Finite Group Databasebeta

← C3×#15084 #10633691

Subgroups — C3×#15084 #10633691

Order 768.

1878 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C212yesmaximal, complemented
2C2×C3×#390C211yesmaximal, characteristic
3C2×C3×#388C214yesmaximal, complemented
4C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#1444C212yesmaximal, complemented
6C3×#1439C212yesmaximal, complemented
7C3×#1438C212yesmaximal
8C3×#1434C211yesmaximal, characteristic, complemented
9#15084C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2214yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2214yes—
18C22×C3×C22⋊C4C2214yes—
19C22×C3×C42C2212yes—
20C22×C3×C42C2211yescharacteristic
21C3×#399—28no—
22C3×#390—24no—
23C3×#388—216no—
24C3×C4×C4⋊C4—24no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2214yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×#399C2×C312yescomplemented
31C2×#390C2×C311yescharacteristic
32C2×#388C2×C314yescomplemented
33C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
34#1444C2×C312yescomplemented
35#1439C2×C312yescomplemented
36#1438C2×C312yes—
37#1434C2×C311yescharacteristic, complemented
38C25×C3C2312yes—
39C22×C3×D4—24no—
40C22×C3×D4—22no—
41C22×C3×D4—24no—
42C22×C3×D4—42no—
43C23×C3×C4C2312yes—
44C23×C3×C4C2314yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—28no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—28no—
57C2×C3×C4⋊C4—44no—
58C2×C3×C4⋊C4—42no—
59C2×C3×C4⋊C4D414yescomplemented
60C2×C3×C22⋊C4—28no—
61C2×C3×C22⋊C4—28no—
62C2×C3×C22⋊C4—28no—
63C2×C3×C22⋊C4—28no—
64C2×C3×C22⋊C4—28no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—28no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—44no—
71C2×C3×C42—24no—
72C2×C3×C42—24no—
73C2×C3×C42—28no—
74C2×C3×C42D414yes—
75C3×#95—24no—
76C3×#95—28no—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95D418yescomplemented
80C23×D4C22×C311yescharacteristic
81C22×C4⋊C4C22×C311yescharacteristic
82C22×C4⋊C4C22×C314yes—
83C22×C4⋊C4C22×C311yescharacteristic
84C22×C4⋊C4C22×C311yescharacteristic
85C22×C22⋊C4C22×C312yes—
86C22×C22⋊C4C22×C314yes—
87C22×C22⋊C4C22×C312yes—
88C22×C22⋊C4C22×C314yes—
89C22×C42C22×C312yes—
90C22×C42C22×C311yescharacteristic
91#399—28no—
92#390—24no—
93#388—216no—
94C4×C4⋊C4—24no—
95C2×#95C22×C312yes—
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C314yes—
99C2×#95C22×C312yes—
100C24×C3—24no—
101C24×C3—22no—
102C24×C3—22no—
103C24×C3—24no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—24no—
107C24×C3—24no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3#4114yes—
111C24×C3C2411yescharacteristic
112C2×C3×D4—28no—
113C2×C3×D4—416no—
114C2×C3×D4—44no—
115C2×C3×D4—44no—
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—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—24no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4C2×D418yes—
142C22×C3×C4C2×D414yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D414yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4#4114yes—
149C22×C3×C4#4112yes—
150C22×C3×C4#4114yes—
151C3×C4⋊C4—28no—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—416no—
154C3×C4⋊C4—44no—
155C3×C22⋊C4—232no—
156C3×C22⋊C4—416no—
157C3×C22⋊C4—48no—
158C3×C22⋊C4—48no—
159C3×C42—28no—
160C3×C42—48no—
161C25C23×C312yes—
162C22×D4—24no—
163C22×D4—22no—
164C22×D4—24no—
165C22×D4—42no—
166C23×C4C23×C312yes—
167C23×C4C23×C312yes—
168C23×C4C23×C314yes—
169C23×C4C23×C312yes—
170C23×C4C23×C311yescharacteristic
171C23×C4C23×C311yescharacteristic
172C23×C4C23×C311yescharacteristic
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C3×D414yescomplemented
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—44no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—28no—
197C2×C42C3×D414yes—
198#95—24no—
199#95—28no—
200#95—24no—
201#95—24no—
202#95C3×D418yescomplemented
203C23×C3—28no—
204C23×C3—24no—
205C23×C3—24no—
206C23×C3—28no—
207C23×C3—24no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—44no—
212C23×C3—44no—
213C23×C3—42no—
214C23×C3—44no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—44no—
218C23×C3—44no—
219C23×C3—44no—
220C23×C3—44no—
221C23×C3—44no—
222C23×C3—42no—
223C23×C3—44no—
224C23×C3—44no—
225C23×C3—42no—
226C23×C3C22×D411yescharacteristic
227C23×C3C22×D411yescharacteristic
228C23×C3C22×D412yes—
229C23×C3C2×#4111yescharacteristic
230C23×C3C2×#4112yes—
231C23×C3C2×#4112yes—
232C23×C3C2×#4112yes—
233C23×C3H(5,𝔽2)12yes—
234C23×C32−1+412yes—
235C3×D4—416no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—28no—
245C2×C3×C4—28no—
246C2×C3×C4—24no—
247C2×C3×C4—28no—
248C2×C3×C4—28no—
249C2×C3×C4—24no—
250C2×C3×C4—28no—
251C2×C3×C4—28no—
252C2×C3×C4—28no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—42no—
259C2×C3×C4—44no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4—44no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—44no—
267C2×C3×C4—44no—
268C2×C3×C4#12114yes—
269C2×C3×C4#12314yes—
270C2×C3×C4#12714yes—
271C24—24no—
272C24—22no—
273C24—22no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24C3×#4114yes—
282C24C24×C311yescharacteristic
283C2×D4—28no—
284C2×D4—416no—
285C2×D4—44no—
286C2×D4—44no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4C2×C3×D418yes—
313C22×C4C2×C3×D414yes—
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D412yes—
317C22×C4C2×C3×D414yes—
318C22×C4C2×C3×D412yes—
319C22×C4C3×#4114yes—
320C22×C4C3×#4112yes—
321C22×C4C3×#4114yes—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—416no—
325C4⋊C4—44no—
326C22⋊C4—232no—
327C22⋊C4—416no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C42—28no—
331C42—48no—
332C22×C3—44no—
333C22×C3—48no—
334C22×C3—48no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—82no—
338C22×C3—82no—
339C22×C3—84no—
340C22×C3—84no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3C2×#12111yescharacteristic
344C22×C3C2×#12311yescharacteristic
345C22×C3#52511yescharacteristic
346C22×C3#52514yes—
347C22×C3#52511yescharacteristic
348C22×C3#52912yes—
349C22×C3C2×#12711yescharacteristic
350C22×C3#53212yes—
351C22×C3#53511yescharacteristic
352C22×C3#53611yescharacteristic
353C22×C3#54214yes—
354C22×C3D4212yes—
355C22×C3#54714yes—
356C22×C3#54812yes—
357C22×C3#55012yes—
358C22×C3#55114yes—
359C22×C3#55212yes—
360C3×C4—24no—
361C3×C4—24no—
362C3×C4—24no—
363C3×C4—44no—
364C3×C4—48no—
365C3×C4—44no—
366C3×C4—82no—
367C23—28no—
368C23—24no—
369C23—24no—
370C23—28no—
371C23—24no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—42no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—42no—
387C23—44no—
388C23—44no—
389C23—42no—
390C23C22×C3×D411yescharacteristic
391C23C22×C3×D411yescharacteristic
392C23C22×C3×D412yes—
393C23C2×C3×#4111yescharacteristic
394C23C2×C3×#4112yes—
395C23C2×C3×#4112yes—
396C23C2×C3×#4112yes—
397C23C3×H(5,𝔽2)12yes—
398C23C3×2−1+412yes—
399D4—416no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4C3×#12114yes—
433C2×C4C3×#12314yes—
434C2×C4C3×#12714yes—
435C2×C3—84no—
436C2×C3#239711yescharacteristic
437C2×C3#246114yes—
438C2×C3#247914yes—
439C2×C3#248211yescharacteristic
440C2×C3#250111yescharacteristic
441C2×C3#253611yescharacteristic
442C2×C3#254111yescharacteristic
443C2×C3#263211yescharacteristic
444C2×C3#285411yescharacteristic
445C22—44no—
446C22—48no—
447C22—48no—
448C22—44no—
449C22—44no—
450C22—82no—
451C22—84no—
452C22—84no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22C2×C3×#12111yescharacteristic
457C22C2×C3×#12311yescharacteristic
458C22C3×#52511yescharacteristic
459C22C3×#52514yes—
460C22C3×#52511yescharacteristic
461C22C3×#52912yes—
462C22C2×C3×#12711yescharacteristic
463C22C3×#53212yes—
464C22C3×#53511yescharacteristic
465C22C3×#53611yescharacteristic
466C22C3×#54214yes—
467C22C3×D4212yes—
468C22C3×#54714yes—
469C22C3×#54812yes—
470C22C3×#55012yes—
471C22C3×#55114yes—
472C22C3×#55212yes—
473C4—24no—
474C4—24no—
475C4—24no—
476C4—44no—
477C4—48no—
478C4—44no—
479C4—82no—
480C3#1508411yescharacteristic, direct product factor, Sylow
481C2—84no—
482C2C3×#239711yescharacteristic
483C2C3×#246114yes—
484C2C3×#247914yes—
485C2C3×#248211yescharacteristic
486C2C3×#250111yescharacteristic
487C2C3×#253611yescharacteristic
488C2C3×#254111yescharacteristic
489C2C3×#263211yescharacteristic
490C2C3×#285411yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#399C2semidirect
3. C2×C3×#388C2semidirect
4. C2×C3×C4×C4⋊C4C2semidirect
5. C3×#1444C2semidirect
6. C3×#1439C2semidirect
8. C3×#1434C2semidirect
9. #15084C3direct (N × H)
30. C2×#399C2×C3semidirect
32. C2×#388C2×C3semidirect
33. C2×C4×C4⋊C4C2×C3semidirect
34. #1444C2×C3semidirect
35. #1439C2×C3semidirect
37. #1434C2×C3semidirect
59. C2×C3×C4⋊C4D4semidirect
79. C3×#95D4semidirect
182. C2×C4⋊C4C3×D4semidirect
202. #95C3×D4semidirect
480. C3#15084direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3902
C2×C3×#3882
C2×C3×C4×C4⋊C42
C3×#14442
C3×#14392
C3×#14382
C3×#14342
#150843