Finite Group Databasebeta

← C3×#25857 #10644464

Subgroups — C3×#25857 #10644464

Order 768.

1390 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×#532C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic, complemented
5C3×#2042C214yesmaximal, complemented
6C3×#1957C211yesmaximal, characteristic, complemented
7C3×#1956C211yesmaximal, characteristic, complemented
8C3×#1788C214yesmaximal, complemented
9C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
10#25857C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C3×#532—24no—
13C3×#532—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×#121—21no—
20C2×C3×#121—21no—
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4—21no—
23C22×C3×C42C2211yescharacteristic
24C3×#487—24no—
25C3×#486—24no—
26C2×C3×#103C2214yes—
27C2×C3×#103C2214yes—
28C2×C3×#102C2214yes—
29C2×C3×#102C2214yes—
30C2×C3×C22⋊C8C2212yes—
31C2×C3×C4×C8C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33C3×#391—21no—
34C3×#389—21no—
35C3×#389C2212yes—
36C3×#384C2211yescharacteristic
37C3×#380—21no—
38C3×#380C2212yes—
39C3×#336C2212yes—
40C2×#532C2×C311yescharacteristic
41C2×#487C2×C311yescharacteristic, complemented
42C2×#486C2×C311yescharacteristic, complemented
43#2857C2×C311yescharacteristic, complemented
44#2042C2×C314yescomplemented
45#1957C2×C311yescharacteristic, complemented
46#1956C2×C311yescharacteristic, complemented
47#1788C2×C314yescomplemented
48C4×C22⋊C8C2×C311yescharacteristic, complemented
49C2×C3×#41—24no—
50C2×C3×#41—24no—
51C2×C3×#41—28no—
52C2×C3×#41—44no—
53C22×C3×Q8—21no—
54C22×C3×Q8C2312yes—
55C22×C3×D4—22no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C23×C3×C4—22no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C3×#127—22no—
70C3×#127D414yes—
71C3×#124—24no—
72C3×#124—24no—
73C3×C22⋊Q8—44no—
74C3×#121—28no—
75C3×#121—28no—
76C3×C4×D4—28no—
77C3×C4×D4—28no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42—22no—
90C2×C3×C42—22no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C3×#103—28no—
96C3×#103—44no—
97C3×#102—28no—
98C3×#102—44no—
99C3×C22⋊C8—24no—
100C3×C4×C8—22no—
101C3×C4×C8—22no—
102C3×#95—22no—
103C3×#95—22no—
104C22×#41C22×C312yes—
105#532—24no—
106#532—24no—
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×C4⋊Q8C22×C311yescharacteristic
109C2×#127C22×C311yescharacteristic
110C2×#124C22×C312yes—
111C2×C22⋊Q8—21no—
112C2×#121—21no—
113C2×#121—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C22×C42C22×C311yescharacteristic
117#487—24no—
118#486—24no—
119C2×#103C22×C314yes—
120C2×#103C22×C314yes—
121C2×#102C22×C314yes—
122C2×#102C22×C314yes—
123C2×C22⋊C8C22×C312yes—
124C2×C4×C8C22×C311yescharacteristic
125C2×C4×C8C22×C311yescharacteristic
126#391—21no—
127#389—21no—
128#389C22×C312yes—
129#384C22×C311yescharacteristic
130#380—21no—
131#380C22×C312yes—
132#336C22×C312yes—
133C24×C3—22no—
134C24×C3C2×D411yescharacteristic
135C3×#41—416no—
136C3×#41—416no—
137C2×C3×Q8—22no—
138C2×C3×Q8—24no—
139C2×C3×Q8—24no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C2×C3×Q8C2×D414yes—
144C2×C3×D4—28no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4C2×D414yes—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—28no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—44no—
171C22×C3×C4—44no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4114yes—
178C22×C3×C4C2411yescharacteristic
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—44no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—44no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—28no—
206C2×#41—44no—
207C22×Q8—21no—
208C22×Q8C23×C312yes—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C23×C312yes—
213C23×C4—22no—
214C23×C4C3×D411yescharacteristic
215C23×C4C3×D411yescharacteristic
216C23×C4C23×C311yescharacteristic
217C22×C8C23×C312yes—
218C22×C8C23×C312yes—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8C3×D414yes—
223#127—22no—
224#127C3×D414yes—
225#124—24no—
226#124—24no—
227C22⋊Q8—44no—
228#121—28no—
229#121—28no—
230C4×D4—28no—
231C4×D4—28no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C23×C312yes—
239C2×C4⋊C4C23×C312yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42C3×D411yescharacteristic
246C2×C42C3×D411yescharacteristic
247C2×C42C23×C311yescharacteristic
248C2×C42C23×C311yescharacteristic
249#103—28no—
250#103—44no—
251#102—28no—
252#102—44no—
253C22⋊C8—24no—
254C4×C8—22no—
255C4×C8—22no—
256#95—22no—
257#95—22no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3#12412yes—
272C23×C3C22×D411yescharacteristic
273C3×Q8—28no—
274C3×Q8—44no—
275C3×Q8—84no—
276C3×D4—28no—
277C3×D4—416no—
278C3×D4—44no—
279C3×D4—48no—
280C3×D4—48no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—416no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
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#12012yes—
304C2×C3×C4#12012yes—
305C2×C3×C4#12412yes—
306C2×C3×C4#13512yes—
307C2×C3×C4#13512yes—
308C2×C3×C4AGL(1,ℤ/8)12yes—
309C2×C3×C4#13712yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4112yes—
313C2×C3×C4H(5,𝔽2)12yes—
314C3×C8—42no—
315C3×C8—42no—
316C24—22no—
317C24C2×C3×D411yescharacteristic
318#41—416no—
319#41—416no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8C2×C3×D414yes—
327C2×D4—28no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×C3×D414yes—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C3×#4114yes—
361C22×C4C24×C311yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C22⋊C4—28no—
378C22⋊C4—28no—
379C22⋊C4—44no—
380C42—22no—
381C42—22no—
382C42—21no—
383C42—21no—
384C42C2×C3×D412yes—
385C42C2×C3×D412yes—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3C2×#12011yescharacteristic
401C22×C3C2×#12411yescharacteristic
402C22×C3#53511yescharacteristic
403C22×C3C2×#13511yescharacteristic
404C22×C3C2×#13511yescharacteristic
405C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C3C2×#13711yescharacteristic
407C3×C4—22no—
408C3×C4—44no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—82no—
412C3×C4—82no—
413C3×C4#44914yes—
414C3×C4#45214yes—
415C3×C4#54612yes—
416C3×C4#54612yes—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23C3×#12412yes—
431C23C22×C3×D411yescharacteristic
432Q8—28no—
433Q8—44no—
434Q8—84no—
435D4—28no—
436D4—416no—
437D4—48no—
438D4—48no—
439D4—44no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—416no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4C3×#12012yes—
463C2×C4C3×#12012yes—
464C2×C4C3×#12412yes—
465C2×C4C3×#13512yes—
466C2×C4C3×#13512yes—
467C2×C4C3×AGL(1,ℤ/8)12yes—
468C2×C4C3×#13712yes—
469C2×C4C22×C3×D411yescharacteristic
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C2×C3×#4112yes—
472C2×C4C3×H(5,𝔽2)12yes—
473C8—42no—
474C8—42no—
475C2×C3—42no—
476C2×C3—84no—
477C2×C3#243511yescharacteristic
478C2×C3C2×#44911yescharacteristic
479C2×C3C2×#45211yescharacteristic
480C2×C3#313111yescharacteristic
481C2×C3#314111yescharacteristic
482C2×C3#320711yescharacteristic
483C2×C3#321011yescharacteristic
484C22—22no—
485C22—22no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22C2×C3×#12011yescharacteristic
499C22C2×C3×#12411yescharacteristic
500C22C3×#53511yescharacteristic
501C22C2×C3×#13511yescharacteristic
502C22C2×C3×#13511yescharacteristic
503C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
504C22C2×C3×#13711yescharacteristic
505C4—22no—
506C4—44no—
507C4—41no—
508C4—41no—
509C4—82no—
510C4—82no—
511C4C3×#44914yes—
512C4C3×#45214yes—
513C4C3×#54612yes—
514C4C3×#54612yes—
515C3#2585711yescharacteristic, direct product factor, Sylow
516C2—42no—
517C2—84no—
518C2C3×#243511yescharacteristic
519C2C2×C3×#44911yescharacteristic
520C2C2×C3×#45211yescharacteristic
521C2C3×#313111yescharacteristic
522C2C3×#314111yescharacteristic
523C2C3×#320711yescharacteristic
524C2C3×#321011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
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
2. C2×C3×#487C2semidirect
3. C2×C3×#486C2semidirect
4. C3×#2857C2semidirect
5. C3×#2042C2semidirect
6. C3×#1957C2semidirect
7. C3×#1956C2semidirect
8. C3×#1788C2semidirect
9. C3×C4×C22⋊C8C2semidirect
10. #25857C3direct (N × H)
41. C2×#487C2×C3semidirect
42. C2×#486C2×C3semidirect
43. #2857C2×C3semidirect
44. #2042C2×C3semidirect
45. #1957C2×C3semidirect
46. #1956C2×C3semidirect
47. #1788C2×C3semidirect
48. C4×C22⋊C8C2×C3semidirect
515. C3#25857direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4872
C2×C3×#4862
C3×#28572
C3×#20422
C3×#19572
C3×#19562
C3×#17882
C3×C4×C22⋊C82
#258573