Finite Group Databasebeta

← C3×#24630 #10643237

Subgroups — C3×#24630 #10643237

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic
2C2×C3×C4×SD16C211yesmaximal, characteristic
3C3×#2546C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic
5C3×#2055C212yesmaximal
6C3×#2013C212yesmaximal
7C3×#1981C212yesmaximal, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1927C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24630C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117—21no—
20C3×#461—24no—
21C3×C4×SD16—24no—
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#102C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32C3×#398—22no—
33C3×#391C2212yes—
34C3×#390C2212yes—
35C3×#389—21no—
36C3×#389—21no—
37C3×#388—22no—
38C3×#384C2211yescharacteristic
39C3×#380—21no—
40C3×C4×C4⋊C4C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#461C2×C311yescharacteristic
45C2×C4×SD16C2×C311yescharacteristic
46#2546C2×C311yescharacteristic, complemented
47#2354C2×C311yescharacteristic
48#2055C2×C312yes—
49#2013C2×C312yes—
50#1981C2×C312yescomplemented
51#1956C2×C311yescharacteristic, complemented
52#1939C2×C311yescharacteristic, complemented
53#1927C2×C312yescomplemented
54#1803C2×C311yescharacteristic, complemented
55C22×C3×Q8C2312yes—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—22no—
60C2×C3×SD16—22no—
61C2×C3×SD16—24no—
62C2×C3×SD16—22no—
63C2×C3×SD16—42no—
64C2×C3×SD16—42no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—42no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C3×#117—44no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C42—22no—
93C2×C3×C42—21no—
94C2×C3×C42C2312yes—
95C2×C3×C42C2311yescharacteristic
96C3×#106—22no—
97C3×C4⋊C8—22no—
98C3×C4⋊C8—42no—
99C3×#103—24no—
100C3×#103—22no—
101C3×#103—24no—
102C3×#103—42no—
103C3×#103—42no—
104C3×#102—22no—
105C3×C4×C8—22no—
106C3×#95—22no—
107C3×#95—22no—
108C3×#95—22no—
109C3×#95D412yes—
110C22×SD16C22×C311yescharacteristic
111C22×SD16C22×C312yes—
112C2×C4⋊Q8C22×C311yescharacteristic
113C2×C4⋊Q8C22×C312yes—
114C2×C4×Q8C22×C311yescharacteristic
115C2×C4×D4C22×C311yescharacteristic
116C2×#117—21no—
117#461—24no—
118C4×SD16—24no—
119C2×#106C22×C311yescharacteristic
120C2×C4⋊C8C22×C311yescharacteristic
121C2×C4⋊C8C22×C312yes—
122C2×#103C22×C311yescharacteristic
123C2×#103C22×C312yes—
124C2×#103C22×C312yes—
125C2×#103C22×C312yes—
126C2×#103C22×C312yes—
127C2×#102C22×C311yescharacteristic
128C2×C4×C8C22×C311yescharacteristic
129#398—22no—
130#391C22×C312yes—
131#390C22×C312yes—
132#389—21no—
133#389—21no—
134#388—22no—
135#384C22×C311yescharacteristic
136#380—21no—
137C4×C4⋊C4C22×C312yes—
138C4×C4⋊C4C22×C311yescharacteristic
139#340C22×C312yes—
140#336C22×C312yes—
141C24×C3—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8—22no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8#4112yes—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4C2×D412yes—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C3×SD16—28no—
174C3×SD16—48no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8#4112yes—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—44no—
191C3×C4⋊C4—44no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—81no—
197C3×C4⋊C4SD1614yescomplemented
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—44no—
201C3×C22⋊C4—82no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—42no—
205C3×C42—44no—
206C3×C42C2×D412yes—
207C3×C42#4112yes—
208C22×Q8C23×C312yes—
209C22×Q8C23×C311yescharacteristic
210C22×D4C23×C311yescharacteristic
211C23×C4—21no—
212C2×SD16—22no—
213C2×SD16—22no—
214C2×SD16—24no—
215C2×SD16—22no—
216C2×SD16—42no—
217C2×SD16—42no—
218C22×C8C23×C312yes—
219C22×C8C23×C311yescharacteristic
220C22×C8C23×C311yescharacteristic
221C4⋊Q8—24no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—42no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×D4—22no—
228C4×D4C3×D414yes—
229#117—44no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C3×D411yescharacteristic
238C2×C4⋊C4C3×D411yescharacteristic
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C312yes—
241C2×C4⋊C4C23×C311yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42C23×C312yes—
248C2×C42C23×C311yescharacteristic
249#106—22no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252#103—24no—
253#103—22no—
254#103—24no—
255#103—42no—
256#103—42no—
257#102—22no—
258C4×C8—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95C3×D412yes—
263C23×C3—22no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3C22×D411yescharacteristic
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—41no—
274C3×Q8—41no—
275C3×Q8—42no—
276C3×Q8—42no—
277C3×Q8—42no—
278C3×Q8—42no—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—41no—
282C3×D4—41no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—48no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4#12112yes—
307C2×C3×C4#12112yes—
308C2×C3×C4C2×SD1614yes—
309C2×C3×C4C2×SD1612yes—
310C2×C3×C4#13712yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C3×C8—22no—
318C3×C8—41no—
319C3×C8—42no—
320C24—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8C3×#4112yes—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4C2×C3×D412yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4C2×C3×D412yes—
348C22×C4C2×C3×D411yescharacteristic
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C24×C311yescharacteristic
352SD16—28no—
353SD16—48no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8C3×#4112yes—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—81no—
376C4⋊C4C3×SD1614yescomplemented
377C4⋊C4C2×C3×D412yes—
378C22⋊C4—24no—
379C22⋊C4—44no—
380C22⋊C4—82no—
381C42—22no—
382C42—22no—
383C42—42no—
384C42—44no—
385C42C2×C3×D412yes—
386C42C3×#4112yes—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3C2×#12111yescharacteristic
395C22×C3#54611yescharacteristic
396C22×C3#55011yescharacteristic
397C22×C3C22×SD1611yescharacteristic
398C22×C3C2×#13711yescharacteristic
399C22×C3#57711yescharacteristic
400C22×C3#57811yescharacteristic
401C3×C4—24no—
402C3×C4—22no—
403C3×C4—42no—
404C3×C4—42no—
405C3×C4—42no—
406C3×C4—82no—
407C3×C4—82no—
408C3×C4—81no—
409C3×C4#46114yes—
410C3×C4#52511yescharacteristic
411C3×C4#53511yescharacteristic
412C3×C4#55311yescharacteristic
413C3×C4#55511yescharacteristic
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23C22×C3×D411yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—41no—
425Q8—41no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430D4—22no—
431D4—22no—
432D4—41no—
433D4—41no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4C3×#12112yes—
458C2×C4C3×#12112yes—
459C2×C4C2×C3×SD1614yes—
460C2×C4C2×C3×SD1612yes—
461C2×C4C3×#13712yes—
462C2×C4C22×C3×D411yescharacteristic
463C2×C4C2×C3×#4111yescharacteristic
464C2×C4C2×C3×#4111yescharacteristic
465C2×C4C2×C3×#4111yescharacteristic
466C2×C4C3×H(5,𝔽2)11yescharacteristic
467C2×C4C3×H(5,𝔽2)11yescharacteristic
468C8—22no—
469C8—42no—
470C8—41no—
471C2×C3—82no—
472C2×C3#241411yescharacteristic
473C2×C3C2×#46111yescharacteristic
474C2×C3#304611yescharacteristic
475C2×C3#329811yescharacteristic
476C2×C3#332911yescharacteristic
477C2×C3#339211yescharacteristic
478C2×C3#339811yescharacteristic
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22C2×C3×#12111yescharacteristic
487C22C3×#54611yescharacteristic
488C22C3×#55011yescharacteristic
489C22C22×C3×SD1611yescharacteristic
490C22C2×C3×#13711yescharacteristic
491C22C3×#57711yescharacteristic
492C22C3×#57811yescharacteristic
493C4—24no—
494C4—22no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—82no—
499C4—82no—
500C4—81no—
501C4C3×#46114yes—
502C4C3×#52511yescharacteristic
503C4C3×#53511yescharacteristic
504C4C3×#55311yescharacteristic
505C4C3×#55511yescharacteristic
506C3#2463011yescharacteristic, direct product factor, Sylow
507C2—82no—
508C2C3×#241411yescharacteristic
509C2C2×C3×#46111yescharacteristic
510C2C3×#304611yescharacteristic
511C2C3×#329811yescharacteristic
512C2C3×#332911yescharacteristic
513C2C3×#339211yescharacteristic
514C2C3×#339811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
3. C3×#2546C2semidirect
7. C3×#1981C2semidirect
8. C3×#1956C2semidirect
9. C3×#1939C2semidirect
10. C3×#1927C2semidirect
11. C3×#1803C2semidirect
12. #24630C3direct (N × H)
46. #2546C2×C3semidirect
50. #1981C2×C3semidirect
51. #1956C2×C3semidirect
52. #1939C2×C3semidirect
53. #1927C2×C3semidirect
54. #1803C2×C3semidirect
197. C3×C4⋊C4SD16semidirect
376. C4⋊C4C3×SD16semidirect
506. C3#24630direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4612
C2×C3×C4×SD162
C3×#25462
C3×#23542
C3×#20552
C3×#20132
C3×#19812
C3×#19562
C3×#19392
C3×#19272
C3×#18032
#246303