Finite Group Databasebeta

← C3×#19079 #10637686

Subgroups — C3×#19079 #10637686

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3545C211yesmaximal, characteristic
2C3×#3181C212yesmaximal, complemented
3C3×#3140C211yesmaximal, characteristic, complemented
4C3×#1664C212yesmaximal, complemented
5C3×#1662C212yesmaximal, complemented
6C3×#1661C212yesmaximal, complemented
7C3×#1659C212yesmaximal, complemented
8C3×#1524C212yesmaximal, complemented
9C2×C3×#329C211yesmaximal, characteristic, complemented
10#19079C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#563C2211yescharacteristic
12C3×#562C2212yes—
13C3×H(3,𝔽4)C2211yescharacteristic
14C3×#555—21no—
15C3×#551—21no—
16C3×#551—22no—
17C3×#546—22no—
18C3×#542—22no—
19C3×#538—22no—
20C3×#536—21no—
21C3×#534—21no—
22C3×#533C2211yescharacteristic, complemented
23C3×#529C2212yes—
24C2×C3×#124C2211yescharacteristic
25C3×#501C2212yescomplemented
26C3×#497C2212yescomplemented
27C3×#491C2212yescomplemented
28C3×#489C2212yes—
29C3×#489C2212yescomplemented
30C3×#489C2212yescomplemented
31C3×#488—21no—
32C3×#488—21no—
33C3×#467—22no—
34C3×#465—22no—
35C3×#441—22no—
36C3×#440—22no—
37C3×#419—21no—
38C3×#418—21no—
39C3×C4×M16C2211yescharacteristic, complemented
40C2×C3×#97C2212yescomplemented
41C3×#330C2212yescomplemented
42C3×#330C2212yescomplemented
43C3×#329C2212yes—
44C3×#329C2211yescharacteristic, complemented
45C3×#329C2211yescharacteristic, complemented
46C3×#329C2212yescomplemented
47C3×#329C2212yescomplemented
48#3545C2×C311yescharacteristic
49#3181C2×C312yescomplemented
50#3140C2×C311yescharacteristic, complemented
51#1664C2×C312yescomplemented
52#1662C2×C312yescomplemented
53#1661C2×C312yescomplemented
54#1659C2×C312yescomplemented
55#1524C2×C312yescomplemented
56C2×#329C2×C311yescharacteristic, complemented
57C2×C3×#41—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4D411yescharacteristic
61C2×C3×Q16—22no—
62C2×C3×SD16—22no—
63C2×C3×M16—21no—
64C22×C3×C8—22no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8C2312yes—
68C3×#126—22no—
69C3×#126—22no—
70C3×#126—22no—
71C3×#125—21no—
72C3×#125C2312yes—
73C3×#124—21no—
74C3×#124—22no—
75C3×#124—21no—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124D411yescharacteristic
79C3×#124D411yescharacteristic
80C3×#124C2312yes—
81C3×#124C2311yescharacteristic
82C3×#124C2311yescharacteristic
83C3×#123—21no—
84C3×#123—22no—
85C3×#123—22no—
86C3×#123—21no—
87C3×#123—22no—
88C3×#123—22no—
89C3×#123—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8D412yes—
96C3×#121—22no—
97C3×#121—22no—
98C3×#121—22no—
99C3×#121—22no—
100C3×#121D412yes—
101C3×#120—21no—
102C3×#120—22no—
103C3×#120—21no—
104C3×#120—22no—
105C3×C4×Q8—22no—
106C3×C4×Q8D412yes—
107C3×C4×D4—22no—
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4D412yes—
111C3×#117—21no—
112C3×#117—22no—
113C3×#117—22no—
114C3×#117—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4—22no—
118C2×C3×C42C2311yescharacteristic
119C3×#107—22no—
120C3×#106—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×#97—22no—
130C3×#97C2312yes—
131C3×#97C2312yes—
132C3×#97C2312yes—
133C3×C4×C8—21no—
134#563C22×C311yescharacteristic
135#562C22×C312yes—
136H(3,𝔽4)C22×C311yescharacteristic
137#555—21no—
138#551—21no—
139#551—22no—
140#546—22no—
141#542—22no—
142#538—22no—
143#536—21no—
144#534—21no—
145#533C22×C311yescharacteristic, complemented
146#529C22×C312yes—
147C2×#124C22×C311yescharacteristic
148#501C22×C312yescomplemented
149#497C22×C312yescomplemented
150#491C22×C312yescomplemented
151#489C22×C312yes—
152#489C22×C312yescomplemented
153#489C22×C312yescomplemented
154#488—21no—
155#488—21no—
156#467—22no—
157#465—22no—
158#441—22no—
159#440—22no—
160#419—21no—
161#418—21no—
162C4×M16C22×C311yescharacteristic, complemented
163C2×#97C22×C312yescomplemented
164#330C22×C312yescomplemented
165#330C22×C312yescomplemented
166#329C22×C312yes—
167#329C22×C312yescomplemented
168#329C22×C312yescomplemented
169#329C22×C311yescharacteristic, complemented
170#329C22×C311yescharacteristic, complemented
171C24×C3—21no—
172C24×C3—21no—
173C3×#41—82no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8—42no—
178C2×C3×Q8C2×D411yescharacteristic
179C2×C3×Q8C2×D411yescharacteristic
180C2×C3×Q8C2×D412yes—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—42no—
188C2×C3×D4—82no—
189C2×C3×D4C2×D411yescharacteristic
190C2×C3×D4C2×D411yescharacteristic
191C2×C3×D4C2×D412yes—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C3×Q16—44no—
200C3×SD16—44no—
201C3×M16—42no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—44no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—41no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—22no—
237C3×C42—41no—
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D412yes—
240C3×C42C2411yescharacteristic
241C2×#41—21no—
242C22×Q8C3×D411yescharacteristic
243C22×D4—21no—
244C22×D4C3×D411yescharacteristic
245C2×Q16—22no—
246C2×SD16—22no—
247C2×M16—21no—
248C22×C8—22no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C23×C312yes—
252#126—22no—
253#126—22no—
254#126—22no—
255#125—21no—
256#125C23×C312yes—
257#124—21no—
258#124—22no—
259#124—21no—
260#124—22no—
261#124—22no—
262#124C3×D411yescharacteristic
263#124C3×D411yescharacteristic
264#124C23×C311yescharacteristic
265#124C23×C311yescharacteristic
266#124C23×C312yes—
267#123—21no—
268#123—22no—
269#123—22no—
270#123—21no—
271#123—22no—
272#123—22no—
273#123—22no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8C3×D412yes—
280#121—22no—
281#121—22no—
282#121—22no—
283#121—22no—
284#121C3×D412yes—
285#120—21no—
286#120—22no—
287#120—21no—
288#120—22no—
289C4×Q8—22no—
290C4×Q8C3×D412yes—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4C3×D412yes—
295#117—21no—
296#117—22no—
297#117—22no—
298#117—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C42C23×C311yescharacteristic
303#107—22no—
304#106—22no—
305#103—22no—
306#103—22no—
307#103—22no—
308#103—22no—
309#103—22no—
310#102—22no—
311#102—22no—
312#102—22no—
313#97—22no—
314#97C23×C312yes—
315#97C23×C312yes—
316#97C23×C312yes—
317C4×C8—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—42no—
322C23×C3—82no—
323C23×C3—81no—
324C23×C3—82no—
325C23×C3#12011yescharacteristic
326C3×Q8—42no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—81no—
332C3×Q8—81no—
333C3×D4—42no—
334C3×D4—41no—
335C3×D4—41no—
336C3×D4—44no—
337C3×D4—81no—
338C3×D4—82no—
339C3×D4—82no—
340C3×D4—82no—
341C3×D4—81no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—82no—
352C2×C3×C4—82no—
353C2×C3×C4—81no—
354C2×C3×C4—82no—
355C2×C3×C4#12012yes—
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4C22×D412yes—
358C2×C3×C4C22×D411yescharacteristic
359C3×C8—24no—
360C3×C8—42no—
361C3×C8—42no—
362C24—21no—
363C24—21no—
364#41—82no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—42no—
369C2×Q8C2×C3×D411yescharacteristic
370C2×Q8C2×C3×D411yescharacteristic
371C2×Q8C2×C3×D412yes—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—42no—
379C2×D4—82no—
380C2×D4C2×C3×D411yescharacteristic
381C2×D4C2×C3×D411yescharacteristic
382C2×D4C2×C3×D412yes—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D412yes—
390Q16—44no—
391SD16—44no—
392M16—42no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—44no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C4⋊C4C2×C3×D412yes—
408C4⋊C4C2×C3×D412yes—
409C22⋊C4—22no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—22no—
413C22⋊C4—22no—
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—41no—
425C42—21no—
426C42—21no—
427C42—22no—
428C42—41no—
429C42C2×C3×D411yescharacteristic
430C42C2×C3×D412yes—
431C42C24×C311yescharacteristic
432C22×C3—22no—
433C22×C3—42no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—162no—
441C22×C3—162no—
442C22×C3#45512yes—
443C22×C3C2×#12011yescharacteristic
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—82no—
454C23—21no—
455C23—22no—
456C23—21no—
457C23—42no—
458C23—82no—
459C23—81no—
460C23—82no—
461C23C3×#12011yescharacteristic
462Q8—42no—
463Q8—41no—
464Q8—41no—
465Q8—42no—
466Q8—42no—
467Q8—81no—
468Q8—81no—
469D4—42no—
470D4—41no—
471D4—41no—
472D4—44no—
473D4—81no—
474D4—82no—
475D4—81no—
476D4—82no—
477D4—82no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4C3×#12012yes—
492C2×C4C3×#12011yescharacteristic
493C2×C4C22×C3×D412yes—
494C2×C4C22×C3×D411yescharacteristic
495C8—24no—
496C8—42no—
497C8—42no—
498C2×C3—22no—
499C2×C3—82no—
500C2×C3—81no—
501C2×C3—81no—
502C2×C3C2×#45511yescharacteristic
503C2×C3#301812yes—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—162no—
513C22—162no—
514C22C3×#45512yes—
515C22C2×C3×#12011yescharacteristic
516C4—21no—
517C4—21no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—82no—
526C3#1907911yescharacteristic, direct product factor, Sylow
527C2—22no—
528C2—81no—
529C2—82no—
530C2—81no—
531C2C2×C3×#45511yescharacteristic
532C2C3×#301812yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3545C2
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×#3181C2semidirect
3. C3×#3140C2semidirect
4. C3×#1664C2semidirect
5. C3×#1662C2semidirect
6. C3×#1661C2semidirect
7. C3×#1659C2semidirect
8. C3×#1524C2semidirect
9. C2×C3×#329C2semidirect
10. #19079C3direct (N × H)
22. C3×#533C22semidirect
25. C3×#501C22semidirect
26. C3×#497C22semidirect
27. C3×#491C22semidirect
29. C3×#489C22semidirect
30. C3×#489C22semidirect
39. C3×C4×M16C22semidirect
40. C2×C3×#97C22semidirect
41. C3×#330C22semidirect
42. C3×#330C22semidirect
44. C3×#329C22semidirect
45. C3×#329C22semidirect
46. C3×#329C22semidirect
47. C3×#329C22semidirect
49. #3181C2×C3semidirect
50. #3140C2×C3semidirect
51. #1664C2×C3semidirect
52. #1662C2×C3semidirect
53. #1661C2×C3semidirect
54. #1659C2×C3semidirect
55. #1524C2×C3semidirect
56. C2×#329C2×C3semidirect
145. #533C22×C3semidirect
148. #501C22×C3semidirect
149. #497C22×C3semidirect
150. #491C22×C3semidirect
152. #489C22×C3semidirect
153. #489C22×C3semidirect
162. C4×M16C22×C3semidirect
163. C2×#97C22×C3semidirect
164. #330C22×C3semidirect
165. #330C22×C3semidirect
167. #329C22×C3semidirect
168. #329C22×C3semidirect
169. #329C22×C3semidirect
170. #329C22×C3semidirect
526. C3#19079direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35452
C3×#31812
C3×#31402
C3×#16642
C3×#16622
C3×#16612
C3×#16592
C3×#15242
C2×C3×#3292
#190793