Finite Group Databasebeta

← C3×#25851 #10644458

Subgroups — C3×#25851 #10644458

Order 768.

1076 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
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2059C212yesmaximal, complemented
6C3×#2058C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal
10C3×#1862C212yesmaximal
11C3×#1762C211yesmaximal, characteristic
12#25851C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×#125C2211yescharacteristic
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C22⋊Q8—21no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C22×C3×C4⋊C4—21no—
28C3×#501—24no—
29C3×#476—24no—
30C2×C3×#107C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×#106C2212yes—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#398—21no—
38C3×#398—21no—
39C3×#391—21no—
40C3×#389C2212yes—
41C3×#389C2212yes—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#554C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#476C2×C311yescharacteristic
51#2546C2×C311yescharacteristic, complemented
52#2059C2×C312yescomplemented
53#2058C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1985C2×C311yescharacteristic, complemented
56#1864C2×C312yes—
57#1862C2×C312yes—
58#1762C2×C311yescharacteristic
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—44no—
73C3×#125—22no—
74C3×#125—22no—
75C3×#125—22no—
76C3×#125D414yes—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—28no—
79C3×C22⋊Q8—28no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4D414yes—
88C2×C3×C4⋊C4—28no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#107—24no—
113C3×#106—24no—
114C3×#106—24no—
115C3×C4⋊C8—22no—
116C3×#102—24no—
117C3×#102Q818yescomplemented
118C3×#97—22no—
119C3×#95—21no—
120C3×#95—22no—
121C3×#95—22no—
122#554—24no—
123#554—24no—
124C2×C4⋊Q8—21no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#125C22×C311yescharacteristic
128C2×#125C22×C311yescharacteristic
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×D4C22×C311yescharacteristic
134C2×C4×D4C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C22×C4⋊C4—21no—
137#501—24no—
138#476—24no—
139C2×#107C22×C312yes—
140C2×#106C22×C312yes—
141C2×#106C22×C312yes—
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#102C22×C312yes—
144C2×#102C22×C312yes—
145C2×#97C22×C311yescharacteristic
146#398—21no—
147#398—21no—
148#391—21no—
149#389C22×C312yes—
150#389C22×C312yes—
151C4×C4⋊C4C22×C311yescharacteristic
152#340C22×C312yes—
153#340C22×C312yes—
154#340C22×C312yes—
155#336C22×C312yes—
156#336C22×C312yes—
157C24×C3—21no—
158C2×C3×Q8—24no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—41no—
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C2×C3×Q8—42no—
165C2×C3×Q8—82no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4C2×D412yes—
169C2×C3×D4C2×Q814yes—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—82no—
181C22×C3×C4—82no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—24no—
193C2×C3×C8C2×Q814yes—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—44no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—44no—
211C3×C4⋊C4C2×D412yes—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D412yes—
214C3×C4⋊C4C2×Q814yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—42no—
223C3×C42C2×D412yes—
224C3×C42C2×D412yes—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8C3×D411yescharacteristic
228C22×D4C23×C311yescharacteristic
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C22×C8C23×C312yes—
233C22×C8C23×C312yes—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—44no—
239#125—22no—
240#125—22no—
241#125—22no—
242#125C3×D414yes—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—28no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4C3×D414yes—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C3×D411yescharacteristic
264C2×C4⋊C4C3×D411yescharacteristic
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42C23×C311yescharacteristic
276C2×C42C23×C311yescharacteristic
277C2×C42C23×C311yescharacteristic
278#107—24no—
279#106—24no—
280#106—24no—
281C4⋊C8—22no—
282#102—24no—
283#102C3×Q818yescomplemented
284#97—22no—
285#95—21no—
286#95—22no—
287#95—22no—
288C23×C3—22no—
289C23×C3—24no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3C22×D411yescharacteristic
294C3×Q8—44no—
295C3×Q8—84no—
296C3×Q8—84no—
297C3×D4—22no—
298C3×D4—24no—
299C3×D4C4⋊Q814yescomplemented
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—48no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—84no—
327C2×C3×C4—84no—
328C2×C3×C4#12012yes—
329C2×C3×C4#12012yes—
330C2×C3×C4AGL(1,ℤ/8)12yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×Q812yes—
334C2×C3×C42−1+412yes—
335C3×C8—24no—
336C3×C8—42no—
337C24—21no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—82no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4C2×C3×D412yes—
349C2×D4C2×C3×Q814yes—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C24×C311yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8C2×C3×Q814yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4C2×C3×D412yes—
392C4⋊C4C2×C3×D412yes—
393C4⋊C4C2×C3×D412yes—
394C4⋊C4C2×C3×Q814yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—42no—
403C42C2×C3×D412yes—
404C42C2×C3×D412yes—
405C22×C3—24no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3C2×#12011yescharacteristic
411C22×C3D4×Q812yes—
412C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C3#57511yescharacteristic
414C22×C3#57712yes—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—44no—
419C3×C4—41no—
420C3×C4—42no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—82no—
426C3×C4C2×C4⋊Q811yescharacteristic
427C3×C4#53611yescharacteristic
428C3×C4D4×Q812yes—
429C23—22no—
430C23—24no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23C22×C3×D411yescharacteristic
435Q8—44no—
436Q8—84no—
437Q8—84no—
438D4—22no—
439D4—24no—
440D4C3×C4⋊Q814yescomplemented
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4C3×#12012yes—
470C2×C4C3×#12012yes—
471C2×C4C3×AGL(1,ℤ/8)12yes—
472C2×C4C22×C3×D411yescharacteristic
473C2×C4C22×C3×D411yescharacteristic
474C2×C4C22×C3×Q812yes—
475C2×C4C3×2−1+412yes—
476C8—24no—
477C8—42no—
478C2×C3—44no—
479C2×C3#243411yescharacteristic
480C2×C3#300611yescharacteristic
481C2×C3#300911yescharacteristic
482C2×C3#338412yes—
483C2×C3#338812yes—
484C22—24no—
485C22—44no—
486C22—44no—
487C22—42no—
488C22—42no—
489C22C2×C3×#12011yescharacteristic
490C22C3×D4×Q812yes—
491C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
492C22C3×#57511yescharacteristic
493C22C3×#57712yes—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—44no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4C2×C3×C4⋊Q811yescharacteristic
506C4C3×#53611yescharacteristic
507C4C3×D4×Q812yes—
508C3#2585111yescharacteristic, direct product factor, Sylow
509C2—44no—
510C2C3×#243411yescharacteristic
511C2C3×#300611yescharacteristic
512C2C3×#300911yescharacteristic
513C2C3×#338412yes—
514C2C3×#338812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
4. C3×#2546C2semidirect
5. C3×#2059C2semidirect
6. C3×#2058C2semidirect
7. C3×#1989C2semidirect
8. C3×#1985C2semidirect
12. #25851C3direct (N × H)
49. C2×#501C2×C3semidirect
51. #2546C2×C3semidirect
52. #2059C2×C3semidirect
53. #2058C2×C3semidirect
54. #1989C2×C3semidirect
55. #1985C2×C3semidirect
117. C3×#102Q8semidirect
283. #102C3×Q8semidirect
299. C3×D4C4⋊Q8semidirect
440. D4C3×C4⋊Q8semidirect
508. C3#25851direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#5012
C2×C3×#4762
C3×#25462
C3×#20592
C3×#20582
C3×#19892
C3×#19852
C3×#18642
C3×#18622
C3×#17622
#258513