Finite Group Databasebeta

← C3×#24678 #10643285

Subgroups — C3×#24678 #10643285

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#475C211yesmaximal, characteristic, complemented
2C2×C3×C4×SD16C211yesmaximal, characteristic
3C3×#2542C211yesmaximal, characteristic
4C3×#2355C211yesmaximal, characteristic, complemented
5C3×#2072C212yesmaximal, complemented
6C3×#2055C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1948C212yesmaximal, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal
11C3×#1802C211yesmaximal, characteristic
12#24678C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#475—24no—
20C3×C4×SD16—24no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#399—22no—
31C3×#391C2212yes—
32C3×#390C2212yes—
33C3×#387—22no—
34C3×#384C2211yescharacteristic
35C3×#384C2212yes—
36C3×#382—21no—
37C3×C4×C4⋊C4—21no—
38C3×C4×C4⋊C4C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#475C2×C311yescharacteristic, complemented
45C2×C4×SD16C2×C311yescharacteristic
46#2542C2×C311yescharacteristic
47#2355C2×C311yescharacteristic, complemented
48#2072C2×C312yescomplemented
49#2055C2×C312yes—
50#1990C2×C311yescharacteristic, complemented
51#1948C2×C312yescomplemented
52#1939C2×C311yescharacteristic, complemented
53#1926C2×C312yes—
54#1802C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—22no—
60C2×C3×SD16—22no—
61C2×C3×SD16—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C22⋊Q8—44no—
68C3×C22⋊Q8—44no—
69C3×C4×Q8—22no—
70C3×C4×Q8D414yes—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×C3×C42—22no—
90C2×C3×C42C2312yes—
91C2×C3×C42C2311yescharacteristic
92C3×#106—22no—
93C3×#106—24no—
94C3×C4⋊C8—24no—
95C3×C4⋊C8—22no—
96C3×#103—22no—
97C3×#103—24no—
98C3×#102—24no—
99C3×#102—22no—
100C3×C4×C8—22no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95D412yes—
105C22×SD16C22×C311yescharacteristic
106C2×C4⋊Q8C22×C311yescharacteristic
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8C22×C311yescharacteristic
110C2×C4×D4C22×C311yescharacteristic
111#475—24no—
112C4×SD16—24no—
113C2×#106C22×C312yes—
114C2×#106C22×C311yescharacteristic
115C2×C4⋊C8C22×C311yescharacteristic
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C312yes—
118C2×#103C22×C311yescharacteristic
119C2×#102C22×C312yes—
120C2×#102C22×C311yescharacteristic
121C2×C4×C8C22×C311yescharacteristic
122#399—22no—
123#391C22×C312yes—
124#390C22×C312yes—
125#387—22no—
126#384C22×C311yescharacteristic
127#384C22×C312yes—
128#382—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C22×C312yes—
131C4×C4⋊C4C22×C311yescharacteristic
132#340C22×C312yes—
133#340C22×C312yes—
134#340C22×C312yes—
135#336C22×C312yes—
136C24×C3—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—42no—
143C2×C3×Q8—42no—
144C2×C3×Q8—42no—
145C2×C3×Q8C2×D412yes—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4#4112yes—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C3×SD16—28no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8#4112yes—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—44no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—81no—
189C3×C4⋊C4SD1614yescomplemented
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—44no—
194C3×C42—24no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—82no—
198C3×C42C2×D412yes—
199C3×C42#4112yes—
200C22×Q8—21no—
201C22×Q8C23×C311yescharacteristic
202C22×D4C23×C311yescharacteristic
203C23×C4—21no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—22no—
207C22×C8C23×C311yescharacteristic
208C22×C8C23×C312yes—
209C22×C8C23×C311yescharacteristic
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C22⋊Q8—44no—
213C22⋊Q8—44no—
214C4×Q8—22no—
215C4×Q8C3×D414yes—
216C4×D4—22no—
217C4×D4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C3×D411yescharacteristic
225C2×C4⋊C4C3×D411yescharacteristic
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C4⋊C4C23×C312yes—
228C2×C4⋊C4C23×C312yes—
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42C23×C312yes—
236C2×C42C23×C311yescharacteristic
237#106—22no—
238#106—24no—
239C4⋊C8—24no—
240C4⋊C8—22no—
241#103—22no—
242#103—24no—
243#102—24no—
244#102—22no—
245C4×C8—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95C3×D412yes—
250C23×C3—22no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3C22×D411yescharacteristic
258C3×Q8—22no—
259C3×Q8—22no—
260C3×Q8—41no—
261C3×Q8—41no—
262C3×Q8—84no—
263C3×D4—22no—
264C3×D4—22no—
265C3×D4—41no—
266C3×D4—41no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—84no—
290C2×C3×C4#12112yes—
291C2×C3×C4#12112yes—
292C2×C3×C4C2×SD1614yes—
293C2×C3×C4C2×SD1612yes—
294C2×C3×C4AGL(1,ℤ/8)12yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C42−1+411yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301C3×C8—22no—
302C3×C8—42no—
303C3×C8—41no—
304C24—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8C2×C3×D412yes—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4C3×#4112yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4C2×C3×D412yes—
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C2×C3×D412yes—
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C24×C311yescharacteristic
333SD16—28no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8C3×#4112yes—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—81no—
357C4⋊C4C3×SD1614yescomplemented
358C4⋊C4C2×C3×D412yes—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C42—24no—
363C42—22no—
364C42—22no—
365C42—82no—
366C42C2×C3×D412yes—
367C42C3×#4112yes—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3C2×#12111yescharacteristic
376C22×C3#54711yescharacteristic
377C22×C3#54811yescharacteristic
378C22×C3C22×SD1611yescharacteristic
379C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
380C22×C3#57711yescharacteristic
381C22×C3#57811yescharacteristic
382C3×C4—24no—
383C3×C4—22no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—42no—
387C3×C4—82no—
388C3×C4—81no—
389C3×C4—82no—
390C3×C4#46014yes—
391C3×C4#52511yescharacteristic
392C3×C4#53611yescharacteristic
393C3×C4#55211yescharacteristic
394C3×C4#55611yescharacteristic
395C23—22no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23C22×C3×D411yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407Q8—84no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4C3×#12112yes—
436C2×C4C3×#12112yes—
437C2×C4C2×C3×SD1614yes—
438C2×C4C2×C3×SD1612yes—
439C2×C4C3×AGL(1,ℤ/8)12yes—
440C2×C4C22×C3×D411yescharacteristic
441C2×C4C2×C3×#4111yescharacteristic
442C2×C4C2×C3×#4111yescharacteristic
443C2×C4C2×C3×#4111yescharacteristic
444C2×C4C3×2−1+411yescharacteristic
445C2×C4C3×2−1+411yescharacteristic
446C8—22no—
447C8—42no—
448C8—41no—
449C2×C3—82no—
450C2×C3#241511yescharacteristic
451C2×C3C2×#46011yescharacteristic
452C2×C3#304611yescharacteristic
453C2×C3#333511yescharacteristic
454C2×C3#335311yescharacteristic
455C2×C3#335911yescharacteristic
456C2×C3#337311yescharacteristic
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C2×C3×#12111yescharacteristic
465C22C3×#54711yescharacteristic
466C22C3×#54811yescharacteristic
467C22C22×C3×SD1611yescharacteristic
468C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
469C22C3×#57711yescharacteristic
470C22C3×#57811yescharacteristic
471C4—22no—
472C4—24no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—82no—
478C4—82no—
479C4C3×#46014yes—
480C4C3×#52511yescharacteristic
481C4C3×#53611yescharacteristic
482C4C3×#55211yescharacteristic
483C4C3×#55611yescharacteristic
484C3#2467811yescharacteristic, direct product factor, Sylow
485C2—82no—
486C2C3×#241511yescharacteristic
487C2C2×C3×#46011yescharacteristic
488C2C3×#304611yescharacteristic
489C2C3×#333511yescharacteristic
490C2C3×#335311yescharacteristic
491C2C3×#335911yescharacteristic
492C2C3×#337311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#475C2semidirect
4. C3×#2355C2semidirect
5. C3×#2072C2semidirect
7. C3×#1990C2semidirect
8. C3×#1948C2semidirect
9. C3×#1939C2semidirect
12. #24678C3direct (N × H)
44. C2×#475C2×C3semidirect
47. #2355C2×C3semidirect
48. #2072C2×C3semidirect
50. #1990C2×C3semidirect
51. #1948C2×C3semidirect
52. #1939C2×C3semidirect
189. C3×C4⋊C4SD16semidirect
357. C4⋊C4C3×SD16semidirect
484. C3#24678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4752
C2×C3×C4×SD162
C3×#25422
C3×#23552
C3×#20722
C3×#20552
C3×#19902
C3×#19482
C3×#19392
C3×#19262
C3×#18022
#246783