Finite Group Databasebeta

← C3×#25710 #10644317

Subgroups — C3×#25710 #10644317

Order 768.

1080 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#478C211yesmaximal, characteristic, complemented
4C3×#2562C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1854C212yesmaximal, complemented
10C3×#1854C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25710C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#555—24no—
14C3×#555—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×#124—22no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C2×C3×#117C2212yes—
24C3×#501—24no—
25C3×#478—24no—
26C2×C3×#107C2212yes—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
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×#555C2×C311yescharacteristic
47C2×#501C2×C311yescharacteristic, complemented
48C2×#478C2×C311yescharacteristic, complemented
49#2562C2×C311yescharacteristic, complemented
50#2034C2×C312yescomplemented
51#2033C2×C312yescomplemented
52#1988C2×C311yescharacteristic, complemented
53#1955C2×C311yescharacteristic, complemented
54#1854C2×C312yescomplemented
55#1854C2×C312yescomplemented
56#1762C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×D4—22no—
59C22×C3×D4D411yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2312yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yes—
68C3×#125—22no—
69C3×#125—22no—
70C3×#124—216no—
71C3×#121—28no—
72C3×#121—24no—
73C3×#121—24no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4D414yes—
78C3×#117—28no—
79C3×#117—24no—
80C3×#117—24no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4D412yes—
96C2×C3×C42—22no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C3×#107—22no—
100C3×#107—24no—
101C3×#106—22no—
102C3×#106—24no—
103C3×C4⋊C8—22no—
104C3×#102—22no—
105C3×#102—22no—
106C3×#102—24no—
107C3×C22⋊C8—24no—
108C3×C22⋊C8—24no—
109C3×#97—22no—
110C3×#95—21no—
111C3×#95—22no—
112C3×#95—22no—
113#555—24no—
114#555—24no—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×#125C22×C311yescharacteristic
117C2×#124—22no—
118C2×#121—21no—
119C2×#121C22×C312yes—
120C2×C4×D4C22×C311yescharacteristic
121C2×C4×D4C22×C311yescharacteristic
122C2×#117—21no—
123C2×#117C22×C312yes—
124#501—24no—
125#478—24no—
126C2×#107C22×C311yescharacteristic
127C2×#107C22×C312yes—
128C2×#106C22×C311yescharacteristic
129C2×#106C22×C312yes—
130C2×C4⋊C8C22×C311yescharacteristic
131C2×#102C22×C311yescharacteristic
132C2×#102C22×C311yescharacteristic
133C2×#102C22×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×C3—21no—
147C24×C3C2×D412yes—
148C2×C3×Q8—24no—
149C2×C3×Q8—81no—
150C2×C3×Q8—81no—
151C2×C3×D4—28no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—42no—
157C2×C3×D4—82no—
158C2×C3×D4—82no—
159C2×C3×D4C2×D412yes—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—44no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8#4114yes—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4C2×D412yes—
197C3×C4⋊C4C2×D414yes—
198C3×C22⋊C4—28no—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—82no—
204C3×C42—28no—
205C3×C42—81no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208C22×Q8—21no—
209C22×D4—22no—
210C22×D4C3×D411yescharacteristic
211C22×D4C23×C311yescharacteristic
212C23×C4—21no—
213C23×C4C23×C312yes—
214C22×C8C23×C311yescharacteristic
215C22×C8C23×C311yescharacteristic
216C22×C8C23×C312yes—
217C4⋊Q8—22no—
218C4⋊Q8C3×D414yes—
219#125—22no—
220#125—22no—
221#124—216no—
222#121—28no—
223#121—24no—
224#121—24no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4C3×D414yes—
229#117—28no—
230#117—24no—
231#117—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C3×D411yescharacteristic
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C312yes—
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C312yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4C3×D412yes—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42C23×C311yescharacteristic
250#107—22no—
251#107—24no—
252#106—22no—
253#106—24no—
254C4⋊C8—22no—
255#102—22no—
256#102—22no—
257#102—24no—
258C22⋊C8—24no—
259C22⋊C8—24no—
260#97—22no—
261#95—21no—
262#95—22no—
263#95—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3C22×D411yescharacteristic
277C3×Q8—84no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4—44no—
281C3×D4—42no—
282C3×D4—88no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—48no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4#12012yes—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C42−1+412yes—
313C3×C8—24no—
314C3×C8—41no—
315C3×C8—41no—
316C24—21no—
317C24C2×C3×D412yes—
318C2×Q8—24no—
319C2×Q8—81no—
320C2×Q8—81no—
321C2×D4—28no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—42no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4C2×C3×D412yes—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—44no—
343C22×C4C2×C3×D412yes—
344C22×C4C2×C3×D411yescharacteristic
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C24×C311yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8C3×#4114yes—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4C2×C3×D412yes—
367C4⋊C4C2×C3×D414yes—
368C22⋊C4—28no—
369C22⋊C4—28no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—44no—
373C22⋊C4—82no—
374C42—28no—
375C42—81no—
376C42C2×C3×D412yes—
377C42C2×C3×D412yes—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3C2×#12011yescharacteristic
391C22×C3#54712yes—
392C22×C3#57512yes—
393C22×C3#57611yescharacteristic
394C22×C3#57711yescharacteristic
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—44no—
400C3×C4—82no—
401C3×C4—81no—
402C3×C4—81no—
403C3×C4#53211yescharacteristic
404C3×C4#53611yescharacteristic
405C3×C4#54812yes—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23C22×C3×D411yescharacteristic
419Q8—84no—
420D4—22no—
421D4—22no—
422D4—44no—
423D4—42no—
424D4—88no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4C3×#12012yes—
450C2×C4C3×#12012yes—
451C2×C4C22×C3×D411yescharacteristic
452C2×C4C22×C3×D411yescharacteristic
453C2×C4C2×C3×#4112yes—
454C2×C4C3×2−1+412yes—
455C8—24no—
456C8—41no—
457C8—41no—
458C2×C3—82no—
459C2×C3—82no—
460C2×C3#243211yescharacteristic
461C2×C3#300511yescharacteristic
462C2×C3#301211yescharacteristic
463C2×C3#334712yes—
464C2×C3#334812yes—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—82no—
476C22—82no—
477C22C2×C3×#12011yescharacteristic
478C22C3×#54712yes—
479C22C3×#57512yes—
480C22C3×#57611yescharacteristic
481C22C3×#57711yescharacteristic
482C4—22no—
483C4—22no—
484C4—42no—
485C4—42no—
486C4—44no—
487C4—81no—
488C4—81no—
489C4—82no—
490C4C3×#53211yescharacteristic
491C4C3×#53611yescharacteristic
492C4C3×#54812yes—
493C3#2571011yescharacteristic, direct product factor, Sylow
494C2—82no—
495C2—82no—
496C2C3×#243211yescharacteristic
497C2C3×#300511yescharacteristic
498C2C3×#301211yescharacteristic
499C2C3×#334712yes—
500C2C3×#334812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#555C2
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×#478C2semidirect
4. C3×#2562C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1988C2semidirect
8. C3×#1955C2semidirect
9. C3×#1854C2semidirect
10. C3×#1854C2semidirect
11. C3×#1762C2semidirect
12. #25710C3direct (N × H)
47. C2×#501C2×C3semidirect
48. C2×#478C2×C3semidirect
49. #2562C2×C3semidirect
50. #2034C2×C3semidirect
51. #2033C2×C3semidirect
52. #1988C2×C3semidirect
53. #1955C2×C3semidirect
54. #1854C2×C3semidirect
55. #1854C2×C3semidirect
56. #1762C2×C3semidirect
493. C3#25710direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×C3×#5012
C2×C3×#4782
C3×#25622
C3×#20342
C3×#20332
C3×#19882
C3×#19552
C3×#18542
C3×#17622
#257103