Finite Group Databasebeta

← C3×#25709 #10644316

Subgroups — C3×#25709 #10644316

Order 768.

1276 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#478C211yesmaximal, characteristic, complemented
4C3×#2562C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1853C212yesmaximal, complemented
10C3×#1852C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25709C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C3×#547—24no—
15C3×#547—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C4⋊C4C2212yes—
24C3×#501—24no—
25C3×#478—24no—
26C2×C3×#107C2212yes—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×C22⋊C8C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#400—21no—
38C3×#394—21no—
39C3×#388—22no—
40C3×#380C2212yes—
41C3×#380C2212yes—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#547C2×C311yescharacteristic
47C2×#501C2×C311yescharacteristic, complemented
48C2×#478C2×C311yescharacteristic, complemented
49#2562C2×C311yescharacteristic, complemented
50#2034C2×C312yescomplemented
51#2033C2×C312yescomplemented
52#1987C2×C311yescharacteristic, complemented
53#1955C2×C311yescharacteristic, complemented
54#1853C2×C312yescomplemented
55#1852C2×C312yescomplemented
56#1762C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8—21no—
60C22×C3×D4—22no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C23×C3×C4C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×#125—22no—
72C3×#125—22no—
73C3×#123—216no—
74C3×C22⋊Q8—28no—
75C3×#121—24no—
76C3×#121—24no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4D414yes—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4D412yes—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C3×#107—22no—
101C3×#107—24no—
102C3×#106—22no—
103C3×#106—24no—
104C3×C4⋊C8—22no—
105C3×#102—22no—
106C3×#102—22no—
107C3×#102—24no—
108C3×C22⋊C8—24no—
109C3×C22⋊C8—24no—
110C3×#97—22no—
111C3×#95—21no—
112C3×#95—22no—
113C3×#95—22no—
114C22×#41—21no—
115#547—24no—
116#547—24no—
117C2×C4⋊Q8C22×C311yescharacteristic
118C2×#125C22×C311yescharacteristic
119C2×#123—22no—
120C2×C22⋊Q8—21no—
121C2×#121C22×C312yes—
122C2×C4×D4C22×C311yescharacteristic
123C2×C4×D4C22×C311yescharacteristic
124C22×C4⋊C4C22×C312yes—
125#501—24no—
126#478—24no—
127C2×#107C22×C311yescharacteristic
128C2×#107C22×C312yes—
129C2×#106C22×C311yescharacteristic
130C2×#106C22×C312yes—
131C2×C4⋊C8C22×C311yescharacteristic
132C2×#102C22×C311yescharacteristic
133C2×#102C22×C311yescharacteristic
134C2×#102C22×C312yes—
135C2×C22⋊C8C22×C312yes—
136C2×C22⋊C8C22×C312yes—
137C2×#97C22×C311yescharacteristic
138#400—21no—
139#394—21no—
140#388—22no—
141#380C22×C312yes—
142#380C22×C312yes—
143C4×C4⋊C4C22×C311yescharacteristic
144#340C22×C312yes—
145#340C22×C312yes—
146#340C22×C312yes—
147C24×C3—21no—
148C24×C3C2×D412yes—
149C3×#41—432no—
150C2×C3×Q8—24no—
151C2×C3×Q8—44no—
152C2×C3×D4—24no—
153C2×C3×D4—28no—
154C2×C3×D4—22no—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—44no—
159C2×C3×D4—44no—
160C2×C3×D4C2×D412yes—
161C22×C3×C4—28no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—44no—
175C22×C3×C4—44no—
176C22×C3×C4—44no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8#4114yescomplemented
188C3×C4⋊C4—28no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4C2×D412yes—
202C3×C4⋊C4C2×D414yes—
203C3×C22⋊C4—28no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—82no—
208C3×C42—81no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211C2×#41—28no—
212C2×#41—48no—
213C22×Q8—21no—
214C22×D4—22no—
215C22×D4C3×D411yescharacteristic
216C22×D4C23×C311yescharacteristic
217C23×C4—22no—
218C23×C4—21no—
219C23×C4C23×C312yes—
220C22×C8C23×C311yescharacteristic
221C22×C8C23×C311yescharacteristic
222C22×C8C23×C312yes—
223C4⋊Q8—22no—
224C4⋊Q8C3×D414yes—
225#125—22no—
226#125—22no—
227#123—216no—
228C22⋊Q8—28no—
229#121—24no—
230#121—24no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4C3×D414yes—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C3×D411yescharacteristic
242C2×C4⋊C4C23×C311yescharacteristic
243C2×C4⋊C4C23×C311yescharacteristic
244C2×C4⋊C4C23×C312yes—
245C2×C4⋊C4C23×C311yescharacteristic
246C2×C4⋊C4C23×C312yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4C3×D412yes—
252C2×C42—21no—
253C2×C42C23×C311yescharacteristic
254#107—22no—
255#107—24no—
256#106—22no—
257#106—24no—
258C4⋊C8—22no—
259#102—22no—
260#102—22no—
261#102—24no—
262C22⋊C8—24no—
263C22⋊C8—24no—
264#97—22no—
265#95—21no—
266#95—22no—
267#95—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3AGL(1,ℤ/8)14yes—
280C23×C3C22×D411yescharacteristic
281C3×Q8—48no—
282C3×D4—28no—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—416no—
286C3×D4—42no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—48no—
296C2×C3×C4—416no—
297C2×C3×C4—42no—
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—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4#12012yes—
313C2×C3×C4#12012yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4112yes—
317C2×C3×C42−1+412yes—
318C3×C8—24no—
319C3×C8—41no—
320C3×C8—41no—
321C24—21no—
322C24C2×C3×D412yes—
323#41—432no—
324C2×Q8—24no—
325C2×Q8—44no—
326C2×D4—24no—
327C2×D4—28no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4C2×C3×D412yes—
335C22×C4—28no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C24×C311yescharacteristic
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8C3×#4114yescomplemented
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4C2×C3×D412yes—
376C4⋊C4C2×C3×D414yes—
377C22⋊C4—28no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—44no—
381C22⋊C4—82no—
382C42—81no—
383C42C2×C3×D412yes—
384C42C2×C3×D412yes—
385C22×C3—24no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3C2×#12011yescharacteristic
398C22×C3#54712yes—
399C22×C3C2×AGL(1,ℤ/8)12yes—
400C22×C3#57711yescharacteristic
401C22×C3#57811yescharacteristic
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—42no—
405C3×C4—42no—
406C3×C4—44no—
407C3×C4—82no—
408C3×C4—81no—
409C3×C4—81no—
410C3×C4#53211yescharacteristic
411C3×C4#53611yescharacteristic
412C3×C4#54812yes—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23C3×AGL(1,ℤ/8)14yes—
425C23C22×C3×D411yescharacteristic
426Q8—48no—
427D4—28no—
428D4—22no—
429D4—22no—
430D4—416no—
431D4—42no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—48no—
441C2×C4—416no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4C3×#12012yes—
458C2×C4C3×#12012yes—
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C22×C3×D411yescharacteristic
461C2×C4C2×C3×#4112yes—
462C2×C4C3×2−1+412yes—
463C8—24no—
464C8—41no—
465C8—41no—
466C2×C3—44no—
467C2×C3—82no—
468C2×C3#243211yescharacteristic
469C2×C3#300311yescharacteristic
470C2×C3#301311yescharacteristic
471C2×C3#334312yes—
472C2×C3#334512yes—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—82no—
484C22—82no—
485C22C2×C3×#12011yescharacteristic
486C22C3×#54712yes—
487C22C2×C3×AGL(1,ℤ/8)12yes—
488C22C3×#57711yescharacteristic
489C22C3×#57811yescharacteristic
490C4—22no—
491C4—22no—
492C4—42no—
493C4—42no—
494C4—44no—
495C4—81no—
496C4—81no—
497C4—82no—
498C4C3×#53211yescharacteristic
499C4C3×#53611yescharacteristic
500C4C3×#54812yes—
501C3#2570911yescharacteristic, direct product factor, Sylow
502C2—44no—
503C2—82no—
504C2C3×#243211yescharacteristic
505C2C3×#300311yescharacteristic
506C2C3×#301311yescharacteristic
507C2C3×#334312yes—
508C2C3×#334512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
3. C2×C3×#478C2semidirect
4. C3×#2562C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1987C2semidirect
8. C3×#1955C2semidirect
9. C3×#1853C2semidirect
10. C3×#1852C2semidirect
11. C3×#1762C2semidirect
12. #25709C3direct (N × H)
47. C2×#501C2×C3semidirect
48. C2×#478C2×C3semidirect
49. #2562C2×C3semidirect
50. #2034C2×C3semidirect
51. #2033C2×C3semidirect
52. #1987C2×C3semidirect
53. #1955C2×C3semidirect
54. #1853C2×C3semidirect
55. #1852C2×C3semidirect
56. #1762C2×C3semidirect
187. C2×C3×C8#41semidirect
361. C2×C8C3×#41semidirect
501. C3#25709direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#5012
C2×C3×#4782
C3×#25622
C3×#20342
C3×#20332
C3×#19872
C3×#19552
C3×#18532
C3×#18522
C3×#17622
#257093