Finite Group Databasebeta

← C3×#14137 #10632744

Subgroups — C3×#14137 #10632744

Order 768.

3158 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#390C212yesmaximal, complemented
4C2×C3×#386C214yesmaximal, complemented
5C2×C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#1439C214yesmaximal, complemented
7C3×C4⋊#95C212yesmaximal, complemented
8#14137C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2212yes—
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4C2212yes—
12C24×C3×C4C2211yescharacteristic
13C2×C3×#121—24no—
14C2×C3×#121—24no—
15C2×C3×#121C418yescomplemented
16C22×C3×C4⋊C4—24no—
17C22×C3×C4⋊C4—28no—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C22⋊C4C2218yes—
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C22⋊C4C2212yes—
24C22×C3×C42C2212yes—
25C3×#390—28no—
26C3×#386—216no—
27C3×C4×C22⋊C4—24no—
28C2×C3×#95C2212yes—
29C2×C3×#95C2218yes—
30C22×#121C2×C311yescharacteristic
31C23×C4⋊C4C2×C311yescharacteristic, complemented
32C2×#390C2×C312yescomplemented
33C2×#386C2×C314yescomplemented
34C2×C4×C22⋊C4C2×C311yescharacteristic, complemented
35#1439C2×C314yescomplemented
36C4⋊#95C2×C312yescomplemented
37C25×C3C2312yes—
38C25×C3C2311yescharacteristic
39C22×C3×D4—28no—
40C22×C3×D4—24no—
41C22×C3×D4—24no—
42C22×C3×D4—28no—
43C22×C3×D4—24no—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—24no—
47C22×C3×D4C2×C418yes—
48C22×C3×D4C2×C414yescomplemented
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—28no—
53C23×C3×C4—28no—
54C23×C3×C4—44no—
55C23×C3×C4C2×C414yes—
56C23×C3×C4D414yes—
57C23×C3×C4D414yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2314yes—
60C23×C3×C4C2314yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C3×#121—232no—
64C2×C3×C4⋊C4—232no—
65C2×C3×C4⋊C4—28no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2×C414yescomplemented
72C2×C3×C4⋊C4D418yescomplemented
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—216no—
78C2×C3×C22⋊C4—216no—
79C2×C3×C22⋊C4—48no—
80C2×C3×C22⋊C4C2×C418yescomplemented
81C2×C3×C22⋊C4D418yescomplemented
82C2×C3×C42—24no—
83C2×C3×C42—24no—
84C2×C3×C42—24no—
85C3×#95—24no—
86C3×#95—216no—
87C23×D4C22×C312yes—
88C23×D4C22×C311yescharacteristic
89C24×C4C22×C312yes—
90C24×C4C22×C311yescharacteristic
91C2×#121—24no—
92C2×#121—24no—
93C2×#121C3×C418yescomplemented
94C22×C4⋊C4—24no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4C22×C312yes—
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C312yes—
99C22×C22⋊C4C22×C318yes—
100C22×C22⋊C4C22×C312yes—
101C22×C22⋊C4C22×C312yes—
102C22×C42C22×C312yes—
103#390—28no—
104#386—216no—
105C4×C22⋊C4—24no—
106C2×#95C22×C318yes—
107C2×#95C22×C312yes—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—24no—
113C24×C3—24no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—24no—
117C24×C3—24no—
118C24×C3—24no—
119C24×C3—24no—
120C24×C3C22×C414yes—
121C24×C3C22×C412yes—
122C24×C3C2×D412yes—
123C24×C3C2×D412yes—
124C24×C3C2×D414yes—
125C24×C3#4114yes—
126C24×C3C2411yescharacteristic
127C2×C3×D4—216no—
128C2×C3×D4—232no—
129C2×C3×D4—28no—
130C2×C3×D4—216no—
131C2×C3×D4—416no—
132C2×C3×D4—48no—
133C2×C3×D4—48no—
134C2×C3×D4—44no—
135C2×C3×D4—48no—
136C2×C3×D4—44no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—216no—
140C22×C3×C4—28no—
141C22×C3×C4—216no—
142C22×C3×C4—28no—
143C22×C3×C4—216no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—28no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—28no—
157C22×C3×C4—24no—
158C22×C3×C4—432no—
159C22×C3×C4—48no—
160C22×C3×C4—48no—
161C22×C3×C4C22⋊C4116yescomplemented
162C22×C3×C4C22×C414yes—
163C22×C3×C4C22×C412yes—
164C22×C3×C4C22×C412yes—
165C22×C3×C4C2×D418yes—
166C22×C3×C4C2×D418yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4#4114yes—
173C3×C4⋊C4—216no—
174C3×C4⋊C4—28no—
175C3×C4⋊C4—216no—
176C3×C22⋊C4—216no—
177C3×C22⋊C4—216no—
178C3×C22⋊C4—432no—
179C3×C42—48no—
180C25C23×C312yes—
181C25C23×C311yescharacteristic
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4C2×C3×C418yes—
191C22×D4C2×C3×C414yescomplemented
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—44no—
198C23×C4C2×C3×C414yes—
199C23×C4C3×D414yes—
200C23×C4C3×D414yes—
201C23×C4C23×C312yes—
202C23×C4C23×C314yes—
203C23×C4C23×C314yes—
204C23×C4C23×C311yescharacteristic
205C23×C4C23×C311yescharacteristic
206#121—232no—
207C2×C4⋊C4—232no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4C2×C3×C414yescomplemented
215C2×C4⋊C4C3×D418yescomplemented
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—216no—
221C2×C22⋊C4—216no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4C2×C3×C418yescomplemented
224C2×C22⋊C4C3×D418yescomplemented
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228#95—24no—
229#95—216no—
230C23×C3—24no—
231C23×C3—24no—
232C23×C3—22no—
233C23×C3—24no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—24no—
238C23×C3—28no—
239C23×C3—28no—
240C23×C3—28no—
241C23×C3—28no—
242C23×C3—24no—
243C23×C3—28no—
244C23×C3—28no—
245C23×C3—28no—
246C23×C3—42no—
247C23×C3—44no—
248C23×C3—44no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—44no—
255C23×C3—48no—
256C23×C3—48no—
257C23×C3—48no—
258C23×C3—48no—
259C23×C3C2×C22⋊C414yes—
260C23×C3C4×D418yes—
261C23×C3#12118yes—
262C23×C3#12414yes—
263C23×C3#12714yes—
264C23×C3C23×C411yescharacteristic
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D411yescharacteristic
267C23×C3C22×D412yes—
268C23×C3C22×D412yes—
269C23×C3C2×#4112yes—
270C23×C3C2×#4112yes—
271C23×C3H(5,𝔽2)12yes—
272C23×C32−1+412yes—
273C3×D4—432no—
274C3×D4—416no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—232no—
280C2×C3×C4—24no—
281C2×C3×C4—28no—
282C2×C3×C4—28no—
283C2×C3×C4—28no—
284C2×C3×C4—28no—
285C2×C3×C4—216no—
286C2×C3×C4—28no—
287C2×C3×C4—432no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—42no—
291C2×C3×C4—44no—
292C2×C3×C4—42no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4—48no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4C2×C22⋊C414yes—
299C2×C3×C4C2×C22⋊C414yes—
300C2×C3×C4C4×D418yes—
301C2×C3×C4#12014yes—
302C2×C3×C4#12118yes—
303C2×C3×C4#12314yes—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24C22×C3×C414yes—
317C24C22×C3×C412yes—
318C24C2×C3×D412yes—
319C24C2×C3×D412yes—
320C24C2×C3×D414yes—
321C24C3×#4114yes—
322C24C24×C311yescharacteristic
323C2×D4—216no—
324C2×D4—232no—
325C2×D4—28no—
326C2×D4—216no—
327C2×D4—416no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—44no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—216no—
336C22×C4—28no—
337C22×C4—216no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—432no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4C3×C22⋊C4116yescomplemented
358C22×C4C22×C3×C414yes—
359C22×C4C22×C3×C412yes—
360C22×C4C22×C3×C412yes—
361C22×C4C2×C3×D418yes—
362C22×C4C2×C3×D418yes—
363C22×C4C2×C3×D412yes—
364C22×C4C2×C3×D412yes—
365C22×C4C2×C3×D412yes—
366C22×C4C2×C3×D412yes—
367C22×C4C2×C3×D414yes—
368C22×C4C3×#4114yes—
369C4⋊C4—216no—
370C4⋊C4—28no—
371C4⋊C4—216no—
372C22⋊C4—216no—
373C22⋊C4—216no—
374C22⋊C4—432no—
375C42—48no—
376C22×C3—24no—
377C22×C3—216no—
378C22×C3—216no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—44no—
385C22×C3—416no—
386C22×C3—44no—
387C22×C3—44no—
388C22×C3—88no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3#39018yescomplemented
394C22×C3C22×C22⋊C411yescharacteristic
395C22×C3C2×C4×D412yes—
396C22×C3C2×C4×D412yes—
397C22×C3#52012yes—
398C22×C3C2×#12011yescharacteristic
399C22×C3C2×#12112yes—
400C22×C3C2×#12112yes—
401C22×C3C2×#12311yescharacteristic
402C22×C3C2×#12411yescharacteristic
403C22×C3C2×#12711yescharacteristic
404C22×C3#53511yescharacteristic
405C22×C3#53611yescharacteristic
406C22×C3#53712yes—
407C22×C3D4214yes—
408C22×C3#54714yes—
409C22×C3#54814yes—
410C22×C3#55314yes—
411C3×C4—24no—
412C3×C4—48no—
413C3×C4—48no—
414C3×C4—44no—
415C3×C4#38618yescomplemented
416C23—24no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—28no—
428C23—24no—
429C23—28no—
430C23—28no—
431C23—28no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—48no—
442C23—48no—
443C23—48no—
444C23—48no—
445C23C2×C3×C22⋊C414yes—
446C23C3×C4×D418yes—
447C23C3×#12118yes—
448C23C3×#12414yes—
449C23C3×#12714yes—
450C23C23×C3×C411yescharacteristic
451C23C22×C3×D411yescharacteristic
452C23C22×C3×D411yescharacteristic
453C23C22×C3×D412yes—
454C23C22×C3×D412yes—
455C23C2×C3×#4112yes—
456C23C2×C3×#4112yes—
457C23C3×H(5,𝔽2)12yes—
458C23C3×2−1+412yes—
459D4—432no—
460D4—416no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—232no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—216no—
472C2×C4—28no—
473C2×C4—432no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4C2×C3×C22⋊C414yes—
485C2×C4C2×C3×C22⋊C414yes—
486C2×C4C3×C4×D418yes—
487C2×C4C3×#12014yes—
488C2×C4C3×#12118yes—
489C2×C4C3×#12314yes—
490C2×C3—28no—
491C2×C3—84no—
492C2×C3C2×#38611yescharacteristic
493C2×C3C2×#39011yescharacteristic
494C2×C3#231111yescharacteristic
495C2×C3#236114yes—
496C2×C3#241614yes—
497C2×C3#243311yescharacteristic
498C2×C3#243611yescharacteristic
499C2×C3#250111yescharacteristic
500C2×C3#250411yescharacteristic
501C22—216no—
502C22—216no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—416no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—88no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C3×#39018yescomplemented
519C22C22×C3×C22⋊C411yescharacteristic
520C22C2×C3×C4×D412yes—
521C22C2×C3×C4×D412yes—
522C22C3×#52012yes—
523C22C2×C3×#12011yescharacteristic
524C22C2×C3×#12112yes—
525C22C2×C3×#12112yes—
526C22C2×C3×#12311yescharacteristic
527C22C2×C3×#12411yescharacteristic
528C22C2×C3×#12711yescharacteristic
529C22C3×#53511yescharacteristic
530C22C3×#53611yescharacteristic
531C22C3×#53712yes—
532C22C3×D4214yes—
533C22C3×#54714yes—
534C22C3×#54814yes—
535C22C3×#55314yes—
536C4—24no—
537C4—48no—
538C4—48no—
539C4—44no—
540C4C3×#38618yescomplemented
541C3#1413711yescharacteristic, direct product factor, Sylow
542C2—28no—
543C2—84no—
544C2C2×C3×#38611yescharacteristic
545C2C2×C3×#39011yescharacteristic
546C2C3×#231111yescharacteristic
547C2C3×#236114yes—
548C2C3×#241614yes—
549C2C3×#243311yescharacteristic
550C2C3×#243611yescharacteristic
551C2C3×#250111yescharacteristic
552C2C3×#250411yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C3×C4⋊C4C2semidirect
3. C2×C3×#390C2semidirect
4. C2×C3×#386C2semidirect
5. C2×C3×C4×C22⋊C4C2semidirect
6. C3×#1439C2semidirect
7. C3×C4⋊#95C2semidirect
8. #14137C3direct (N × H)
15. C2×C3×#121C4semidirect
31. C23×C4⋊C4C2×C3semidirect
32. C2×#390C2×C3semidirect
33. C2×#386C2×C3semidirect
34. C2×C4×C22⋊C4C2×C3semidirect
35. #1439C2×C3semidirect
36. C4⋊#95C2×C3semidirect
48. C22×C3×D4C2×C4semidirect
71. C2×C3×C4⋊C4C2×C4semidirect
72. C2×C3×C4⋊C4D4semidirect
80. C2×C3×C22⋊C4C2×C4semidirect
81. C2×C3×C22⋊C4D4semidirect
93. C2×#121C3×C4semidirect
161. C22×C3×C4C22⋊C4semidirect
191. C22×D4C2×C3×C4semidirect
214. C2×C4⋊C4C2×C3×C4semidirect
215. C2×C4⋊C4C3×D4semidirect
223. C2×C22⋊C4C2×C3×C4semidirect
224. C2×C22⋊C4C3×D4semidirect
357. C22×C4C3×C22⋊C4semidirect
393. C22×C3#390semidirect
415. C3×C4#386semidirect
518. C22C3×#390semidirect
540. C4C3×#386semidirect
541. C3#14137direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C23×C3×C4⋊C42
C2×C3×#3902
C2×C3×#3862
C2×C3×C4×C22⋊C42
C3×#14392
C3×C4⋊#952
#141373