Finite Group Databasebeta

← C3×#24702 #10643309

Subgroups — C3×#24702 #10643309

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#477C211yesmaximal, characteristic, complemented
2C2×C3×#431C211yesmaximal, characteristic
3C3×#2857C211yesmaximal, characteristic
4C3×C4×C22⋊Q8C211yesmaximal, characteristic
5C3×#2076C212yesmaximal
6C3×#2036C212yesmaximal
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1947C211yesmaximal, characteristic, complemented
9C3×#1942C212yesmaximal, complemented
10C3×#1933C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24702C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C3×#477—24no—
19C3×#431—24no—
20C2×C3×#107C2212yes—
21C2×C3×#107C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×C22⋊C8C2212yes—
27C2×C3×C4×C8C2212yes—
28C3×#391—21no—
29C3×#391C2211yescharacteristic
30C3×#389—21no—
31C3×#389C2211yescharacteristic
32C3×#389C2212yes—
33C3×#387—21no—
34C3×#385—21no—
35C3×#384C2211yescharacteristic
36C3×#380—21no—
37C3×#380C2212yes—
38C3×#380C2211yescharacteristic
39C3×C4×C4⋊C4—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#477C2×C311yescharacteristic, complemented
47C2×#431C2×C311yescharacteristic
48#2857C2×C311yescharacteristic
49C4×C22⋊Q8C2×C311yescharacteristic
50#2076C2×C312yes—
51#2036C2×C312yes—
52#1987C2×C311yescharacteristic, complemented
53#1947C2×C311yescharacteristic, complemented
54#1942C2×C312yescomplemented
55#1933C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8—22no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—44no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42—21no—
89C2×C3×C42—21no—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×#107—24no—
95C3×#107—24no—
96C3×C4⋊C8—22no—
97C3×C4⋊C8D414yes—
98C3×#103—24no—
99C3×#103—24no—
100C3×C22⋊C8—24no—
101C3×C4×C8—24no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—22no—
108C3×#95—22no—
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C22×C311yescharacteristic
112C2×C4×Q8C22×C311yescharacteristic
113C22×C42C22×C311yescharacteristic
114#477—24no—
115#431—24no—
116C2×#107C22×C312yes—
117C2×#107C22×C312yes—
118C2×C4⋊C8C22×C311yescharacteristic
119C2×C4⋊C8C22×C311yescharacteristic
120C2×#103C22×C312yes—
121C2×#103C22×C312yes—
122C2×C22⋊C8C22×C312yes—
123C2×C4×C8C22×C312yes—
124#391—21no—
125#391C22×C311yescharacteristic
126#389—21no—
127#389C22×C312yes—
128#389C22×C311yescharacteristic
129#387—21no—
130#385—21no—
131#384C22×C311yescharacteristic
132#380—21no—
133#380C22×C311yescharacteristic
134#380C22×C312yes—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C22×C311yescharacteristic
137C4×C22⋊C4—21no—
138#340C22×C312yes—
139#340C22×C312yes—
140#340C22×C312yes—
141#340C22×C312yes—
142C24×C3C2×D411yescharacteristic
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—42no—
149C2×C3×Q8—42no—
150C2×C3×Q8—42no—
151C2×C3×Q8#4112yes—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—41no—
169C22×C3×C4Q1614yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8C2×D414yes—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—44no—
198C3×C4⋊C4#4112yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—44no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—21no—
205C3×C42—21no—
206C3×C42—42no—
207C3×C42—42no—
208C3×C42C2×D412yes—
209C3×C42#4112yes—
210C22×Q8—21no—
211C22×Q8C23×C311yescharacteristic
212C23×C4C3×D411yescharacteristic
213C23×C4C3×D411yescharacteristic
214C23×C4C23×C311yescharacteristic
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—44no—
222C4×Q8—22no—
223C4×Q8—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C312yes—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42C3×D411yescharacteristic
244C2×C42C3×D411yescharacteristic
245C2×C42C23×C311yescharacteristic
246C2×C42C23×C311yescharacteristic
247#107—24no—
248#107—24no—
249C4⋊C8—22no—
250C4⋊C8C3×D414yes—
251#103—24no—
252#103—24no—
253C22⋊C8—24no—
254C4×C8—24no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—22no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3C2×Q1612yes—
269C23×C3C22×D411yescharacteristic
270C3×Q8—22no—
271C3×Q8—22no—
272C3×Q8—41no—
273C3×Q8—41no—
274C3×Q8—84no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—28no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—44no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4#12112yes—
309C2×C3×C4#12112yes—
310C2×C3×C4C2×Q1612yes—
311C2×C3×C4C2×Q1612yes—
312C2×C3×C4#13512yes—
313C2×C3×C4AGL(1,ℤ/8)12yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C42−1+411yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320C3×C8—24no—
321C3×C8—42no—
322C24C2×C3×D411yescharacteristic
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8C3×#4112yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4C3×Q1614yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C24×C311yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8C2×C3×D414yes—
362C4⋊C4—24no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4C3×#4112yes—
379C22⋊C4—24no—
380C22⋊C4—44no—
381C22⋊C4—42no—
382C42—22no—
383C42—22no—
384C42—21no—
385C42—21no—
386C42—42no—
387C42—42no—
388C42C2×C3×D412yes—
389C42C3×#4112yes—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3C2×#12111yescharacteristic
398C22×C3#52511yescharacteristic
399C22×C3#53611yescharacteristic
400C22×C3C22×Q1611yescharacteristic
401C22×C3C2×#13511yescharacteristic
402C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C3#57511yescharacteristic
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—42no—
408C3×C4—41no—
409C3×C4—42no—
410C3×C4—82no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4—82no—
414C3×C4#46714yes—
415C3×C4#54711yescharacteristic
416C3×C4#54811yescharacteristic
417C3×C4#55211yescharacteristic
418C3×C4#55611yescharacteristic
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23C2×C3×Q1612yes—
426C23C22×C3×D411yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—84no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4C3×#12112yes—
466C2×C4C3×#12112yes—
467C2×C4C2×C3×Q1612yes—
468C2×C4C2×C3×Q1612yes—
469C2×C4C3×#13512yes—
470C2×C4C3×AGL(1,ℤ/8)12yes—
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C2×C3×#4111yescharacteristic
473C2×C4C2×C3×#4111yescharacteristic
474C2×C4C2×C3×#4111yescharacteristic
475C2×C4C3×2−1+411yescharacteristic
476C2×C4C3×2−1+411yescharacteristic
477C8—24no—
478C8—42no—
479C2×C3—42no—
480C2×C3#241511yescharacteristic
481C2×C3C2×#46711yescharacteristic
482C2×C3#305511yescharacteristic
483C2×C3#310411yescharacteristic
484C2×C3#310611yescharacteristic
485C2×C3#323011yescharacteristic
486C2×C3#323311yescharacteristic
487C22—22no—
488C22—22no—
489C22—22no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22C2×C3×#12111yescharacteristic
495C22C3×#52511yescharacteristic
496C22C3×#53611yescharacteristic
497C22C22×C3×Q1611yescharacteristic
498C22C2×C3×#13511yescharacteristic
499C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
500C22C3×#57511yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—82no—
509C4—81no—
510C4—82no—
511C4C3×#46714yes—
512C4C3×#54711yescharacteristic
513C4C3×#54811yescharacteristic
514C4C3×#55211yescharacteristic
515C4C3×#55611yescharacteristic
516C3#2470211yescharacteristic, direct product factor, Sylow
517C2—42no—
518C2C3×#241511yescharacteristic
519C2C2×C3×#46711yescharacteristic
520C2C3×#305511yescharacteristic
521C2C3×#310411yescharacteristic
522C2C3×#310611yescharacteristic
523C2C3×#323011yescharacteristic
524C2C3×#323311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#477C2semidirect
7. C3×#1987C2semidirect
8. C3×#1947C2semidirect
9. C3×#1942C2semidirect
10. C3×#1933C2semidirect
11. C3×#1803C2semidirect
12. #24702C3direct (N × H)
46. C2×#477C2×C3semidirect
52. #1987C2×C3semidirect
53. #1947C2×C3semidirect
54. #1942C2×C3semidirect
55. #1933C2×C3semidirect
56. #1803C2×C3semidirect
516. C3#24702direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4772
C2×C3×#4312
C3×#28572
C3×C4×C22⋊Q82
C3×#20762
C3×#20362
C3×#19872
C3×#19472
C3×#19422
C3×#19332
C3×#18032
#247023