Finite Group Databasebeta

← C3×#27678 #10646285

Subgroups — C3×#27678 #10646285

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C2×C3×#384C211yesmaximal, characteristic, complemented
3C3×#2100C212yesmaximal, complemented
4C3×#2094C212yesmaximal, complemented
5C3×#1948C212yesmaximal, complemented
6C3×#1874C212yesmaximal
7C3×#1853C212yesmaximal, complemented
8C3×#1802C212yesmaximal, complemented
9C3×#1797C211yesmaximal, characteristic, complemented
10#27678C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#121C2211yescharacteristic
12C2×C3×C4×D4—21no—
13C2×C3×C4×D4C2211yescharacteristic
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4—21no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C42C2211yescharacteristic
18C2×C3×#106C2214yes—
19C2×C3×C4⋊C8C2212yes—
20C2×C3×#102C2214yes—
21C2×C3×C22⋊C8C2212yes—
22C3×#390C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#388—21no—
25C3×#386—21no—
26C3×#384—21no—
27C3×#384—21no—
28C3×#384C2212yes—
29C3×#384C2212yes—
30C3×#380C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C22⋊C4—21no—
33C3×#340C2214yes—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C4×#121C2×C311yescharacteristic
37C2×#384C2×C311yescharacteristic, complemented
38#2100C2×C312yescomplemented
39#2094C2×C312yescomplemented
40#1948C2×C312yescomplemented
41#1874C2×C312yes—
42#1853C2×C312yescomplemented
43#1802C2×C312yescomplemented
44#1797C2×C311yescharacteristic, complemented
45C22×C3×D4—21no—
46C22×C3×D4C2311yescharacteristic
47C23×C3×C4—22no—
48C23×C3×C4—21no—
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×C3×C8C2314yes—
53C3×#121—22no—
54C3×#121Q814yes—
55C3×C4×D4—24no—
56C3×C4×D4—44no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—44no—
64C2×C3×C4⋊C4—44no—
65C2×C3×C4⋊C4—44no—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2314yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—24no—
76C2×C3×C42—21no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C3×#106—44no—
82C3×C4⋊C8—24no—
83C3×#102—44no—
84C3×C22⋊C8—24no—
85C3×#95—21no—
86C3×#95—21no—
87C3×#95—21no—
88C2×#121C22×C311yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4C22×C311yescharacteristic
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4C22×C312yes—
94C22×C42C22×C311yescharacteristic
95C2×#106C22×C314yes—
96C2×C4⋊C8C22×C312yes—
97C2×#102C22×C314yes—
98C2×C22⋊C8C22×C312yes—
99#390C22×C311yescharacteristic
100#389C22×C311yescharacteristic
101#388—21no—
102#386—21no—
103#384—21no—
104#384—21no—
105#384C22×C312yes—
106#384C22×C312yes—
107#380C22×C311yescharacteristic
108C4×C4⋊C4C22×C311yescharacteristic
109C4×C22⋊C4—21no—
110#340C22×C314yes—
111#340C22×C314yes—
112#336C22×C312yes—
113C24×C3—21no—
114C24×C3C2×D411yescharacteristic
115C2×C3×D4—24no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—22no—
119C2×C3×D4—42no—
120C2×C3×D4—42no—
121C2×C3×D4C2×Q812yes—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—44no—
134C22×C3×C4—44no—
135C22×C3×C4—44no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4SD1614yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×Q812yes—
146C22×C3×C4#4112yes—
147C22×C3×C4C2411yescharacteristic
148C2×C3×C8—24no—
149C2×C3×C8—24no—
150C2×C3×C8—24no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—44no—
157C3×C4⋊C4—44no—
158C3×C4⋊C4—44no—
159C3×C4⋊C4—44no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—44no—
162C3×C4⋊C4C2×Q812yes—
163C3×C22⋊C4—24no—
164C3×C22⋊C4—42no—
165C3×C22⋊C4—42no—
166C3×C22⋊C4—42no—
167C3×C22⋊C4—42no—
168C3×C42—22no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—42no—
172C3×C42#4114yes—
173C22×D4—21no—
174C22×D4C23×C311yescharacteristic
175C23×C4—22no—
176C23×C4—21no—
177C23×C4C3×D411yescharacteristic
178C23×C4C3×D411yescharacteristic
179C23×C4C23×C311yescharacteristic
180C22×C8C23×C314yes—
181#121—22no—
182#121C3×Q814yes—
183C4×D4—24no—
184C4×D4—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4C23×C314yes—
195C2×C4⋊C4C23×C311yescharacteristic
196C2×C4⋊C4C23×C311yescharacteristic
197C2×C4⋊C4C23×C311yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—24no—
204C2×C42—21no—
205C2×C42C3×D411yescharacteristic
206C2×C42C3×D411yescharacteristic
207C2×C42C23×C311yescharacteristic
208C2×C42C23×C311yescharacteristic
209#106—44no—
210C4⋊C8—24no—
211#102—44no—
212C22⋊C8—24no—
213#95—21no—
214#95—21no—
215#95—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—22no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3C22⋊Q812yes—
228C23×C3C2×SD1612yes—
229C23×C3AGL(1,ℤ/8)12yes—
230C23×C3C22×D411yescharacteristic
231C3×D4—24no—
232C3×D4—48no—
233C3×D4—42no—
234C2×C3×C4—28no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—28no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—44no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—88no—
259C2×C3×C4—84no—
260C2×C3×C4C22⋊Q812yes—
261C2×C3×C4#12312yes—
262C2×C3×C4#12312yes—
263C2×C3×C4C2×SD1612yes—
264C2×C3×C4C2×SD1612yes—
265C2×C3×C4AGL(1,ℤ/8)12yes—
266C2×C3×C4#13712yes—
267C2×C3×C4C22×Q811yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269C2×C3×C4C2×#4112yes—
270C2×C3×C4H(5,𝔽2)12yes—
271C3×C8—44no—
272C24—21no—
273C24C2×C3×D411yescharacteristic
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—21no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4C2×C3×Q812yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C3×SD1614yes—
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C2×C3×D411yescharacteristic
301C22×C4C2×C3×D411yescharacteristic
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×Q812yes—
305C22×C4C3×#4112yes—
306C22×C4C24×C311yescharacteristic
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4C2×C3×Q812yes—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—42no—
331C42C3×#4114yes—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3#47614yes—
346C22×C3C2×C22⋊Q811yescharacteristic
347C22×C3C2×#12311yescharacteristic
348C22×C3#53511yescharacteristic
349C22×C3C22×SD1611yescharacteristic
350C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
351C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
352C22×C3C2×#13711yescharacteristic
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—42no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4—84no—
361C3×C4#48114yes—
362C3×C4#55312yes—
363C3×C4#55412yes—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23C3×C22⋊Q812yes—
376C23C2×C3×SD1612yes—
377C23C3×AGL(1,ℤ/8)12yes—
378C23C22×C3×D411yescharacteristic
379D4—24no—
380D4—48no—
381D4—42no—
382C2×C4—28no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—88no—
407C2×C4—84no—
408C2×C4C3×C22⋊Q812yes—
409C2×C4C3×#12312yes—
410C2×C4C3×#12312yes—
411C2×C4C2×C3×SD1612yes—
412C2×C4C2×C3×SD1612yes—
413C2×C4C3×AGL(1,ℤ/8)12yes—
414C2×C4C3×#13712yes—
415C2×C4C22×C3×Q811yescharacteristic
416C2×C4C2×C3×#4111yescharacteristic
417C2×C4C2×C3×#4112yes—
418C2×C4C3×H(5,𝔽2)12yes—
419C8—44no—
420C2×C3—24no—
421C2×C3—82no—
422C2×C3#248711yescharacteristic
423C2×C3C2×#47611yescharacteristic
424C2×C3#307911yescharacteristic
425C2×C3C2×#48111yescharacteristic
426C2×C3#309211yescharacteristic
427C2×C3#320611yescharacteristic
428C2×C3#321711yescharacteristic
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22C3×#47614yes—
443C22C2×C3×C22⋊Q811yescharacteristic
444C22C2×C3×#12311yescharacteristic
445C22C3×#53511yescharacteristic
446C22C22×C3×SD1611yescharacteristic
447C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
448C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
449C22C2×C3×#13711yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—42no—
455C4—81no—
456C4—84no—
457C4—81no—
458C4C3×#48114yes—
459C4C3×#55312yes—
460C4C3×#55412yes—
461C3#2767811yescharacteristic, direct product factor, Sylow
462C2—24no—
463C2—82no—
464C2C3×#248711yescharacteristic
465C2C2×C3×#47611yescharacteristic
466C2C3×#307911yescharacteristic
467C2C2×C3×#48111yescharacteristic
468C2C3×#309211yescharacteristic
469C2C3×#320611yescharacteristic
470C2C3×#321711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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×#384C2semidirect
3. C3×#2100C2semidirect
4. C3×#2094C2semidirect
5. C3×#1948C2semidirect
7. C3×#1853C2semidirect
8. C3×#1802C2semidirect
9. C3×#1797C2semidirect
10. #27678C3direct (N × H)
37. C2×#384C2×C3semidirect
38. #2100C2×C3semidirect
39. #2094C2×C3semidirect
40. #1948C2×C3semidirect
42. #1853C2×C3semidirect
43. #1802C2×C3semidirect
44. #1797C2×C3semidirect
461. C3#27678direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C2×C3×#3842
C3×#21002
C3×#20942
C3×#19482
C3×#18742
C3×#18532
C3×#18022
C3×#17972
#276783