Finite Group Databasebeta

← C3×#25685 #10644292

Subgroups — C3×#25685 #10644292

Order 768.

1096 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1991C211yesmaximal, characteristic, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25685C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#540—24no—
14C3×#540—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×#123—21no—
20C2×C3×#121—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#501—24no—
28C3×#486—24no—
29C2×C3×#107C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
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×#540C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#486C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic, complemented
52#2034C2×C312yescomplemented
53#2033C2×C312yescomplemented
54#1991C2×C311yescharacteristic, complemented
55#1961C2×C311yescharacteristic, complemented
56#1864C2×C312yescomplemented
57#1862C2×C312yescomplemented
58#1750C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—22no—
73C3×#127—22no—
74C3×#127D414yes—
75C3×#125—22no—
76C3×#125D414yes—
77C3×#123—28no—
78C3×#123—28no—
79C3×#121—28no—
80C3×#121—28no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—22no—
84C3×#121—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#106—24no—
112C3×#102—24no—
113C3×#102—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#540—24no—
124#540—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#127C22×C311yescharacteristic
127C2×#125C22×C311yescharacteristic
128C2×#123—21no—
129C2×#123—21no—
130C2×#121—21no—
131C2×#121—21no—
132C2×#121C22×C311yescharacteristic
133C2×#121C22×C311yescharacteristic
134C2×C4×D4C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137#501—24no—
138#486—24no—
139C2×#107C22×C312yes—
140C2×#106C22×C312yes—
141C2×#102C22×C312yes—
142C2×#102C22×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×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—42no—
162C2×C3×Q8—82no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—82no—
174C2×C3×D4—82no—
175C2×C3×D4—82no—
176C2×C3×D4—82no—
177C2×C3×D4C2×D412yes—
178C2×C3×D4C2×D412yes—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—82no—
190C22×C3×C4—82no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8#4114yes—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—82no—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—82no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42C2×D412yes—
227C3×C42C2×D412yes—
228C22×Q8—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C23×C311yescharacteristic
234C22×D4C23×C311yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C23×C311yescharacteristic
238C22×C8C23×C312yes—
239C22×C8C23×C312yes—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242#127—22no—
243#127C3×D414yes—
244#125—22no—
245#125C3×D414yes—
246#123—28no—
247#123—28no—
248#121—28no—
249#121—28no—
250#121—22no—
251#121—22no—
252#121—22no—
253#121—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258#117—22no—
259#117—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C3×D411yescharacteristic
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C312yes—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C3×D411yescharacteristic
276C2×C22⋊C4C3×D411yescharacteristic
277C2×C42C23×C311yescharacteristic
278C2×C42C23×C311yescharacteristic
279#107—24no—
280#106—24no—
281#102—24no—
282#102—24no—
283#102—24no—
284#102—24no—
285C22⋊C8—24no—
286#97—22no—
287C4×C8—22no—
288#95—22no—
289#95—21no—
290#95—21no—
291#95—22no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—84no—
309C3×D4—24no—
310C3×D4—24no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—84no—
314C3×D4—84no—
315C3×D4—84no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—84no—
339C2×C3×C4—84no—
340C2×C3×C4#12012yes—
341C2×C3×C4#12012yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4112yes—
345C2×C3×C42−1+412yes—
346C3×C8—24no—
347C3×C8—42no—
348C24—21no—
349C24—21no—
350C24C2×C3×D411yescharacteristic
351C2×Q8—42no—
352C2×Q8—82no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4C2×C3×D412yes—
368C2×D4C2×C3×D412yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C24×C311yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8C3×#4114yes—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—82no—
405C4⋊C4C2×C3×D412yes—
406C4⋊C4C2×C3×D412yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—82no—
414C42—21no—
415C42—21no—
416C42C2×C3×D412yes—
417C42C2×C3×D412yes—
418C22×C3—42no—
419C22×C3—41no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—41no—
428C22×C3—42no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#53211yescharacteristic
435C22×C3#53611yescharacteristic
436C22×C3#57612yes—
437C22×C3#57712yes—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4—41no—
444C3×C4—42no—
445C3×C4—82no—
446C3×C4—82no—
447C3×C4#54712yes—
448C3×C4#54812yes—
449C23—22no—
450C23—22no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C22×C3×D411yescharacteristic
465Q8—84no—
466D4—24no—
467D4—24no—
468D4—42no—
469D4—42no—
470D4—84no—
471D4—84no—
472D4—84no—
473D4—84no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4C3×#12012yes—
498C2×C4C3×#12012yes—
499C2×C4C22×C3×D411yescharacteristic
500C2×C4C22×C3×D411yescharacteristic
501C2×C4C2×C3×#4112yes—
502C2×C4C3×2−1+412yes—
503C8—24no—
504C8—42no—
505C2×C3—81no—
506C2×C3—82no—
507C2×C3—82no—
508C2×C3#243211yescharacteristic
509C2×C3#300811yescharacteristic
510C2×C3#300911yescharacteristic
511C2×C3#317912yes—
512C2×C3#324112yes—
513C22—42no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22C2×C3×#12011yescharacteristic
529C22C3×#53211yescharacteristic
530C22C3×#53611yescharacteristic
531C22C3×#57612yes—
532C22C3×#57712yes—
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—42no—
538C4—41no—
539C4—42no—
540C4—82no—
541C4—82no—
542C4C3×#54712yes—
543C4C3×#54812yes—
544C3#2568511yescharacteristic, direct product factor, Sylow
545C2—81no—
546C2—82no—
547C2—82no—
548C2C3×#243211yescharacteristic
549C2C3×#300811yescharacteristic
550C2C3×#300911yescharacteristic
551C2C3×#317912yes—
552C2C3×#324112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#540C2
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×#501C2semidirect
3. C2×C3×#486C2semidirect
4. C3×#2628C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1991C2semidirect
8. C3×#1961C2semidirect
9. C3×#1864C2semidirect
10. C3×#1862C2semidirect
11. C3×#1750C2semidirect
12. #25685C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#486C2×C3semidirect
51. #2628C2×C3semidirect
52. #2034C2×C3semidirect
53. #2033C2×C3semidirect
54. #1991C2×C3semidirect
55. #1961C2×C3semidirect
56. #1864C2×C3semidirect
57. #1862C2×C3semidirect
58. #1750C2×C3semidirect
544. C3#25685direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#5012
C2×C3×#4862
C3×#26282
C3×#20342
C3×#20332
C3×#19912
C3×#19612
C3×#18642
C3×#18622
C3×#17502
#256853