Finite Group Databasebeta

← C3×#35757 #10654364

Subgroups — C3×#35757 #10654364

Order 768.

5054 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×#550C211yesmaximal, characteristic, complemented
3C2×C3×#548C212yesmaximal, complemented
4C3×#2615C213yesmaximal, complemented
5C3×#2436C213yesmaximal, complemented
6C3×#2431C216yesmaximal, complemented
7C3×#2409C213yesmaximal, complemented
8C3×#2407C216yesmaximal, complemented
9C3×#2388C213yesmaximal, complemented
10C3×#2341C211yesmaximal, characteristic, complemented
11C3×Q8×C22⋊C4C212yesmaximal, complemented
12#35757C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4—28no—
14C2×C3×2−1+4—216no—
15C22×C3×#41C2213yes—
16C22×C3×#41C2216yes—
17C22×C3×#41C2211yescharacteristic
18C22×C3×#41C2211yescharacteristic
19C23×C3×Q8C2213yes—
20C23×C3×Q8C2212yes—
21C3×#550—28no—
22C3×#548—216no—
23C2×C3×#127C2213yes—
24C2×C3×#124C2216yes—
25C2×C3×C22⋊Q8C2216yes—
26C2×C3×#121C2213yes—
27C2×C3×#121C2216yes—
28C2×C3×#121C2213yes—
29C2×C3×C4×Q8C2212yes—
30C2×C3×C4×Q8C2211yescharacteristic
31C2×C3×C4×D4C2216yes—
32C2×C3×C4×D4C2213yes—
33C3×#395C2213yes—
34C3×#394C2211yescharacteristic
35C3×#394C2216yes—
36C3×#393C2213yes—
37C3×#393C2216yes—
38C3×#392C2216yes—
39C3×#392C2213yes—
40C3×#392C22112yes—
41C3×#391C2216yes—
42C3×#391C2216yes—
43C3×#390C2216yes—
44C3×#386C22112yes—
45C3×#386C2216yes—
46C3×#385C2216yes—
47C3×#380C2216yes—
48C3×#380C2213yes—
49C3×C4×C22⋊C4C2216yes—
50C3×C4×C22⋊C4C2213yes—
51C22×2−1+4C2×C311yescharacteristic, complemented
52C2×#550C2×C311yescharacteristic, complemented
53C2×#548C2×C312yescomplemented
54#2615C2×C313yescomplemented
55#2436C2×C313yescomplemented
56#2431C2×C316yescomplemented
57#2409C2×C313yescomplemented
58#2407C2×C316yescomplemented
59#2388C2×C313yescomplemented
60#2341C2×C311yescharacteristic, complemented
61Q8×C22⋊C4C2×C312yescomplemented
62C3×2−1+4—2128no—
63C2×C3×#41—224no—
64C2×C3×#41—28no—
65C2×C3×#41—28no—
66C2×C3×#41—24no—
67C2×C3×#41—224no—
68C2×C3×#41—212no—
69C2×C3×#41—224no—
70C2×C3×#41—224no—
71C2×C3×#41D418yescomplemented
72C22×C3×Q8—224no—
73C22×C3×Q8—28no—
74C22×C3×Q8—28no—
75C22×C3×Q8—212no—
76C22×C3×Q8D4116yescomplemented
77C22×C3×Q8C2313yes—
78C22×C3×Q8C2316yes—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C22×C3×D4C2316yes—
82C22×C3×D4C2313yes—
83C22×C3×D4C2313yes—
84C22×C3×D4C2313yes—
85C22×C3×D4C2316yes—
86C22×C3×D4C2316yes—
87C23×C3×C4C2316yes—
88C23×C3×C4C2313yes—
89C23×C3×C4C2313yes—
90C23×C3×C4C2316yes—
91C3×#127—212no—
92C3×#124—224no—
93C3×C22⋊Q8—224no—
94C3×#121—212no—
95C3×#121—224no—
96C3×#121—212no—
97C3×C4×Q8—28no—
98C3×C4×Q8—24no—
99C3×C4×D4—224no—
100C3×C4×D4—212no—
101C2×C3×C4⋊C4C2316yes—
102C2×C3×C4⋊C4C2313yes—
103C2×C3×C4⋊C4C2316yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C22⋊C4C2313yes—
107C2×C3×C22⋊C4C2316yes—
108C2×C3×C22⋊C4C23112yes—
109C2×C3×C22⋊C4C2313yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2316yes—
112C2×C3×C22⋊C4C2316yes—
113C2×C3×C22⋊C4C2316yes—
114C2×C3×C42C2316yes—
115C2×C3×C42C2313yes—
116C3×#95C2311yescharacteristic
117C3×#95C2313yes—
118C3×#95C2316yes—
119C3×#95C2316yes—
120C3×#95C23112yes—
121C2×2−1+4—28no—
122C2×2−1+4—216no—
123C22×#41C22×C313yes—
124C22×#41C22×C316yes—
125C22×#41C22×C311yescharacteristic
126C22×#41C22×C311yescharacteristic
127C23×Q8C22×C313yes—
128C23×Q8C22×C312yes—
129#550—28no—
130#548—216no—
131C2×#127C22×C313yes—
132C2×#124C22×C316yes—
133C2×C22⋊Q8C22×C316yes—
134C2×#121C22×C316yes—
135C2×#121C22×C313yes—
136C2×#121C22×C313yes—
137C2×C4×Q8C22×C311yescharacteristic
138C2×C4×Q8C22×C312yes—
139C2×C4×D4C22×C316yes—
140C2×C4×D4C22×C313yes—
141#395C22×C313yes—
142#394C22×C311yescharacteristic
143#394C22×C316yes—
144#393C22×C316yes—
145#393C22×C313yes—
146#392C22×C313yes—
147#392C22×C3112yes—
148#392C22×C316yes—
149#391C22×C316yes—
150#391C22×C316yes—
151#390C22×C316yes—
152#386C22×C3112yes—
153#386C22×C316yes—
154#385C22×C316yes—
155#380C22×C316yes—
156#380C22×C313yes—
157C4×C22⋊C4C22×C316yes—
158C4×C22⋊C4C22×C313yes—
159C24×C3C2413yes—
160C24×C3C2412yes—
161C24×C3C2413yes—
162C3×#41—2192no—
163C3×#41—296no—
164C3×#41—232no—
165C3×#41—416no—
166C2×C3×Q8—296no—
167C2×C3×Q8—264no—
168C2×C3×Q8—212no—
169C2×C3×Q8—26no—
170C2×C3×Q8—212no—
171C2×C3×Q8—212no—
172C2×C3×Q8C2×D4124yes—
173C2×C3×Q8C2×D414yes—
174C2×C3×Q8C2×D418yes—
175C2×C3×D4—212no—
176C2×C3×D4—24no—
177C2×C3×D4—212no—
178C2×C3×D4—212no—
179C2×C3×D4—26no—
180C2×C3×D4—212no—
181C2×C3×D4—212no—
182C2×C3×D4—22no—
183C2×C3×D4—212no—
184C2×C3×D4—212no—
185C2×C3×D4—26no—
186C2×C3×D4—26no—
187C2×C3×D4—43no—
188C2×C3×D4—46no—
189C2×C3×D4—46no—
190C2×C3×D4—412no—
191C2×C3×D4C2×D4112yes—
192C22×C3×C4—212no—
193C22×C3×C4—224no—
194C22×C3×C4—212no—
195C22×C3×C4—212no—
196C22×C3×C4—212no—
197C22×C3×C4—26no—
198C22×C3×C4—212no—
199C22×C3×C4C2×D4112yes—
200C22×C3×C4C2×D4124yes—
201C22×C3×C4C2413yes—
202C22×C3×C4C2412yes—
203C22×C3×C4C2416yes—
204C22×C3×C4C2411yescharacteristic
205C22×C3×C4C2416yes—
206C22×C3×C4C2411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×C3×C4C2413yes—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—212no—
211C3×C4⋊C4—26no—
212C3×C4⋊C4—212no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—26no—
215C3×C22⋊C4—26no—
216C3×C22⋊C4—212no—
217C3×C22⋊C4—224no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—46no—
220C3×C22⋊C4—46no—
221C3×C22⋊C4—46no—
222C3×C42—212no—
223C3×C42—26no—
2242−1+4—2128no—
225C2×#41—224no—
226C2×#41—28no—
227C2×#41—28no—
228C2×#41—24no—
229C2×#41—224no—
230C2×#41—212no—
231C2×#41—224no—
232C2×#41—224no—
233C2×#41C3×D418yescomplemented
234C22×Q8—224no—
235C22×Q8—28no—
236C22×Q8—28no—
237C22×Q8—212no—
238C22×Q8C3×D4116yescomplemented
239C22×Q8C23×C313yes—
240C22×Q8C23×C316yes—
241C22×Q8C23×C311yescharacteristic
242C22×D4C23×C313yes—
243C22×D4C23×C311yescharacteristic
244C22×D4C23×C316yes—
245C22×D4C23×C313yes—
246C22×D4C23×C313yes—
247C22×D4C23×C316yes—
248C22×D4C23×C316yes—
249C23×C4C23×C313yes—
250C23×C4C23×C316yes—
251C23×C4C23×C313yes—
252C23×C4C23×C316yes—
253#127—212no—
254#124—224no—
255C22⋊Q8—224no—
256#121—212no—
257#121—224no—
258#121—212no—
259C4×Q8—28no—
260C4×Q8—24no—
261C4×D4—224no—
262C4×D4—212no—
263C2×C4⋊C4C23×C311yescharacteristic
264C2×C4⋊C4C23×C316yes—
265C2×C4⋊C4C23×C313yes—
266C2×C4⋊C4C23×C316yes—
267C2×C22⋊C4C23×C312yes—
268C2×C22⋊C4C23×C313yes—
269C2×C22⋊C4C23×C313yes—
270C2×C22⋊C4C23×C316yes—
271C2×C22⋊C4C23×C3112yes—
272C2×C22⋊C4C23×C312yes—
273C2×C22⋊C4C23×C316yes—
274C2×C22⋊C4C23×C316yes—
275C2×C22⋊C4C23×C316yes—
276C2×C42C23×C316yes—
277C2×C42C23×C313yes—
278#95C23×C311yescharacteristic
279#95C23×C313yes—
280#95C23×C316yes—
281#95C23×C316yes—
282#95C23×C3112yes—
283C23×C3—26no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—23no—
288C23×C3—22no—
289C23×C3—23no—
290C23×C3—26no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—26no—
294C23×C3—23no—
295C23×C3—23no—
296C23×C3—23no—
297C23×C3—26no—
298C23×C3C22×D416yes—
299C23×C3C22×D414yes—
300C23×C3C2511yescharacteristic
301C3×Q8—224no—
302C3×Q8—248no—
303C3×Q8#120116yes—
304C3×D4—248no—
305C3×D4—28no—
306C3×D4—224no—
307C3×D4—412no—
308C3×D4—424no—
309C3×D4—412no—
310C3×D4—812no—
311C2×C3×C4—248no—
312C2×C3×C4—21no—
313C2×C3×C4—26no—
314C2×C3×C4—22no—
315C2×C3×C4—26no—
316C2×C3×C4—21no—
317C2×C3×C4—224no—
318C2×C3×C4—248no—
319C2×C3×C4—22no—
320C2×C3×C4—26no—
321C2×C3×C4—23no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—26no—
327C2×C3×C4—26no—
328C2×C3×C4—26no—
329C2×C3×C4—412no—
330C2×C3×C4C22×D4112yes—
331C2×C3×C4C22×D412yes—
332C2×C3×C4C22×D4112yes—
333C2×C3×C4C22×D416yes—
334C2×C3×C42−1+412yescomplemented
335C24C24×C313yes—
336C24C24×C312yes—
337C24C24×C313yes—
338#41—2192no—
339#41—296no—
340#41—232no—
341#41—416no—
342C2×Q8—296no—
343C2×Q8—264no—
344C2×Q8—212no—
345C2×Q8—26no—
346C2×Q8—212no—
347C2×Q8—212no—
348C2×Q8C2×C3×D4124yes—
349C2×Q8C2×C3×D414yes—
350C2×Q8C2×C3×D418yes—
351C2×D4—212no—
352C2×D4—24no—
353C2×D4—212no—
354C2×D4—212no—
355C2×D4—26no—
356C2×D4—212no—
357C2×D4—212no—
358C2×D4—22no—
359C2×D4—212no—
360C2×D4—212no—
361C2×D4—26no—
362C2×D4—26no—
363C2×D4—43no—
364C2×D4—46no—
365C2×D4—46no—
366C2×D4—412no—
367C2×D4C2×C3×D4112yes—
368C22×C4—212no—
369C22×C4—224no—
370C22×C4—212no—
371C22×C4—212no—
372C22×C4—212no—
373C22×C4—26no—
374C22×C4—212no—
375C22×C4C2×C3×D4112yes—
376C22×C4C2×C3×D4124yes—
377C22×C4C24×C313yes—
378C22×C4C24×C312yes—
379C22×C4C24×C316yes—
380C22×C4C24×C311yescharacteristic
381C22×C4C24×C316yes—
382C22×C4C24×C311yescharacteristic
383C22×C4C24×C311yescharacteristic
384C22×C4C24×C313yes—
385C4⋊C4—22no—
386C4⋊C4—212no—
387C4⋊C4—26no—
388C4⋊C4—212no—
389C22⋊C4—24no—
390C22⋊C4—26no—
391C22⋊C4—26no—
392C22⋊C4—212no—
393C22⋊C4—224no—
394C22⋊C4—24no—
395C22⋊C4—46no—
396C22⋊C4—46no—
397C22⋊C4—46no—
398C42—212no—
399C42—26no—
400C22×C3—26no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—26no—
404C22×C3—24no—
405C22×C3—26no—
406C22×C3—42no—
407C22×C3—46no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—46no—
411C22×C3—42no—
412C22×C3—43no—
413C22×C3—46no—
414C22×C3—43no—
415C22×C3—46no—
416C22×C3—43no—
417C22×C3C23×D412yes—
418C22×C3C23×D411yescharacteristic
419C22×C3C2×2−1+411yescharacteristic
420C22×C3#58511yescharacteristic
421C22×C3#58512yes—
422C3×C4—212no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—41no—
426C3×C4—46no—
427C3×C4—42no—
428C3×C4—43no—
429C3×C4C2×#120112yes—
430C23—26no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—23no—
435C23—22no—
436C23—23no—
437C23—26no—
438C23—22no—
439C23—22no—
440C23—26no—
441C23—23no—
442C23—23no—
443C23—23no—
444C23—26no—
445C23C22×C3×D416yes—
446C23C22×C3×D414yes—
447C23C25×C311yescharacteristic
448Q8—224no—
449Q8—248no—
450Q8C3×#120116yes—
451D4—248no—
452D4—28no—
453D4—224no—
454D4—412no—
455D4—424no—
456D4—412no—
457D4—812no—
458C2×C4—248no—
459C2×C4—21no—
460C2×C4—26no—
461C2×C4—22no—
462C2×C4—26no—
463C2×C4—21no—
464C2×C4—224no—
465C2×C4—248no—
466C2×C4—22no—
467C2×C4—26no—
468C2×C4—23no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—26no—
474C2×C4—26no—
475C2×C4—26no—
476C2×C4—412no—
477C2×C4C22×C3×D4112yes—
478C2×C4C22×C3×D412yes—
479C2×C4C22×C3×D4112yes—
480C2×C4C22×C3×D416yes—
481C2×C4C3×2−1+412yescomplemented
482C2×C3—44no—
483C2×C3—46no—
484C2×C3—83no—
485C2×C3C22×#12011yescharacteristic
486C2×C3#348712yes—
487C2×C3#348911yescharacteristic
488C2×C3#349712yes—
489C2×C3#350011yescharacteristic
490C22—26no—
491C22—24no—
492C22—24no—
493C22—26no—
494C22—24no—
495C22—26no—
496C22—42no—
497C22—46no—
498C22—42no—
499C22—42no—
500C22—46no—
501C22—42no—
502C22—43no—
503C22—46no—
504C22—43no—
505C22—46no—
506C22—43no—
507C22C23×C3×D412yes—
508C22C23×C3×D411yescharacteristic
509C22C2×C3×2−1+411yescharacteristic
510C22C3×#58511yescharacteristic
511C22C3×#58512yes—
512C4—212no—
513C4—22no—
514C4—22no—
515C4—41no—
516C4—46no—
517C4—42no—
518C4—43no—
519C4C2×C3×#120112yes—
520C3#3575711yescharacteristic, direct product factor, Sylow
521C2—44no—
522C2—46no—
523C2—83no—
524C2C22×C3×#12011yescharacteristic
525C2C3×#348712yes—
526C2C3×#348911yescharacteristic
527C2C3×#349712yes—
528C2C3×#350011yescharacteristic
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×2−1+4C2semidirect
2. C2×C3×#550C2semidirect
3. C2×C3×#548C2semidirect
4. C3×#2615C2semidirect
5. C3×#2436C2semidirect
6. C3×#2431C2semidirect
7. C3×#2409C2semidirect
8. C3×#2407C2semidirect
9. C3×#2388C2semidirect
10. C3×#2341C2semidirect
11. C3×Q8×C22⋊C4C2semidirect
12. #35757C3direct (N × H)
51. C22×2−1+4C2×C3semidirect
52. C2×#550C2×C3semidirect
53. C2×#548C2×C3semidirect
54. #2615C2×C3semidirect
55. #2436C2×C3semidirect
56. #2431C2×C3semidirect
57. #2409C2×C3semidirect
58. #2407C2×C3semidirect
59. #2388C2×C3semidirect
60. #2341C2×C3semidirect
61. Q8×C22⋊C4C2×C3semidirect
71. C2×C3×#41D4semidirect
76. C22×C3×Q8D4semidirect
233. C2×#41C3×D4semidirect
238. C22×Q8C3×D4semidirect
334. C2×C3×C42−1+4semidirect
481. C2×C4C3×2−1+4semidirect
520. C3#35757direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C2×C3×#5502
C2×C3×#5482
C3×#26152
C3×#24362
C3×#24312
C3×#24092
C3×#24072
C3×#23882
C3×#23412
C3×Q8×C22⋊C42
#357573