Finite Group Databasebeta

← C3×#24677 #10643284

Subgroups — C3×#24677 #10643284

Order 768.

870 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#476C211yesmaximal, characteristic, complemented
2C2×C3×C4×D8C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic, complemented
5C3×#2073C212yesmaximal, complemented
6C3×#2054C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24677C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×#117—21no—
18C3×#476—24no—
19C3×C4×D8—24no—
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C4×C8C2211yescharacteristic
29C3×#393—22no—
30C3×#390C2212yes—
31C3×#390C2212yes—
32C3×#388—22no—
33C3×#384C2211yescharacteristic
34C3×#382C2212yes—
35C3×#380—21no—
36C3×#380—21no—
37C3×#380—21no—
38C3×C4×C4⋊C4C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#476C2×C311yescharacteristic, complemented
45C2×C4×D8C2×C311yescharacteristic
46#2563C2×C311yescharacteristic, complemented
47#2354C2×C311yescharacteristic, complemented
48#2073C2×C312yescomplemented
49#2054C2×C312yes—
50#1990C2×C311yescharacteristic, complemented
51#1949C2×C312yescomplemented
52#1939C2×C311yescharacteristic, complemented
53#1926C2×C312yescomplemented
54#1802C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C2×C3×D8—22no—
61C2×C3×D8—22no—
62C2×C3×D8—22no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C3×#117—44no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C42—22no—
89C2×C3×C42C2312yes—
90C2×C3×C42C2311yescharacteristic
91C3×#107—22no—
92C3×#106—24no—
93C3×C4⋊C8—24no—
94C3×C4⋊C8—22no—
95C3×#102—22no—
96C3×#102—24no—
97C3×#102—24no—
98C3×#102—22no—
99C3×C4×C8—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95D412yes—
104C22×D8C22×C311yescharacteristic
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107C2×C4×D4C22×C311yescharacteristic
108C2×#117—21no—
109#476—24no—
110C4×D8—24no—
111C2×#107C22×C311yescharacteristic
112C2×#106C22×C312yes—
113C2×C4⋊C8C22×C312yes—
114C2×C4⋊C8C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×#102C22×C311yescharacteristic
118C2×#102C22×C311yescharacteristic
119C2×C4×C8C22×C311yescharacteristic
120#393—22no—
121#390C22×C312yes—
122#390C22×C312yes—
123#388—22no—
124#384C22×C311yescharacteristic
125#382C22×C312yes—
126#380—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4C22×C312yes—
130C4×C4⋊C4C22×C311yescharacteristic
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135C24×C3—21no—
136C24×C3—21no—
137C2×C3×Q8—42no—
138C2×C3×Q8—81no—
139C2×C3×Q8—81no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4C2×D412yes—
151C2×C3×D4#4112yes—
152C22×C3×C4—21no—
153C22×C3×C4—24no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C3×D8—28no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8#4112yes—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—81no—
189C3×C4⋊C4C2×D412yes—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—82no—
194C3×C22⋊C4—82no—
195C3×C22⋊C4—82no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—44no—
199C3×C42C2×D412yes—
200C3×C42#4112yes—
201C22×Q8—21no—
202C22×D4C23×C311yescharacteristic
203C22×D4C23×C311yescharacteristic
204C23×C4—21no—
205C23×C4—21no—
206C2×D8—22no—
207C2×D8—22no—
208C2×D8—22no—
209C22×C8C23×C311yescharacteristic
210C22×C8C23×C312yes—
211C22×C8C23×C311yescharacteristic
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4C3×D414yes—
218#117—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C3×D411yescharacteristic
223C2×C4⋊C4C3×D411yescharacteristic
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C4⋊C4C23×C312yes—
226C2×C4⋊C4C23×C312yes—
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C42—22no—
235C2×C42C23×C312yes—
236C2×C42C23×C311yescharacteristic
237#107—22no—
238#106—24no—
239C4⋊C8—24no—
240C4⋊C8—22no—
241#102—22no—
242#102—24no—
243#102—24no—
244#102—22no—
245C4×C8—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95C3×D412yes—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3C22×D411yescharacteristic
265C3×Q8—84no—
266C3×D4—22no—
267C3×D4—22no—
268C3×D4—22no—
269C3×D4—22no—
270C3×D4—41no—
271C3×D4—41no—
272C3×D4—41no—
273C3×D4—41no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—48no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—84no—
296C2×C3×C4#12112yes—
297C2×C3×C4#12112yes—
298C2×C3×C4#13512yes—
299C2×C3×C4AGL(1,ℤ/8)12yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×C3×C42−1+411yescharacteristic
306C3×C8—22no—
307C3×C8—42no—
308C3×C8—41no—
309C24—21no—
310C24—21no—
311C2×Q8—42no—
312C2×Q8—81no—
313C2×Q8—81no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4C2×C3×D412yes—
325C2×D4C3×#4112yes—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C24×C311yescharacteristic
341D8—28no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8C3×#4112yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—81no—
363C4⋊C4C2×C3×D412yes—
364C22⋊C4—24no—
365C22⋊C4—44no—
366C22⋊C4—42no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C42—22no—
371C42—22no—
372C42—44no—
373C42C2×C3×D412yes—
374C42C3×#4112yes—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3C2×#12111yescharacteristic
390C22×C3#54711yescharacteristic
391C22×C3#54811yescharacteristic
392C22×C3C2×#13511yescharacteristic
393C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C3#57611yescharacteristic
395C22×C3#57711yescharacteristic
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—82no—
401C3×C4—81no—
402C3×C4—82no—
403C3×C4#46314yes—
404C3×C4#52511yescharacteristic
405C3×C4#53611yescharacteristic
406C3×C4#55211yescharacteristic
407C3×C4#55611yescharacteristic
408C23—22no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C22×C3×D411yescharacteristic
423Q8—84no—
424D4—22no—
425D4—22no—
426D4—22no—
427D4—22no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—48no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—84no—
454C2×C4C3×#12112yes—
455C2×C4C3×#12112yes—
456C2×C4C3×#13512yes—
457C2×C4C3×AGL(1,ℤ/8)12yes—
458C2×C4C22×C3×D411yescharacteristic
459C2×C4C2×C3×#4111yescharacteristic
460C2×C4C2×C3×#4111yescharacteristic
461C2×C4C2×C3×#4111yescharacteristic
462C2×C4C3×2−1+411yescharacteristic
463C2×C4C3×2−1+411yescharacteristic
464C8—22no—
465C8—42no—
466C8—41no—
467C2×C3—82no—
468C2×C3—82no—
469C2×C3#241511yescharacteristic
470C2×C3C2×#46311yescharacteristic
471C2×C3#304511yescharacteristic
472C2×C3#334111yescharacteristic
473C2×C3#335411yescharacteristic
474C2×C3#336011yescharacteristic
475C2×C3#336611yescharacteristic
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22C2×C3×#12111yescharacteristic
491C22C3×#54711yescharacteristic
492C22C3×#54811yescharacteristic
493C22C2×C3×#13511yescharacteristic
494C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
495C22C3×#57611yescharacteristic
496C22C3×#57711yescharacteristic
497C4—22no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—81no—
502C4—82no—
503C4—82no—
504C4C3×#46314yes—
505C4C3×#52511yescharacteristic
506C4C3×#53611yescharacteristic
507C4C3×#55211yescharacteristic
508C4C3×#55611yescharacteristic
509C3#2467711yescharacteristic, direct product factor, Sylow
510C2—82no—
511C2—82no—
512C2C3×#241511yescharacteristic
513C2C2×C3×#46311yescharacteristic
514C2C3×#304511yescharacteristic
515C2C3×#334111yescharacteristic
516C2C3×#335411yescharacteristic
517C2C3×#336011yescharacteristic
518C2C3×#336611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#476C2semidirect
3. C3×#2563C2semidirect
4. C3×#2354C2semidirect
5. C3×#2073C2semidirect
7. C3×#1990C2semidirect
8. C3×#1949C2semidirect
9. C3×#1939C2semidirect
10. C3×#1926C2semidirect
11. C3×#1802C2semidirect
12. #24677C3direct (N × H)
44. C2×#476C2×C3semidirect
46. #2563C2×C3semidirect
47. #2354C2×C3semidirect
48. #2073C2×C3semidirect
50. #1990C2×C3semidirect
51. #1949C2×C3semidirect
52. #1939C2×C3semidirect
53. #1926C2×C3semidirect
54. #1802C2×C3semidirect
509. C3#24677direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4762
C2×C3×C4×D82
C3×#25632
C3×#23542
C3×#20732
C3×#20542
C3×#19902
C3×#19492
C3×#19392
C3×#19262
C3×#18022
#246773