Finite Group Databasebeta

← C3×#25649 #10644256

Subgroups — C3×#25649 #10644256

Order 768.

1346 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×C3×#486C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2041C214yesmaximal, complemented
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1982C211yesmaximal, characteristic, complemented
8C3×#1864C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25649C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C3×#550—24no—
13C3×#550—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2212yes—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#121—21no—
18C2×C3×#121—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C3×#486—24no—
23C3×#476—24no—
24C2×C3×#106C2212yes—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2214yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2214yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#395—22no—
32C3×#391—21no—
33C3×#389C2214yes—
34C3×#384—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#550C2×C311yescharacteristic
39C2×#486C2×C311yescharacteristic, complemented
40C2×#476C2×C311yescharacteristic, complemented
41#2567C2×C311yescharacteristic, complemented
42#2041C2×C314yescomplemented
43#1984C2×C311yescharacteristic, complemented
44#1982C2×C311yescharacteristic, complemented
45#1864C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic, complemented
47C2×C3×#41—28no—
48C2×C3×#41—44no—
49C2×C3×#41—44no—
50C22×C3×Q8—21no—
51C22×C3×Q8D411yescharacteristic
52C22×C3×D4—22no—
53C22×C3×D4—21no—
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4C2312yes—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—22no—
59C23×C3×C4—21no—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×#127—24no—
65C3×#127—24no—
66C3×#127—22no—
67C3×#127D414yes—
68C3×#121—28no—
69C3×#121—216no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C3×#106—24no—
89C3×C4⋊C8—22no—
90C3×#102—24no—
91C3×#102—28no—
92C3×#102—24no—
93C3×#102—28no—
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—24no—
97C22×#41—21no—
98#550—24no—
99#550—24no—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×#127C22×C312yes—
102C2×#127C22×C311yescharacteristic
103C2×#121—21no—
104C2×#121—22no—
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×D4C22×C312yes—
107C2×C4×D4C22×C311yescharacteristic
108#486—24no—
109#476—24no—
110C2×#106C22×C312yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#102C22×C314yes—
113C2×#102C22×C314yes—
114C2×#102C22×C312yes—
115C2×#102C22×C312yes—
116C2×C4×C8C22×C311yescharacteristic
117#395—22no—
118#391—21no—
119#389C22×C314yes—
120#384—21no—
121C4×C4⋊C4C22×C311yescharacteristic
122#340C22×C312yes—
123#336C22×C314yes—
124C24×C3—22no—
125C24×C3—21no—
126C3×#41—432no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—42no—
130C2×C3×Q8—82no—
131C2×C3×Q8SD1614yes—
132C2×C3×D4—28no—
133C2×C3×D4—28no—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—22no—
137C2×C3×D4—21no—
138C2×C3×D4—22no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—24no—
142C2×C3×D4—44no—
143C2×C3×D4—44no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4—82no—
147C2×C3×D4—84no—
148C2×C3×D4C2×D414yes—
149C2×C3×D4C2×D412yes—
150C22×C3×C4—28no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—24no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—84no—
177C3×C4⋊C4C2×D412yes—
178C3×C4⋊C4#4114yescomplemented
179C3×C22⋊C4—28no—
180C3×C22⋊C4—24no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—42no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186C2×#41—28no—
187C2×#41—44no—
188C2×#41—44no—
189C22×Q8—21no—
190C22×Q8C3×D411yescharacteristic
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4C23×C312yes—
196C22×D4C23×C311yescharacteristic
197C23×C4—22no—
198C23×C4—21no—
199C22×C8C23×C312yes—
200C22×C8C23×C312yes—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203#127—24no—
204#127—24no—
205#127—22no—
206#127C3×D414yes—
207#121—28no—
208#121—216no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—22no—
214C4×D4C3×D414yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4C3×D411yescharacteristic
218C2×C4⋊C4C3×D412yes—
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C312yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42C23×C312yes—
226C2×C42C23×C311yescharacteristic
227#106—24no—
228C4⋊C8—22no—
229#102—24no—
230#102—28no—
231#102—28no—
232#102—24no—
233C4×C8—22no—
234#95—21no—
235#95—24no—
236C23×C3—24no—
237C23×C3—22no—
238C23×C3—42no—
239C23×C3—44no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—41no—
244C23×C3—44no—
245C23×C3—42no—
246C23×C3C22×D411yescharacteristic
247C3×Q8—28no—
248C3×Q8—84no—
249C3×D4—22no—
250C3×D4—22no—
251C3×D4—28no—
252C3×D4—416no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—48no—
257C3×D4—84no—
258C3×D4—88no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—416no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—48no—
272C2×C3×C4—42no—
273C2×C3×C4—44no—
274C2×C3×C4#12012yes—
275C2×C3×C4#12012yes—
276C2×C3×C4C2×SD1614yes—
277C2×C3×C4C2×SD1612yes—
278C2×C3×C4AGL(1,ℤ/8)12yes—
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4112yes—
282C2×C3×C42−1+412yes—
283C3×C8—42no—
284C3×C8—42no—
285C24—22no—
286C24—21no—
287#41—432no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—82no—
292C2×Q8C3×SD1614yes—
293C2×D4—28no—
294C2×D4—28no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—82no—
308C2×D4—84no—
309C2×D4C2×C3×D414yes—
310C2×D4C2×C3×D412yes—
311C22×C4—28no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D412yes—
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C24×C311yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—22no—
329C2×C8—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—84no—
338C4⋊C4C2×C3×D412yes—
339C4⋊C4C3×#4114yescomplemented
340C22⋊C4—28no—
341C22⋊C4—24no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42C2×C3×D412yes—
346C42C2×C3×D412yes—
347C22×C3—42no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—42no—
351C22×C3—44no—
352C22×C3—42no—
353C22×C3—82no—
354C22×C3—84no—
355C22×C3—84no—
356C22×C3—82no—
357C22×C3C2×#12011yescharacteristic
358C22×C3#54812yes—
359C22×C3C22×SD1611yescharacteristic
360C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
361C22×C3#57612yes—
362C3×C4—24no—
363C3×C4—22no—
364C3×C4—44no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—82no—
368C3×C4#45014yes—
369C3×C4#53211yescharacteristic
370C3×C4#53611yescharacteristic
371C3×C4#54712yes—
372C23—24no—
373C23—22no—
374C23—42no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—44no—
382C23C22×C3×D411yescharacteristic
383Q8—28no—
384Q8—84no—
385D4—22no—
386D4—22no—
387D4—28no—
388D4—416no—
389D4—48no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—84no—
394D4—88no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—416no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4C3×#12012yes—
411C2×C4C3×#12012yes—
412C2×C4C2×C3×SD1614yes—
413C2×C4C2×C3×SD1612yes—
414C2×C4C3×AGL(1,ℤ/8)12yes—
415C2×C4C22×C3×D411yescharacteristic
416C2×C4C22×C3×D411yescharacteristic
417C2×C4C2×C3×#4112yes—
418C2×C4C3×2−1+412yes—
419C8—42no—
420C8—42no—
421C2×C3—82no—
422C2×C3—84no—
423C2×C3#243211yescharacteristic
424C2×C3C2×#45011yescharacteristic
425C2×C3#300811yescharacteristic
426C2×C3#336212yes—
427C2×C3#336612yes—
428C22—42no—
429C22—44no—
430C22—44no—
431C22—42no—
432C22—44no—
433C22—42no—
434C22—82no—
435C22—84no—
436C22—84no—
437C22—82no—
438C22C2×C3×#12011yescharacteristic
439C22C3×#54812yes—
440C22C22×C3×SD1611yescharacteristic
441C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
442C22C3×#57612yes—
443C4—24no—
444C4—22no—
445C4—42no—
446C4—42no—
447C4—44no—
448C4—82no—
449C4C3×#45014yes—
450C4C3×#53211yescharacteristic
451C4C3×#53611yescharacteristic
452C4C3×#54712yes—
453C3#2564911yescharacteristic, direct product factor, Sylow
454C2—82no—
455C2—84no—
456C2C3×#243211yescharacteristic
457C2C2×C3×#45011yescharacteristic
458C2C3×#300811yescharacteristic
459C2C3×#336212yes—
460C2C3×#336612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#486C2semidirect
3. C2×C3×#476C2semidirect
4. C3×#2567C2semidirect
5. C3×#2041C2semidirect
6. C3×#1984C2semidirect
7. C3×#1982C2semidirect
8. C3×#1864C2semidirect
9. C3×C4×#102C2semidirect
10. #25649C3direct (N × H)
39. C2×#486C2×C3semidirect
40. C2×#476C2×C3semidirect
41. #2567C2×C3semidirect
42. #2041C2×C3semidirect
43. #1984C2×C3semidirect
44. #1982C2×C3semidirect
45. #1864C2×C3semidirect
46. C4×#102C2×C3semidirect
178. C3×C4⋊C4#41semidirect
339. C4⋊C4C3×#41semidirect
453. C3#25649direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4862
C2×C3×#4762
C3×#25672
C3×#20412
C3×#19842
C3×#19822
C3×#18642
C3×C4×#1022
#256493