Finite Group Databasebeta

← C3×#15965 #10634572

Subgroups — C3×#15965 #10634572

Order 768.

1298 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C212yesmaximal
2C2×C3×#387C212yesmaximal
3C3×#1446C211yesmaximal, characteristic
4C3×#1444C211yesmaximal, characteristic
5C3×#1443C212yesmaximal
6C3×#1442C212yesmaximal
7C3×#1442C211yesmaximal, characteristic
8C3×#1441C212yesmaximal
9C3×#1436C212yesmaximal
10#15965C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2212yes—
17C3×#391—28no—
18C3×#387—28no—
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×#391C2×C312yes—
35C2×#387C2×C312yes—
36#1446C2×C311yescharacteristic
37#1444C2×C311yescharacteristic
38#1443C2×C312yes—
39#1442C2×C312yes—
40#1442C2×C311yescharacteristic
41#1441C2×C312yes—
42#1436C2×C312yes—
43C22×C3×Q8—24no—
44C22×C3×Q8—24no—
45C22×C3×Q8—41no—
46C22×C3×Q8—41no—
47C22×C3×Q8Q814yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—42no—
65C2×C3×C4⋊C4—41no—
66C2×C3×C4⋊C4—41no—
67C2×C3×C42—24no—
68C2×C3×C42—24no—
69C2×C3×C42—42no—
70C3×#95—24no—
71C3×#95—24no—
72C3×#95—24no—
73C3×#95—24no—
74C3×#95—24no—
75C3×#95—24no—
76C3×#95—22no—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—22no—
81C3×#95—41no—
82C3×#95—42no—
83C3×#95—41no—
84C3×#95D418yes—
85C23×Q8C22×C311yescharacteristic
86C22×C4⋊C4C22×C312yes—
87C22×C4⋊C4C22×C311yescharacteristic
88C22×C4⋊C4C22×C312yes—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C42C22×C312yes—
91#391—28no—
92#387—28no—
93C2×#95C22×C312yes—
94C2×#95C22×C312yes—
95C2×#95C22×C312yes—
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C311yescharacteristic
99C2×#95C22×C312yes—
100C2×#95C22×C311yescharacteristic
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C311yescharacteristic
106C2×#95C22×C311yescharacteristic
107C2×#95C22×C312yes—
108C24×C3C2411yescharacteristic
109C2×C3×Q8—216no—
110C2×C3×Q8—44no—
111C2×C3×Q8—44no—
112C2×C3×Q8—48no—
113C2×C3×Q8—44no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×Q812yes—
166C22×C3×C4C2×Q814yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4114yes—
170C22×C3×C4#4114yes—
171C3×C4⋊C4—216no—
172C3×C4⋊C4—48no—
173C3×C4⋊C4—44no—
174C3×C4⋊C4—44no—
175C3×C42—48no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—41no—
179C22×Q8—41no—
180C22×Q8C3×Q814yes—
181C23×C4C23×C312yes—
182C23×C4C23×C312yes—
183C23×C4C23×C312yes—
184C23×C4C23×C312yes—
185C23×C4C23×C311yescharacteristic
186C23×C4C23×C311yescharacteristic
187C23×C4C23×C312yes—
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C312yes—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—42no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—41no—
215#95—42no—
216#95—41no—
217#95C3×D418yes—
218C23×C3C22×D412yes—
219C23×C3C22×Q811yescharacteristic
220C23×C3C2×#4112yes—
221C23×C3C2×#4111yescharacteristic
222C23×C3C2×#4112yes—
223C23×C3C2×#4111yescharacteristic
224C23×C3H(5,𝔽2)11yescharacteristic
225C23×C3H(5,𝔽2)11yescharacteristic
226C23×C32−1+411yescharacteristic
227C23×C32−1+412yes—
228C23×C32−1+411yescharacteristic
229C3×Q8—416no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4C22⋊Q818yes—
287C24C24×C311yescharacteristic
288C2×Q8—216no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—48no—
292C2×Q8—44no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4C2×C3×D414yes—
342C22×C4C2×C3×D414yes—
343C22×C4C2×C3×D414yes—
344C22×C4C2×C3×Q812yes—
345C22×C4C2×C3×Q814yes—
346C22×C4C3×#4112yes—
347C22×C4C3×#4112yes—
348C22×C4C3×#4114yes—
349C22×C4C3×#4114yes—
350C4⋊C4—216no—
351C4⋊C4—48no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C42—48no—
355C22×C3C2×C22⋊Q812yes—
356C22×C3#52512yes—
357C22×C3#52912yes—
358C22×C3#53612yes—
359C22×C3#53911yescharacteristic
360C22×C3#53911yescharacteristic
361C22×C3#54112yes—
362C22×C3#54212yes—
363C22×C3#54311yescharacteristic
364C22×C3D4211yescharacteristic
365C22×C3#54612yes—
366C22×C3#54712yes—
367C22×C3#54812yes—
368C22×C3#55111yescharacteristic
369C22×C3#55111yescharacteristic
370C22×C3#55511yescharacteristic
371C22×C3#55612yes—
372C22×C3#55612yes—
373C22×C3#55711yescharacteristic
374C22×C3#55712yes—
375C22×C3#55711yescharacteristic
376C22×C3#56311yescharacteristic
377C22×C3#56311yescharacteristic
378C3×C4—28no—
379C3×C4—44no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—44no—
383C3×C4—44no—
384C3×C4—42no—
385C3×C4—82no—
386C3×C4—82no—
387C23C22×C3×D412yes—
388C23C22×C3×Q811yescharacteristic
389C23C2×C3×#4112yes—
390C23C2×C3×#4111yescharacteristic
391C23C2×C3×#4112yes—
392C23C2×C3×#4111yescharacteristic
393C23C3×H(5,𝔽2)11yescharacteristic
394C23C3×H(5,𝔽2)11yescharacteristic
395C23C3×2−1+411yescharacteristic
396C23C3×2−1+412yes—
397C23C3×2−1+411yescharacteristic
398Q8—416no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4C3×C22⋊Q818yes—
456C2×C3#243711yescharacteristic
457C2×C3#245312yes—
458C2×C3#246311yescharacteristic
459C2×C3#247412yes—
460C2×C3#252311yescharacteristic
461C2×C3#258112yes—
462C2×C3#268011yescharacteristic
463C2×C3#271612yes—
464C2×C3#278911yescharacteristic
465C2×C3#279911yescharacteristic
466C2×C3#280911yescharacteristic
467C22C2×C3×C22⋊Q812yes—
468C22C3×#52512yes—
469C22C3×#52912yes—
470C22C3×#53612yes—
471C22C3×#53911yescharacteristic
472C22C3×#53911yescharacteristic
473C22C3×#54112yes—
474C22C3×#54212yes—
475C22C3×#54311yescharacteristic
476C22C3×D4211yescharacteristic
477C22C3×#54612yes—
478C22C3×#54712yes—
479C22C3×#54812yes—
480C22C3×#55111yescharacteristic
481C22C3×#55111yescharacteristic
482C22C3×#55511yescharacteristic
483C22C3×#55612yes—
484C22C3×#55612yes—
485C22C3×#55711yescharacteristic
486C22C3×#55712yes—
487C22C3×#55711yescharacteristic
488C22C3×#56311yescharacteristic
489C22C3×#56311yescharacteristic
490C4—28no—
491C4—44no—
492C4—42no—
493C4—42no—
494C4—44no—
495C4—44no—
496C4—42no—
497C4—82no—
498C4—82no—
499C3#1596511yescharacteristic, direct product factor, Sylow
500C2C3×#243711yescharacteristic
501C2C3×#245312yes—
502C2C3×#246311yescharacteristic
503C2C3×#247412yes—
504C2C3×#252311yescharacteristic
505C2C3×#258112yes—
506C2C3×#268011yescharacteristic
507C2C3×#271612yes—
508C2C3×#278911yescharacteristic
509C2C3×#279911yescharacteristic
510C2C3×#280911yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
10. #15965C3direct (N × H)
499. C3#15965direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C2×C3×#3872
C3×#14462
C3×#14442
C3×#14432
C3×#14422
C3×#14412
C3×#14362
#159653