Finite Group Databasebeta

← C3×#24652 #10643259

Subgroups — C3×#24652 #10643259

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#462C211yesmaximal, characteristic, complemented
2C2×C3×#431C211yesmaximal, characteristic
3C3×#2857C211yesmaximal, characteristic
4C3×C4×C22⋊Q8C211yesmaximal, characteristic
5C3×#2036C212yesmaximal
6C3×#2001C212yesmaximal
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1969C212yesmaximal, complemented
9C3×#1947C211yesmaximal, characteristic, complemented
10C3×#1933C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24652C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C3×#462—24no—
20C3×#431—24no—
21C2×C3×#107C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×C4×C8C2212yes—
30C3×#391C2211yescharacteristic
31C3×#391C2212yes—
32C3×#389—21no—
33C3×#389—21no—
34C3×#389C2211yescharacteristic
35C3×#387—21no—
36C3×#385—21no—
37C3×#384C2211yescharacteristic
38C3×#380—21no—
39C3×#380—21no—
40C3×#380C2211yescharacteristic
41C3×C4×C4⋊C4—21no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C22⋊C4—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#462C2×C311yescharacteristic, complemented
47C2×#431C2×C311yescharacteristic
48#2857C2×C311yescharacteristic
49C4×C22⋊Q8C2×C311yescharacteristic
50#2036C2×C312yes—
51#2001C2×C312yes—
52#1981C2×C311yescharacteristic, complemented
53#1969C2×C312yescomplemented
54#1947C2×C311yescharacteristic, complemented
55#1933C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×Q16—24no—
63C2×C3×Q16—24no—
64C2×C3×Q16—42no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—42no—
72C3×C22⋊Q8—42no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C42—22no—
91C2×C3×C42—22no—
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×C3×C42D411yescharacteristic
95C2×C3×C42D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×#107—42no—
99C3×C4⋊C8—22no—
100C3×C4⋊C8D414yes—
101C3×#103—24no—
102C3×#103—24no—
103C3×#103—42no—
104C3×#103—42no—
105C3×C22⋊C8—24no—
106C3×C4×C8—24no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—22no—
113C3×#95—22no—
114C22×Q16C22×C312yes—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×C22⋊Q8C22×C311yescharacteristic
117C2×C22⋊Q8C22×C312yes—
118C2×C4×Q8C22×C311yescharacteristic
119C22×C42C22×C311yescharacteristic
120#462—24no—
121#431—24no—
122C2×#107C22×C312yes—
123C2×C4⋊C8C22×C311yescharacteristic
124C2×C4⋊C8C22×C311yescharacteristic
125C2×#103C22×C312yes—
126C2×#103C22×C312yes—
127C2×#103C22×C312yes—
128C2×#103C22×C312yes—
129C2×C22⋊C8C22×C312yes—
130C2×C4×C8C22×C312yes—
131#391C22×C312yes—
132#391C22×C311yescharacteristic
133#389—21no—
134#389—21no—
135#389C22×C311yescharacteristic
136#387—21no—
137#385—21no—
138#384C22×C311yescharacteristic
139#380—21no—
140#380—21no—
141#380C22×C311yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4C22×C311yescharacteristic
144C4×C22⋊C4—21no—
145#340C22×C312yes—
146#340C22×C312yes—
147C24×C3C2×D411yescharacteristic
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8#4112yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—41no—
177C22×C3×C4Q1614yescomplemented
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C3×Q16—48no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8C2×D414yes—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4—44no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4#4112yes—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—44no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—42no—
215C3×C42—42no—
216C3×C42C2×D412yes—
217C3×C42#4112yes—
218C22×Q8C23×C312yes—
219C22×Q8C23×C311yescharacteristic
220C23×C4C3×D411yescharacteristic
221C23×C4C3×D411yescharacteristic
222C23×C4C23×C311yescharacteristic
223C2×Q16—24no—
224C2×Q16—24no—
225C2×Q16—42no—
226C22×C8C23×C312yes—
227C22×C8C23×C312yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—42no—
233C22⋊Q8—42no—
234C4×Q8—22no—
235C4×Q8—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C23×C311yescharacteristic
245C2×C4⋊C4C23×C312yes—
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C4⋊C4C23×C311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C3×D411yescharacteristic
256C2×C42C3×D411yescharacteristic
257C2×C42C23×C311yescharacteristic
258C2×C42C23×C311yescharacteristic
259#107—42no—
260C4⋊C8—22no—
261C4⋊C8C3×D414yes—
262#103—24no—
263#103—24no—
264#103—42no—
265#103—42no—
266C22⋊C8—24no—
267C4×C8—24no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3C2×Q1612yes—
282C23×C3C22×D411yescharacteristic
283C3×Q8—22no—
284C3×Q8—22no—
285C3×Q8—41no—
286C3×Q8—41no—
287C3×Q8—42no—
288C3×Q8—42no—
289C3×Q8—42no—
290C3×Q8—42no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—28no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—44no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4#12112yes—
325C2×C3×C4#12112yes—
326C2×C3×C4C2×Q1612yes—
327C2×C3×C4C2×Q1612yes—
328C2×C3×C4#13512yes—
329C2×C3×C4#13712yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335C2×C3×C4H(5,𝔽2)11yescharacteristic
336C3×C8—24no—
337C3×C8—42no—
338C24C2×C3×D411yescharacteristic
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8C3×#4112yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4C3×Q1614yescomplemented
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C24×C311yescharacteristic
375Q16—48no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8C2×C3×D414yes—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4C3×#4112yes—
398C22⋊C4—24no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C42—22no—
402C42—22no—
403C42—21no—
404C42—21no—
405C42—42no—
406C42—42no—
407C42C2×C3×D412yes—
408C42C3×#4112yes—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3C2×#12111yescharacteristic
417C22×C3#52511yescharacteristic
418C22×C3#53511yescharacteristic
419C22×C3C22×Q1611yescharacteristic
420C22×C3C2×#13511yescharacteristic
421C22×C3C2×#13711yescharacteristic
422C22×C3#57511yescharacteristic
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—42no—
429C3×C4—82no—
430C3×C4—82no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4#46714yes—
434C3×C4#54611yescharacteristic
435C3×C4#55011yescharacteristic
436C3×C4#55311yescharacteristic
437C3×C4#55511yescharacteristic
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23C2×C3×Q1612yes—
445C23C22×C3×D411yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—41no—
449Q8—41no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—28no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4C3×#12112yes—
488C2×C4C3×#12112yes—
489C2×C4C2×C3×Q1612yes—
490C2×C4C2×C3×Q1612yes—
491C2×C4C3×#13512yes—
492C2×C4C3×#13712yes—
493C2×C4C22×C3×D411yescharacteristic
494C2×C4C2×C3×#4111yescharacteristic
495C2×C4C2×C3×#4111yescharacteristic
496C2×C4C2×C3×#4111yescharacteristic
497C2×C4C3×H(5,𝔽2)11yescharacteristic
498C2×C4C3×H(5,𝔽2)11yescharacteristic
499C8—24no—
500C8—42no—
501C2×C3—42no—
502C2×C3#241411yescharacteristic
503C2×C3C2×#46711yescharacteristic
504C2×C3#305611yescharacteristic
505C2×C3#310411yescharacteristic
506C2×C3#310711yescharacteristic
507C2×C3#320911yescharacteristic
508C2×C3#321211yescharacteristic
509C22—22no—
510C22—22no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22C2×C3×#12111yescharacteristic
517C22C3×#52511yescharacteristic
518C22C3×#53511yescharacteristic
519C22C22×C3×Q1611yescharacteristic
520C22C2×C3×#13511yescharacteristic
521C22C2×C3×#13711yescharacteristic
522C22C3×#57511yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—41no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—81no—
531C4—82no—
532C4—81no—
533C4C3×#46714yes—
534C4C3×#54611yescharacteristic
535C4C3×#55011yescharacteristic
536C4C3×#55311yescharacteristic
537C4C3×#55511yescharacteristic
538C3#2465211yescharacteristic, direct product factor, Sylow
539C2—42no—
540C2C3×#241411yescharacteristic
541C2C2×C3×#46711yescharacteristic
542C2C3×#305611yescharacteristic
543C2C3×#310411yescharacteristic
544C2C3×#310711yescharacteristic
545C2C3×#320911yescharacteristic
546C2C3×#321211yescharacteristic
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×#462C2semidirect
7. C3×#1981C2semidirect
8. C3×#1969C2semidirect
9. C3×#1947C2semidirect
10. C3×#1933C2semidirect
11. C3×#1803C2semidirect
12. #24652C3direct (N × H)
46. C2×#462C2×C3semidirect
52. #1981C2×C3semidirect
53. #1969C2×C3semidirect
54. #1947C2×C3semidirect
55. #1933C2×C3semidirect
56. #1803C2×C3semidirect
177. C22×C3×C4Q16semidirect
368. C22×C4C3×Q16semidirect
538. C3#24652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4622
C2×C3×#4312
C3×#28572
C3×C4×C22⋊Q82
C3×#20362
C3×#20012
C3×#19812
C3×#19692
C3×#19472
C3×#19332
C3×#18032
#246523