Finite Group Databasebeta

← C3×#33002 #10651609

Subgroups — C3×#33002 #10651609

Order 768.

1808 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic
4C3×#2854C211yesmaximal, characteristic, complemented
5C3×#1879C214yesmaximal, complemented
6C3×#1862C214yesmaximal, complemented
7C3×#1823C211yesmaximal, characteristic, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9C3×#1786C211yesmaximal, characteristic, complemented
10#33002C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4—21no—
13C22×C3×D8C2212yes—
14C2×C3×#123—21no—
15C2×C3×#121C2211yescharacteristic
16C2×C3×C4×D4—24no—
17C2×C3×C4×D4—24no—
18C2×C3×C4×D4—22no—
19C2×C3×C4×D4C2214yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#480—24no—
24C3×#447—24no—
25C2×C3×#107C2212yescomplemented
26C2×C3×#102C2214yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#390C2212yescomplemented
32C3×#389C2212yescomplemented
33C3×#388—22no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×#340C2214yescomplemented
37C3×#336C2214yescomplemented
38C22×C4×D4C2×C311yescharacteristic, complemented
39C2×#480C2×C311yescharacteristic, complemented
40C2×#447C2×C311yescharacteristic
41#2854C2×C311yescharacteristic, complemented
42#1879C2×C314yescomplemented
43#1862C2×C314yescomplemented
44#1823C2×C311yescharacteristic, complemented
45#1817C2×C311yescharacteristic, complemented
46#1786C2×C311yescharacteristic, complemented
47C25×C3—21no—
48C22×C3×D4—28no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4D414yes—
53C22×C3×D4C2311yescharacteristic
54C22×C3×D4C2312yes—
55C23×C3×C4—28no—
56C23×C3×C4—21no—
57C23×C3×C4—24no—
58C23×C3×C4—44no—
59C23×C3×C4—44no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C2×C3×D8—24no—
64C2×C3×D8—28no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×#123—44no—
68C3×#121—22no—
69C3×#121—22no—
70C3×C4×D4—216no—
71C3×C4×D4—28no—
72C3×C4×D4—28no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D414yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—28no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42D412yes—
89C2×C3×C42C2312yes—
90C3×#107—24no—
91C3×#102—24no—
92C3×#102—28no—
93C3×#102—24no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—22no—
96C3×#95—21no—
97C3×#95—22no—
98C23×D4C22×C311yescharacteristic
99C24×C4—21no—
100C22×D8C22×C312yes—
101C2×#123—21no—
102C2×#121C22×C311yescharacteristic
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4C22×C314yes—
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C42C22×C311yescharacteristic
110#480—24no—
111#447—24no—
112C2×#107C22×C312yescomplemented
113C2×#102C22×C312yes—
114C2×#102C22×C314yes—
115C2×#102C22×C312yescomplemented
116C2×C22⋊C8C22×C311yescharacteristic
117C2×C22⋊C8C22×C311yescharacteristic
118#390C22×C312yescomplemented
119#389C22×C312yescomplemented
120#388—22no—
121#386—21no—
122#380C22×C311yescharacteristic
123#340C22×C314yescomplemented
124#336C22×C314yescomplemented
125C24×C3—28no—
126C24×C3—24no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—44no—
130C24×C3—42no—
131C24×C3—42no—
132C24×C3C2×D411yescharacteristic
133C2×C3×D4—28no—
134C2×C3×D4—216no—
135C2×C3×D4—216no—
136C2×C3×D4—24no—
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4C2×D414yes—
146C2×C3×D4#4118yes—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—216no—
151C22×C3×C4—22no—
152C22×C3×C4—24no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—42no—
157C22×C3×C4—48no—
158C22×C3×C4—432no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—48no—
162C22×C3×C4D814yescomplemented
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2411yescharacteristic
169C3×D8—48no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—24no—
173C2×C3×C8—24no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4C2×D414yes—
181C3×C22⋊C4—216no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—44no—
186C3×C42—24no—
187C3×C42—22no—
188C3×C42—22no—
189C25—21no—
190C22×D4—28no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—22no—
194C22×D4C3×D414yes—
195C22×D4C23×C311yescharacteristic
196C22×D4C23×C312yes—
197C23×C4—28no—
198C23×C4—21no—
199C23×C4—24no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C3×D411yescharacteristic
203C23×C4C3×D411yescharacteristic
204C23×C4C23×C311yescharacteristic
205C2×D8—24no—
206C2×D8—28no—
207C22×C8C23×C312yes—
208C22×C8C23×C312yes—
209#123—44no—
210#121—22no—
211#121—22no—
212C4×D4—216no—
213C4×D4—28no—
214C4×D4—28no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C3×D414yes—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C312yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42C3×D412yes—
231C2×C42C23×C312yes—
232#107—24no—
233#102—24no—
234#102—28no—
235#102—24no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238#95—21no—
239#95—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—216no—
243C23×C3—24no—
244C23×C3—28no—
245C23×C3—416no—
246C23×C3—416no—
247C23×C3—48no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—44no—
251C23×C3—48no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—42no—
257C23×C3#12012yes—
258C23×C3C2×D812yes—
259C23×C3AGL(1,ℤ/8)12yes—
260C23×C3C22×D411yescharacteristic
261C3×D4—216no—
262C3×D4—24no—
263C3×D4—28no—
264C3×D4—42no—
265C3×D4—42no—
266C3×D4—84no—
267C3×D4#12318yescomplemented
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—28no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—432no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—48no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4#12012yes—
290C2×C3×C4C2×D814yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C2×#4114yes—
294C3×C8—42no—
295C3×C8—42no—
296C24—28no—
297C24—24no—
298C24—22no—
299C24—21no—
300C24—44no—
301C24—42no—
302C24—42no—
303C24C2×C3×D411yescharacteristic
304C2×D4—28no—
305C2×D4—216no—
306C2×D4—216no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4C2×C3×D414yes—
317C2×D4C3×#4118yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—216no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—48no—
329C22×C4—432no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4C3×D814yescomplemented
334C22×C4C2×C3×D412yes—
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D412yes—
339C22×C4C24×C311yescharacteristic
340D8—48no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—24no—
350C4⋊C4—44no—
351C4⋊C4C2×C3×D414yes—
352C22⋊C4—216no—
353C22⋊C4—44no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C42—24no—
358C42—22no—
359C42—22no—
360C22×C3—24no—
361C22×C3—22no—
362C22×C3—24no—
363C22×C3—22no—
364C22×C3—28no—
365C22×C3—432no—
366C22×C3—48no—
367C22×C3—48no—
368C22×C3—44no—
369C22×C3—44no—
370C22×C3—82no—
371C22×C3—81no—
372C22×C3—82no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3#44714yes—
376C22×C3C2×#12011yescharacteristic
377C22×C3#52512yes—
378C22×C3C22×D811yescharacteristic
379C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
380C22×C3#57512yes—
381C3×C4—22no—
382C3×C4—24no—
383C3×C4—44no—
384C3×C4—42no—
385C3×C4—82no—
386C3×C4—81no—
387C3×C4C2×#12312yes—
388C3×C4#52512yes—
389C23—22no—
390C23—22no—
391C23—216no—
392C23—24no—
393C23—28no—
394C23—48no—
395C23—416no—
396C23—416no—
397C23—42no—
398C23—42no—
399C23—44no—
400C23—48no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—42no—
406C23C3×#12012yes—
407C23C2×C3×D812yes—
408C23C3×AGL(1,ℤ/8)12yes—
409C23C22×C3×D411yescharacteristic
410D4—216no—
411D4—24no—
412D4—28no—
413D4—42no—
414D4—42no—
415D4—84no—
416D4C3×#12318yescomplemented
417C2×C4—22no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—432no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—48no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4C3×#12012yes—
439C2×C4C2×C3×D814yes—
440C2×C4C22×C3×D411yescharacteristic
441C2×C4C22×C3×D411yescharacteristic
442C2×C4C2×C3×#4114yes—
443C8—42no—
444C8—42no—
445C2×C3—24no—
446C2×C3—48no—
447C2×C3—161no—
448C2×C3#284911yescharacteristic
449C2×C3C2×#44711yescharacteristic
450C2×C3#300511yescharacteristic
451C2×C3#310012yes—
452C2×C3#311412yes—
453C22—22no—
454C22—24no—
455C22—22no—
456C22—24no—
457C22—28no—
458C22—432no—
459C22—44no—
460C22—44no—
461C22—48no—
462C22—48no—
463C22—81no—
464C22—81no—
465C22—82no—
466C22—81no—
467C22—82no—
468C22C3×#44714yes—
469C22C2×C3×#12011yescharacteristic
470C22C3×#52512yes—
471C22C22×C3×D811yescharacteristic
472C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
473C22C3×#57512yes—
474C4—24no—
475C4—22no—
476C4—44no—
477C4—42no—
478C4—81no—
479C4—82no—
480C4C2×C3×#12312yes—
481C4C3×#52512yes—
482C3#3300211yescharacteristic, direct product factor, Sylow
483C2—24no—
484C2—48no—
485C2—161no—
486C2C3×#284911yescharacteristic
487C2C2×C3×#44711yescharacteristic
488C2C3×#300511yescharacteristic
489C2C3×#310012yes—
490C2C3×#311412yes—
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×C4×D4C2semidirect
2. C2×C3×#480C2semidirect
4. C3×#2854C2semidirect
5. C3×#1879C2semidirect
6. C3×#1862C2semidirect
7. C3×#1823C2semidirect
8. C3×#1817C2semidirect
9. C3×#1786C2semidirect
10. #33002C3direct (N × H)
25. C2×C3×#107C22semidirect
28. C2×C3×#102C22semidirect
31. C3×#390C22semidirect
32. C3×#389C22semidirect
36. C3×#340C22semidirect
37. C3×#336C22semidirect
38. C22×C4×D4C2×C3semidirect
39. C2×#480C2×C3semidirect
41. #2854C2×C3semidirect
42. #1879C2×C3semidirect
43. #1862C2×C3semidirect
44. #1823C2×C3semidirect
45. #1817C2×C3semidirect
46. #1786C2×C3semidirect
112. C2×#107C22×C3semidirect
115. C2×#102C22×C3semidirect
118. #390C22×C3semidirect
119. #389C22×C3semidirect
123. #340C22×C3semidirect
124. #336C22×C3semidirect
162. C22×C3×C4D8semidirect
267. C3×D4#123semidirect
333. C22×C4C3×D8semidirect
416. D4C3×#123semidirect
482. C3#33002direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#4802
C2×C3×#4472
C3×#28542
C3×#18792
C3×#18622
C3×#18232
C3×#18172
C3×#17862
#330023