Finite Group Databasebeta

← C3×#25659 #10644266

Subgroups — C3×#25659 #10644266

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×C3×#489C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1864C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25659C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×#123—21no—
22C2×C3×#123—21no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C3×#489—24no—
29C3×#476—24no—
30C2×C3×#106C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#398—21no—
40C3×#398—21no—
41C3×#391—21no—
42C3×#389C2212yes—
43C3×#389C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#559C2×C311yescharacteristic
49C2×#489C2×C311yescharacteristic, complemented
50C2×#476C2×C311yescharacteristic, complemented
51#2546C2×C311yescharacteristic, complemented
52#2042C2×C312yescomplemented
53#2041C2×C312yes—
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic, complemented
56#1865C2×C312yescomplemented
57#1864C2×C312yescomplemented
58#1762C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8D411yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—44no—
73C3×#127—22no—
74C3×#127—22no—
75C3×#124—28no—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124—22no—
79C3×#124D414yes—
80C3×#123—28no—
81C3×#123—28no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×D4—28no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4D414yes—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C42—21no—
108C2×C3×C42—21no—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#106—24no—
113C3×C4⋊C8—22no—
114C3×#103—24no—
115C3×#103—24no—
116C3×#102—24no—
117C3×#102—24no—
118C3×#102—24no—
119C3×#102—24no—
120C3×#97—22no—
121C3×#95—21no—
122C3×#95—22no—
123C3×#95—22no—
124#559—24no—
125#559—24no—
126C2×C4⋊Q8—21no—
127C2×C4⋊Q8C22×C311yescharacteristic
128C2×#127C22×C311yescharacteristic
129C2×#124—21no—
130C2×#124C22×C311yescharacteristic
131C2×#124C22×C311yescharacteristic
132C2×#123—21no—
133C2×#123—21no—
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×Q8C22×C311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4C22×C311yescharacteristic
138C2×C4×D4C22×C311yescharacteristic
139#489—24no—
140#476—24no—
141C2×#106C22×C312yes—
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×#102C22×C312yes—
148C2×#102C22×C312yes—
149C2×#97C22×C311yescharacteristic
150#398—21no—
151#398—21no—
152#391—21no—
153#389C22×C312yes—
154#389C22×C312yes—
155C4×C4⋊C4C22×C311yescharacteristic
156#340C22×C312yes—
157#336C22×C312yes—
158#336C22×C312yes—
159C24×C3—21no—
160C24×C3—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—41no—
167C2×C3×Q8—42no—
168C2×C3×Q8—42no—
169C2×C3×Q8—42no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—24no—
172C2×C3×D4—24no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—42no—
180C2×C3×D4—42no—
181C2×C3×D4—82no—
182C2×C3×D4C2×D412yes—
183C2×C3×D4C2×D412yes—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—82no—
191C22×C3×C4—82no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—44no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—44no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4#4114yes—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C42—24no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—42no—
230C3×C42C2×D412yes—
231C3×C42C2×D412yes—
232C22×Q8—21no—
233C22×Q8C3×D411yescharacteristic
234C22×Q8C23×C311yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C23×C311yescharacteristic
238C22×D4C23×C311yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C22×C8C23×C312yes—
242C22×C8C23×C312yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—44no—
246#127—22no—
247#127—22no—
248#124—28no—
249#124—22no—
250#124—22no—
251#124—22no—
252#124C3×D414yes—
253#123—28no—
254#123—28no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—28no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4C3×D414yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C3×D411yescharacteristic
269C2×C4⋊C4C3×D411yescharacteristic
270C2×C4⋊C4C3×D411yescharacteristic
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C312yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42C23×C311yescharacteristic
283C2×C42C23×C311yescharacteristic
284C2×C42C23×C311yescharacteristic
285#106—24no—
286C4⋊C8—22no—
287#103—24no—
288#103—24no—
289#102—24no—
290#102—24no—
291#102—24no—
292#102—24no—
293#97—22no—
294#95—21no—
295#95—22no—
296#95—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—24no—
309C3×Q8—44no—
310C3×Q8—41no—
311C3×Q8—41no—
312C3×Q8—84no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—24no—
316C3×D4—48no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—42no—
320C3×D4—84no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—24no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—84no—
344C2×C3×C4—84no—
345C2×C3×C4#12012yes—
346C2×C3×C4#12012yes—
347C2×C3×C4AGL(1,ℤ/8)12yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4112yes—
351C2×C3×C42−1+412yes—
352C3×C8—42no—
353C3×C8—42no—
354C24—21no—
355C24—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8C2×C3×D412yes—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—82no—
377C2×D4C2×C3×D412yes—
378C2×D4C2×C3×D412yes—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C24×C311yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4C2×C3×D412yes—
412C4⋊C4C3×#4114yes—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C42—24no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—42no—
425C42C2×C3×D412yes—
426C42C2×C3×D412yes—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54812yes—
439C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
440C22×C3#57511yescharacteristic
441C22×C3#57712yes—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—82no—
452C3×C4#53211yescharacteristic
453C3×C4#53611yescharacteristic
454C3×C4#54712yes—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C22×C3×D411yescharacteristic
466Q8—24no—
467Q8—44no—
468Q8—41no—
469Q8—41no—
470Q8—84no—
471D4—22no—
472D4—22no—
473D4—24no—
474D4—48no—
475D4—41no—
476D4—41no—
477D4—42no—
478D4—84no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4C3×#12012yes—
504C2×C4C3×#12012yes—
505C2×C4C3×AGL(1,ℤ/8)12yes—
506C2×C4C22×C3×D411yescharacteristic
507C2×C4C22×C3×D411yescharacteristic
508C2×C4C2×C3×#4112yes—
509C2×C4C3×2−1+412yes—
510C8—42no—
511C8—42no—
512C2×C3—82no—
513C2×C3—82no—
514C2×C3#243211yescharacteristic
515C2×C3#300611yescharacteristic
516C2×C3#300911yescharacteristic
517C2×C3#337112yes—
518C2×C3#337312yes—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×C3×#12011yescharacteristic
530C22C3×#54812yes—
531C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
532C22C3×#57511yescharacteristic
533C22C3×#57712yes—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—42no—
542C4—44no—
543C4—82no—
544C4C3×#53211yescharacteristic
545C4C3×#53611yescharacteristic
546C4C3×#54712yes—
547C3#2565911yescharacteristic, direct product factor, Sylow
548C2—82no—
549C2—82no—
550C2C3×#243211yescharacteristic
551C2C3×#300611yescharacteristic
552C2C3×#300911yescharacteristic
553C2C3×#337112yes—
554C2C3×#337312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#489C2semidirect
3. C2×C3×#476C2semidirect
4. C3×#2546C2semidirect
5. C3×#2042C2semidirect
7. C3×#1985C2semidirect
8. C3×#1983C2semidirect
9. C3×#1865C2semidirect
10. C3×#1864C2semidirect
11. C3×#1762C2semidirect
12. #25659C3direct (N × H)
49. C2×#489C2×C3semidirect
50. C2×#476C2×C3semidirect
51. #2546C2×C3semidirect
52. #2042C2×C3semidirect
54. #1985C2×C3semidirect
55. #1983C2×C3semidirect
56. #1865C2×C3semidirect
57. #1864C2×C3semidirect
58. #1762C2×C3semidirect
547. C3#25659direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4892
C2×C3×#4762
C3×#25462
C3×#20422
C3×#20412
C3×#19852
C3×#19832
C3×#18652
C3×#18642
C3×#17622
#256593