Finite Group Databasebeta

← C3×#29623 #10648230

Subgroups — C3×#29623 #10648230

Order 768.

1516 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
1C22×C3×#124C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic
4C3×#2854C211yesmaximal, characteristic, complemented
5C3×#2042C214yesmaximal, complemented
6C3×#2012C214yesmaximal, complemented
7C3×#1817C211yesmaximal, characteristic, complemented
8C3×#1787C211yesmaximal, characteristic, complemented
9C3×#1786C211yesmaximal, characteristic, complemented
10#29623C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C22×C3×SD16C2212yescomplemented
14C22×C3×D8C2212yes—
15C2×C3×#124—24no—
16C2×C3×#124—24no—
17C2×C3×#124—22no—
18C2×C3×#124C2214yes—
19C2×C3×#123—21no—
20C2×C3×#121C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C22⋊C4—21no—
23C22×C3×C42C2211yescharacteristic
24C3×#448—24no—
25C3×#447—24no—
26C2×C3×#103C2212yescomplemented
27C2×C3×#103C2214yescomplemented
28C2×C3×#102C2212yes—
29C2×C3×#102C2214yes—
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C3×#390C2212yescomplemented
33C3×#389C2212yescomplemented
34C3×#388—22no—
35C3×#386—21no—
36C3×#380C2211yescharacteristic
37C3×#336C2214yescomplemented
38C22×#124C2×C311yescharacteristic, complemented
39C2×#448C2×C311yescharacteristic, complemented
40C2×#447C2×C311yescharacteristic
41#2854C2×C311yescharacteristic, complemented
42#2042C2×C314yescomplemented
43#2012C2×C314yescomplemented
44#1817C2×C311yescharacteristic, complemented
45#1787C2×C311yescharacteristic, complemented
46#1786C2×C311yescharacteristic, complemented
47C25×C3—21no—
48C22×C3×Q8—22no—
49C22×C3×Q8—21no—
50C22×C3×Q8—44no—
51C22×C3×Q8D414yes—
52C22×C3×Q8C2312yes—
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4—22no—
56C22×C3×D4—44no—
57C22×C3×D4D414yes—
58C22×C3×D4C2311yescharacteristic
59C22×C3×D4C2312yes—
60C23×C3×C4—21no—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C2×C3×SD16—24no—
65C2×C3×SD16—44no—
66C2×C3×D8—24no—
67C2×C3×D8—44no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×#124—216no—
71C3×#124—28no—
72C3×#124—28no—
73C3×#123—44no—
74C3×#121—22no—
75C3×#121—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—28no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42D412yes—
90C2×C3×C42C2312yes—
91C3×#103—24no—
92C3×#103—44no—
93C3×#102—24no—
94C3×#102—44no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×#95—21no—
98C3×#95—22no—
99C23×Q8C22×C311yescharacteristic
100C23×D4C22×C311yescharacteristic
101C22×SD16C22×C312yescomplemented
102C22×D8C22×C312yes—
103C2×#124—24no—
104C2×#124—24no—
105C2×#124—22no—
106C2×#124C22×C314yes—
107C2×#123—21no—
108C2×#121C22×C311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42C22×C311yescharacteristic
112#448—24no—
113#447—24no—
114C2×#103C22×C312yescomplemented
115C2×#103C22×C314yescomplemented
116C2×#102C22×C312yes—
117C2×#102C22×C314yes—
118C2×C22⋊C8C22×C311yescharacteristic
119C2×C22⋊C8C22×C311yescharacteristic
120#390C22×C312yescomplemented
121#389C22×C312yescomplemented
122#388—22no—
123#386—21no—
124#380C22×C311yescharacteristic
125#336C22×C314yescomplemented
126C24×C3—24no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—44no—
130C24×C3—44no—
131C24×C3—42no—
132C24×C3—42no—
133C24×C3C2×D411yescharacteristic
134C2×C3×Q8—24no—
135C2×C3×Q8—22no—
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—48no—
139C2×C3×Q8—44no—
140C2×C3×Q8—48no—
141C2×C3×Q8C2×D414yes—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—48no—
150C2×C3×D4—44no—
151C2×C3×D4—48no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4C2×D414yes—
156C22×C3×C4—24no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—42no—
166C22×C3×C4—44no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C3×SD16—48no—
174C3×D8—48no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—24no—
178C2×C3×C8—24no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—44no—
183C3×C22⋊C4—216no—
184C3×C22⋊C4—216no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—44no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—42no—
189C3×C42—24no—
190C3×C42—22no—
191C3×C42—22no—
192C25—21no—
193C22×Q8—22no—
194C22×Q8—21no—
195C22×Q8—44no—
196C22×Q8C3×D414yes—
197C22×Q8C23×C312yes—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—44no—
202C22×D4C3×D414yes—
203C22×D4C23×C311yescharacteristic
204C22×D4C23×C312yes—
205C23×C4—21no—
206C23×C4C3×D411yescharacteristic
207C23×C4C3×D411yescharacteristic
208C23×C4C23×C311yescharacteristic
209C2×SD16—24no—
210C2×SD16—44no—
211C2×D8—24no—
212C2×D8—44no—
213C22×C8C23×C312yes—
214C22×C8C23×C312yes—
215#124—216no—
216#124—28no—
217#124—28no—
218#123—44no—
219#121—22no—
220#121—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C4⋊C4C23×C312yes—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C3×D412yes—
235C2×C42C23×C312yes—
236#103—24no—
237#103—44no—
238#102—24no—
239#102—44no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242#95—21no—
243#95—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—24no—
247C23×C3—48no—
248C23×C3—48no—
249C23×C3—44no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—44no—
253C23×C3—48no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—42no—
259C23×C3—88no—
260C23×C3—88no—
261C23×C3#12012yes—
262C23×C3#13512yes—
263C23×C3AGL(1,ℤ/8)12yes—
264C23×C3C22×D411yescharacteristic
265C3×Q8—24no—
266C3×Q8—24no—
267C3×Q8—48no—
268C3×Q8—44no—
269C3×D4—24no—
270C3×D4—24no—
271C3×D4—48no—
272C3×D4—44no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—84no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—44no—
287C2×C3×C4—48no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4#12012yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4H(5,𝔽2)14yes—
300C3×C8—42no—
301C3×C8—42no—
302C24—24no—
303C24—22no—
304C24—21no—
305C24—44no—
306C24—44no—
307C24—42no—
308C24—42no—
309C24C2×C3×D411yescharacteristic
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—48no—
315C2×Q8—44no—
316C2×Q8—48no—
317C2×Q8C2×C3×D414yes—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—48no—
326C2×D4—44no—
327C2×D4—48no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4C2×C3×D414yes—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4C2×C3×D412yes—
344C22×C4C2×C3×D412yes—
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C2×C3×D412yes—
347C22×C4C2×C3×D412yes—
348C22×C4C24×C311yescharacteristic
349SD16—48no—
350D8—48no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—24no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—24no—
358C4⋊C4—44no—
359C22⋊C4—216no—
360C22⋊C4—216no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C42—24no—
366C42—22no—
367C42—22no—
368C22×C3—22no—
369C22×C3—24no—
370C22×C3—24no—
371C22×C3—22no—
372C22×C3—44no—
373C22×C3—44no—
374C22×C3—44no—
375C22×C3—816no—
376C22×C3—84no—
377C22×C3—84no—
378C22×C3—81no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3#44914yes—
384C22×C3C2×#12011yescharacteristic
385C22×C3#53512yes—
386C22×C3C2×#13511yescharacteristic
387C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
388C22×C3#57512yes—
389C3×C4—22no—
390C3×C4—44no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—82no—
394C3×C4—81no—
395C3×C4#53412yes—
396C3×C4#53412yes—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—48no—
401C23—48no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—48no—
406C23—44no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23—88no—
413C23—88no—
414C23C3×#12012yes—
415C23C3×#13512yes—
416C23C3×AGL(1,ℤ/8)12yes—
417C23C22×C3×D411yescharacteristic
418Q8—24no—
419Q8—24no—
420Q8—48no—
421Q8—44no—
422D4—24no—
423D4—24no—
424D4—48no—
425D4—44no—
426D4—42no—
427D4—42no—
428D4—84no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—48no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4C3×#12012yes—
450C2×C4C22×C3×D411yescharacteristic
451C2×C4C22×C3×D411yescharacteristic
452C2×C4C3×H(5,𝔽2)14yes—
453C8—42no—
454C8—42no—
455C2×C3—24no—
456C2×C3—84no—
457C2×C3—161no—
458C2×C3#261311yescharacteristic
459C2×C3C2×#44911yescharacteristic
460C2×C3#300511yescharacteristic
461C2×C3#321112yes—
462C2×C3#321312yes—
463C22—24no—
464C22—24no—
465C22—22no—
466C22—22no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—816no—
471C22—84no—
472C22—84no—
473C22—82no—
474C22—82no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22C3×#44914yes—
479C22C2×C3×#12011yescharacteristic
480C22C3×#53512yes—
481C22C2×C3×#13511yescharacteristic
482C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
483C22C3×#57512yes—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—44no—
488C4—82no—
489C4—81no—
490C4C3×#53412yes—
491C4C3×#53412yes—
492C3#2962311yescharacteristic, direct product factor, Sylow
493C2—24no—
494C2—84no—
495C2—161no—
496C2C3×#261311yescharacteristic
497C2C2×C3×#44911yescharacteristic
498C2C3×#300511yescharacteristic
499C2C3×#321112yes—
500C2C3×#321312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
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. C22×C3×#124C2semidirect
2. C2×C3×#448C2semidirect
4. C3×#2854C2semidirect
5. C3×#2042C2semidirect
6. C3×#2012C2semidirect
7. C3×#1817C2semidirect
8. C3×#1787C2semidirect
9. C3×#1786C2semidirect
10. #29623C3direct (N × H)
13. C22×C3×SD16C22semidirect
26. C2×C3×#103C22semidirect
27. C2×C3×#103C22semidirect
32. C3×#390C22semidirect
33. C3×#389C22semidirect
37. C3×#336C22semidirect
38. C22×#124C2×C3semidirect
39. C2×#448C2×C3semidirect
41. #2854C2×C3semidirect
42. #2042C2×C3semidirect
43. #2012C2×C3semidirect
44. #1817C2×C3semidirect
45. #1787C2×C3semidirect
46. #1786C2×C3semidirect
101. C22×SD16C22×C3semidirect
114. C2×#103C22×C3semidirect
115. C2×#103C22×C3semidirect
120. #390C22×C3semidirect
121. #389C22×C3semidirect
125. #336C22×C3semidirect
492. C3#29623direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1242
C2×C3×#4482
C2×C3×#4472
C3×#28542
C3×#20422
C3×#20122
C3×#18172
C3×#17872
C3×#17862
#296233