Finite Group Databasebeta

← C3×#25571 #10644178

Subgroups — C3×#25571 #10644178

Order 768.

1386 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C3×#2542C211yesmaximal, characteristic, complemented
5C3×#2000C214yesmaximal, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic, complemented
8C3×#1872C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25571C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C22×C3×SD16C2212yes—
13C3×#547—24no—
14C3×#547—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2214yes—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C4⋊C4C2212yes—
24C3×#487—24no—
25C3×#461—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2214yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#399—22no—
33C3×#390—21no—
34C3×#384—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2214yes—
37C3×#340C2212yes—
38C2×#547C2×C311yescharacteristic
39C2×#487C2×C311yescharacteristic, complemented
40C2×#461C2×C311yescharacteristic, complemented
41#2542C2×C311yescharacteristic, complemented
42#2000C2×C314yescomplemented
43#1988C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic, complemented
45#1872C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C2×C3×#41—28no—
48C2×C3×#41—44no—
49C2×C3×#41—44no—
50C22×C3×Q8—21no—
51C22×C3×Q8C2312yes—
52C22×C3×D4—22no—
53C22×C3×D4D411yescharacteristic
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4—21no—
57C23×C3×C4C2312yes—
58C2×C3×SD16—24no—
59C2×C3×SD16—44no—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×#123—216no—
67C3×C22⋊Q8—28no—
68C3×C22⋊Q8—28no—
69C3×C22⋊Q8—44no—
70C3×#121—24no—
71C3×#121—24no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—24no—
90C2×C3×C22⋊C4D412yes—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C3×C4⋊C8—22no—
94C3×#103—24no—
95C3×#103—28no—
96C3×#102—24no—
97C3×C22⋊C8—44no—
98C3×C4×C8—22no—
99C3×#95—21no—
100C22×#41—21no—
101C22×SD16C22×C312yes—
102#547—24no—
103#547—24no—
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C22×C314yes—
109C2×#121C22×C312yes—
110C2×C4×D4C22×C311yescharacteristic
111C2×C4×D4C22×C311yescharacteristic
112C22×C4⋊C4C22×C312yes—
113#487—24no—
114#461—24no—
115C2×C4⋊C8C22×C311yescharacteristic
116C2×#103C22×C314yes—
117C2×#103C22×C312yes—
118C2×#102C22×C312yes—
119C2×C22⋊C8C22×C314yes—
120C2×C4×C8C22×C311yescharacteristic
121#399—22no—
122#390—21no—
123#384—21no—
124C4×C4⋊C4C22×C311yescharacteristic
125#340C22×C312yes—
126#340C22×C314yes—
127C24×C3—21no—
128C24×C3C2×D412yes—
129C3×#41—432no—
130C2×C3×Q8—24no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—24no—
135C2×C3×Q8—42no—
136C2×C3×Q8—42no—
137C2×C3×D4—28no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4SD1614yescomplemented
146C2×C3×D4C2×D412yes—
147C22×C3×C4—28no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—44no—
158C22×C3×C4—44no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C3×SD16—48no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—24no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—48no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4C2×D412yes—
181C3×C4⋊C4C2×D414yes—
182C3×C4⋊C4#4114yescomplemented
183C3×C22⋊C4—28no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—48no—
187C3×C42—42no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190C2×#41—28no—
191C2×#41—44no—
192C2×#41—44no—
193C22×Q8—21no—
194C22×Q8C23×C312yes—
195C22×D4—22no—
196C22×D4C3×D411yescharacteristic
197C22×D4C23×C311yescharacteristic
198C23×C4—22no—
199C23×C4—21no—
200C23×C4C23×C312yes—
201C2×SD16—24no—
202C2×SD16—44no—
203C22×C8C23×C312yes—
204C22×C8C23×C312yes—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8C3×D414yes—
209#123—216no—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—44no—
213#121—24no—
214#121—24no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4C3×D414yes—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C3×D411yescharacteristic
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C4⋊C4C23×C312yes—
229C2×C4⋊C4C23×C312yes—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4C3×D412yes—
234C2×C42—21no—
235C2×C42C23×C311yescharacteristic
236C4⋊C8—22no—
237#103—24no—
238#103—28no—
239#102—24no—
240C22⋊C8—44no—
241C4×C8—22no—
242#95—21no—
243C23×C3—22no—
244C23×C3—24no—
245C23×C3—22no—
246C23×C3—24no—
247C23×C3—22no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3C2×SD1614yes—
253C23×C3C22×D411yescharacteristic
254C3×Q8—48no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×Q8—44no—
258C3×D4—22no—
259C3×D4—22no—
260C3×D4—28no—
261C3×D4—416no—
262C3×D4—42no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—416no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—44no—
279C2×C3×C4—48no—
280C2×C3×C4—42no—
281C2×C3×C4—44no—
282C2×C3×C4#12012yes—
283C2×C3×C4#12012yes—
284C2×C3×C4C2×SD1612yes—
285C2×C3×C4AGL(1,ℤ/8)12yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C4H(5,𝔽2)12yes—
290C3×C8—42no—
291C3×C8—42no—
292C24—21no—
293C24C2×C3×D412yes—
294#41—432no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×D4—28no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4C3×SD1614yescomplemented
311C2×D4C2×C3×D412yes—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4C2×C3×D412yes—
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C24×C311yescharacteristic
330SD16—48no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C4⋊C4—28no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4C2×C3×D412yes—
346C4⋊C4C2×C3×D414yes—
347C4⋊C4C3×#4114yescomplemented
348C22⋊C4—28no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—48no—
352C42—42no—
353C42C2×C3×D412yes—
354C42C2×C3×D412yes—
355C22×C3—24no—
356C22×C3—28no—
357C22×C3—44no—
358C22×C3—42no—
359C22×C3—42no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3C2×#12011yescharacteristic
366C22×C3#54612yes—
367C22×C3C22×SD1611yescharacteristic
368C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
369C22×C3#57812yes—
370C3×C4—22no—
371C3×C4—44no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—44no—
375C3×C4—82no—
376C3×C4#45014yes—
377C3×C4#53211yescharacteristic
378C3×C4#53511yescharacteristic
379C3×C4#55012yes—
380C23—22no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23C2×C3×SD1614yes—
390C23C22×C3×D411yescharacteristic
391Q8—48no—
392Q8—42no—
393Q8—42no—
394Q8—44no—
395D4—28no—
396D4—22no—
397D4—22no—
398D4—416no—
399D4—42no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—416no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4C3×#12012yes—
420C2×C4C3×#12012yes—
421C2×C4C2×C3×SD1612yes—
422C2×C4C3×AGL(1,ℤ/8)12yes—
423C2×C4C22×C3×D411yescharacteristic
424C2×C4C22×C3×D411yescharacteristic
425C2×C4C2×C3×#4112yes—
426C2×C4C3×H(5,𝔽2)12yes—
427C8—42no—
428C8—42no—
429C2×C3—44no—
430C2×C3—82no—
431C2×C3#243111yescharacteristic
432C2×C3C2×#45011yescharacteristic
433C2×C3#301111yescharacteristic
434C2×C3#329812yes—
435C2×C3#333012yes—
436C22—24no—
437C22—28no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22C2×C3×#12011yescharacteristic
447C22C3×#54612yes—
448C22C22×C3×SD1611yescharacteristic
449C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
450C22C3×#57812yes—
451C4—22no—
452C4—42no—
453C4—42no—
454C4—44no—
455C4—44no—
456C4—82no—
457C4C3×#45014yes—
458C4C3×#53211yescharacteristic
459C4C3×#53511yescharacteristic
460C4C3×#55012yes—
461C3#2557111yescharacteristic, direct product factor, Sylow
462C2—44no—
463C2—82no—
464C2C3×#243111yescharacteristic
465C2C2×C3×#45011yescharacteristic
466C2C3×#301111yescharacteristic
467C2C3×#329812yes—
468C2C3×#333012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#487C2semidirect
3. C2×C3×#461C2semidirect
4. C3×#2542C2semidirect
5. C3×#2000C2semidirect
6. C3×#1988C2semidirect
7. C3×#1954C2semidirect
8. C3×#1872C2semidirect
9. C3×C4×#102C2semidirect
10. #25571C3direct (N × H)
39. C2×#487C2×C3semidirect
40. C2×#461C2×C3semidirect
41. #2542C2×C3semidirect
42. #2000C2×C3semidirect
43. #1988C2×C3semidirect
44. #1954C2×C3semidirect
45. #1872C2×C3semidirect
46. C4×#102C2×C3semidirect
145. C2×C3×D4SD16semidirect
182. C3×C4⋊C4#41semidirect
310. C2×D4C3×SD16semidirect
347. C4⋊C4C3×#41semidirect
461. C3#25571direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4872
C2×C3×#4612
C3×#25422
C3×#20002
C3×#19882
C3×#19542
C3×#18722
C3×C4×#1022
#255713