Finite Group Databasebeta

← C3×#85301 #11922585

Subgroups — C3×#85301 #11922585

Order 1152.

802 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 88 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×GL(2,ℤ/4)C211yesmaximal, characteristic
2C22×C3×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C3×A4×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C212yesmaximal, complemented
5C3×C4⋊SL(2,ℤ/4)C212yesmaximal, complemented
6#85301C311yesmaximal, characteristic, direct product factor
7C3×#2412—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C22×C3×S4C2211yescharacteristic
10C2×C3×C4×A4C2212yes—
11C3×GL(2,ℤ/4)—22no—
12C2×C3×SL(2,ℤ/4)—22no—
13C2×C3×SL(2,ℤ/4)C2211yescharacteristic
14C2×C3×SL(2,ℤ/4)C2212yes—
15C3×#886—41nomaximal
16C2×GL(2,ℤ/4)C2×C311yescharacteristic
17C22×SL(2,ℤ/4)C2×C311yescharacteristic, complemented
18C2×C3×#123—31no—
19C2×C3×#120—31no—
20C22×C3×C22⋊C4—31no—
21C22×C3×C22⋊C4D311yescharacteristic, complemented
22A4×C22⋊C4—21no—
23A4×C22⋊C4C2×C311yescharacteristic, complemented
24(C2×S4)⋊C4C2×C312yescomplemented
25C4⋊SL(2,ℤ/4)C2×C312yescomplemented
26C3×#397—32no—
27C3×#394—32no—
28C3×#386—31no—
29C3×#386—32no—
30C3×#385—31no—
31C3×#381—31no—
32C3×#380—32no—
33C22×C3×A4—22no—
34C22×C3×A4D412yes—
35C22×C3×A4C2311yescharacteristic
36C2×C3×S4—21no—
37C2×C3×S4—22no—
38C2×C3×#67—41no—
39C22×C3×Dic3—41no—
40C3×C4×A4—22no—
41C3×SL(2,ℤ/4)—22no—
42C3×SL(2,ℤ/4)—21no—
43C32×C22⋊C4—41no—
44C3×(C2×D3)⋊C4—42no—
45C3×C4⋊Dic3—42no—
46#2412—31noSylow
47C25×C3C2×D311yescharacteristic
48C23×A4—21no—
49C23×A4C22×C311yescharacteristic
50C22×S4C22×C311yescharacteristic
51C22×C3×D4—31no—
52C22×C3×D4—31no—
53C22×C3×D4—31no—
54C23×C3×C4—31no—
55C23×C3×C4—31no—
56C23×C3×C4C2×D312yes—
57C2×C4×A4—22no—
58C2×C4×A4C22×C312yes—
59GL(2,ℤ/4)—22no—
60C2×SL(2,ℤ/4)—22no—
61C2×SL(2,ℤ/4)C22×C311yescharacteristic
62C2×SL(2,ℤ/4)C22×C312yes—
63C3×#123—61no—
64C3×#123—62no—
65C3×#123—61no—
66C3×#120—62no—
67C3×#120—62no—
68C2×C3×C4⋊C4—32no—
69C2×C3×C4⋊C4—32no—
70C2×C3×C22⋊C4—31no—
71C2×C3×C22⋊C4—31no—
72C2×C3×C22⋊C4—32no—
73C2×C3×C22⋊C4—31no—
74C2×C3×C22⋊C4—32no—
75C2×C3×C22⋊C4—32no—
76C2×C3×C22⋊C4—31no—
77C2×C3×C22⋊C4—31no—
78C2×C3×C22⋊C4—32no—
79C2×C3×C22⋊C4—32no—
80C2×C3×C22⋊C4—62no—
81C2×C3×C22⋊C4—62no—
82#886—41no—
83C3×#95—32no—
84C3×#95—32no—
85C3×#95—32no—
86C3×#95—32no—
87C23×C32—41no—
88C22×C3×D3—41no—
89C2×C3×A4—22no—
90C2×C3×A4C2×D411yescharacteristic
91C2×C3×A4#4112yes—
92C3×S4—41no—
93C2×C32×C4—42no—
94C3×#67—82no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—42no—
97C2×C3×Dic3—82no—
98C2×#123—31no—
99C2×#120—31no—
100C22×C22⋊C4—31no—
101C22×C22⋊C4C3×D311yescharacteristic, complemented
102#397—32no—
103#394—32no—
104#386—31no—
105#386—32no—
106#385—31no—
107#381—31no—
108#380—32no—
109C24×C3—22no—
110C24×C3—31no—
111C24×C3—31no—
112C24×C3—32no—
113C24×C3—31no—
114C24×C3—62no—
115C24×C3D1212yes—
116C24×C3C22×D311yescharacteristic
117C22×A4—22no—
118C22×A4—22no—
119C22×A4—21no—
120C22×A4—42no—
121C22×A4C3×D412yes—
122C22×A4C23×C311yescharacteristic
123C2×S4—21no—
124C2×S4—22no—
125C2×C3×D4—61no—
126C2×C3×D4—62no—
127C2×C3×D4—62no—
128C2×C3×D4—61no—
129C2×C3×D4—62no—
130C2×C3×D4—62no—
131C2×C3×D4—61no—
132C2×C3×D4—62no—
133C2×C3×D4—61no—
134C2×C3×D4—122no—
135C22×C3×C4—22no—
136C22×C3×C4—32no—
137C22×C3×C4—32no—
138C22×C3×C4—31no—
139C22×C3×C4—32no—
140C22×C3×C4—32no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—62no—
144C22×C3×C4—62no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147C22×C3×C4—62no—
148C22×C3×C4—61no—
149C22×C3×C4—122no—
150C2×#67—41no—
151C22×Dic3—41no—
152C4×A4—22no—
153C4×A4—42no—
154SL(2,ℤ/4)—22no—
155SL(2,ℤ/4)—21no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—62no—
158C3×C4⋊C4—122no—
159C3×C22⋊C4—31no—
160C3×C22⋊C4—32no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—61no—
163C3×C22⋊C4—62no—
164C3×C22⋊C4—62no—
165C3×C22⋊C4—64no—
166C3×C22⋊C4—61no—
167C3×C22⋊C4—62no—
168C3×C22⋊C4—62no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—61no—
171C3×C22⋊C4—62no—
172C3×C22⋊C4—81no—
173C3×C22⋊C4—122no—
174C3×C22⋊C4S411yescharacteristic, complemented
175(C2×D3)⋊C4—42no—
176C4⋊Dic3—42no—
177C22×C32—42no—
178C22×C32—41no—
179C22×C32—82no—
180C2×C3×D3—81no—
181C2×C3×D3—82no—
182C3×A4#12311yescharacteristic, complemented
183C32×C4—82no—
184C3×Dic3—82no—
185C3×Dic3—81no—
186C25C2×C3×D311yescharacteristic
187C22×D4—31no—
188C22×D4—31no—
189C22×D4—31no—
190C23×C4—31no—
191C23×C4—31no—
192C23×C4C2×C3×D312yes—
193#123—61no—
194#123—62no—
195#123—61no—
196#120—62no—
197#120—62no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—32no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—32no—
203C2×C22⋊C4—32no—
204C2×C22⋊C4—31no—
205C2×C22⋊C4—31no—
206C2×C22⋊C4—32no—
207C2×C22⋊C4—32no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—32no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212#95—32no—
213#95—32no—
214#95—32no—
215#95—32no—
216C23×C3—22no—
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—32no—
220C23×C3—32no—
221C23×C3—32no—
222C23×C3—32no—
223C23×C3—31no—
224C23×C3—41no—
225C23×C3—62no—
226C23×C3—62no—
227C23×C3—61no—
228C23×C3—62no—
229C23×C3—62no—
230C23×C3—64no—
231C23×C3—62no—
232C23×C3—62no—
233C23×C3—61no—
234C23×C3—61no—
235C23×C3—62no—
236C23×C3—61no—
237C23×C3—81no—
238C23×C3—122no—
239C23×C3C2×D1211yescharacteristic
240C23×C3#23712yes—
241C23×C3C2×S411yescharacteristic
242C22×D3—41no—
243C2×A4—22no—
244C2×A4—21no—
245C2×A4—22no—
246C2×A4—42no—
247C2×A4C2×C3×D411yescharacteristic
248C2×A4C3×#4112yes—
249S4—41no—
250C3×D4—121no—
251C3×D4—122no—
252C3×D4—122no—
253C3×D4—124no—
254C3×D4—121no—
255C3×D4—122no—
256C2×C3×C4—32no—
257C2×C3×C4—42no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—61no—
273C2×C3×C4—62no—
274C2×C3×C4—61no—
275C2×C3×C4—82no—
276C2×C3×C4—122no—
277C2×C3×C4—124no—
278C2×C3×C4—122no—
279C2×C3×C4C2×S412yes—
280#67—82no—
281C2×Dic3—41no—
282C2×Dic3—42no—
283C2×Dic3—82no—
284C2×C32—41no—
285C2×C32—42no—
286C2×C32—82no—
287C3×D3—161no—
288C24—22no—
289C24—31no—
290C24—31no—
291C24—32no—
292C24—31no—
293C24—62no—
294C24C3×D1212yes—
295C24C22×C3×D311yescharacteristic
296C2×D4—61no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—61no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—62no—
305C2×D4—122no—
306C22×C4—22no—
307C22×C4—32no—
308C22×C4—32no—
309C22×C4—31no—
310C22×C4—32no—
311C22×C4—32no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—61no—
318C22×C4—62no—
319C22×C4—61no—
320C22×C4—122no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—122no—
324C22⋊C4—31no—
325C22⋊C4—32no—
326C22⋊C4—61no—
327C22⋊C4—62no—
328C22⋊C4—62no—
329C22⋊C4—64no—
330C22⋊C4—61no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—61no—
335C22⋊C4—62no—
336C22⋊C4—122no—
337C22⋊C4C3×S411yescharacteristic, complemented
338C22×C3—22no—
339C22×C3—31no—
340C22×C3—32no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—32no—
344C22×C3—32no—
345C22×C3—32no—
346C22×C3—41no—
347C22×C3—42no—
348C22×C3—62no—
349C22×C3—62no—
350C22×C3—64no—
351C22×C3—61no—
352C22×C3—62no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—82no—
356C22×C3—81no—
357C22×C3—82no—
358C22×C3—122no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—122no—
362C22×C3—121no—
363C22×C3—122no—
364C22×C3—162no—
365C22×C3#88611yescharacteristic, complemented
366C22×C3C4⋊S412yes—
367C22×C3C22×S411yescharacteristic
368C2×D3—81no—
369C2×D3—82no—
370A4—21no—
371A4C3×#12311yescharacteristic, complemented
372C3×C4—22no—
373C3×C4—62no—
374C3×C4—82no—
375C3×C4—122no—
376C3×C4—121no—
377C3×C4—121no—
378C3×C4—122no—
379C3×C4—121no—
380C3×C4—162no—
381Dic3—82no—
382Dic3—81no—
383C32—41noSylow
384C23—22no—
385C23—31no—
386C23—31no—
387C23—32no—
388C23—32no—
389C23—31no—
390C23—32no—
391C23—32no—
392C23—62no—
393C23—62no—
394C23—61no—
395C23—62no—
396C23—62no—
397C23—64no—
398C23—62no—
399C23—62no—
400C23—61no—
401C23—61no—
402C23—62no—
403C23—61no—
404C23—122no—
405C23C2×C3×D1211yescharacteristic
406C23C3×#23712yes—
407C23C2×C3×S411yescharacteristic
408D4—121no—
409D4—122no—
410D4—122no—
411D4—124no—
412D4—121no—
413D4—122no—
414C2×C4—32no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—61no—
432C2×C4—122no—
433C2×C4—124no—
434C2×C4—122no—
435C2×C4C2×C3×S412yes—
436C2×C3—22no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—32no—
440C2×C3—41no—
441C2×C3—42no—
442C2×C3—62no—
443C2×C3—82no—
444C2×C3—81no—
445C2×C3—82no—
446C2×C3—162no—
447C2×C3—241no—
448C2×C3C2×C4⋊S411yescharacteristic
449C2×C3#591412yes—
450D3—161no—
451C22—22no—
452C22—31no—
453C22—32no—
454C22—31no—
455C22—31no—
456C22—32no—
457C22—32no—
458C22—32no—
459C22—62no—
460C22—62no—
461C22—64no—
462C22—61no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—122no—
467C22—121no—
468C22—121no—
469C22—122no—
470C22—121no—
471C22—122no—
472C22C3×#88611yescharacteristic, complemented
473C22C3×C4⋊S412yes—
474C22C22×C3×S411yescharacteristic
475C4—22no—
476C4—62no—
477C4—122no—
478C4—121no—
479C4—121no—
480C4—122no—
481C4—121no—
482C3—41no—
483C3—81no—
484C3#8530111yescharacteristic, direct product factor
485C2—22no—
486C2—31no—
487C2—31no—
488C2—32no—
489C2—62no—
490C2—241no—
491C2C2×C3×C4⋊S411yescharacteristic
492C2C3×#591412yes—
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×C22⋊C4D3
socleC24×C3C22×D3
prime radicalC2×C3×GL(2,ℤ/4)C2
hypercenterC3×C22⋊C4S4
hypocenterA4C3×#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×SL(2,ℤ/4)C2semidirect
3. C3×A4×C22⋊C4C2semidirect
4. C3×(C2×S4)⋊C4C2semidirect
5. C3×C4⋊SL(2,ℤ/4)C2semidirect
6. #85301C3direct (N × H)
17. C22×SL(2,ℤ/4)C2×C3semidirect
21. C22×C3×C22⋊C4D3semidirect
23. A4×C22⋊C4C2×C3semidirect
24. (C2×S4)⋊C4C2×C3semidirect
25. C4⋊SL(2,ℤ/4)C2×C3semidirect
101. C22×C22⋊C4C3×D3semidirect
174. C3×C22⋊C4S4semidirect
182. C3×A4#123semidirect
337. C22⋊C4C3×S4semidirect
365. C22×C3#886semidirect
371. A4C3×#123semidirect
472. C22C3×#886semidirect
484. C3#85301direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×GL(2,ℤ/4)2
C22×C3×SL(2,ℤ/4)2
C3×A4×C22⋊C42
C3×(C2×S4)⋊C42
C3×C4⋊SL(2,ℤ/4)2
#853013
C3×#24123
C3×#8864