Finite Group Databasebeta

← C3×#85506 #11922790

Subgroups — C3×#85506 #11922790

Order 1152.

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

normal subgroups only

This group has 768 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×Q8×A4C211yesmaximal, characteristic, complemented
2C2×C3×C4×S4C211yesmaximal, characteristic
3C3×(C2×S4)⋊C4C212yesmaximal
4C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C3×SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6#85506C311yesmaximal, characteristic, direct product factor
7C3×#2409—31nomaximal
8C22×C3×S4C2211yescharacteristic
9C3×Q8×A4—22no—
10C2×C3×C4×A4C2212yes—
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C2×C3×SL(2,ℤ/4)C2212yes—
14C3×C4×S4—22no—
15C3×#946—41nomaximal
16C23×C3×Q8D311yescharacteristic, complemented
17C2×Q8×A4—21no—
18C2×Q8×A4C2×C311yescharacteristic, complemented
19C2×C4×S4C2×C311yescharacteristic
20C2×C3×C22⋊Q8—31no—
21C2×C3×C22⋊Q8—31no—
22C2×C3×C4×D4—31no—
23(C2×S4)⋊C4C2×C312yes—
24C4⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
25SL(2,ℤ/4)⋊C4C2×C312yescomplemented
26C3×#395—32no—
27C3×#393—32no—
28C3×#391—31no—
29C3×#391—31no—
30C3×#386—32no—
31C3×#385—32no—
32C3×#380—31no—
33C22×C3×A4C2311yescharacteristic
34C2×C3×S4—21no—
35C2×C3×S4—21no—
36C2×C3×S4Q812yescomplemented
37C2×C32×Q8—41no—
38C2×C3×C4×D3—41no—
39C3×C4×A4—22no—
40C3×C4×A4D412yes—
41C3×SL(2,ℤ/4)—21no—
42C3×SL(2,ℤ/4)—22no—
43C3×(C2×D3)⋊C4—42no—
44C3×C4⋊Dic3—41no—
45C3×Dic3⋊C4—42no—
46#2409—31noSylow
47C22×S4C22×C311yescharacteristic
48C22×C3×Q8—22no—
49C22×C3×Q8—31no—
50C22×C3×Q8—31no—
51C22×C3×Q8—32no—
52C22×C3×Q8—62no—
53C22×C3×D4—31no—
54C23×C3×C4—31no—
55C23×C3×C4C2×D312yes—
56C23×C3×C4C2×D311yescharacteristic
57Q8×A4—24no—
58Q8×A4—22no—
59C2×C4×A4—22no—
60C2×C4×A4—21no—
61C2×C4×A4C22×C312yes—
62C2×C4×A4C22×C311yescharacteristic
63C2×SL(2,ℤ/4)C22×C311yescharacteristic
64C2×SL(2,ℤ/4)C22×C312yes—
65C4×S4—22no—
66C3×C22⋊Q8—64no—
67C3×C22⋊Q8—61no—
68C3×C22⋊Q8—62no—
69C3×C22⋊Q8—61no—
70C3×C4×D4—62no—
71C3×C4×D4—62no—
72C2×C3×C4⋊C4—31no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C4⋊C4—32no—
75C2×C3×C4⋊C4—32no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C22⋊C4—32no—
78C2×C3×C22⋊C4—32no—
79C2×C3×C22⋊C4—31no—
80C2×C3×C22⋊C4—32no—
81C2×C3×C22⋊C4—31no—
82C2×C3×C42—31no—
83#946—41no—
84C3×#95—32no—
85C3×#95—32no—
86C3×#95—32no—
87C3×#95—32no—
88C3×#95—32no—
89C22×C3×D3—41no—
90C2×C3×A4C2×D411yescharacteristic
91C2×C3×A4C2×Q811yescharacteristic
92C2×C3×A4#4111yescharacteristic
93C3×S4—22no—
94C32×Q8—82no—
95C2×C32×C4—41no—
96C2×C32×C4—42no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—42no—
99C3×C4×D3—82no—
100C23×Q8C3×D311yescharacteristic, complemented
101C2×C22⋊Q8—31no—
102C2×C22⋊Q8—31no—
103C2×C4×D4—31no—
104#395—32no—
105#393—32no—
106#391—31no—
107#391—31no—
108#386—32no—
109#385—32no—
110#380—31no—
111C24×C3—31no—
112C24×C3C22×D311yescharacteristic
113C22×A4—21no—
114C22×A4C23×C311yescharacteristic
115C2×S4—21no—
116C2×S4—21no—
117C2×S4C3×Q812yescomplemented
118C2×C3×Q8—34no—
119C2×C3×Q8—32no—
120C2×C3×Q8—31no—
121C2×C3×Q8—41no—
122C2×C3×Q8—61no—
123C2×C3×Q8—62no—
124C2×C3×Q8—62no—
125C2×C3×Q8—62no—
126C2×C3×Q8—64no—
127C2×C3×Q8—64no—
128C2×C3×Q8—81no—
129C2×C3×Q8S411yescharacteristic, complemented
130C2×C3×D4—32no—
131C2×C3×D4—32no—
132C2×C3×D4—61no—
133C2×C3×D4—61no—
134C2×C3×D4—61no—
135C2×C3×D4—61no—
136C22×C3×C4—22no—
137C22×C3×C4—31no—
138C22×C3×C4—31no—
139C22×C3×C4—32no—
140C22×C3×C4—32no—
141C22×C3×C4—31no—
142C22×C3×C4—32no—
143C22×C3×C4—31no—
144C22×C3×C4—32no—
145C22×C3×C4—32no—
146C22×C3×C4—31no—
147C22×C3×C4—62no—
148C22×C3×C4—62no—
149C22×C3×C4—61no—
150C22×C3×C4—62no—
151C22×C3×C4—61no—
152C22×C3×C4#6712yes—
153C2×C4×D3—41no—
154C4×A4—24no—
155C4×A4—22no—
156C4×A4—22no—
157C4×A4C3×D412yes—
158SL(2,ℤ/4)—21no—
159SL(2,ℤ/4)—22no—
160C3×C4⋊C4—62no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—62no—
163C3×C4⋊C4—64no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—62no—
166C3×C4⋊C4—61no—
167C3×C22⋊C4—62no—
168C3×C22⋊C4—64no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—62no—
171C3×C22⋊C4—61no—
172C3×C22⋊C4—62no—
173C3×C22⋊C4—61no—
174C3×C22⋊C4—62no—
175C3×C42—62no—
176(C2×D3)⋊C4—42no—
177C4⋊Dic3—41no—
178Dic3⋊C4—42no—
179C22×C32—41no—
180C2×C3×D3—42no—
181C2×C3×D3—81no—
182C2×C3×D3—81no—
183C3×A4C22⋊Q811yescharacteristic, complemented
184C32×C4—42no—
185C32×C4—82no—
186C3×Dic3—81no—
187C3×Dic3—82no—
188C22×Q8—22no—
189C22×Q8—31no—
190C22×Q8—31no—
191C22×Q8—32no—
192C22×Q8—62no—
193C22×D4—31no—
194C23×C4—31no—
195C23×C4C2×C3×D312yes—
196C23×C4C2×C3×D311yescharacteristic
197C22⋊Q8—64no—
198C22⋊Q8—61no—
199C22⋊Q8—62no—
200C22⋊Q8—61no—
201C4×D4—62no—
202C4×D4—62no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—32no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—32no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—31no—
212C2×C22⋊C4—32no—
213C2×C42—31no—
214#95—32no—
215#95—32no—
216#95—32no—
217#95—32no—
218#95—32no—
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—31no—
222C23×C3—31no—
223C23×C3—32no—
224C23×C3—61no—
225C23×C3—61no—
226C23×C3—61no—
227C23×C3—61no—
228C23×C3—61no—
229C23×C3—61no—
230C23×C3D3×Q811yescharacteristic
231C23×C3#23911yescharacteristic
232C23×C3C2×#6711yescharacteristic
233C22×D3—41no—
234C2×A4—21no—
235C2×A4—21no—
236C2×A4—21no—
237C2×A4C2×C3×D411yescharacteristic
238C2×A4C2×C3×Q811yescharacteristic
239C2×A4C3×#4111yescharacteristic
240S4—22no—
241C3×Q8—22no—
242C3×Q8—64no—
243C3×Q8—64no—
244C3×Q8—62no—
245C3×Q8—82no—
246C3×Q8—84no—
247C3×D4—62no—
248C3×D4—62no—
249C3×D4—62no—
250C3×D4—62no—
251C2×C3×C4—31no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—32no—
255C2×C3×C4—34no—
256C2×C3×C4—32no—
257C2×C3×C4—32no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—61no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—62no—
274C2×C3×C4—61no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—61no—
278C2×C3×C4—82no—
279C2×C3×C4—81no—
280C2×C3×C4—122no—
281C2×C3×C4—122no—
282C2×C3×C4C2×S412yes—
283C2×C3×C4C2×S411yescharacteristic
284C2×Dic3—41no—
285C2×Dic3—42no—
286C4×D3—82no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C3×D3—82no—
291C24—31no—
292C24C22×C3×D311yescharacteristic
293C2×Q8—34no—
294C2×Q8—32no—
295C2×Q8—31no—
296C2×Q8—61no—
297C2×Q8—62no—
298C2×Q8—62no—
299C2×Q8—62no—
300C2×Q8—64no—
301C2×Q8—64no—
302C2×Q8C3×S411yescharacteristic, complemented
303C2×D4—32no—
304C2×D4—32no—
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C22×C4—22no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—31no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—31no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—61no—
325C22×C4C3×#6712yes—
326C4⋊C4—62no—
327C4⋊C4—61no—
328C4⋊C4—62no—
329C4⋊C4—64no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4—61no—
333C22⋊C4—62no—
334C22⋊C4—64no—
335C22⋊C4—62no—
336C22⋊C4—61no—
337C22⋊C4—62no—
338C22⋊C4—62no—
339C22⋊C4—61no—
340C22⋊C4—62no—
341C42—62no—
342C22×C3—31no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—41no—
352C22×C3—61no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—62no—
356C22×C3—81no—
357C22×C3—121no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3#94611yescharacteristic, complemented
362C22×C3C22×S411yescharacteristic
363C2×D3—42no—
364C2×D3—81no—
365C2×D3—81no—
366A4—21no—
367A4C3×C22⋊Q811yescharacteristic, complemented
368C3×C4—22no—
369C3×C4—32no—
370C3×C4—42no—
371C3×C4—62no—
372C3×C4—62no—
373C3×C4—82no—
374C3×C4—84no—
375C3×C4—82no—
376C3×C4—122no—
377C3×C4—121no—
378C3×C4—122no—
379C3×C4—121no—
380C3×C4GL(2,ℤ/4)12yes—
381Dic3—81no—
382Dic3—82no—
383C32—41noSylow
384C23—31no—
385C23—31no—
386C23—31no—
387C23—31no—
388C23—32no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23C3×D3×Q811yescharacteristic
396C23C3×#23911yescharacteristic
397C23C2×C3×#6711yescharacteristic
398Q8—22no—
399Q8—64no—
400Q8—64no—
401Q8—62no—
402D4—62no—
403D4—62no—
404D4—62no—
405D4—62no—
406C2×C4—31no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—32no—
410C2×C4—34no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—62no—
419C2×C4—61no—
420C2×C4—62no—
421C2×C4—61no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—61no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—61no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4C2×C3×S412yes—
434C2×C4C2×C3×S411yescharacteristic
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—81no—
443C2×C3—81no—
444C2×C3—81no—
445C2×C3—122no—
446C2×C3Q8×S411yescharacteristic
447C2×C3#591911yescharacteristic
448C2×C3C2×GL(2,ℤ/4)11yescharacteristic
449D3—82no—
450C22—31no—
451C22—31no—
452C22—31no—
453C22—31no—
454C22—31no—
455C22—31no—
456C22—31no—
457C22—31no—
458C22—31no—
459C22—61no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—121no—
464C22—121no—
465C22—121no—
466C22—121no—
467C22C3×#94611yescharacteristic, complemented
468C22C22×C3×S411yescharacteristic
469C4—22no—
470C4—32no—
471C4—62no—
472C4—62no—
473C4—122no—
474C4—121no—
475C4—122no—
476C4—121no—
477C4C3×GL(2,ℤ/4)12yes—
478C3—41no—
479C3—81no—
480C3#8550611yescharacteristic, direct product factor
481C2—31no—
482C2—31no—
483C2—31no—
484C2—31no—
485C2—122no—
486C2C3×Q8×S411yescharacteristic
487C2C3×#591911yescharacteristic
488C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC2×C3×Q8S4
hypocenterA4C3×C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
4. C3×C4⋊SL(2,ℤ/4)C2semidirect
5. C3×SL(2,ℤ/4)⋊C4C2semidirect
6. #85506C3direct (N × H)
16. C23×C3×Q8D3semidirect
18. C2×Q8×A4C2×C3semidirect
24. C4⋊SL(2,ℤ/4)C2×C3semidirect
25. SL(2,ℤ/4)⋊C4C2×C3semidirect
36. C2×C3×S4Q8semidirect
100. C23×Q8C3×D3semidirect
117. C2×S4C3×Q8semidirect
129. C2×C3×Q8S4semidirect
183. C3×A4C22⋊Q8semidirect
302. C2×Q8C3×S4semidirect
361. C22×C3#946semidirect
367. A4C3×C22⋊Q8semidirect
467. C22C3×#946semidirect
480. C3#85506direct (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×C4×S42
C3×(C2×S4)⋊C42
C3×C4⋊SL(2,ℤ/4)2
C3×SL(2,ℤ/4)⋊C42
#855063
C3×#24093
C3×#9464