Finite Group Databasebeta

← C3×#29620 #10648227

Subgroups — C3×#29620 #10648227

Order 768.

1324 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3449C211yesmaximal, characteristic, complemented
2C3×#3006C212yesmaximal, complemented
3C3×#2500C211yesmaximal, characteristic, complemented
4C3×#2040C212yesmaximal, complemented
5C3×#2039C212yesmaximal, complemented
6C3×#2006C212yesmaximal, complemented
7C3×#2006C212yesmaximal, complemented
8C3×#1822C211yesmaximal, characteristic, complemented
9C3×#1789C212yesmaximal, complemented
10#29620C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C2×C3×AGL(1,ℤ/8)C2214yescomplemented
13C3×#537C2212yes—
14C3×#537C2212yes—
15C3×#536—22no—
16C3×#535—22no—
17C3×#534—22no—
18C3×#532—22no—
19C3×#525—24no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C3×#449—24no—
23C3×#448—22no—
24C3×#447—22no—
25C2×C3×#104C2214yescomplemented
26C2×C3×#104C2214yescomplemented
27C3×#417C2214yescomplemented
28C3×#407C2212yes—
29C3×#394—22no—
30C3×#390C2211yescharacteristic, complemented
31C3×#390C2211yescharacteristic, complemented
32C3×#388—22no—
33C3×#384C2211yescharacteristic, complemented
34C3×#382C2211yescharacteristic, complemented
35C3×C4×C22⋊C4C2211yescharacteristic
36C3×#343C2212yescomplemented
37C3×#343C2212yescomplemented
38#3449C2×C311yescharacteristic, complemented
39#3006C2×C312yescomplemented
40#2500C2×C311yescharacteristic, complemented
41#2040C2×C312yescomplemented
42#2039C2×C312yescomplemented
43#2006C2×C312yescomplemented
44#2006C2×C312yescomplemented
45#1822C2×C311yescharacteristic, complemented
46#1789C2×C312yescomplemented
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41—44no—
50C2×C3×#41—44no—
51C2×C3×#41D414yes—
52C2×C3×#41C2314yes—
53C22×C3×Q8D412yes—
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4D412yes—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—22no—
59C23×C3×C4C2311yescharacteristic
60C3×AGL(1,ℤ/8)—44no—
61C3×AGL(1,ℤ/8)—44no—
62C2×C3×SD16—24no—
63C2×C3×D8—24no—
64C2×C3×M16C2314yes—
65C3×C4⋊Q8—22no—
66C3×#127—22no—
67C3×#124—24no—
68C3×#123—28no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—24no—
71C3×#121—24no—
72C3×#121—24no—
73C3×#121—24no—
74C3×#121—24no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#120—28no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×#117—22no—
81C3×#117—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D412yes—
91C2×C3×C42D412yes—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×#104—44no—
95C3×#104—44no—
96C3×#103—24no—
97C3×#102—24no—
98C3×C22⋊C8—22no—
99C3×C22⋊C8—22no—
100C3×#95—21no—
101C3×#95—22no—
102C22×#41C22×C312yes—
103C2×AGL(1,ℤ/8)C22×C314yescomplemented
104#537C22×C312yes—
105#537C22×C312yes—
106#536—22no—
107#535—22no—
108#534—22no—
109#532—22no—
110#525—24no—
111C2×#121C22×C311yescharacteristic
112C2×#117C22×C311yescharacteristic
113#449—24no—
114#448—22no—
115#447—22no—
116C2×#104C22×C314yescomplemented
117C2×#104C22×C314yescomplemented
118#417C22×C314yescomplemented
119#407C22×C312yes—
120#394—22no—
121#390C22×C311yescharacteristic, complemented
122#390C22×C311yescharacteristic, complemented
123#388—22no—
124#384C22×C311yescharacteristic, complemented
125#382C22×C311yescharacteristic, complemented
126C4×C22⋊C4C22×C311yescharacteristic
127#343C22×C312yescomplemented
128#343C22×C312yescomplemented
129C24×C3—22no—
130C24×C3—21no—
131C24×C3C2×D411yescharacteristic
132C3×#41—24no—
133C3×#41—24no—
134C3×#41—24no—
135C3×#41—24no—
136C3×#41—44no—
137C3×#41—44no—
138C3×#41—44no—
139C3×#41—44no—
140C3×#41—48no—
141C2×C3×Q8—22no—
142C2×C3×Q8—44no—
143C2×C3×Q8C2×D414yes—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—44no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—42no—
156C2×C3×D4—82no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—48no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C3×SD16—48no—
174C3×D8—48no—
175C3×M16—24no—
176C3×M16—24no—
177C2×C3×C8—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—82no—
184C3×C22⋊C4—28no—
185C3×C22⋊C4—28no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—41no—
190C3×C22⋊C4—41no—
191C3×C22⋊C4—82no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—22no—
196C2×#41—22no—
197C2×#41—22no—
198C2×#41—44no—
199C2×#41—44no—
200C2×#41C3×D414yes—
201C2×#41C23×C314yes—
202C22×Q8C3×D412yes—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4C3×D412yes—
206C22×D4C23×C311yescharacteristic
207C23×C4—22no—
208C23×C4C23×C311yescharacteristic
209AGL(1,ℤ/8)—44no—
210AGL(1,ℤ/8)—44no—
211C2×SD16—24no—
212C2×D8—24no—
213C2×M16C23×C314yes—
214C4⋊Q8—22no—
215#127—22no—
216#124—24no—
217#123—28no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220#121—24no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—22no—
225#121—22no—
226#120—28no—
227C4×D4—24no—
228C4×D4—24no—
229#117—22no—
230#117—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4C3×D412yes—
240C2×C42C3×D412yes—
241C2×C42C23×C311yescharacteristic
242C2×C42C23×C311yescharacteristic
243#104—44no—
244#104—44no—
245#103—24no—
246#102—24no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249#95—21no—
250#95—22no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—24no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—44no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—82no—
263C23×C3—82no—
264C23×C3#12012yes—
265C23×C3C22×D411yescharacteristic
266C3×Q8—24no—
267C3×Q8—24no—
268C3×Q8—44no—
269C3×D4—24no—
270C3×D4—24no—
271C3×D4—44no—
272C3×D4—48no—
273C3×D4—44no—
274C3×D4—44no—
275C3×D4—44no—
276C3×D4—84no—
277C3×D4—84no—
278C3×D4—84no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—48no—
288C2×C3×C4—41no—
289C2×C3×C4—44no—
290C2×C3×C4—41no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4C22×D412yes—
303C2×C3×C4H(5,𝔽2)12yes—
304C2×C3×C4H(5,𝔽2)12yes—
305C3×C8—44no—
306C24—22no—
307C24—21no—
308C24C2×C3×D411yescharacteristic
309#41—24no—
310#41—24no—
311#41—24no—
312#41—24no—
313#41—44no—
314#41—44no—
315#41—44no—
316#41—44no—
317#41—48no—
318C2×Q8—22no—
319C2×Q8—44no—
320C2×Q8C2×C3×D414yes—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—82no—
334C2×D4C2×C3×D414yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—48no—
344C22×C4C2×C3×D411yescharacteristic
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D412yes—
347C22×C4C2×C3×D412yes—
348C22×C4C2×C3×D412yes—
349C22×C4C24×C311yescharacteristic
350SD16—48no—
351D8—48no—
352M16—24no—
353M16—24no—
354C2×C8—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—82no—
361C22⋊C4—28no—
362C22⋊C4—28no—
363C22⋊C4—24no—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—82no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—44no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—88no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3C2×#12011yescharacteristic
390C22×C3#53511yescharacteristic
391C22×C3#53511yescharacteristic
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—42no—
395C3×C4—44no—
396C3×C4—42no—
397C3×C4—42no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4#53414yes—
401C23—21no—
402C23—21no—
403C23—24no—
404C23—41no—
405C23—41no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—82no—
413C23—82no—
414C23C3×#12012yes—
415C23C22×C3×D411yescharacteristic
416Q8—24no—
417Q8—24no—
418Q8—44no—
419D4—24no—
420D4—24no—
421D4—44no—
422D4—48no—
423D4—44no—
424D4—44no—
425D4—44no—
426D4—84no—
427D4—84no—
428D4—84no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—48no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4C3×#12011yescharacteristic
451C2×C4C3×#12011yescharacteristic
452C2×C4C22×C3×D412yes—
453C2×C4C3×H(5,𝔽2)12yes—
454C2×C4C3×H(5,𝔽2)12yes—
455C8—44no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—42no—
459C2×C3—84no—
460C2×C3—161no—
461C2×C3#261311yescharacteristic
462C2×C3#301811yescharacteristic
463C2×C3#302111yescharacteristic
464C22—21no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22—88no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22C2×C3×#12011yescharacteristic
481C22C3×#53511yescharacteristic
482C22C3×#53511yescharacteristic
483C4—21no—
484C4—21no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—44no—
489C4—81no—
490C4—81no—
491C4C3×#53414yes—
492C3#2962011yescharacteristic, direct product factor, Sylow
493C2—21no—
494C2—21no—
495C2—42no—
496C2—84no—
497C2—161no—
498C2C3×#261311yescharacteristic
499C2C3×#301811yescharacteristic
500C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3449C2semidirect
2. C3×#3006C2semidirect
3. C3×#2500C2semidirect
4. C3×#2040C2semidirect
5. C3×#2039C2semidirect
6. C3×#2006C2semidirect
7. C3×#2006C2semidirect
8. C3×#1822C2semidirect
9. C3×#1789C2semidirect
10. #29620C3direct (N × H)
12. C2×C3×AGL(1,ℤ/8)C22semidirect
25. C2×C3×#104C22semidirect
26. C2×C3×#104C22semidirect
27. C3×#417C22semidirect
30. C3×#390C22semidirect
31. C3×#390C22semidirect
33. C3×#384C22semidirect
34. C3×#382C22semidirect
36. C3×#343C22semidirect
37. C3×#343C22semidirect
38. #3449C2×C3semidirect
39. #3006C2×C3semidirect
40. #2500C2×C3semidirect
41. #2040C2×C3semidirect
42. #2039C2×C3semidirect
43. #2006C2×C3semidirect
44. #2006C2×C3semidirect
45. #1822C2×C3semidirect
46. #1789C2×C3semidirect
103. C2×AGL(1,ℤ/8)C22×C3semidirect
116. C2×#104C22×C3semidirect
117. C2×#104C22×C3semidirect
118. #417C22×C3semidirect
121. #390C22×C3semidirect
122. #390C22×C3semidirect
124. #384C22×C3semidirect
125. #382C22×C3semidirect
127. #343C22×C3semidirect
128. #343C22×C3semidirect
492. C3#29620direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34492
C3×#30062
C3×#25002
C3×#20402
C3×#20392
C3×#20062
C3×#18222
C3×#17892
#296203