Finite Group Databasebeta

← C3×#25760 #10644367

Subgroups — C3×#25760 #10644367

Order 768.

1514 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#494C211yesmaximal, characteristic, complemented
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C2×C3×#451C214yesmaximal, complemented
5C3×#2542C211yesmaximal, characteristic
6C3×#2001C214yesmaximal, complemented
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1954C211yesmaximal, characteristic
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25760C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2212yes—
12C22×C3×Q16C2212yes—
13C22×C3×Q16C2214yes—
14C3×D4×Q8—24no—
15C3×D4×Q8—24no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8—22no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C22⋊Q8C2214yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#494—24no—
26C3×#462—24no—
27C3×#451—216no—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×C3×#103C2214yes—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2214yes—
33C2×C3×C4×C8C2211yescharacteristic
34C3×#399—22no—
35C3×#390—21no—
36C3×#384—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#340C2212yes—
39C2×D4×Q8C2×C311yescharacteristic
40C2×#494C2×C311yescharacteristic, complemented
41C2×#462C2×C311yescharacteristic, complemented
42C2×#451C2×C314yescomplemented
43#2542C2×C311yescharacteristic
44#2001C2×C314yescomplemented
45#1981C2×C311yescharacteristic, complemented
46#1954C2×C311yescharacteristic
47C4×#103C2×C311yescharacteristic, complemented
48C22×C3×Q8—24no—
49C22×C3×Q8—24no—
50C22×C3×Q8—22no—
51C22×C3×Q8—28no—
52C22×C3×Q8—44no—
53C22×C3×Q8C2312yes—
54C22×C3×Q8C2312yes—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4D411yescharacteristic
57C23×C3×C4—22no—
58C23×C3×C4C2312yes—
59C2×C3×Q16—28no—
60C2×C3×Q16—24no—
61C2×C3×Q16—28no—
62C2×C3×Q16—28no—
63C2×C3×Q16—44no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—28no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×C22⋊Q8—216no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—28no—
75C3×C22⋊Q8—44no—
76C3×C4×Q8—22no—
77C3×C4×Q8D414yes—
78C3×C4×D4—28no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—24no—
93C2×C3×C22⋊C4D412yes—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—22no—
98C3×#103—28no—
99C3×#103—24no—
100C3×#103D418yescomplemented
101C3×C22⋊C8—28no—
102C3×C4×C8—22no—
103C3×#95—21no—
104C23×Q8C22×C312yes—
105C22×Q16C22×C312yes—
106C22×Q16C22×C314yes—
107D4×Q8—24no—
108D4×Q8—24no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C22×C311yescharacteristic
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C22×C312yes—
114C2×C22⋊Q8C22×C314yes—
115C2×C4×Q8C22×C311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4C22×C311yescharacteristic
118#494—24no—
119#462—24no—
120#451—216no—
121C2×C4⋊C8C22×C311yescharacteristic
122C2×#103C22×C314yes—
123C2×#103C22×C312yes—
124C2×#103C22×C312yes—
125C2×C22⋊C8C22×C314yes—
126C2×C4×C8C22×C311yescharacteristic
127#399—22no—
128#390—21no—
129#384—21no—
130C4×C4⋊C4C22×C311yescharacteristic
131#340C22×C312yes—
132C24×C3C2×D412yes—
133C2×C3×Q8—28no—
134C2×C3×Q8—21no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C2×C3×Q8—24no—
140C2×C3×Q8—22no—
141C2×C3×Q8—24no—
142C2×C3×Q8—24no—
143C2×C3×Q8—48no—
144C2×C3×Q8—416no—
145C2×C3×Q8—48no—
146C2×C3×Q8—48no—
147C2×C3×Q8C2×D414yes—
148C2×C3×Q8C2×D414yes—
149C2×C3×Q8C2×D412yes—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4Q1614yescomplemented
153C22×C3×C4—21no—
154C22×C3×C4—28no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—48no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C3×Q16—416no—
169C3×Q16—48no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—24no—
173C2×C3×C8—22no—
174C2×C3×C8C2×D414yes—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—48no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4C2×D414yes—
186C3×C4⋊C4C2×D412yes—
187C3×C22⋊C4—28no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—48no—
190C3×C42—24no—
191C3×C42—42no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—28no—
198C22×Q8—44no—
199C22×Q8C23×C312yes—
200C22×Q8C23×C312yes—
201C22×Q8C23×C311yescharacteristic
202C22×D4C3×D411yescharacteristic
203C23×C4—22no—
204C23×C4C23×C312yes—
205C2×Q16—28no—
206C2×Q16—24no—
207C2×Q16—28no—
208C2×Q16—28no—
209C2×Q16—44no—
210C22×C8C23×C312yes—
211C22×C8C23×C312yes—
212C4⋊Q8—28no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8C3×D414yes—
217C22⋊Q8—216no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—28no—
221C22⋊Q8—44no—
222C4×Q8—22no—
223C4×Q8C3×D414yes—
224C4×D4—28no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C3×D411yescharacteristic
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C4⋊C4C23×C312yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4C3×D412yes—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C23×C311yescharacteristic
243C4⋊C8—22no—
244#103—28no—
245#103—24no—
246#103C3×D418yescomplemented
247C22⋊C8—28no—
248C4×C8—22no—
249#95—21no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3C2×Q1614yes—
255C23×C3C22×D411yescharacteristic
256C3×Q8—22no—
257C3×Q8—24no—
258C3×Q8—28no—
259C3×Q8—416no—
260C3×Q8—42no—
261C3×Q8—42no—
262C3×Q8—42no—
263C3×Q8—42no—
264C3×Q8—44no—
265C3×Q8#12714yes—
266C3×D4—28no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—416no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—44no—
283C2×C3×C4—42no—
284C2×C3×C4—44no—
285C2×C3×C4#12012yes—
286C2×C3×C4#12012yes—
287C2×C3×C4C2×Q1612yes—
288C2×C3×C4#13712yes—
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×D412yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4H(5,𝔽2)12yes—
293C3×C8—24no—
294C3×C8—42no—
295C24C2×C3×D412yes—
296C2×Q8—28no—
297C2×Q8—21no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—48no—
307C2×Q8—416no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8C2×C3×D414yes—
311C2×Q8C2×C3×D414yes—
312C2×Q8C2×C3×D412yes—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4C3×Q1614yescomplemented
316C22×C4—21no—
317C22×C4—28no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—48no—
327C22×C4C2×C3×D412yes—
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C24×C311yescharacteristic
331Q16—416no—
332Q16—48no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—22no—
337C2×C8C2×C3×D414yes—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4C2×C3×D414yes—
349C4⋊C4C2×C3×D412yes—
350C22⋊C4—28no—
351C22⋊C4—24no—
352C22⋊C4—48no—
353C42—24no—
354C42—42no—
355C42C2×C3×D412yes—
356C42C2×C3×D412yes—
357C22×C3—24no—
358C22×C3—28no—
359C22×C3—42no—
360C22×C3—42no—
361C22×C3—44no—
362C22×C3C2×#12011yescharacteristic
363C22×C3D4212yes—
364C22×C3C22×Q1611yescharacteristic
365C22×C3C2×#13711yescharacteristic
366C22×C3#57812yes—
367C3×C4—24no—
368C3×C4—22no—
369C3×C4—42no—
370C3×C4—44no—
371C3×C4—42no—
372C3×C4—44no—
373C3×C4—82no—
374C3×C4#45114yescomplemented
375C3×C4C2×#12711yescharacteristic
376C3×C4#53511yescharacteristic
377C3×C4D4212yes—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—24no—
382C23C2×C3×Q1614yes—
383C23C22×C3×D411yescharacteristic
384Q8—22no—
385Q8—24no—
386Q8—28no—
387Q8—416no—
388Q8—42no—
389Q8—42no—
390Q8—42no—
391Q8—42no—
392Q8—44no—
393Q8C3×#12714yes—
394D4—28no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—416no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4C3×#12012yes—
414C2×C4C3×#12012yes—
415C2×C4C2×C3×Q1612yes—
416C2×C4C3×#13712yes—
417C2×C4C22×C3×D411yescharacteristic
418C2×C4C22×C3×D412yes—
419C2×C4C22×C3×D411yescharacteristic
420C2×C4C3×H(5,𝔽2)12yes—
421C8—24no—
422C8—42no—
423C2×C3—44no—
424C2×C3#243311yescharacteristic
425C2×C3C2×#45111yescharacteristic
426C2×C3#301111yescharacteristic
427C2×C3#327612yes—
428C2×C3D4×Q1612yes—
429C22—24no—
430C22—28no—
431C22—42no—
432C22—42no—
433C22—44no—
434C22C2×C3×#12011yescharacteristic
435C22C3×D4212yes—
436C22C22×C3×Q1611yescharacteristic
437C22C2×C3×#13711yescharacteristic
438C22C3×#57812yes—
439C4—24no—
440C4—22no—
441C4—42no—
442C4—44no—
443C4—42no—
444C4—44no—
445C4—82no—
446C4C3×#45114yescomplemented
447C4C2×C3×#12711yescharacteristic
448C4C3×#53511yescharacteristic
449C4C3×D4212yes—
450C3#2576011yescharacteristic, direct product factor, Sylow
451C2—44no—
452C2C3×#243311yescharacteristic
453C2C2×C3×#45111yescharacteristic
454C2C3×#301111yescharacteristic
455C2C3×#327612yes—
456C2C3×D4×Q1612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
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×#494C2semidirect
3. C2×C3×#462C2semidirect
4. C2×C3×#451C2semidirect
6. C3×#2001C2semidirect
7. C3×#1981C2semidirect
9. C3×C4×#103C2semidirect
10. #25760C3direct (N × H)
40. C2×#494C2×C3semidirect
41. C2×#462C2×C3semidirect
42. C2×#451C2×C3semidirect
44. #2001C2×C3semidirect
45. #1981C2×C3semidirect
47. C4×#103C2×C3semidirect
100. C3×#103D4semidirect
152. C2×C3×D4Q16semidirect
246. #103C3×D4semidirect
315. C2×D4C3×Q16semidirect
374. C3×C4#451semidirect
446. C4C3×#451semidirect
450. C3#25760direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#4942
C2×C3×#4622
C2×C3×#4512
C3×#25422
C3×#20012
C3×#19812
C3×#19542
C3×C4×#1032
#257603