Finite Group Databasebeta

← C3×#85312 #11922596

Subgroups — C3×#85312 #11922596

Order 1152.

700 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 92 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×C4×S4C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic, complemented
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C212yesmaximal
5C3×C4⋊SL(2,ℤ/4)C212yesmaximal, complemented
6#85312C311yesmaximal, characteristic, direct product factor
7C3×#2417—31nomaximal
8C22×C3×S4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)C2211yescharacteristic
12C2×C3×SL(2,ℤ/4)C2212yes—
13C3×C4×S4—22no—
14C3×A4⋊Q8—22no—
15C3×#897—41nomaximal
16C2×C4×S4C2×C311yescharacteristic
17C2×A4⋊Q8C2×C311yescharacteristic, complemented
18C2×C3×C22⋊Q8—31no—
19C2×C3×C22⋊Q8—31no—
20C2×C3×C4×D4—31no—
21C22×C3×C4⋊C4D311yescharacteristic, complemented
22A4×C4⋊C4—21no—
23A4×C4⋊C4C2×C311yescharacteristic, complemented
24(C2×S4)⋊C4C2×C312yes—
25C4⋊SL(2,ℤ/4)C2×C312yescomplemented
26C3×#398—32no—
27C3×#396—32no—
28C3×#391—31no—
29C3×#389—31no—
30C3×#386—32no—
31C3×#380—31no—
32C3×#380—32no—
33C22×C3×A4C2311yescharacteristic
34C2×C3×S4—21no—
35C2×C3×S4—21no—
36C2×C3×S4Q812yescomplemented
37C2×C3×C4×D3—41no—
38C2×C3×Dic6—41no—
39C3×C4×A4—22no—
40C3×C4×A4D412yes—
41C3×SL(2,ℤ/4)—22no—
42C3×SL(2,ℤ/4)—21no—
43C32×C4⋊C4—41no—
44C3×(C2×D3)⋊C4—42no—
45C3×C4⋊Dic3—42no—
46#2417—31noSylow
47C22×S4C22×C311yescharacteristic
48C22×C3×Q8—31no—
49C22×C3×D4—31no—
50C23×C3×C4—31no—
51C23×C3×C4C2×D312yes—
52C23×C3×C4C2×D311yescharacteristic
53C2×C4×A4—21no—
54C2×C4×A4—22no—
55C2×C4×A4C22×C312yes—
56C2×C4×A4C22×C311yescharacteristic
57C2×SL(2,ℤ/4)C22×C312yes—
58C2×SL(2,ℤ/4)C22×C311yescharacteristic
59C4×S4—22no—
60A4⋊Q8—22no—
61C3×C22⋊Q8—61no—
62C3×C22⋊Q8—62no—
63C3×C22⋊Q8—62no—
64C3×C22⋊Q8—62no—
65C3×C22⋊Q8—61no—
66C3×C4×D4—62no—
67C3×C4×D4—62no—
68C2×C3×C4⋊C4—31no—
69C2×C3×C4⋊C4—31no—
70C2×C3×C4⋊C4—32no—
71C2×C3×C4⋊C4—31no—
72C2×C3×C4⋊C4—32no—
73C2×C3×C4⋊C4—32no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—32no—
76C2×C3×C22⋊C4—32no—
77C2×C3×C22⋊C4—31no—
78C2×C3×C22⋊C4—31no—
79C2×C3×C22⋊C4—32no—
80C2×C3×C22⋊C4—32no—
81C2×C3×C42—31no—
82#897—41no—
83C3×#95—32no—
84C3×#95—32no—
85C3×#95—32no—
86C3×#95—32no—
87C22×C3×D3—41no—
88C2×C3×A4C2×D411yescharacteristic
89C2×C3×A4C2×Q811yescharacteristic
90C2×C3×A4#4111yescharacteristic
91C3×S4—22no—
92C2×C32×C4—41no—
93C2×C32×C4—42no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—42no—
96C3×C4×D3—82no—
97C3×Dic6—82no—
98C2×C22⋊Q8—31no—
99C2×C22⋊Q8—31no—
100C2×C4×D4—31no—
101C22×C4⋊C4C3×D311yescharacteristic, complemented
102#398—32no—
103#396—32no—
104#391—31no—
105#389—31no—
106#386—32no—
107#380—31no—
108#380—32no—
109C24×C3—31no—
110C24×C3C22×D311yescharacteristic
111C22×A4—21no—
112C22×A4C23×C311yescharacteristic
113C2×S4—21no—
114C2×S4—21no—
115C2×S4C3×Q812yescomplemented
116C2×C3×Q8—61no—
117C2×C3×Q8—62no—
118C2×C3×Q8—62no—
119C2×C3×Q8—61no—
120C2×C3×D4—32no—
121C2×C3×D4—32no—
122C2×C3×D4—61no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—61no—
126C22×C3×C4—22no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—32no—
130C22×C3×C4—31no—
131C22×C3×C4—32no—
132C22×C3×C4—32no—
133C22×C3×C4—31no—
134C22×C3×C4—32no—
135C22×C3×C4—32no—
136C22×C3×C4—31no—
137C22×C3×C4—62no—
138C22×C3×C4—61no—
139C22×C3×C4—62no—
140C22×C3×C4—62no—
141C22×C3×C4—61no—
142C22×C3×C4D1212yes—
143C2×C4×D3—41no—
144C2×Dic6—41no—
145C4×A4—22no—
146C4×A4—22no—
147C4×A4—42no—
148C4×A4C3×D412yes—
149SL(2,ℤ/4)—22no—
150SL(2,ℤ/4)—21no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—61no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—64no—
158C3×C4⋊C4—62no—
159C3×C4⋊C4—62no—
160C3×C4⋊C4—81no—
161C3×C4⋊C4—122no—
162C3×C4⋊C4S411yescharacteristic, complemented
163C3×C22⋊C4—62no—
164C3×C22⋊C4—62no—
165C3×C22⋊C4—61no—
166C3×C22⋊C4—62no—
167C3×C22⋊C4—61no—
168C3×C22⋊C4—62no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—122no—
171C3×C42—62no—
172(C2×D3)⋊C4—42no—
173C4⋊Dic3—42no—
174C22×C32—41no—
175C2×C3×D3—42no—
176C2×C3×D3—81no—
177C2×C3×D3—81no—
178C3×A4C22⋊Q811yescharacteristic, complemented
179C32×C4—42no—
180C32×C4—82no—
181C3×Dic3—82no—
182C3×Dic3—81no—
183C22×Q8—31no—
184C22×D4—31no—
185C23×C4—31no—
186C23×C4C2×C3×D312yes—
187C23×C4C2×C3×D311yescharacteristic
188C22⋊Q8—61no—
189C22⋊Q8—62no—
190C22⋊Q8—62no—
191C22⋊Q8—62no—
192C22⋊Q8—61no—
193C4×D4—62no—
194C4×D4—62no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—32no—
203C2×C22⋊C4—31no—
204C2×C22⋊C4—32no—
205C2×C22⋊C4—31no—
206C2×C22⋊C4—32no—
207C2×C22⋊C4—32no—
208C2×C42—31no—
209#95—32no—
210#95—32no—
211#95—32no—
212#95—32no—
213C23×C3—31no—
214C23×C3—31no—
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—32no—
218C23×C3—61no—
219C23×C3—61no—
220C23×C3—61no—
221C23×C3—61no—
222C23×C3—61no—
223C23×C3—61no—
224C23×C3C2×D1211yescharacteristic
225C23×C3#23711yescharacteristic
226C23×C3D3×Q811yescharacteristic
227C22×D3—41no—
228C2×A4—21no—
229C2×A4—21no—
230C2×A4—21no—
231C2×A4C2×C3×D411yescharacteristic
232C2×A4C2×C3×Q811yescharacteristic
233C2×A4C3×#4111yescharacteristic
234S4—22no—
235C3×Q8—122no—
236C3×Q8—122no—
237C3×D4—62no—
238C3×D4—62no—
239C3×D4—62no—
240C3×D4—62no—
241C2×C3×C4—31no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—61no—
257C2×C3×C4—62no—
258C2×C3×C4—61no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—61no—
268C2×C3×C4—81no—
269C2×C3×C4—82no—
270C2×C3×C4—122no—
271C2×C3×C4—122no—
272C2×C3×C4C2×S412yes—
273C2×C3×C4C2×S411yescharacteristic
274C2×Dic3—41no—
275C2×Dic3—42no—
276C4×D3—82no—
277Dic6—82no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C3×D3—82no—
282C24—31no—
283C24C22×C3×D311yescharacteristic
284C2×Q8—61no—
285C2×Q8—62no—
286C2×Q8—62no—
287C2×Q8—61no—
288C2×D4—32no—
289C2×D4—32no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—61no—
294C22×C4—22no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—32no—
298C22×C4—31no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—31no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—61no—
310C22×C4C3×D1212yes—
311C4⋊C4—32no—
312C4⋊C4—31no—
313C4⋊C4—61no—
314C4⋊C4—62no—
315C4⋊C4—62no—
316C4⋊C4—64no—
317C4⋊C4—62no—
318C4⋊C4—62no—
319C4⋊C4—122no—
320C4⋊C4C3×S411yescharacteristic, complemented
321C22⋊C4—62no—
322C22⋊C4—62no—
323C22⋊C4—61no—
324C22⋊C4—62no—
325C22⋊C4—61no—
326C22⋊C4—62no—
327C22⋊C4—62no—
328C22⋊C4—122no—
329C42—62no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—41no—
340C22×C3—61no—
341C22×C3—62no—
342C22×C3—62no—
343C22×C3—62no—
344C22×C3—81no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3#89711yescharacteristic, complemented
350C22×C3C22×S411yescharacteristic
351C2×D3—42no—
352C2×D3—81no—
353C2×D3—81no—
354A4—21no—
355A4C3×C22⋊Q811yescharacteristic, complemented
356C3×C4—22no—
357C3×C4—32no—
358C3×C4—42no—
359C3×C4—62no—
360C3×C4—62no—
361C3×C4—82no—
362C3×C4—82no—
363C3×C4—121no—
364C3×C4—122no—
365C3×C4—121no—
366C3×C4—122no—
367C3×C4—162no—
368C3×C4C4⋊S412yes—
369Dic3—82no—
370Dic3—81no—
371C32—41noSylow
372C23—31no—
373C23—31no—
374C23—31no—
375C23—31no—
376C23—32no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23C2×C3×D1211yescharacteristic
384C23C3×#23711yescharacteristic
385C23C3×D3×Q811yescharacteristic
386Q8—122no—
387Q8—122no—
388D4—62no—
389D4—62no—
390D4—62no—
391D4—62no—
392C2×C4—31no—
393C2×C4—32no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—61no—
406C2×C4—62no—
407C2×C4—61no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—62no—
413C2×C4—61no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—61no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4C2×C3×S412yes—
420C2×C4C2×C3×S411yescharacteristic
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—31no—
424C2×C3—31no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—81no—
429C2×C3—81no—
430C2×C3—81no—
431C2×C3—122no—
432C2×C3C2×C4⋊S411yescharacteristic
433C2×C3#591411yescharacteristic
434C2×C3Q8×S411yescharacteristic
435D3—82no—
436C22—31no—
437C22—31no—
438C22—31no—
439C22—31no—
440C22—31no—
441C22—31no—
442C22—31no—
443C22—31no—
444C22—31no—
445C22—61no—
446C22—62no—
447C22—62no—
448C22—62no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22C3×#89711yescharacteristic, complemented
454C22C22×C3×S411yescharacteristic
455C4—22no—
456C4—32no—
457C4—62no—
458C4—62no—
459C4—121no—
460C4—122no—
461C4—121no—
462C4—122no—
463C4C3×C4⋊S412yes—
464C3—41no—
465C3—81no—
466C3#8531211yescharacteristic, direct product factor
467C2—31no—
468C2—31no—
469C2—31no—
470C2—31no—
471C2—122no—
472C2C2×C3×C4⋊S411yescharacteristic
473C2C3×#591411yescharacteristic
474C2C3×Q8×S411yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC3×C4⋊C4S4
hypocenterA4C3×C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×A4⋊Q8C2semidirect
3. C3×A4×C4⋊C4C2semidirect
5. C3×C4⋊SL(2,ℤ/4)C2semidirect
6. #85312C3direct (N × H)
17. C2×A4⋊Q8C2×C3semidirect
21. C22×C3×C4⋊C4D3semidirect
23. A4×C4⋊C4C2×C3semidirect
25. C4⋊SL(2,ℤ/4)C2×C3semidirect
36. C2×C3×S4Q8semidirect
101. C22×C4⋊C4C3×D3semidirect
115. C2×S4C3×Q8semidirect
162. C3×C4⋊C4S4semidirect
178. C3×A4C22⋊Q8semidirect
320. C4⋊C4C3×S4semidirect
349. C22×C3#897semidirect
355. A4C3×C22⋊Q8semidirect
453. C22C3×#897semidirect
466. C3#85312direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
C2×C3×A4⋊Q82
C3×A4×C4⋊C42
C3×(C2×S4)⋊C42
C3×C4⋊SL(2,ℤ/4)2
#853123
C3×#24173
C3×#8974