Finite Group Databasebeta

← C3×#85496 #11922776

Subgroups — C3×#85496 #11922776

Order 1152.

1200 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 96 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, complemented
2C2×C3×GL(2,ℤ/4)C214yesmaximal, complemented
3C2×C3×C4⋊S4C211yesmaximal, characteristic, complemented
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5#85496C311yesmaximal, characteristic, direct product factor
6C3×#2433—31nomaximal
7C23×C3×A4C2212yes—
8C22×C3×S4C2212yes—
9C3×D4×A4—22no—
10C2×C3×C4×A4C2211yescharacteristic
11C3×GL(2,ℤ/4)—28no—
12C2×C3×SL(2,ℤ/4)C2212yes—
13C3×C4⋊S4—22no—
14C3×#940—41nomaximal
15C23×C3×D4D311yescharacteristic, complemented
16C2×D4×A4—21no—
17C2×D4×A4C2×C311yescharacteristic, complemented
18C2×GL(2,ℤ/4)C2×C314yescomplemented
19C2×C4⋊S4C2×C311yescharacteristic, complemented
20C2×C3×#127—31no—
21C2×C3×#127—31no—
22C2×C3×#121—31no—
23C2×C3×#120—34no—
24C4×SL(2,ℤ/4)C2×C311yescharacteristic, complemented
25C3×#392—34no—
26C3×#390—31no—
27C3×#390—31no—
28C3×C4×C22⋊C4—31no—
29C22×C3×A4—22no—
30C22×C3×A4—24no—
31C22×C3×A4C2311yescharacteristic
32C2×C3×S4—24no—
33C2×C3×S4—22no—
34C2×C32×D4—41no—
35C2×C3×#67—44no—
36C2×C3×D12—41no—
37C3×C4×A4D412yescomplemented
38C3×SL(2,ℤ/4)D414yescomplemented
39C3×C4×Dic3—41no—
40#2433—31noSylow
41C25×C3C2×D312yes—
42C23×A4—22no—
43C23×A4C22×C312yes—
44C22×S4C22×C312yes—
45C22×C3×D4—22no—
46C22×C3×D4—31no—
47C22×C3×D4—31no—
48C22×C3×D4—32no—
49C22×C3×D4—31no—
50C22×C3×D4—32no—
51C22×C3×D4—34no—
52C22×C3×D4—34no—
53C22×C3×D4—62no—
54C23×C3×C4C2×D311yescharacteristic
55D4×A4—24no—
56D4×A4—22no—
57C2×C4×A4—21no—
58C2×C4×A4C22×C311yescharacteristic
59GL(2,ℤ/4)—28no—
60C2×SL(2,ℤ/4)C22×C312yes—
61C4⋊S4—22no—
62C3×#127—64no—
63C3×#127—62no—
64C3×#127—61no—
65C3×#127—61no—
66C3×#121—62no—
67C3×#121—62no—
68C3×#120—68no—
69C3×#120—68no—
70C2×C3×C4⋊C4—31no—
71C2×C3×C22⋊C4—32no—
72C2×C3×C22⋊C4—34no—
73C2×C3×C22⋊C4—34no—
74C2×C3×C22⋊C4—32no—
75C2×C3×C42—31no—
76C2×C3×C42—31no—
77#940—41no—
78C3×#95—32no—
79C23×C32—42no—
80C22×C3×D3—42no—
81C2×C3×A4—42no—
82C2×C3×A4C2×D412yes—
83C2×C3×A4C2×D411yescharacteristic
84C3×S4—42no—
85C32×D4—82no—
86C2×C32×C4—41no—
87C3×#67—88no—
88C2×C3×Dic3—42no—
89C3×D12—82no—
90C23×D4C3×D311yescharacteristic, complemented
91C2×#127—31no—
92C2×#127—31no—
93C2×#121—31no—
94C2×#120—34no—
95#392—34no—
96#390—31no—
97#390—31no—
98C4×C22⋊C4—31no—
99C24×C3—24no—
100C24×C3—22no—
101C24×C3—32no—
102C24×C3—32no—
103C24×C3—32no—
104C24×C3—62no—
105C24×C3—64no—
106C24×C3C22×D311yescharacteristic
107C22×A4—24no—
108C22×A4—24no—
109C22×A4—22no—
110C22×A4—21no—
111C22×A4—44no—
112C22×A4C23×C311yescharacteristic
113C2×S4—22no—
114C2×S4—24no—
115C2×C3×D4—34no—
116C2×C3×D4—32no—
117C2×C3×D4—31no—
118C2×C3×D4—41no—
119C2×C3×D4—62no—
120C2×C3×D4—62no—
121C2×C3×D4—61no—
122C2×C3×D4—62no—
123C2×C3×D4—64no—
124C2×C3×D4—68no—
125C2×C3×D4—68no—
126C2×C3×D4—62no—
127C2×C3×D4—61no—
128C2×C3×D4—62no—
129C2×C3×D4—64no—
130C2×C3×D4—64no—
131C2×C3×D4—62no—
132C2×C3×D4—68no—
133C2×C3×D4—64no—
134C2×C3×D4—64no—
135C2×C3×D4—62no—
136C2×C3×D4—64no—
137C2×C3×D4—61no—
138C2×C3×D4—81no—
139C2×C3×D4—124no—
140C2×C3×D4—124no—
141C2×C3×D4S411yescharacteristic, complemented
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—32no—
145C22×C3×C4—32no—
146C22×C3×C4—32no—
147C22×C3×C4—32no—
148C22×C3×C4#6712yescomplemented
149C2×#67—44no—
150C2×D12—41no—
151C4×A4—22no—
152C4×A4C3×D412yescomplemented
153SL(2,ℤ/4)C3×D414yescomplemented
154C3×C4⋊C4—62no—
155C3×C22⋊C4—34no—
156C3×C22⋊C4—34no—
157C3×C22⋊C4—68no—
158C3×C22⋊C4—68no—
159C3×C22⋊C4—64no—
160C3×C42—61no—
161C3×C42—62no—
162C3×C42—61no—
163C4×Dic3—41no—
164C22×C32—41no—
165C22×C32—82no—
166C22×C32—84no—
167C2×C3×D3—82no—
168C2×C3×D3—84no—
169C3×A4#12711yescharacteristic, complemented
170C32×C4—42no—
171C3×Dic3—44no—
172C25C2×C3×D312yes—
173C22×D4—22no—
174C22×D4—31no—
175C22×D4—31no—
176C22×D4—32no—
177C22×D4—31no—
178C22×D4—32no—
179C22×D4—34no—
180C22×D4—34no—
181C22×D4—62no—
182C23×C4C2×C3×D311yescharacteristic
183#127—64no—
184#127—62no—
185#127—61no—
186#127—61no—
187#121—62no—
188#121—62no—
189#120—68no—
190#120—68no—
191C2×C4⋊C4—31no—
192C2×C22⋊C4—32no—
193C2×C22⋊C4—34no—
194C2×C22⋊C4—34no—
195C2×C22⋊C4—32no—
196C2×C42—31no—
197C2×C42—31no—
198#95—32no—
199C23×C3—31no—
200C23×C3—32no—
201C23×C3—31no—
202C23×C3—34no—
203C23×C3—32no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—64no—
207C23×C3—62no—
208C23×C3—62no—
209C23×C3—64no—
210C23×C3—64no—
211C23×C3—64no—
212C23×C3—62no—
213C23×C3—62no—
214C23×C3—64no—
215C23×C3—62no—
216C23×C3—64no—
217C23×C3—62no—
218C23×C3—82no—
219C23×C3—124no—
220C23×C3—122no—
221C23×C3—124no—
222C23×C3D3×D412yes—
223C23×C3C2×#6711yescharacteristic
224C23×C3C2×S412yes—
225C22×D3—42no—
226C2×A4—22no—
227C2×A4—21no—
228C2×A4—44no—
229C2×A4—42no—
230C2×A4C2×C3×D412yes—
231C2×A4C2×C3×D411yescharacteristic
232S4—42no—
233C3×D4—22no—
234C3×D4—64no—
235C3×D4—64no—
236C3×D4—62no—
237C3×D4—82no—
238C3×D4—84no—
239C3×D4—122no—
240C3×D4—128no—
241C3×D4—1216no—
242C3×D4—128no—
243C3×D4—124no—
244C3×D4—122no—
245C3×D4—122no—
246C3×D4—122no—
247C2×C3×C4—31no—
248C2×C3×C4—32no—
249C2×C3×C4—32no—
250C2×C3×C4—32no—
251C2×C3×C4—34no—
252C2×C3×C4—34no—
253C2×C3×C4—41no—
254C2×C3×C4—61no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—64no—
260C2×C3×C4—62no—
261C2×C3×C4—81no—
262C2×C3×C4C2×S411yescharacteristic
263#67—88no—
264C2×Dic3—42no—
265D12—82no—
266C2×C32—42no—
267C2×C32—41no—
268C2×C32—162no—
269C3×D3—162no—
270C24—24no—
271C24—22no—
272C24—32no—
273C24—32no—
274C24—32no—
275C24—62no—
276C24—64no—
277C24C22×C3×D311yescharacteristic
278C2×D4—34no—
279C2×D4—32no—
280C2×D4—31no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—61no—
284C2×D4—62no—
285C2×D4—64no—
286C2×D4—68no—
287C2×D4—68no—
288C2×D4—62no—
289C2×D4—61no—
290C2×D4—62no—
291C2×D4—64no—
292C2×D4—64no—
293C2×D4—62no—
294C2×D4—68no—
295C2×D4—64no—
296C2×D4—64no—
297C2×D4—64no—
298C2×D4—62no—
299C2×D4—61no—
300C2×D4—124no—
301C2×D4—124no—
302C2×D4C3×S411yescharacteristic, complemented
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—32no—
309C22×C4C3×#6712yescomplemented
310C4⋊C4—62no—
311C22⋊C4—34no—
312C22⋊C4—34no—
313C22⋊C4—68no—
314C22⋊C4—68no—
315C22⋊C4—64no—
316C42—61no—
317C42—62no—
318C42—61no—
319C22×C3—24no—
320C22×C3—22no—
321C22×C3—32no—
322C22×C3—31no—
323C22×C3—32no—
324C22×C3—32no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—41no—
328C22×C3—61no—
329C22×C3—68no—
330C22×C3—64no—
331C22×C3—62no—
332C22×C3—84no—
333C22×C3—82no—
334C22×C3—84no—
335C22×C3—81no—
336C22×C3—122no—
337C22×C3—124no—
338C22×C3—122no—
339C22×C3—122no—
340C22×C3—122no—
341C22×C3—124no—
342C22×C3—122no—
343C22×C3—122no—
344C22×C3—124no—
345C22×C3—122no—
346C22×C3—164no—
347C22×C3#94011yescharacteristic, complemented
348C22×C3C22×S411yescharacteristic
349C2×D3—82no—
350C2×D3—84no—
351A4—21no—
352A4C3×#12711yescharacteristic, complemented
353C3×C4—32no—
354C3×C4—42no—
355C3×C4—64no—
356C3×C4—64no—
357C3×C4—82no—
358C3×C4—122no—
359C3×C4GL(2,ℤ/4)12yescomplemented
360Dic3—44no—
361C32—41noSylow
362C23—31no—
363C23—32no—
364C23—31no—
365C23—34no—
366C23—32no—
367C23—42no—
368C23—64no—
369C23—62no—
370C23—62no—
371C23—64no—
372C23—64no—
373C23—64no—
374C23—62no—
375C23—62no—
376C23—64no—
377C23—62no—
378C23—64no—
379C23—62no—
380C23—124no—
381C23—122no—
382C23—124no—
383C23C3×D3×D412yes—
384C23C2×C3×#6711yescharacteristic
385C23C2×C3×S412yes—
386D4—22no—
387D4—64no—
388D4—64no—
389D4—62no—
390D4—122no—
391D4—128no—
392D4—1216no—
393D4—128no—
394D4—124no—
395D4—122no—
396D4—122no—
397D4—122no—
398C2×C4—31no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—34no—
403C2×C4—34no—
404C2×C4—61no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—64no—
410C2×C4—62no—
411C2×C4C2×C3×S411yescharacteristic
412C2×C3—32no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—42no—
416C2×C3—42no—
417C2×C3—41no—
418C2×C3—82no—
419C2×C3—81no—
420C2×C3—122no—
421C2×C3—162no—
422C2×C3—164no—
423C2×C3—242no—
424C2×C3D4×S412yes—
425C2×C3C2×GL(2,ℤ/4)11yescharacteristic
426D3—162no—
427C22—24no—
428C22—22no—
429C22—32no—
430C22—31no—
431C22—32no—
432C22—32no—
433C22—31no—
434C22—31no—
435C22—61no—
436C22—68no—
437C22—64no—
438C22—62no—
439C22—122no—
440C22—124no—
441C22—122no—
442C22—122no—
443C22—122no—
444C22—124no—
445C22—122no—
446C22—122no—
447C22—124no—
448C22—122no—
449C22C3×#94011yescharacteristic, complemented
450C22C22×C3×S411yescharacteristic
451C4—32no—
452C4—64no—
453C4—64no—
454C4—122no—
455C4C3×GL(2,ℤ/4)12yescomplemented
456C3—41no—
457C3—81no—
458C3#8549611yescharacteristic, direct product factor
459C2—32no—
460C2—31no—
461C2—31no—
462C2—42no—
463C2—122no—
464C2—242no—
465C2C3×D4×S412yes—
466C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×D4D3
socleC24×C3C22×D3
hypercenterC2×C3×D4S4
hypocenterA4C3×#127
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×A4C2semidirect
2. C2×C3×GL(2,ℤ/4)C2semidirect
3. C2×C3×C4⋊S4C2semidirect
4. C3×C4×SL(2,ℤ/4)C2semidirect
5. #85496C3direct (N × H)
15. C23×C3×D4D3semidirect
17. C2×D4×A4C2×C3semidirect
18. C2×GL(2,ℤ/4)C2×C3semidirect
19. C2×C4⋊S4C2×C3semidirect
24. C4×SL(2,ℤ/4)C2×C3semidirect
37. C3×C4×A4D4semidirect
38. C3×SL(2,ℤ/4)D4semidirect
90. C23×D4C3×D3semidirect
141. C2×C3×D4S4semidirect
148. C22×C3×C4#67semidirect
152. C4×A4C3×D4semidirect
153. SL(2,ℤ/4)C3×D4semidirect
169. C3×A4#127semidirect
302. C2×D4C3×S4semidirect
309. C22×C4C3×#67semidirect
347. C22×C3#940semidirect
352. A4C3×#127semidirect
359. C3×C4GL(2,ℤ/4)semidirect
449. C22C3×#940semidirect
455. C4C3×GL(2,ℤ/4)semidirect
458. C3#85496direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
C2×C3×GL(2,ℤ/4)2
C2×C3×C4⋊S42
C3×C4×SL(2,ℤ/4)2
#854963
C3×#24333
C3×#9404