4—31no—
103C22×C3×C4—32no—
104C22×C3×C4—32no—
105C22×C3×C4—32no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—32no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—32no—
113C22×C3×C4—32no—
114C22×C3×C4—32no—
115C22×C3×C4—31no—
116C22×C3×C4—32no—
117C22×C3×C4—31no—
118C22×C3×C4—32no—
119C22×C3×C4—41no—
120C22×C3×C4—61no—
121C22×C3×C4—62no—
122C22×C3×C4—61no—
123C22×C3×C4—61no—
124C22×C3×C4—62no—
125C22×C3×C4—81no—
126C22×C3×C4C2×Dic311yescharacteristic
127C22×C3×C4C2×Dic312yes—
128C22×C3×C4S411yescharacteristic
129C22×C3×C4C22×D311yescharacteristic
130C4×A4—21no—
131C4×A4—22no—
132C4×A4—21no—
133C4×A4C2×C3×C411yescharacteristic
134C4×A4C2×C3×C412yes—
135C4×A4C23×C311yescharacteristic
136C3×M16—64no—
137C3×M16—64no—
138C3×M16—64no—
139C3×M16—64no—
140C2×C3×C8—34no—
141C2×C3×C8—34no—
142C2×C3×C8—62no—
143C2×C3×C8—62no—
144C2×C3×C8—62no—
145C2×C3×C8—62no—
146#208—44no—
147C2×C3⋊C8—42no—
148C22×C32—41no—
149C22×C32—42no—
150C22×C32—82no—
151C3×A4C2×M1611yescharacteristic, complemented
152C32×C4—42no—
153C32×C4—41no—
154C32×C4—41no—
155C25C3×Dic311yescharacteristic
156C23×C4—31no—
157C23×C4—31no—
158C23×C4—32no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—32no—
162C23×C4C3×Dic312yes—
163C23×C4C3×Dic311yescharacteristic
164C23×C4C2×C3×D311yescharacteristic
165C23×C4C2×C3×D312yes—
166C2×M16—34no—
167C2×M16—34no—
168C2×M16—61no—
169C2×M16—61no—
170C2×M16—62no—
171C2×M16—62no—
172C2×M16—61no—
173C2×M16—61no—
174C22×C8—32no—
175C22×C8—32no—
176C22⋊C8—38no—
177C22⋊C8—34no—
178C22⋊C8—34no—
179C23×C3—22no—
180C23×C3—31no—
181C23×C3—31no—
182C23×C3—32no—
183C23×C3—32no—
184C23×C3—34no—
185C23×C3—32no—
186C23×C3—32no—
187C23×C3—31no—
188C23×C3—41no—
189C23×C3—62no—
190C23×C3—62no—
191C23×C3—62no—
192C23×C3—61no—
193C23×C3—64no—
194C23×C3—62no—
195C23×C3—64no—
196C23×C3—81no—
197C23×C3#20812yescomplemented
198C23×C3SL(2,ℤ/4)11yescharacteristic
199C23×C3C22×Dic311yescharacteristic
200C2×A4—24no—
201C2×A4—22no—
202C2×A4—22no—
203C2×A4—21no—
204C2×A4C3×M1612yescomplemented
205C2×A4C22×C3×C411yescharacteristic
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—32no—
212C2×C3×C4—32no—
213C2×C3×C4—31no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—32no—
220C2×C3×C4—31no—
221C2×C3×C4—32no—
222C2×C3×C4—31no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—61no—
228C2×C3×C4—62no—
229C2×C3×C4—61no—
230C2×C3×C4—62no—
231C2×C3×C4—81no—
232C2×C3×C4—82no—
233C2×C3×C4—82no—
234C2×C3×C4—81no—
235C2×C3×C4SL(2,ℤ/4)11yescharacteristic
236C2×C3×C4SL(2,ℤ/4)12yes—
237C2×C3×C4C2×S412yes—
238C2×C3×C4C2×S411yescharacteristic
239C3×C8—64no—
240C3×C8—64no—
241C3⋊C8—44no—
242C2×C32—42no—
243C2×C32—41no—
244C2×C32—82no—
245C24—22no—
246C24—31no—
247C24—31no—
248C24—32no—
249C24—62no—
250C24C2×C3×Dic312yes—
251C24C2×C3×Dic311yescharacteristic
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—31no—
260C22×C4—32no—
261C22×C4—31no—
262C22×C4—32no—
263C22×C4—31no—
264C22×C4—32no—
265C22×C4—32no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—31no—
269C22×C4—32no—
270C22×C4—31no—
271C22×C4—32no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—62no—
277C22×C4C2×C3×Dic311yescharacteristic
278C22×C4C2×C3×Dic312yes—
279C22×C4C3×S411yescharacteristic
280C22×C4C22×C3×D311yescharacteristic
281M16—64no—
282M16—64no—
283M16—64no—
284M16—64no—
285C2×C8—34no—
286C2×C8—34no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—62no—
291C22×C3—22no—
292C22×C3—32no—
293C22×C3—31no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—34no—
299C22×C3—32no—
300C22×C3—41no—
301C22×C3—42no—
302C22×C3—62no—
303C22×C3—62no—
304C22×C3—64no—
305C22×C3—61no—
306C22×C3—62no—
307C22×C3—64no—
308C22×C3—62no—
309C22×C3—82no—
310C22×C3—84no—
311C22×C3—82no—
312C22×C3—81no—
313C22×C3C2×#20811yescharacteristic, complemented
314C22×C3C2×SL(2,ℤ/4)12yes—
315C22×C3C2×SL(2,ℤ/4)11yescharacteristic
316A4—21no—
317A4C2×C3×M1611yescharacteristic, complemented
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—32no—
321C3×C4—41no—
322C3×C4—42no—
323C3×C4—41no—
324C3×C4—81no—
325C3×C4—82no—
326C3×C4—81no—
327C3×C4C2×SL(2,ℤ/4)12yes—
328C3×C4C2×SL(2,ℤ/4)11yescharacteristic
329C3×C4C22×S411yescharacteristic
330C32—41noSylow
331C23—22no—
332C23—31no—
333C23—31no—
334C23—32no—
335C23—32no—
336C23—34no—
337C23—32no—
338C23—32no—
339C23—31no—
340C23—62no—
341C23—62no—
342C23—62no—
343C23—61no—
344C23—64no—
345C23—62no—
346C23—64no—
347C23C3×#20812yescomplemented
348C23C3×SL(2,ℤ/4)11yescharacteristic
349C23C22×C3×Dic311yescharacteristic
350C2×C4—31no—
351C2×C4—32no—
352C2×C4—32no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—32no—
356C2×C4—31no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—32no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—32no—
364C2×C4—31no—
365C2×C4—32no—
366C2×C4—31no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—61no—
370C2×C4—62no—
371C2×C4C3×SL(2,ℤ/4)12yes—
372C2×C4C3×SL(2,ℤ/4)11yescharacteristic
373C2×C4C2×C3×S411yescharacteristic
374C2×C4C2×C3×S412yes—
375C8—64no—
376C8—64no—
377C2×C3—22no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—32no—
381C2×C3—42no—
382C2×C3—41no—
383C2×C3—62no—
384C2×C3—82no—
385C2×C3—84no—
386C2×C3—82no—
387C2×C3—81no—
388C2×C3#540912yesdirect product factor
389C2×C3C22×SL(2,ℤ/4)11yescharacteristic
390C22—22no—
391C22—32no—
392C22—31no—
393C22—32no—
394C22—32no—
395C22—31no—
396C22—31no—
397C22—34no—
398C22—32no—
399C22—62no—
400C22—62no—
401C22—64no—
402C22—61no—
403C22—62no—
404C22—64no—
405C22—62no—
406C22C2×C3×#20811yescharacteristic, complemented
407C22C2×C3×SL(2,ℤ/4)12yes—
408C22C2×C3×SL(2,ℤ/4)11yescharacteristic
409C4—31no—
410C4—31no—
411C4—32no—
412C4C2×C3×SL(2,ℤ/4)11yescharacteristic
413C4C2×C3×SL(2,ℤ/4)12yes—
414C4C22×C3×S411yescharacteristic
415C3—41no—
416C3—81no—
417C3C2×#540911yescharacteristic, direct product factor
418C2—22no—
419C2—31no—
420C2—31no—
421C2—32no—
422C2—62no—
423C2C3×#540912yesdirect product factor
424C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutatorC2×A4C22×C3×C4
FrattiniC4C22×C3×S4
FittingC24×C3×C4D3
socleC24×C3C2×Dic3
prime radicalC23×C3×A4C4
cosocleC4×A4C23×C3
hypercenterC22×C3×C4S4
hypocenterA4C2×C3×M16
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#5409C2direct (N × H)
3. C2×C3×A4⋊C8C2semidirect
4. C2×#5409C3direct (N × H)
12. C3×A4⋊C8C22semidirect
18. #5409C2×C3direct (N × H)
19. C2×A4⋊C8C2×C3semidirect
66. A4⋊C8C22×C3semidirect
72. C2×C3×A4M16semidirect
151. C3×A4C2×M16semidirect
197. C23×C3#208semidirect
204. C2×A4C3×M16semidirect
313. C22×C3C2×#208semidirect
317. A4C2×C3×M16semidirect
347. C23C3×#208semidirect
388. C2×C3#5409direct (N × H)
406. C22C2×C3×#208semidirect
417. C3C2×#5409direct (N × H)
423. C2C3×#5409direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×A42
C3×#54092
C2×C3×A4⋊C82
C2×#54093
C2×C3×#4073
C2×C3×#2084