Finite Group Databasebeta

← C3×#18342 #10636949

Subgroups — C3×#18342 #10636949

Order 768.

998 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3459C211yesmaximal, characteristic
2C3×#3238C212yesmaximal, complemented
3C3×#2547C211yesmaximal, characteristic, complemented
4C3×#1612C214yesmaximal, complemented
5C3×#1608C212yesmaximal, complemented
6C3×#1607C212yesmaximal, complemented
7C3×#1509C212yesmaximal, complemented
8C3×#1466C211yesmaximal, characteristic, complemented
9#18342C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#543C2211yescharacteristic
11C3×#539—22no—
12C3×#539—21no—
13C3×#538—22no—
14C3×#538C2212yes—
15C3×#534—21no—
16C3×#534C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic, complemented
18C3×#529—22no—
19C3×#529C2212yes—
20C2×C3×#124—21no—
21C3×#525—21no—
22C3×#525—22no—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#501—22no—
26C3×#489—22no—
27C3×#483C2214yescomplemented
28C3×#480C2214yes—
29C3×#477—22no—
30C3×#476—22no—
31C3×#432C2212yescomplemented
32C3×#397C2212yes—
33C3×#397C2212yescomplemented
34C3×#391—21no—
35C3×#380—21no—
36C3×#380—21no—
37C3×C4×C4⋊C4C2211yescharacteristic, complemented
38C3×#328C2214yescomplemented
39C3×#327C2214yes—
40C3×#323C2212yes—
41C3×#323C2212yescomplemented
42#3459C2×C311yescharacteristic
43#3238C2×C312yescomplemented
44#2547C2×C311yescharacteristic, complemented
45#1612C2×C314yescomplemented
46#1608C2×C312yescomplemented
47#1607C2×C312yescomplemented
48#1509C2×C312yescomplemented
49#1466C2×C311yescharacteristic, complemented
50C2×C3×#41—21no—
51C22×C3×Q8—21no—
52C22×C3×D4—21no—
53C22×C3×D4D411yescharacteristic
54C23×C3×C4—21no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—22no—
59C3×#126—22no—
60C3×#126—22no—
61C3×#126—22no—
62C3×#125—22no—
63C3×#125D412yes—
64C3×#124—22no—
65C3×#124—44no—
66C3×#124D412yes—
67C3×#123—22no—
68C3×#123—22no—
69C3×#123—22no—
70C3×#123—22no—
71C3×#123—22no—
72C3×#123—22no—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8C2312yes—
79C3×#121—22no—
80C3×#121—21no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—21no—
84C3×#121C2312yes—
85C3×#120—22no—
86C3×#120—22no—
87C3×#120—22no—
88C3×C4×Q8D412yes—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4D412yes—
97C3×#117—21no—
98C3×#117—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2311yescharacteristic
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C3×#107—24no—
114C3×#106—24no—
115C3×C4⋊C8—22no—
116C3×#103—24no—
117C3×#102—24no—
118C3×C22⋊C8C2314yes—
119C3×#97—22no—
120C3×#95—22no—
121C3×#95—22no—
122C3×#95C2312yes—
123#543C22×C311yescharacteristic
124#539—22no—
125#539—21no—
126#538—22no—
127#538C22×C312yes—
128#534—21no—
129#534C22×C311yescharacteristic
130C2×C4⋊Q8C22×C311yescharacteristic, complemented
131#529—22no—
132#529C22×C312yes—
133C2×#124—21no—
134#525—21no—
135#525—22no—
136C2×C4×D4—21no—
137C2×C4×D4C22×C311yescharacteristic
138#501—22no—
139#489—22no—
140#483C22×C314yescomplemented
141#480C22×C314yes—
142#477—22no—
143#476—22no—
144#432C22×C312yescomplemented
145#397C22×C312yes—
146#397C22×C312yescomplemented
147#391—21no—
148#380—21no—
149#380—21no—
150C4×C4⋊C4C22×C311yescharacteristic, complemented
151#328C22×C314yescomplemented
152#327C22×C314yes—
153#323C22×C312yes—
154#323C22×C312yescomplemented
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C24×C3C2×D411yescharacteristic
158C3×#41—82no—
159C2×C3×Q8—21no—
160C2×C3×Q8—42no—
161C2×C3×Q8—42no—
162C2×C3×Q8—42no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—41no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4—82no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×C3×C8—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4C2×D412yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—44no—
223C3×C22⋊C4—44no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—42no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229C2×#41—21no—
230C22×Q8—21no—
231C22×D4—21no—
232C22×D4C3×D411yescharacteristic
233C23×C4—21no—
234C23×C4C3×D411yescharacteristic
235C23×C4C3×D411yescharacteristic
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238#126—22no—
239#126—22no—
240#126—22no—
241#125—22no—
242#125C3×D412yes—
243#124—22no—
244#124—44no—
245#124C3×D412yes—
246#123—22no—
247#123—22no—
248#123—22no—
249#123—22no—
250#123—22no—
251#123—22no—
252C22⋊Q8—21no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—21no—
256C22⋊Q8—22no—
257C22⋊Q8C23×C312yes—
258#121—22no—
259#121—21no—
260#121—22no—
261#121—22no—
262#121—21no—
263#121C23×C312yes—
264#120—22no—
265#120—22no—
266#120—22no—
267C4×Q8C3×D412yes—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4C3×D412yes—
276#117—21no—
277#117—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C3×D411yescharacteristic
282C2×C4⋊C4C23×C312yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4C23×C311yescharacteristic
288C2×C22⋊C4C23×C311yescharacteristic
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C23×C311yescharacteristic
292#107—24no—
293#106—24no—
294C4⋊C8—22no—
295#103—24no—
296#102—24no—
297C22⋊C8C23×C314yes—
298#97—22no—
299#95—22no—
300#95—22no—
301#95C23×C312yes—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—42no—
313C23×C3—81no—
314C23×C3—81no—
315C23×C3AGL(1,ℤ/8)12yes—
316C23×C3C22×D411yescharacteristic
317C3×Q8—42no—
318C3×Q8—84no—
319C3×Q8—81no—
320C3×D4—44no—
321C3×D4—42no—
322C3×D4—44no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—82no—
326C3×D4—81no—
327C3×D4—82no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—81no—
350C2×C3×C4—81no—
351C2×C3×C4—81no—
352C2×C3×C4—82no—
353C2×C3×C4—81no—
354C2×C3×C4#12012yes—
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4C22×D412yes—
358C3×C8—44no—
359C24—21no—
360C24C2×C3×D411yescharacteristic
361C24C2×C3×D411yescharacteristic
362#41—82no—
363C2×Q8—21no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8C2×C3×D412yes—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4C2×C3×D412yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—82no—
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C24×C311yescharacteristic
398C2×C8—24no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4C2×C3×D412yes—
413C4⋊C4C2×C3×D412yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C42—22no—
429C42—22no—
430C42—42no—
431C42C2×C3×D412yes—
432C42C2×C3×D412yes—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—22no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—41no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—41no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—84no—
449C22×C3C2×#12011yescharacteristic
450C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
451C22×C3#57511yescharacteristic
452C3×C4—22no—
453C3×C4—21no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—82no—
461C3×C4—82no—
462C3×C4#45512yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—21no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—81no—
475C23—81no—
476C23C3×AGL(1,ℤ/8)12yes—
477C23C22×C3×D411yescharacteristic
478Q8—42no—
479Q8—84no—
480Q8—81no—
481D4—44no—
482D4—42no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—81no—
487D4—82no—
488D4—82no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4C3×#12012yes—
516C2×C4C3×#12011yescharacteristic
517C2×C4C3×#12011yescharacteristic
518C2×C4C22×C3×D412yes—
519C8—44no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—42no—
523C2×C3—81no—
524C2×C3—82no—
525C2×C3#300611yescharacteristic
526C2×C3C2×#45511yescharacteristic
527C2×C3#302811yescharacteristic
528C22—21no—
529C22—21no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—81no—
542C22—81no—
543C22—84no—
544C22C2×C3×#12011yescharacteristic
545C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
546C22C3×#57511yescharacteristic
547C4—22no—
548C4—21no—
549C4—22no—
550C4—42no—
551C4—41no—
552C4—41no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4C3×#45512yes—
558C3#1834211yescharacteristic, direct product factor, Sylow
559C2—21no—
560C2—21no—
561C2—42no—
562C2—81no—
563C2—82no—
564C2C3×#300611yescharacteristic
565C2C2×C3×#45511yescharacteristic
566C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3459C2
LCSG ▷ C22×C4 ▷ 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×#3238C2semidirect
3. C3×#2547C2semidirect
4. C3×#1612C2semidirect
5. C3×#1608C2semidirect
6. C3×#1607C2semidirect
7. C3×#1509C2semidirect
8. C3×#1466C2semidirect
9. #18342C3direct (N × H)
17. C2×C3×C4⋊Q8C22semidirect
27. C3×#483C22semidirect
31. C3×#432C22semidirect
33. C3×#397C22semidirect
37. C3×C4×C4⋊C4C22semidirect
38. C3×#328C22semidirect
41. C3×#323C22semidirect
43. #3238C2×C3semidirect
44. #2547C2×C3semidirect
45. #1612C2×C3semidirect
46. #1608C2×C3semidirect
47. #1607C2×C3semidirect
48. #1509C2×C3semidirect
49. #1466C2×C3semidirect
130. C2×C4⋊Q8C22×C3semidirect
140. #483C22×C3semidirect
144. #432C22×C3semidirect
146. #397C22×C3semidirect
150. C4×C4⋊C4C22×C3semidirect
151. #328C22×C3semidirect
154. #323C22×C3semidirect
558. C3#18342direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34592
C3×#32382
C3×#25472
C3×#16122
C3×#16082
C3×#16072
C3×#15092
C3×#14662
#183423