Finite Group Databasebeta

← C3×#18679 #10637286

Subgroups — C3×#18679 #10637286

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic
2C3×#3249C211yesmaximal, characteristic, complemented
3C3×#3167C211yesmaximal, characteristic, complemented
4C3×#3077C211yesmaximal, characteristic, complemented
5C3×#1652C212yesmaximal, complemented
6C3×#1641C212yesmaximal, complemented
7C3×#1638C212yesmaximal, complemented
8C3×#1631C212yesmaximal, complemented
9C3×#1547C211yesmaximal, characteristic, complemented
10C3×#1493C211yesmaximal, characteristic, complemented
11C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
12#18679C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—22no—
14C3×#556C2212yes—
15C3×#555—21no—
16C3×#553—22no—
17C3×#550C2212yes—
18C3×#547—21no—
19C3×#538—22no—
20C3×#535—21no—
21C3×#533C2211yescharacteristic
22C3×#533C2211yescharacteristic
23C3×#532C2211yescharacteristic
24C3×#525—22no—
25C3×#520—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C3×#498C2211yescharacteristic, complemented
28C3×#492C2211yescharacteristic, complemented
29C3×#487C2211yescharacteristic, complemented
30C3×#486C2211yescharacteristic, complemented
31C3×#478—22no—
32C3×#477—21no—
33C3×#477C2212yescomplemented
34C3×#476C2212yes—
35C3×#474C2212yes—
36C3×#467—22no—
37C3×#465—22no—
38C3×#462—21no—
39C3×#462C2212yescomplemented
40C3×C4×SD16—22no—
41C3×#428—21no—
42C3×#423C2211yescharacteristic
43C2×C3×C4⋊C8C2211yescharacteristic
44C3×#418—21no—
45C2×C3×C4×C8C2211yescharacteristic
46C3×#332C2212yescomplemented
47C3×#331C2212yescomplemented
48C3×Q8⋊C8C2212yescomplemented
49C3×Q8⋊C8C2212yescomplemented
50C3×D4⋊C8C2212yes—
51C3×D4⋊C8C2212yes—
52#3506C2×C311yescharacteristic
53#3249C2×C311yescharacteristic, complemented
54#3167C2×C311yescharacteristic, complemented
55#3077C2×C311yescharacteristic, complemented
56#1652C2×C312yescomplemented
57#1641C2×C312yescomplemented
58#1638C2×C312yescomplemented
59#1631C2×C312yescomplemented
60#1547C2×C311yescharacteristic, complemented
61#1493C2×C311yescharacteristic, complemented
62C2×D4⋊C8C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41—21no—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C2×C3×Q16—22no—
68C2×C3×SD16—22no—
69C2×C3×M16—21no—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C3×C4⋊Q8D411yescharacteristic
73C3×C4⋊Q8D411yescharacteristic
74C3×C4⋊Q8C2311yescharacteristic
75C3×C4⋊Q8C2311yescharacteristic
76C3×C4⋊Q8C2311yescharacteristic
77C3×#127—22no—
78C3×#127C2311yescharacteristic
79C3×#126—24no—
80C3×#125—21no—
81C3×#125—21no—
82C3×#124—24no—
83C3×#124—22no—
84C3×#124—21no—
85C3×#123—24no—
86C3×#123—24no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8D412yes—
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—24no—
95C3×#121—21no—
96C3×#121—22no—
97C3×#121D412yes—
98C3×#120—24no—
99C3×C4×Q8—22no—
100C3×C4×Q8—21no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×Q8C2312yes—
104C3×C4×D4—24no—
105C3×C4×D4—24no—
106C3×C4×D4—22no—
107C3×C4×D4—21no—
108C3×C4×D4—21no—
109C3×C4×D4—21no—
110C3×C4×D4D412yes—
111C3×C4×D4D412yes—
112C3×C4×D4C2312yes—
113C3×#117—22no—
114C3×#117—21no—
115C3×#117—22no—
116C3×#117—21no—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4D411yescharacteristic
119C2×C3×C22⋊C4—21no—
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#107—22no—
123C3×#106—22no—
124C3×#106—22no—
125C3×C4⋊C8—21no—
126C3×C4⋊C8—21no—
127C3×C4⋊C8C2312yes—
128C3×C4⋊C8C2312yes—
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#559—22no—
139#556C22×C312yes—
140#555—21no—
141#553—22no—
142#550C22×C312yes—
143#547—21no—
144#538—22no—
145#535—21no—
146#533C22×C311yescharacteristic
147#533C22×C311yescharacteristic
148#532C22×C311yescharacteristic
149#525—22no—
150#520—21no—
151C2×C4×D4C22×C311yescharacteristic
152#498C22×C311yescharacteristic, complemented
153#492C22×C311yescharacteristic, complemented
154#487C22×C311yescharacteristic, complemented
155#486C22×C311yescharacteristic, complemented
156#478—22no—
157#477—21no—
158#477C22×C312yescomplemented
159#476C22×C312yes—
160#474C22×C312yes—
161#467—22no—
162#465—22no—
163#462—21no—
164#462C22×C312yescomplemented
165C4×SD16—22no—
166#428—21no—
167#423C22×C311yescharacteristic
168C2×C4⋊C8C22×C311yescharacteristic
169#418—21no—
170C2×C4×C8C22×C311yescharacteristic
171#332C22×C312yescomplemented
172#331C22×C312yescomplemented
173Q8⋊C8C22×C312yescomplemented
174Q8⋊C8C22×C312yescomplemented
175D4⋊C8C22×C312yes—
176D4⋊C8C22×C312yes—
177C24×C3—21no—
178C3×#41—42no—
179C3×#41—42no—
180C3×#41—82no—
181C3×#41—82no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8C2×D412yes—
187C2×C3×D4—22no—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C2×D412yes—
196C2×C3×D4C2×D412yes—
197C22×C3×C4—24no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—81no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C3×Q16—82no—
208C3×SD16—44no—
209C3×M16—42no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—42no—
215C2×C3×C8—81no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—41no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4C2×D412yes—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—41no—
242C3×C42—24no—
243C3×C42—22no—
244C3×C42—41no—
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247C3×C42C2×D411yescharacteristic
248C3×C42C2411yescharacteristic
249C2×#41—22no—
250C2×#41—21no—
251C22×D4C3×D411yescharacteristic
252C23×C4—21no—
253C2×Q16—22no—
254C2×SD16—22no—
255C2×M16—21no—
256C22×C8—21no—
257C22×C8—21no—
258C4⋊Q8C3×D411yescharacteristic
259C4⋊Q8C3×D411yescharacteristic
260C4⋊Q8C23×C311yescharacteristic
261C4⋊Q8C23×C311yescharacteristic
262C4⋊Q8C23×C311yescharacteristic
263#127—22no—
264#127C23×C311yescharacteristic
265#126—24no—
266#125—21no—
267#125—21no—
268#124—24no—
269#124—22no—
270#124—21no—
271#123—24no—
272#123—24no—
273C22⋊Q8—22no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8C3×D412yes—
278#121—22no—
279#121—22no—
280#121—24no—
281#121—21no—
282#121—22no—
283#121C3×D412yes—
284#120—24no—
285C4×Q8—22no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8C23×C312yes—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4C3×D412yes—
297C4×D4C3×D412yes—
298C4×D4C23×C312yes—
299#117—22no—
300#117—21no—
301#117—22no—
302#117—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C3×D411yescharacteristic
305C2×C22⋊C4—21no—
306C2×C42C23×C311yescharacteristic
307#107—22no—
308#107—22no—
309#106—22no—
310#106—22no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8C23×C312yes—
314C4⋊C8C23×C312yes—
315#103—22no—
316#103—22no—
317#103—22no—
318#102—22no—
319#102—22no—
320#102—22no—
321C4×C8—21no—
322C4×C8C23×C311yescharacteristic
323C4×C8C23×C311yescharacteristic
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—42no—
327C23×C3—81no—
328C23×C3—81no—
329C23×C3#12011yescharacteristic
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8—42no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×Q8—81no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4—42no—
341C3×D4—44no—
342C3×D4—41no—
343C3×D4—41no—
344C3×D4—41no—
345C3×D4—41no—
346C3×D4—42no—
347C3×D4—84no—
348C3×D4—84no—
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—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—82no—
362C2×C3×C4—81no—
363C2×C3×C4—84no—
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12011yescharacteristic
366C2×C3×C4#12011yescharacteristic
367C2×C3×C4#13512yes—
368C2×C3×C4#13712yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C3×C8—22no—
373C3×C8—41no—
374C3×C8—42no—
375C3×C8—42no—
376C24—21no—
377#41—42no—
378#41—42no—
379#41—82no—
380#41—82no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8C2×C3×D412yes—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4C2×C3×D412yes—
395C2×D4C2×C3×D412yes—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406Q16—82no—
407SD16—44no—
408M16—42no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—42no—
414C2×C8—81no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4C2×C3×D412yes—
430C4⋊C4C2×C3×D412yes—
431C4⋊C4C2×C3×D412yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—41no—
441C42—24no—
442C42—22no—
443C42—41no—
444C42C2×C3×D411yescharacteristic
445C42C2×C3×D411yescharacteristic
446C42C2×C3×D411yescharacteristic
447C42C24×C311yescharacteristic
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—84no—
457C22×C3#45212yes—
458C22×C3C2×#12011yescharacteristic
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4#45212yes—
469C3×C4C2×#13511yescharacteristic
470C3×C4C2×#13711yescharacteristic
471C3×C4#57611yescharacteristic
472C3×C4#57711yescharacteristic
473C23—22no—
474C23—22no—
475C23—42no—
476C23—81no—
477C23—81no—
478C23C3×#12011yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487D4—22no—
488D4—22no—
489D4—42no—
490D4—44no—
491D4—41no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—42no—
496D4—84no—
497D4—84no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—84no—
513C2×C4C3×#12011yescharacteristic
514C2×C4C3×#12011yescharacteristic
515C2×C4C3×#12011yescharacteristic
516C2×C4C3×#13512yes—
517C2×C4C3×#13712yes—
518C2×C4C22×C3×D411yescharacteristic
519C2×C4C22×C3×D411yescharacteristic
520C2×C4C22×C3×D411yescharacteristic
521C8—22no—
522C8—41no—
523C8—42no—
524C8—42no—
525C2×C3—22no—
526C2×C3—82no—
527C2×C3—82no—
528C2×C3C2×#45211yescharacteristic
529C2×C3#301211yescharacteristic
530C2×C3#301811yescharacteristic
531C22—21no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—84no—
540C22C3×#45212yes—
541C22C2×C3×#12011yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—81no—
550C4—81no—
551C4C3×#45212yes—
552C4C2×C3×#13511yescharacteristic
553C4C2×C3×#13711yescharacteristic
554C4C3×#57611yescharacteristic
555C4C3×#57711yescharacteristic
556C3#1867911yescharacteristic, direct product factor, Sylow
557C2—22no—
558C2—82no—
559C2—82no—
560C2C2×C3×#45211yescharacteristic
561C2C3×#301211yescharacteristic
562C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3506C2
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×#3249C2semidirect
3. C3×#3167C2semidirect
4. C3×#3077C2semidirect
5. C3×#1652C2semidirect
6. C3×#1641C2semidirect
7. C3×#1638C2semidirect
8. C3×#1631C2semidirect
9. C3×#1547C2semidirect
10. C3×#1493C2semidirect
11. C2×C3×D4⋊C8C2semidirect
12. #18679C3direct (N × H)
27. C3×#498C22semidirect
28. C3×#492C22semidirect
29. C3×#487C22semidirect
30. C3×#486C22semidirect
33. C3×#477C22semidirect
39. C3×#462C22semidirect
46. C3×#332C22semidirect
47. C3×#331C22semidirect
48. C3×Q8⋊C8C22semidirect
49. C3×Q8⋊C8C22semidirect
53. #3249C2×C3semidirect
54. #3167C2×C3semidirect
55. #3077C2×C3semidirect
56. #1652C2×C3semidirect
57. #1641C2×C3semidirect
58. #1638C2×C3semidirect
59. #1631C2×C3semidirect
60. #1547C2×C3semidirect
61. #1493C2×C3semidirect
62. C2×D4⋊C8C2×C3semidirect
152. #498C22×C3semidirect
153. #492C22×C3semidirect
154. #487C22×C3semidirect
155. #486C22×C3semidirect
158. #477C22×C3semidirect
164. #462C22×C3semidirect
171. #332C22×C3semidirect
172. #331C22×C3semidirect
173. Q8⋊C8C22×C3semidirect
174. Q8⋊C8C22×C3semidirect
556. C3#18679direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#32492
C3×#31672
C3×#30772
C3×#16522
C3×#16412
C3×#16382
C3×#16312
C3×#15472
C3×#14932
C2×C3×D4⋊C82
#186793