Finite Group Databasebeta

← C3×#25680 #10644287

Subgroups — C3×#25680 #10644287

Order 768.

1032 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic
2C2×C3×#498C211yesmaximal, characteristic, complemented
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic, complemented
5C3×#2035C212yesmaximal, complemented
6C3×#2034C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1958C211yesmaximal, characteristic, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1864C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25680C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#542—24no—
14C3×#542—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#124—21no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×#123—21no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#498—24no—
28C3×#489—24no—
29C2×C3×#106C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×#97C2211yescharacteristic
37C2×C3×C4×C8C2211yescharacteristic
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yes—
43C3×#382C2211yescharacteristic
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#542C2×C311yescharacteristic
49C2×#498C2×C311yescharacteristic, complemented
50C2×#489C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic, complemented
52#2035C2×C312yescomplemented
53#2034C2×C312yes—
54#1993C2×C311yescharacteristic, complemented
55#1958C2×C311yescharacteristic, complemented
56#1865C2×C312yescomplemented
57#1864C2×C312yescomplemented
58#1750C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yes—
72C3×#126—28no—
73C3×#124—28no—
74C3×#124—22no—
75C3×#124D414yes—
76C3×#123—28no—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#106—24no—
111C3×#106—24no—
112C3×#103—24no—
113C3×#103—24no—
114C3×#102—24no—
115C3×#102—24no—
116C3×C22⋊C8—24no—
117C3×#97—22no—
118C3×C4×C8—22no—
119C3×#95—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—22no—
123#542—24no—
124#542—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#124—21no—
129C2×#124C22×C311yescharacteristic
130C2×#123—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×#121C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137#498—24no—
138#489—24no—
139C2×#106C22×C312yes—
140C2×#106C22×C312yes—
141C2×#103C22×C312yes—
142C2×#103C22×C312yes—
143C2×#102C22×C312yes—
144C2×#102C22×C312yes—
145C2×C22⋊C8C22×C312yes—
146C2×#97C22×C311yescharacteristic
147C2×C4×C8C22×C311yescharacteristic
148#400—21no—
149#399—21no—
150#395—21no—
151#393—21no—
152#389C22×C312yes—
153#382C22×C311yescharacteristic
154#380C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—24no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—42no—
165C2×C3×Q8—82no—
166C2×C3×Q8—82no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—24no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—82no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—24no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—82no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8#4114yes—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—82no—
210C3×C4⋊C4C2×D412yes—
211C3×C4⋊C4C2×D412yes—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—82no—
220C3×C42—24no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42C2×D412yes—
224C3×C42C2×D412yes—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8C23×C311yescharacteristic
228C22×D4—21no—
229C22×D4C23×C311yescharacteristic
230C23×C4—21no—
231C23×C4C23×C311yescharacteristic
232C22×C8C23×C312yes—
233C22×C8C23×C312yes—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8C3×D414yes—
238#126—28no—
239#124—28no—
240#124—22no—
241#124C3×D414yes—
242#123—28no—
243C22⋊Q8—28no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246#121—22no—
247#121—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×D4—22no—
251C4×D4—22no—
252#117—22no—
253#117—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4C3×D411yescharacteristic
260C2×C4⋊C4C3×D411yescharacteristic
261C2×C4⋊C4C23×C312yes—
262C2×C4⋊C4C23×C312yes—
263C2×C4⋊C4C23×C311yescharacteristic
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4C3×D411yescharacteristic
272C2×C22⋊C4C3×D411yescharacteristic
273C2×C42—21no—
274C2×C42C23×C311yescharacteristic
275C2×C42C23×C311yescharacteristic
276#106—24no—
277#106—24no—
278#103—24no—
279#103—24no—
280#102—24no—
281#102—24no—
282C22⋊C8—24no—
283#97—22no—
284C4×C8—22no—
285#95—22no—
286#95—21no—
287#95—21no—
288#95—22no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C3×Q8—24no—
301C3×Q8—42no—
302C3×Q8—84no—
303C3×Q8—84no—
304C3×D4—24no—
305C3×D4—42no—
306C3×D4—84no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—84no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12012yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4112yes—
337C2×C3×C42−1+412yes—
338C3×C8—24no—
339C3×C8—42no—
340C24—21no—
341C24C2×C3×D411yescharacteristic
342C2×Q8—24no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—42no—
347C2×Q8—82no—
348C2×Q8—82no—
349C2×Q8C2×C3×D412yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—82no—
355C2×D4C2×C3×D412yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—82no—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C24×C311yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8C3×#4114yes—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—42no—
391C4⋊C4—82no—
392C4⋊C4C2×C3×D412yes—
393C4⋊C4C2×C3×D412yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—82no—
402C42—24no—
403C42—21no—
404C42—21no—
405C42C2×C3×D412yes—
406C42C2×C3×D412yes—
407C22×C3—42no—
408C22×C3—41no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3C2×#12011yescharacteristic
418C22×C3#53211yescharacteristic
419C22×C3#53611yescharacteristic
420C22×C3#57712yes—
421C22×C3#57712yes—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—41no—
425C3×C4—42no—
426C3×C4—41no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—41no—
430C3×C4—82no—
431C3×C4—82no—
432C3×C4#54712yes—
433C3×C4#54812yes—
434C23—22no—
435C23—22no—
436C23—21no—
437C23—22no—
438C23—21no—
439C23—21no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23C22×C3×D411yescharacteristic
445Q8—24no—
446Q8—42no—
447Q8—84no—
448Q8—84no—
449D4—24no—
450D4—42no—
451D4—84no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—84no—
477C2×C4C3×#12012yes—
478C2×C4C3×#12012yes—
479C2×C4C22×C3×D411yescharacteristic
480C2×C4C22×C3×D411yescharacteristic
481C2×C4C2×C3×#4112yes—
482C2×C4C3×2−1+412yes—
483C8—24no—
484C8—42no—
485C2×C3—81no—
486C2×C3—82no—
487C2×C3#243211yescharacteristic
488C2×C3#300911yescharacteristic
489C2×C3#301011yescharacteristic
490C2×C3#317312yes—
491C2×C3#324412yes—
492C22—42no—
493C22—41no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—82no—
501C22—82no—
502C22C2×C3×#12011yescharacteristic
503C22C3×#53211yescharacteristic
504C22C3×#53611yescharacteristic
505C22C3×#57712yes—
506C22C3×#57712yes—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—42no—
511C4—41no—
512C4—42no—
513C4—42no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4C3×#54712yes—
518C4C3×#54812yes—
519C3#2568011yescharacteristic, direct product factor, Sylow
520C2—81no—
521C2—82no—
522C2C3×#243211yescharacteristic
523C2C3×#300911yescharacteristic
524C2C3×#301011yescharacteristic
525C2C3×#317312yes—
526C2C3×#324412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#498C2semidirect
3. C2×C3×#489C2semidirect
4. C3×#2628C2semidirect
5. C3×#2035C2semidirect
7. C3×#1993C2semidirect
8. C3×#1958C2semidirect
9. C3×#1865C2semidirect
10. C3×#1864C2semidirect
11. C3×#1750C2semidirect
12. #25680C3direct (N × H)
49. C2×#498C2×C3semidirect
50. C2×#489C2×C3semidirect
51. #2628C2×C3semidirect
52. #2035C2×C3semidirect
54. #1993C2×C3semidirect
55. #1958C2×C3semidirect
56. #1865C2×C3semidirect
57. #1864C2×C3semidirect
58. #1750C2×C3semidirect
519. C3#25680direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×C3×#4982
C2×C3×#4892
C3×#26282
C3×#20352
C3×#20342
C3×#19932
C3×#19582
C3×#18652
C3×#18642
C3×#17502
#256803