Finite Group Databasebeta

← C3×#25711 #10644318

Subgroups — C3×#25711 #10644318

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#479C211yesmaximal, characteristic, complemented
4C3×#2562C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic
9C3×#1854C212yesmaximal, complemented
10C3×#1854C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25711C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#117C2212yes—
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×#556C2×C311yescharacteristic
47C2×#501C2×C311yescharacteristic, complemented
48C2×#479C2×C311yescharacteristic, complemented
49#2562C2×C311yescharacteristic
50#2036C2×C312yescomplemented
51#2035C2×C312yescomplemented
52#1987C2×C311yescharacteristic, complemented
53#1955C2×C311yescharacteristic
54#1854C2×C312yescomplemented
55#1854C2×C312yescomplemented
56#1763C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×#126—216no—
67C3×#125—22no—
68C3×#125—22no—
69C3×#124—28no—
70C3×C22⋊Q8—24no—
71C3×C22⋊Q8—24no—
72C3×C4×Q8—28no—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×#117—24no—
78C3×#117—24no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
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—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4D412yes—
95C2×C3×C42—22no—
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#556—24no—
114#556—24no—
115C2×C4⋊Q8C22×C311yescharacteristic
116C2×#126—22no—
117C2×#125C22×C311yescharacteristic
118C2×#124—21no—
119C2×C22⋊Q8C22×C312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×D4C22×C311yescharacteristic
123C2×#117C22×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—44no—
152C2×C3×Q8C2×D412yes—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8#4114yes—
178C3×C4⋊C4—28no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4C2×D412yes—
193C3×C4⋊C4C2×D414yes—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—82no—
199C3×C42—28no—
200C3×C42—24no—
201C3×C42—81no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204C22×Q8—21no—
205C22×Q8C23×C311yescharacteristic
206C22×D4C3×D411yescharacteristic
207C23×C4C23×C312yes—
208C22×C8C23×C311yescharacteristic
209C22×C8C23×C311yescharacteristic
210C22×C8C23×C312yes—
211C4⋊Q8—22no—
212C4⋊Q8C3×D414yes—
213#126—216no—
214#125—22no—
215#125—22no—
216#124—28no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219C4×Q8—28no—
220C4×Q8—22no—
221C4×Q8C3×D414yes—
222C4×D4—22no—
223C4×D4—22no—
224#117—24no—
225#117—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C3×D411yescharacteristic
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C4⋊C4C23×C312yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4C3×D412yes—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C23×C311yescharacteristic
246#107—22no—
247#107—24no—
248#106—22no—
249#106—24no—
250C4⋊C8—22no—
251#103—22no—
252#103—22no—
253#103—24no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256#97—22no—
257#95—21no—
258#95—22no—
259#95—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3C22×D411yescharacteristic
268C3×Q8—22no—
269C3×Q8—22no—
270C3×Q8—48no—
271C3×Q8—42no—
272C3×D4—44no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—42no—
283C2×C3×C4—44no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4#12012yes—
299C2×C3×C4#12012yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C2×#4112yes—
303C2×C3×C42−1+412yes—
304C3×C8—24no—
305C3×C8—41no—
306C3×C8—41no—
307C24C2×C3×D412yes—
308C2×Q8—24no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—44no—
313C2×Q8C2×C3×D412yes—
314C2×D4—22no—
315C2×D4—42no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4C2×C3×D412yes—
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C24×C311yescharacteristic
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8C3×#4114yes—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4C2×C3×D412yes—
354C4⋊C4C2×C3×D414yes—
355C22⋊C4—28no—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—44no—
359C22⋊C4—82no—
360C42—28no—
361C42—24no—
362C42—81no—
363C42C2×C3×D412yes—
364C42C2×C3×D412yes—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3C2×#12011yescharacteristic
373C22×C3#54712yes—
374C22×C3#57512yes—
375C22×C3#57711yescharacteristic
376C22×C3#57811yescharacteristic
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—44no—
382C3×C4—42no—
383C3×C4—82no—
384C3×C4—81no—
385C3×C4—81no—
386C3×C4#53211yescharacteristic
387C3×C4#53611yescharacteristic
388C3×C4#54812yes—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23C22×C3×D411yescharacteristic
397Q8—22no—
398Q8—22no—
399Q8—48no—
400Q8—42no—
401D4—44no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4C3×#12012yes—
428C2×C4C3×#12012yes—
429C2×C4C22×C3×D411yescharacteristic
430C2×C4C22×C3×D411yescharacteristic
431C2×C4C2×C3×#4112yes—
432C2×C4C3×2−1+412yes—
433C8—24no—
434C8—41no—
435C8—41no—
436C2×C3—82no—
437C2×C3#243211yescharacteristic
438C2×C3#300511yescharacteristic
439C2×C3#301311yescharacteristic
440C2×C3#334912yes—
441C2×C3#335012yes—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22C2×C3×#12011yescharacteristic
450C22C3×#54712yes—
451C22C3×#57512yes—
452C22C3×#57711yescharacteristic
453C22C3×#57811yescharacteristic
454C4—22no—
455C4—22no—
456C4—42no—
457C4—42no—
458C4—44no—
459C4—42no—
460C4—81no—
461C4—81no—
462C4—82no—
463C4C3×#53211yescharacteristic
464C4C3×#53611yescharacteristic
465C4C3×#54812yes—
466C3#2571111yescharacteristic, direct product factor, Sylow
467C2—82no—
468C2C3×#243211yescharacteristic
469C2C3×#300511yescharacteristic
470C2C3×#301311yescharacteristic
471C2C3×#334912yes—
472C2C3×#335012yes—
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×#1987C2semidirect
9. C3×#1854C2semidirect
10. C3×#1854C2semidirect
11. C3×#1763C2semidirect
12. #25711C3direct (N × H)
47. C2×#501C2×C3semidirect
48. C2×#479C2×C3semidirect
50. #2036C2×C3semidirect
51. #2035C2×C3semidirect
52. #1987C2×C3semidirect
54. #1854C2×C3semidirect
55. #1854C2×C3semidirect
56. #1763C2×C3semidirect
466. C3#25711direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#5012
C2×C3×#4792
C3×#25622
C3×#20362
C3×#20352
C3×#19872
C3×#19552
C3×#18542
C3×#17632
#257113