Finite Group Databasebeta

← C3×#14223 #10632830

Subgroups — C3×#14223 #10632830

Order 768.

1390 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#398C211yesmaximal, characteristic
2C2×C3×#387C212yesmaximal
3C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
4C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
5C3×#1445C211yesmaximal, characteristic
6C3×#1442C211yesmaximal, characteristic
7C3×#1441C211yesmaximal, characteristic
8C3×#1436C212yesmaximal
9C3×#1434C212yesmaximal
10C3×#1434C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#14223C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C3×#398C418yescomplemented
23C3×#387—28no—
24C3×C4×C4⋊C4—24no—
25C3×C4×C4⋊C4—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2211yescharacteristic
38C2×C3×#95C2212yes—
39C2×C3×#95C2211yescharacteristic
40C2×#398C2×C311yescharacteristic
41C2×#387C2×C312yes—
42C2×C4×C4⋊C4C2×C311yescharacteristic
43C2×C4×C4⋊C4C2×C311yescharacteristic
44#1445C2×C311yescharacteristic
45#1442C2×C311yescharacteristic
46#1441C2×C311yescharacteristic
47#1436C2×C312yes—
48#1434C2×C311yescharacteristic
49#1434C2×C312yes—
50#1434C2×C312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2×C418yes—
73C2×C3×C4⋊C4C2×C414yes—
74C2×C3×C4⋊C4C2×C414yes—
75C2×C3×C4⋊C4D414yes—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42—22no—
79C2×C3×C42—24no—
80C2×C3×C42—24no—
81C2×C3×C42—22no—
82C2×C3×C42—24no—
83C2×C3×C42—24no—
84C2×C3×C42—24no—
85C2×C3×C42D414yes—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—24no—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—24no—
97C3×#95—22no—
98C3×#95C2×C418yes—
99C3×#95C2×C414yes—
100C22×C4⋊C4C22×C311yescharacteristic
101C22×C4⋊C4C22×C312yes—
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C42C22×C311yescharacteristic
106C22×C42C22×C312yes—
107C22×C42C22×C311yescharacteristic
108C22×C42C22×C311yescharacteristic
109#398C3×C418yescomplemented
110#387—28no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—24no—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C311yescharacteristic
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C2×#95C22×C311yescharacteristic
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C311yescharacteristic
125C2×#95C22×C312yes—
126C2×#95C22×C311yescharacteristic
127C24×C3C2411yescharacteristic
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
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—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
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—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4C22×C412yes—
177C22×C3×C4C22×C414yes—
178C22×C3×C4C22×C414yes—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C22×C412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4114yes—
189C22×C3×C4#4112yes—
190C22×C3×C4#4112yes—
191C22×C3×C4#4112yes—
192C22×C3×C4#4114yes—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—216no—
197C3×C4⋊C4—28no—
198C3×C42—28no—
199C3×C42—28no—
200C3×C42—48no—
201C3×C42—44no—
202C23×C4C23×C312yes—
203C23×C4C23×C312yes—
204C23×C4C23×C312yes—
205C23×C4C23×C311yescharacteristic
206C23×C4C23×C311yescharacteristic
207C23×C4C23×C312yes—
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C311yescharacteristic
210C23×C4C23×C311yescharacteristic
211C23×C4C23×C311yescharacteristic
212C23×C4C23×C311yescharacteristic
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C2×C3×C418yes—
224C2×C4⋊C4C2×C3×C414yes—
225C2×C4⋊C4C2×C3×C414yes—
226C2×C4⋊C4C3×D414yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42C3×D414yes—
237#95—22no—
238#95—22no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95C2×C3×C418yes—
250#95C2×C3×C414yes—
251C23×C3C23×C411yescharacteristic
252C23×C3C22×D411yescharacteristic
253C23×C3C22×D411yescharacteristic
254C23×C3C2×#4111yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4111yescharacteristic
257C23×C3C2×#4112yes—
258C23×C3C2×#4112yes—
259C23×C3C2×#4111yescharacteristic
260C23×C32−1+412yes—
261C23×C32−1+412yes—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4C4×D414yes—
324C2×C3×C4C4×D414yes—
325C2×C3×C4#12114yes—
326C2×C3×C4#12314yes—
327C24C24×C311yescharacteristic
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4C22×C3×C412yes—
377C22×C4C22×C3×C414yes—
378C22×C4C22×C3×C414yes—
379C22×C4C22×C3×C412yes—
380C22×C4C22×C3×C412yes—
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D412yes—
385C22×C4C2×C3×D412yes—
386C22×C4C2×C3×D412yes—
387C22×C4C3×#4112yes—
388C22×C4C3×#4114yes—
389C22×C4C3×#4112yes—
390C22×C4C3×#4112yes—
391C22×C4C3×#4112yes—
392C22×C4C3×#4114yes—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—216no—
397C4⋊C4—28no—
398C42—28no—
399C42—28no—
400C42—48no—
401C42—44no—
402C22×C3C2×C4×D411yescharacteristic
403C22×C3C2×C4×D411yescharacteristic
404C22×C3C4×#4112yes—
405C22×C3C4×#4111yescharacteristic
406C22×C3#51912yes—
407C22×C3C2×#12111yescharacteristic
408C22×C3C2×#12311yescharacteristic
409C22×C3#52511yescharacteristic
410C22×C3#52511yescharacteristic
411C22×C3#52912yes—
412C22×C3#52911yescharacteristic
413C22×C3#53211yescharacteristic
414C22×C3#53611yescharacteristic
415C22×C3#53611yescharacteristic
416C22×C3#54112yes—
417C22×C3#54712yes—
418C22×C3#54812yes—
419C22×C3#55212yes—
420C22×C3#55212yes—
421C22×C3#55212yes—
422C22×C3#55212yes—
423C22×C3#55212yes—
424C22×C3#55612yes—
425C3×C4—24no—
426C3×C4—24no—
427C3×C4—24no—
428C3×C4—24no—
429C3×C4—42no—
430C3×C4—44no—
431C3×C4—42no—
432C3×C4—44no—
433C3×C4—44no—
434C3×C4—42no—
435C3×C4—82no—
436C23C23×C3×C411yescharacteristic
437C23C22×C3×D411yescharacteristic
438C23C22×C3×D411yescharacteristic
439C23C2×C3×#4111yescharacteristic
440C23C2×C3×#4111yescharacteristic
441C23C2×C3×#4111yescharacteristic
442C23C2×C3×#4112yes—
443C23C2×C3×#4112yes—
444C23C2×C3×#4111yescharacteristic
445C23C3×2−1+412yes—
446C23C3×2−1+412yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4C3×C4×D414yes—
509C2×C4C3×C4×D414yes—
510C2×C4C3×#12114yes—
511C2×C4C3×#12314yes—
512C2×C3C4×#12111yescharacteristic
513C2×C3C4×#12311yescharacteristic
514C2×C3#232711yescharacteristic
515C2×C3#236212yes—
516C2×C3#237112yes—
517C2×C3#248411yescharacteristic
518C2×C3#248611yescharacteristic
519C2×C3#251912yes—
520C2×C3#252412yes—
521C2×C3#253611yescharacteristic
522C2×C3#259911yescharacteristic
523C22C2×C3×C4×D411yescharacteristic
524C22C2×C3×C4×D411yescharacteristic
525C22C3×C4×#4112yes—
526C22C3×C4×#4111yescharacteristic
527C22C3×#51912yes—
528C22C2×C3×#12111yescharacteristic
529C22C2×C3×#12311yescharacteristic
530C22C3×#52511yescharacteristic
531C22C3×#52511yescharacteristic
532C22C3×#52912yes—
533C22C3×#52911yescharacteristic
534C22C3×#53211yescharacteristic
535C22C3×#53611yescharacteristic
536C22C3×#53611yescharacteristic
537C22C3×#54112yes—
538C22C3×#54712yes—
539C22C3×#54812yes—
540C22C3×#55212yes—
541C22C3×#55212yes—
542C22C3×#55212yes—
543C22C3×#55212yes—
544C22C3×#55212yes—
545C22C3×#55612yes—
546C4—24no—
547C4—24no—
548C4—24no—
549C4—24no—
550C4—44no—
551C4—42no—
552C4—44no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—82no—
557C3#1422311yescharacteristic, direct product factor, Sylow
558C2C3×C4×#12111yescharacteristic
559C2C3×C4×#12311yescharacteristic
560C2C3×#232711yescharacteristic
561C2C3×#236212yes—
562C2C3×#237112yes—
563C2C3×#248411yescharacteristic
564C2C3×#248611yescharacteristic
565C2C3×#251912yes—
566C2C3×#252412yes—
567C2C3×#253611yescharacteristic
568C2C3×#259911yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
12. #14223C3direct (N × H)
22. C3×#398C4semidirect
109. #398C3×C4semidirect
557. C3#14223direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3982
C2×C3×#3872
C2×C3×C4×C4⋊C42
C3×#14452
C3×#14422
C3×#14412
C3×#14362
C3×#14342
#142233