Finite Group Databasebeta

← C3×#29678 #10648285

Subgroups — C3×#29678 #10648285

Order 768.

1274 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic, complemented
2C3×#3038C212yesmaximal, complemented
3C3×#2541C211yesmaximal, characteristic, complemented
4C3×#2027C212yesmaximal, complemented
5C3×#2006C212yesmaximal, complemented
6C3×#2002C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9#29678C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×AGL(1,ℤ/8)C2214yes—
11C3×#544C2211yescharacteristic
12C3×#542—22no—
13C3×#540—22no—
14C3×#538—22no—
15C3×#537C2211yescharacteristic
16C3×#535C2212yes—
17C3×#532—22no—
18C2×C3×#127C2212yes—
19C2×C3×#127C2211yescharacteristic
20C3×#529—22no—
21C3×#525—22no—
22C2×C3×#121—22no—
23C3×C4×#41C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#460—22no—
26C3×#460—22no—
27C3×#459—22no—
28C3×#459—22no—
29C3×#423C2212yes—
30C2×C3×#104C2214yes—
31C3×#418C2214yes—
32C2×C3×#100C2212yes—
33C3×#392—22no—
34C3×#390C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3460C2×C311yescharacteristic, complemented
39#3038C2×C312yescomplemented
40#2541C2×C311yescharacteristic, complemented
41#2027C2×C312yescomplemented
42#2006C2×C312yescomplemented
43#2002C2×C314yescomplemented
44#1806C2×C312yescomplemented
45#1800C2×C311yescharacteristic, complemented
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C22×C3×D4—24no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4C2312yes—
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4D412yes—
54C3×AGL(1,ℤ/8)—44no—
55C3×AGL(1,ℤ/8)—44no—
56C2×C3×SD16—24no—
57C2×C3×D8—24no—
58C2×C3×M16C2314yes—
59C3×C4⋊Q8—22no—
60C3×#127—22no—
61C3×#127—22no—
62C3×#127—22no—
63C3×#127—44no—
64C3×#126—24no—
65C3×#125—22no—
66C3×#124—24no—
67C3×#124—22no—
68C3×#123—24no—
69C3×#123—24no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×#121—22no—
74C3×#121—24no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—48no—
80C3×#120—24no—
81C3×C4×Q8D412yescomplemented
82C3×C4×Q8D412yescomplemented
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D412yescomplemented
89C3×C4×D4D412yescomplemented
90C3×#117—22no—
91C3×#117—22no—
92C3×#117C2312yes—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—24no—
96C2×C3×C22⋊C4D412yes—
97C2×C3×C42—22no—
98C2×C3×C42C2312yes—
99C2×C3×C42C2311yescharacteristic
100C3×C4⋊C8—22no—
101C3×C4⋊C8—22no—
102C3×#104—44no—
103C3×#102—24no—
104C3×#102—24no—
105C3×#100—42no—
106C3×#95—21no—
107C2×AGL(1,ℤ/8)C22×C314yes—
108#544C22×C311yescharacteristic
109#542—22no—
110#540—22no—
111#538—22no—
112#537C22×C311yescharacteristic
113#535C22×C312yes—
114#532—22no—
115C2×#127C22×C311yescharacteristic
116C2×#127C22×C312yes—
117#529—22no—
118#525—22no—
119C2×#121—22no—
120C4×#41C22×C312yes—
121C2×C4×D4C22×C311yescharacteristic
122#460—22no—
123#460—22no—
124#459—22no—
125#459—22no—
126#423C22×C312yes—
127C2×#104C22×C314yes—
128#418C22×C314yes—
129C2×#100C22×C312yes—
130#392—22no—
131#390C22×C312yes—
132C4×C4⋊C4C22×C311yescharacteristic
133#343C22×C312yes—
134#339C22×C314yes—
135C24×C3—22no—
136C24×C3C2×D412yes—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—84no—
142C2×C3×Q8—22no—
143C2×C3×Q8C2×D412yes—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—44no—
157C2×C3×D4—48no—
158C2×C3×D4—48no—
159C2×C3×D4—42no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—82no—
163C2×C3×D4—84no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—44no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C3×SD16—48no—
177C3×D8—48no—
178C3×M16—24no—
179C3×M16—24no—
180C2×C3×C8—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4C2×D412yes—
189C3×C4⋊C4C2×D412yes—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—48no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—24no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204C2×#41—22no—
205C2×#41C23×C312yes—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C23×C312yes—
210C22×D4C23×C311yescharacteristic
211C23×C4C3×D412yes—
212AGL(1,ℤ/8)—44no—
213AGL(1,ℤ/8)—44no—
214C2×SD16—24no—
215C2×D8—24no—
216C2×M16C23×C314yes—
217C4⋊Q8—22no—
218#127—22no—
219#127—22no—
220#127—22no—
221#127—44no—
222#126—24no—
223#125—22no—
224#124—24no—
225#124—22no—
226#123—24no—
227#123—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231#121—22no—
232#121—24no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—48no—
238#120—24no—
239C4×Q8C3×D412yescomplemented
240C4×Q8C3×D412yescomplemented
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4C3×D412yescomplemented
247C4×D4C3×D412yescomplemented
248#117—22no—
249#117—22no—
250#117C23×C312yes—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C22⋊C4—24no—
254C2×C22⋊C4C3×D412yes—
255C2×C42—22no—
256C2×C42C23×C312yes—
257C2×C42C23×C311yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—22no—
260#104—44no—
261#102—24no—
262#102—24no—
263#100—42no—
264#95—21no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3C22×D411yescharacteristic
277C3×Q8—22no—
278C3×Q8—22no—
279C3×Q8—82no—
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—42no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—44no—
286C3×D4—42no—
287C3×D4—44no—
288C3×D4—82no—
289C3×D4—84no—
290C3×D4—84no—
291C3×D4—82no—
292C3×D4—84no—
293C3×D4—816no—
294C3×D4—82no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4—42no—
309C2×C3×C4—82no—
310C2×C3×C4—82no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12011yescharacteristic
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4C22×D412yes—
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C4H(5,𝔽2)12yes—
317C2×C3×C42−1+411yescharacteristic
318C3×C8—44no—
319C24—22no—
320C24C2×C3×D412yes—
321#41—22no—
322#41—22no—
323#41—22no—
324#41—22no—
325#41—84no—
326C2×Q8—22no—
327C2×Q8C2×C3×D412yes—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—82no—
347C2×D4—84no—
348C2×D4C2×C3×D412yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—44no—
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D412yes—
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C24×C311yescharacteristic
360SD16—48no—
361D8—48no—
362M16—24no—
363M16—24no—
364C2×C8—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4C2×C3×D412yes—
373C4⋊C4C2×C3×D412yes—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—48no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42C2×C3×D412yes—
387C42C2×C3×D412yes—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—82no—
398C22×C3—84no—
399C22×C3—84no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#53511yescharacteristic
405C22×C3#53711yescharacteristic
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—44no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4#45312yes—
414C3×C4#53512yes—
415C3×C4#53712yes—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23C22×C3×D411yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—82no—
431D4—22no—
432D4—22no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—44no—
437D4—42no—
438D4—44no—
439D4—82no—
440D4—84no—
441D4—82no—
442D4—84no—
443D4—84no—
444D4—816no—
445D4—82no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4C3×#12012yes—
463C2×C4C3×#12011yescharacteristic
464C2×C4C3×#12011yescharacteristic
465C2×C4C22×C3×D412yes—
466C2×C4C3×H(5,𝔽2)11yescharacteristic
467C2×C4C3×H(5,𝔽2)12yes—
468C2×C4C3×2−1+411yescharacteristic
469C8—44no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—82no—
473C2×C3—82no—
474C2×C3—162no—
475C2×C3#261511yescharacteristic
476C2×C3C2×#45311yescharacteristic
477C2×C3#302111yescharacteristic
478C22—21no—
479C22—21no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—82no—
488C22—84no—
489C22—84no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22C2×C3×#12011yescharacteristic
494C22C3×#53511yescharacteristic
495C22C3×#53711yescharacteristic
496C4—21no—
497C4—22no—
498C4—44no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4C3×#45312yes—
504C4C3×#53512yes—
505C4C3×#53712yes—
506C3#2967811yescharacteristic, direct product factor, Sylow
507C2—21no—
508C2—21no—
509C2—82no—
510C2—82no—
511C2—162no—
512C2C3×#261511yescharacteristic
513C2C2×C3×#45311yescharacteristic
514C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3460C2semidirect
2. C3×#3038C2semidirect
3. C3×#2541C2semidirect
4. C3×#2027C2semidirect
5. C3×#2006C2semidirect
6. C3×#2002C2semidirect
7. C3×#1806C2semidirect
8. C3×#1800C2semidirect
9. #29678C3direct (N × H)
38. #3460C2×C3semidirect
39. #3038C2×C3semidirect
40. #2541C2×C3semidirect
41. #2027C2×C3semidirect
42. #2006C2×C3semidirect
43. #2002C2×C3semidirect
44. #1806C2×C3semidirect
45. #1800C2×C3semidirect
81. C3×C4×Q8D4semidirect
82. C3×C4×Q8D4semidirect
88. C3×C4×D4D4semidirect
89. C3×C4×D4D4semidirect
239. C4×Q8C3×D4semidirect
240. C4×Q8C3×D4semidirect
246. C4×D4C3×D4semidirect
247. C4×D4C3×D4semidirect
506. C3#29678direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×#30382
C3×#25412
C3×#20272
C3×#20062
C3×#20022
C3×#18062
C3×#18002
#296783