Finite Group Databasebeta

← C3×#25712 #10644319

Subgroups — C3×#25712 #10644319

Order 768.

1060 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#479C211yesmaximal, characteristic, complemented
4C3×#2562C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic
9C3×#1853C212yesmaximal, complemented
10C3×#1852C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25712C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×C22⋊Q8—22no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C4⋊C4C2212yes—
24C3×#501—24no—
25C3×#479—24no—
26C2×C3×#107C2212yes—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×C22⋊C8C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#400—21no—
38C3×#394—21no—
39C3×#388—22no—
40C3×#380C2212yes—
41C3×#380C2212yes—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#554C2×C311yescharacteristic
47C2×#501C2×C311yescharacteristic, complemented
48C2×#479C2×C311yescharacteristic, complemented
49#2562C2×C311yescharacteristic
50#2036C2×C312yescomplemented
51#2035C2×C312yescomplemented
52#1988C2×C311yescharacteristic, complemented
53#1955C2×C311yescharacteristic
54#1853C2×C312yescomplemented
55#1852C2×C312yescomplemented
56#1763C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—22no—
61C23×C3×C4C2312yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C3×#125—28no—
68C3×#125—22no—
69C3×#125—22no—
70C3×C22⋊Q8—216no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C3×C4×D4—28no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4D412yes—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C3×#107—22no—
100C3×#107—24no—
101C3×#106—22no—
102C3×#106—24no—
103C3×C4⋊C8—22no—
104C3×#103—22no—
105C3×#103—22no—
106C3×#103—24no—
107C3×C22⋊C8—24no—
108C3×C22⋊C8—24no—
109C3×#97—22no—
110C3×#95—21no—
111C3×#95—22no—
112C3×#95—22no—
113#554—24no—
114#554—24no—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×#125—21no—
117C2×#125C22×C311yescharacteristic
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8C22×C312yes—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4C22×C311yescharacteristic
123C22×C4⋊C4C22×C312yes—
124#501—24no—
125#479—24no—
126C2×#107C22×C311yescharacteristic
127C2×#107C22×C312yes—
128C2×#106C22×C311yescharacteristic
129C2×#106C22×C312yes—
130C2×C4⋊C8C22×C311yescharacteristic
131C2×#103C22×C311yescharacteristic
132C2×#103C22×C311yescharacteristic
133C2×#103C22×C312yes—
134C2×C22⋊C8C22×C312yes—
135C2×C22⋊C8C22×C312yes—
136C2×#97C22×C311yescharacteristic
137#400—21no—
138#394—21no—
139#388—22no—
140#380C22×C312yes—
141#380C22×C312yes—
142C4×C4⋊C4C22×C311yescharacteristic
143#340C22×C312yes—
144#340C22×C312yes—
145#340C22×C312yes—
146C24×C3C2×D412yes—
147C2×C3×Q8—24no—
148C2×C3×Q8—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—81no—
152C2×C3×Q8—81no—
153C2×C3×Q8C2×D412yes—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—28no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—82no—
170C22×C3×C4—82no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8#4114yes—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4C2×D414yes—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—82no—
203C3×C42—24no—
204C3×C42—81no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207C22×Q8—21no—
208C22×Q8C23×C311yescharacteristic
209C22×D4C3×D411yescharacteristic
210C23×C4—22no—
211C23×C4C23×C312yes—
212C22×C8C23×C311yescharacteristic
213C22×C8C23×C311yescharacteristic
214C22×C8C23×C312yes—
215C4⋊Q8—22no—
216C4⋊Q8C3×D414yes—
217#125—28no—
218#125—22no—
219#125—22no—
220C22⋊Q8—216no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C4×Q8—22no—
224C4×Q8C3×D414yes—
225C4×D4—28no—
226C4×D4—22no—
227C4×D4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C3×D411yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C4⋊C4C23×C312yes—
240C2×C4⋊C4C23×C311yescharacteristic
241C2×C4⋊C4C23×C312yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C3×D412yes—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C23×C311yescharacteristic
249#107—22no—
250#107—24no—
251#106—22no—
252#106—24no—
253C4⋊C8—22no—
254#103—22no—
255#103—22no—
256#103—24no—
257C22⋊C8—24no—
258C22⋊C8—24no—
259#97—22no—
260#95—21no—
261#95—22no—
262#95—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3#13714yes—
270C23×C3C22×D411yescharacteristic
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—42no—
274C3×Q8—84no—
275C3×D4—28no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—88no—
303C2×C3×C4#12012yes—
304C2×C3×C4#12012yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C42−1+412yes—
309C3×C8—24no—
310C3×C8—41no—
311C3×C8—41no—
312C24C2×C3×D412yes—
313C2×Q8—24no—
314C2×Q8—21no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—81no—
318C2×Q8—81no—
319C2×Q8C2×C3×D412yes—
320C2×D4—24no—
321C2×D4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C24×C311yescharacteristic
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8C3×#4114yes—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4C2×C3×D412yes—
364C4⋊C4C2×C3×D414yes—
365C22⋊C4—28no—
366C22⋊C4—24no—
367C22⋊C4—44no—
368C22⋊C4—82no—
369C42—24no—
370C42—81no—
371C42C2×C3×D412yes—
372C42C2×C3×D412yes—
373C22×C3—24no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3C2×#12011yescharacteristic
381C22×C3#54712yes—
382C22×C3C2×#13712yes—
383C22×C3#57611yescharacteristic
384C22×C3#57711yescharacteristic
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—44no—
391C3×C4—82no—
392C3×C4—81no—
393C3×C4—81no—
394C3×C4#53211yescharacteristic
395C3×C4#53611yescharacteristic
396C3×C4#54812yes—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23C3×#13714yes—
404C23C22×C3×D411yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—42no—
408Q8—84no—
409D4—28no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—88no—
437C2×C4C3×#12012yes—
438C2×C4C3×#12012yes—
439C2×C4C22×C3×D411yescharacteristic
440C2×C4C22×C3×D411yescharacteristic
441C2×C4C2×C3×#4112yes—
442C2×C4C3×2−1+412yes—
443C8—24no—
444C8—41no—
445C8—41no—
446C2×C3—44no—
447C2×C3#243211yescharacteristic
448C2×C3#300411yescharacteristic
449C2×C3#301211yescharacteristic
450C2×C3#334412yes—
451C2×C3#334612yes—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22C2×C3×#12011yescharacteristic
460C22C3×#54712yes—
461C22C2×C3×#13712yes—
462C22C3×#57611yescharacteristic
463C22C3×#57711yescharacteristic
464C4—22no—
465C4—22no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—81no—
471C4—81no—
472C4—82no—
473C4C3×#53211yescharacteristic
474C4C3×#53611yescharacteristic
475C4C3×#54812yes—
476C3#2571211yescharacteristic, direct product factor, Sylow
477C2—44no—
478C2C3×#243211yescharacteristic
479C2C3×#300411yescharacteristic
480C2C3×#301211yescharacteristic
481C2C3×#334412yes—
482C2C3×#334612yes—
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×#501C2semidirect
3. C2×C3×#479C2semidirect
5. C3×#2036C2semidirect
6. C3×#2035C2semidirect
7. C3×#1988C2semidirect
9. C3×#1853C2semidirect
10. C3×#1852C2semidirect
11. C3×#1763C2semidirect
12. #25712C3direct (N × H)
47. C2×#501C2×C3semidirect
48. C2×#479C2×C3semidirect
50. #2036C2×C3semidirect
51. #2035C2×C3semidirect
52. #1988C2×C3semidirect
54. #1853C2×C3semidirect
55. #1852C2×C3semidirect
56. #1763C2×C3semidirect
476. C3#25712direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#5012
C2×C3×#4792
C3×#25622
C3×#20362
C3×#20352
C3×#19882
C3×#19552
C3×#18532
C3×#18522
C3×#17632
#257123