Finite Group Databasebeta

← C3×#85512 #11922796

Subgroups — C3×#85512 #11922796

Order 1152.

644 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C3×A4×#41C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#5416C212yesmaximal, complemented
4C3×#5414C212yesmaximal, complemented
5C3×#5409C211yesmaximal, characteristic, complemented
6#85512C311yesmaximal, characteristic, direct product factor
7C3×#3007—31nomaximal
8C3×Q8×A4C2211yescharacteristic
9C3×D4×A4—21no—
10C3×D4×A4C2211yescharacteristic
11C2×C3×C4×A4—21no—
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3×SL(2,ℤ/4)—21no—
14C3×A4⋊Q8—21no—
15C3×A4⋊Q8C2212yes—
16C3×#951—41nomaximal
17C3×A4⋊C8C2212yes—
18C22×C3×#41D311yescharacteristic
19A4×#41—21no—
20A4×#41C2×C311yescharacteristic
21C2×A4⋊Q8C2×C311yescharacteristic, complemented
22C2×C3×#137—31no—
23C2×C3×#137—31no—
24C2×C3×C22⋊Q8—31no—
25#5416C2×C312yescomplemented
26#5414C2×C312yescomplemented
27#5409C2×C311yescharacteristic, complemented
28C3×#452—32no—
29C3×#452—32no—
30C3×#451—32no—
31C3×#450—32no—
32C3×#417—31no—
33C3×#417—31no—
34C3×#407—31no—
35C22×C3×A4—21no—
36C22×C3×A4D411yescharacteristic
37C32×#41—41no—
38C2×C3×Dic6—41no—
39C3×C4×A4—21no—
40C3×C4×A4D411yescharacteristic
41C3×C4×A4C2311yescharacteristic
42C3×SL(2,ℤ/4)—22no—
43C3×#216—42no—
44C3×#214—42no—
45C3×#208—41no—
46#3007—31noSylow
47C2×C3×#41—31no—
48C2×C3×#41—31no—
49C2×C3×#41—31no—
50C2×C3×#41—31no—
51C2×C3×#41—61no—
52C2×C3×#41—61no—
53C22×C3×Q8—31no—
54C22×C3×Q8C2×D311yescharacteristic
55C22×C3×D4—21no—
56C22×C3×D4C2×D311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4C2×D311yescharacteristic
59Q8×A4—21no—
60Q8×A4C22×C311yescharacteristic
61D4×A4—22no—
62D4×A4—21no—
63D4×A4—21no—
64D4×A4C22×C311yescharacteristic
65C2×C4×A4—22no—
66C2×C4×A4—21no—
67C2×C4×A4—21no—
68C2×C4×A4C22×C311yescharacteristic
69C2×SL(2,ℤ/4)—21no—
70A4⋊Q8—21no—
71A4⋊Q8C22×C312yes—
72C3×#137—62no—
73C3×#137—62no—
74C3×#137—61no—
75C3×#137—61no—
76C3×#137—61no—
77C3×#137—61no—
78C2×C3×Q16—32no—
79C2×C3×Q16—32no—
80C2×C3×SD16—32no—
81C2×C3×SD16—32no—
82C2×C3×M16—31no—
83C2×C3×M16—31no—
84C3×C22⋊Q8—32no—
85C3×C22⋊Q8—32no—
86C3×C22⋊Q8—61no—
87C3×C22⋊Q8—61no—
88C2×C3×C4⋊C4—31no—
89C2×C3×C4⋊C4—61no—
90C2×C3×C22⋊C4—61no—
91#951—41no—
92A4⋊C8C22×C312yes—
93C3×#103—32no—
94C3×#103—32no—
95C3×#102—32no—
96C3×#102—32no—
97C3×C22⋊C8—32no—
98C3×C22⋊C8—32no—
99C2×C3×A4—21no—
100C2×C3×A4—41no—
101C2×C3×A4C2×D411yescharacteristic
102C32×Q8—41no—
103C32×D4—41no—
104C32×D4—81no—
105C2×C32×C4—41no—
106C2×C32×C4—81no—
107C2×C3×Dic3—81no—
108C3×Dic6—42no—
109C3×Dic6—81no—
110C3×C3⋊C8—42no—
111C22×#41C3×D311yescharacteristic
112C2×#137—31no—
113C2×#137—31no—
114C2×C22⋊Q8—31no—
115#452—32no—
116#452—32no—
117#451—32no—
118#450—32no—
119#417—31no—
120#417—31no—
121#407—31no—
122C24×C3—21no—
123C24×C3#6711yescharacteristic
124C22×A4—22no—
125C22×A4—21no—
126C22×A4—21no—
127C22×A4C3×D411yescharacteristic
128C3×#41—32no—
129C3×#41—32no—
130C3×#41—31no—
131C3×#41—31no—
132C3×#41—31no—
133C3×#41—41no—
134C3×#41—61no—
135C3×#41—61no—
136C3×#41—61no—
137C3×#41—62no—
138C3×#41—62no—
139C3×#41—81no—
140C3×#41S411yescharacteristic
141C2×C3×Q8—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—32no—
144C2×C3×Q8—61no—
145C2×C3×Q8—61no—
146C2×C3×Q8—61no—
147C2×C3×Q8—61no—
148C2×C3×Q8—61no—
149C2×C3×Q8—61no—
150C2×C3×D4—31no—
151C2×C3×D4—31no—
152C2×C3×D4—61no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4—61no—
156C2×C3×D4—61no—
157C22×C3×C4—21no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4—61no—
165C22×C3×C4—61no—
166C22×C3×C4—61no—
167C22×C3×C4—61no—
168C22×C3×C4#6711yescharacteristic
169C22×C3×C4C22×D311yescharacteristic
170C2×Dic6—41no—
171C4×A4—22no—
172C4×A4—21no—
173C4×A4—21no—
174C4×A4—21no—
175C4×A4C3×D411yescharacteristic
176C4×A4C23×C311yescharacteristic
177SL(2,ℤ/4)—22no—
178C3×Q16—64no—
179C3×Q16—62no—
180C3×Q16—62no—
181C3×SD16—64no—
182C3×SD16—62no—
183C3×SD16—62no—
184C3×M16—61no—
185C3×M16—61no—
186C3×M16—61no—
187C3×M16—61no—
188C2×C3×C8—32no—
189C2×C3×C8—32no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—62no—
193C3×C4⋊C4—62no—
194C3×C22⋊C4—62no—
195C3×C22⋊C4—62no—
196#216—42no—
197#214—42no—
198#208—41no—
199C22×C32—41no—
200C22×C32—81no—
201C3×A4#13711yescharacteristic, complemented
202C32×C4—41no—
203C32×C4—41no—
204C32×C4—81no—
205C3×Dic3—82no—
206C2×#41—31no—
207C2×#41—31no—
208C2×#41—31no—
209C2×#41—31no—
210C2×#41—61no—
211C2×#41—61no—
212C22×Q8—31no—
213C22×Q8C2×C3×D311yescharacteristic
214C22×D4—21no—
215C22×D4C2×C3×D311yescharacteristic
216C23×C4—21no—
217C23×C4C2×C3×D311yescharacteristic
218#137—62no—
219#137—62no—
220#137—61no—
221#137—61no—
222#137—61no—
223#137—61no—
224C2×Q16—32no—
225C2×Q16—32no—
226C2×SD16—32no—
227C2×SD16—32no—
228C2×M16—31no—
229C2×M16—31no—
230C22⋊Q8—32no—
231C22⋊Q8—32no—
232C22⋊Q8—61no—
233C22⋊Q8—61no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—61no—
236C2×C22⋊C4—61no—
237#103—32no—
238#103—32no—
239#102—32no—
240#102—32no—
241C22⋊C8—32no—
242C22⋊C8—32no—
243C23×C3—21no—
244C23×C3—31no—
245C23×C3—31no—
246C23×C3—41no—
247C23×C3—61no—
248C23×C3—61no—
249C23×C3—61no—
250C23×C3—121no—
251C23×C3C2×#6711yescharacteristic
252C2×A4—21no—
253C2×A4—21no—
254C2×A4—41no—
255C2×A4—42no—
256C2×A4—41no—
257C2×A4C2×C3×D411yescharacteristic
258C3×Q8—32no—
259C3×Q8—31no—
260C3×Q8—41no—
261C3×Q8—61no—
262C3×Q8—62no—
263C3×Q8—62no—
264C3×Q8—81no—
265C3×Q8—121no—
266C3×Q8—121no—
267C3×Q8C2×S411yescharacteristic
268C3×D4—21no—
269C3×D4—32no—
270C3×D4—31no—
271C3×D4—41no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—62no—
276C3×D4—61no—
277C3×D4—81no—
278C3×D4—82no—
279C3×D4—81no—
280C3×D4C2×S411yescharacteristic
281C2×C3×C4—21no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—41no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—62no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—62no—
298C2×C3×C4—62no—
299C2×C3×C4—81no—
300C2×C3×C4—82no—
301C2×C3×C4—81no—
302C2×C3×C4—121no—
303C2×C3×C4—122no—
304C2×C3×C4—121no—
305C2×C3×C4C2×S411yescharacteristic
306C2×Dic3—81no—
307Dic6—42no—
308Dic6—81no—
309C3×C8—62no—
310C3×C8—62no—
311C3⋊C8—42no—
312C2×C32—41no—
313C2×C32—81no—
314C2×C32—161no—
315C24—21no—
316C24C3×#6711yescharacteristic
317#41—32no—
318#41—32no—
319#41—31no—
320#41—31no—
321#41—31no—
322#41—61no—
323#41—61no—
324#41—61no—
325#41—62no—
326#41—62no—
327#41C3×S411yescharacteristic
328C2×Q8—31no—
329C2×Q8—31no—
330C2×Q8—32no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×D4—31no—
338C2×D4—31no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—61no—
342C2×D4—61no—
343C2×D4—61no—
344C22×C4—21no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4C3×#6711yescharacteristic
356C22×C4C22×C3×D311yescharacteristic
357Q16—64no—
358Q16—62no—
359Q16—62no—
360SD16—64no—
361SD16—62no—
362SD16—62no—
363M16—61no—
364M16—61no—
365M16—61no—
366M16—61no—
367C2×C8—32no—
368C2×C8—32no—
369C4⋊C4—32no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C22⋊C4—62no—
374C22⋊C4—62no—
375C22×C3—21no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—41no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—61no—
384C22×C3—61no—
385C22×C3—81no—
386C22×C3—82no—
387C22×C3—81no—
388C22×C3—121no—
389C22×C3—121no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3#95111yescharacteristic, complemented
393C22×C3GL(2,ℤ/4)11yescharacteristic
394A4—21no—
395A4C3×#13711yescharacteristic, complemented
396C3×C4—21no—
397C3×C4—31no—
398C3×C4—31no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—61no—
402C3×C4—81no—
403C3×C4—82no—
404C3×C4—81no—
405C3×C4—81no—
406C3×C4—122no—
407C3×C4—122no—
408C3×C4GL(2,ℤ/4)11yescharacteristic
409C3×C4C22×S411yescharacteristic
410Dic3—82no—
411C32—41noSylow
412C23—21no—
413C23—31no—
414C23—31no—
415C23—41no—
416C23—61no—
417C23—61no—
418C23—61no—
419C23—121no—
420C23C2×C3×#6711yescharacteristic
421Q8—32no—
422Q8—31no—
423Q8—61no—
424Q8—62no—
425Q8—62no—
426Q8—121no—
427Q8—121no—
428Q8C2×C3×S411yescharacteristic
429D4—21no—
430D4—32no—
431D4—31no—
432D4—61no—
433D4—61no—
434D4—61no—
435D4—62no—
436D4—61no—
437D4C2×C3×S411yescharacteristic
438C2×C4—21no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—62no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—121no—
456C2×C4—122no—
457C2×C4—121no—
458C2×C4C2×C3×S411yescharacteristic
459C8—62no—
460C8—62no—
461C2×C3—21no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—61no—
467C2×C3—81no—
468C2×C3—81no—
469C2×C3—121no—
470C2×C3—161no—
471C2×C3—161no—
472C2×C3—162no—
473C2×C3C2×GL(2,ℤ/4)11yescharacteristic
474C22—21no—
475C22—31no—
476C22—31no—
477C22—31no—
478C22—61no—
479C22—61no—
480C22—61no—
481C22—61no—
482C22—61no—
483C22—121no—
484C22—121no—
485C22—121no—
486C22—121no—
487C22C3×#95111yescharacteristic, complemented
488C22C3×GL(2,ℤ/4)11yescharacteristic
489C4—21no—
490C4—31no—
491C4—31no—
492C4—61no—
493C4—122no—
494C4—122no—
495C4C3×GL(2,ℤ/4)11yescharacteristic
496C4C22×C3×S411yescharacteristic
497C3—41no—
498C3—81no—
499C3#8551211yescharacteristic, direct product factor
500C2—21no—
501C2—31no—
502C2—31no—
503C2—41no—
504C2—61no—
505C2—121no—
506C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C3C2×GL(2,ℤ/4)
commutator, cosocleC4×A4C23×C3
FrattiniC4C22×C3×S4
FittingC22×C3×#41D3
socleC23×C3C2×#67
prime radicalC3×A4×#41C2
hypercenterC3×#41S4
hypocenterA4C3×#137
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×A4⋊Q8C2semidirect
3. C3×#5416C2semidirect
4. C3×#5414C2semidirect
5. C3×#5409C2semidirect
6. #85512C3direct (N × H)
21. C2×A4⋊Q8C2×C3semidirect
25. #5416C2×C3semidirect
26. #5414C2×C3semidirect
27. #5409C2×C3semidirect
201. C3×A4#137semidirect
392. C22×C3#951semidirect
395. A4C3×#137semidirect
487. C22C3×#951semidirect
499. C3#85512direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4×#412
C2×C3×A4⋊Q82
C3×#54162
C3×#54142
C3×#54092
#855123
C3×#30073
C3×#9514