Finite Group Databasebeta

← C3×#18692 #10637299

Subgroups — C3×#18692 #10637299

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3548C211yesmaximal, characteristic
2C3×#3255C211yesmaximal, characteristic, complemented
3C3×#3254C211yesmaximal, characteristic, complemented
4C3×#3131C211yesmaximal, characteristic, complemented
5C3×#1644C212yesmaximal, complemented
6C3×#1643C212yesmaximal, complemented
7C3×#1642C212yesmaximal, complemented
8C3×#1641C212yesmaximal, complemented
9C3×#1560C211yesmaximal, characteristic, complemented
10C3×#1493C211yesmaximal, characteristic, complemented
11C3×#1492C211yesmaximal, characteristic, complemented
12#18692C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#562—22no—
14C3×#554—21no—
15C3×#554C2212yes—
16C3×#550—21no—
17C3×#550C2212yes—
18C3×#547—21no—
19C3×#547—21no—
20C3×#540—22no—
21C3×#537—22no—
22C3×#537—22no—
23C3×#533C2211yescharacteristic
24C3×#533C2211yescharacteristic
25C3×#532C2211yescharacteristic
26C3×#532C2211yescharacteristic
27C3×#488—22no—
28C3×#487C2211yescharacteristic
29C3×#487C2211yescharacteristic
30C3×#486C2211yescharacteristic
31C3×#486C2211yescharacteristic
32C3×#477—21no—
33C3×#475—21no—
34C3×#475C2212yes—
35C3×#474C2212yes—
36C3×#470—22no—
37C3×#468—22no—
38C3×#462—21no—
39C3×#462C2212yes—
40C3×#461C2212yes—
41C3×#460—21no—
42C3×#423C2211yescharacteristic
43C3×#423C2211yescharacteristic
44C3×#416—22no—
45C2×C3×C4×C8C2211yescharacteristic
46C3×#333C2212yes—
47C3×#331C2212yes—
48C3×Q8⋊C8C2212yes—
49C3×Q8⋊C8C2212yes—
50C3×D4⋊C8C2212yes—
51C3×D4⋊C8C2212yes—
52#3548C2×C311yescharacteristic
53#3255C2×C311yescharacteristic, complemented
54#3254C2×C311yescharacteristic, complemented
55#3131C2×C311yescharacteristic, complemented
56#1644C2×C312yescomplemented
57#1643C2×C312yescomplemented
58#1642C2×C312yescomplemented
59#1641C2×C312yescomplemented
60#1560C2×C311yescharacteristic, complemented
61#1493C2×C311yescharacteristic, complemented
62#1492C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—21no—
66C2×C3×#41D412yes—
67C2×C3×Q16—22no—
68C2×C3×SD16—22no—
69C2×C3×M16—21no—
70C2×C3×M16—21no—
71C22×C3×C8—21no—
72C3×C4⋊Q8D411yescharacteristic
73C3×C4⋊Q8D411yescharacteristic
74C3×C4⋊Q8D411yescharacteristic
75C3×C4⋊Q8C2311yescharacteristic
76C3×C4⋊Q8C2311yescharacteristic
77C3×C4⋊Q8C2311yescharacteristic
78C3×#127—22no—
79C3×#127D411yescharacteristic
80C3×#127C2311yescharacteristic
81C3×#125—22no—
82C3×#125—21no—
83C3×#125—21no—
84C3×#124—21no—
85C3×#124D412yes—
86C3×#123—24no—
87C3×#123—24no—
88C3×C22⋊Q8—24no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—24no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8D412yes—
96C3×#121—24no—
97C3×#121—24no—
98C3×#121—24no—
99C3×#121—22no—
100C3×#121—24no—
101C3×#121—21no—
102C3×#121—21no—
103C3×#121—21no—
104C3×#121—22no—
105C3×#121D412yes—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8C2312yes—
110C3×C4×D4—22no—
111C3×C4×D4—22no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4C2312yes—
116C3×#117—21no—
117C3×#117—21no—
118C2×C3×C4⋊C4—22no—
119C2×C3×C4⋊C4—22no—
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#106—22no—
123C3×C4⋊C8—21no—
124C3×C4⋊C8—21no—
125C3×C4⋊C8C2312yes—
126C3×C4⋊C8C2312yes—
127C3×#103—22no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#103—22no—
131C3×#103—22no—
132C3×#102—22no—
133C3×#102—22no—
134C3×#102—22no—
135C3×C4×C8—21no—
136C3×C4×C8C2311yescharacteristic
137C3×C4×C8C2311yescharacteristic
138#562—22no—
139#554—21no—
140#554C22×C312yes—
141#550—21no—
142#550C22×C312yes—
143#547—21no—
144#547—21no—
145#540—22no—
146#537—22no—
147#537—22no—
148#533C22×C311yescharacteristic
149#533C22×C311yescharacteristic
150#532C22×C311yescharacteristic
151#532C22×C311yescharacteristic
152#488—22no—
153#487C22×C311yescharacteristic
154#487C22×C311yescharacteristic
155#486C22×C311yescharacteristic
156#486C22×C311yescharacteristic
157#477—21no—
158#475—21no—
159#475C22×C312yes—
160#474C22×C312yes—
161#470—22no—
162#468—22no—
163#462—21no—
164#462C22×C312yes—
165#461C22×C312yes—
166#460—21no—
167#423C22×C311yescharacteristic
168#423C22×C311yescharacteristic
169#416—22no—
170C2×C4×C8C22×C311yescharacteristic
171#333C22×C312yes—
172#331C22×C312yes—
173Q8⋊C8C22×C312yes—
174Q8⋊C8C22×C312yes—
175D4⋊C8C22×C312yes—
176D4⋊C8C22×C312yes—
177C3×#41—44no—
178C3×#41—81no—
179C3×#41—81no—
180C3×#41—82no—
181C3×#41—82no—
182C3×#41—84no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8C2×D412yes—
188C2×C3×Q8C2×D412yes—
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4C2×D412yes—
198C2×C3×D4C2×D412yes—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C3×Q16—82no—
207C3×SD16—82no—
208C3×M16—81no—
209C3×M16—81no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—42no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—41no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—41no—
241C3×C42—22no—
242C3×C42—41no—
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42C2411yescharacteristic
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—21no—
250C2×#41C3×D412yes—
251C2×Q16—22no—
252C2×SD16—22no—
253C2×M16—21no—
254C2×M16—21no—
255C22×C8—21no—
256C4⋊Q8C3×D411yescharacteristic
257C4⋊Q8C3×D411yescharacteristic
258C4⋊Q8C3×D411yescharacteristic
259C4⋊Q8C23×C311yescharacteristic
260C4⋊Q8C23×C311yescharacteristic
261C4⋊Q8C23×C311yescharacteristic
262#127—22no—
263#127C3×D411yescharacteristic
264#127C23×C311yescharacteristic
265#125—22no—
266#125—21no—
267#125—21no—
268#124—21no—
269#124C3×D412yes—
270#123—24no—
271#123—24no—
272C22⋊Q8—22no—
273C22⋊Q8—24no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8C3×D412yes—
280#121—24no—
281#121—24no—
282#121—22no—
283#121—24no—
284#121—24no—
285#121—21no—
286#121—21no—
287#121—21no—
288#121—22no—
289#121C3×D412yes—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8C23×C312yes—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4C23×C312yes—
300#117—21no—
301#117—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C42C23×C311yescharacteristic
305#107—22no—
306#106—22no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8C23×C312yes—
310C4⋊C8C23×C312yes—
311#103—22no—
312#103—22no—
313#103—22no—
314#103—22no—
315#103—22no—
316#102—22no—
317#102—22no—
318#102—22no—
319C4×C8—21no—
320C4×C8C23×C311yescharacteristic
321C4×C8C23×C311yescharacteristic
322C23×C3—22no—
323C23×C3—22no—
324C23×C3#12011yescharacteristic
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×Q8—41no—
333C3×Q8—81no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—42no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—42no—
342C3×D4—81no—
343C3×D4—82no—
344C3×D4—82no—
345C3×D4—84no—
346C3×D4—84no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—84no—
362C2×C3×C4—84no—
363C2×C3×C4#12011yescharacteristic
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12011yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C3×C8—41no—
370C3×C8—41no—
371C3×C8—42no—
372C3×C8—42no—
373#41—44no—
374#41—81no—
375#41—81no—
376#41—82no—
377#41—82no—
378#41—84no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8C2×C3×D412yes—
384C2×Q8C2×C3×D412yes—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4C2×C3×D412yes—
394C2×D4C2×C3×D412yes—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402Q16—82no—
403SD16—82no—
404M16—81no—
405M16—81no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—42no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4C2×C3×D412yes—
428C4⋊C4C2×C3×D412yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—41no—
437C42—22no—
438C42—41no—
439C42C2×C3×D411yescharacteristic
440C42C2×C3×D411yescharacteristic
441C42C2×C3×D411yescharacteristic
442C42C24×C311yescharacteristic
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3C2×#12011yescharacteristic
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4#45512yes—
463C3×C4#57611yescharacteristic
464C3×C4#57711yescharacteristic
465C3×C4#57711yescharacteristic
466C3×C4#57811yescharacteristic
467C23—22no—
468C23—22no—
469C23C3×#12011yescharacteristic
470Q8—22no—
471Q8—22no—
472Q8—42no—
473Q8—42no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477Q8—41no—
478Q8—81no—
479D4—22no—
480D4—22no—
481D4—42no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—42no—
487D4—81no—
488D4—82no—
489D4—82no—
490D4—84no—
491D4—84no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4C3×#12011yescharacteristic
509C2×C4C3×#12011yescharacteristic
510C2×C4C3×#12011yescharacteristic
511C2×C4C22×C3×D411yescharacteristic
512C2×C4C22×C3×D411yescharacteristic
513C2×C4C22×C3×D411yescharacteristic
514C8—41no—
515C8—41no—
516C8—42no—
517C8—42no—
518C2×C3—41no—
519C2×C3—82no—
520C2×C3—82no—
521C2×C3#301211yescharacteristic
522C2×C3#301311yescharacteristic
523C2×C3C2×#45511yescharacteristic
524C22—21no—
525C22—21no—
526C22—21no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22C2×C3×#12011yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4C3×#45512yes—
544C4C3×#57611yescharacteristic
545C4C3×#57711yescharacteristic
546C4C3×#57711yescharacteristic
547C4C3×#57811yescharacteristic
548C3#1869211yescharacteristic, direct product factor, Sylow
549C2—41no—
550C2—82no—
551C2—82no—
552C2C3×#301211yescharacteristic
553C2C3×#301311yescharacteristic
554C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3255C2semidirect
3. C3×#3254C2semidirect
4. C3×#3131C2semidirect
5. C3×#1644C2semidirect
6. C3×#1643C2semidirect
7. C3×#1642C2semidirect
8. C3×#1641C2semidirect
9. C3×#1560C2semidirect
10. C3×#1493C2semidirect
11. C3×#1492C2semidirect
12. #18692C3direct (N × H)
53. #3255C2×C3semidirect
54. #3254C2×C3semidirect
55. #3131C2×C3semidirect
56. #1644C2×C3semidirect
57. #1643C2×C3semidirect
58. #1642C2×C3semidirect
59. #1641C2×C3semidirect
60. #1560C2×C3semidirect
61. #1493C2×C3semidirect
62. #1492C2×C3semidirect
548. C3#18692direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35482
C3×#32552
C3×#32542
C3×#31312
C3×#16442
C3×#16432
C3×#16422
C3×#16412
C3×#15602
C3×#14932
C3×#14922
#186923