Finite Group Databasebeta

← C3×#14024 #10632631

Subgroups — C3×#14024 #10632631

Order 768.

2054 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C22×C3×C4×Q8C211yesmaximal, characteristic
3C2×C3×#391C211yesmaximal, characteristic
4C2×C3×#391C211yesmaximal, characteristic
5C2×C3×#391C211yesmaximal, characteristic
6C3×#1441C214yesmaximal
7C3×#1436C214yesmaximal
8C3×C4⋊#95C211yesmaximal, characteristic
9C3×C4×#95C211yesmaximal, characteristic
10#14024C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C2×C3×C4⋊Q8—24no—
14C2×C3×C4⋊Q8—24no—
15C2×C3×C4⋊Q8C418yescomplemented
16C2×C3×C4×Q8—24no—
17C2×C3×C4×Q8—24no—
18C2×C3×C4×Q8—24no—
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2214yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C42C2212yes—
24C22×C3×C42C2211yescharacteristic
25C22×C3×C42C2211yescharacteristic
26C22×C3×C42C2211yescharacteristic
27C3×#391—24no—
28C3×#391—24no—
29C3×#391—24no—
30C2×C3×#95C2212yes—
31C2×C3×#95C2214yes—
32C2×C3×#95C2214yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2214yes—
36C2×C3×#95C2212yes—
37C22×C4⋊Q8C2×C311yescharacteristic
38C22×C4×Q8C2×C311yescharacteristic
39C2×#391C2×C311yescharacteristic
40C2×#391C2×C311yescharacteristic
41C2×#391C2×C311yescharacteristic
42#1441C2×C314yes—
43#1436C2×C314yes—
44C4⋊#95C2×C311yescharacteristic
45C4×#95C2×C311yescharacteristic
46C22×C3×Q8—22no—
47C22×C3×Q8—22no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8C2×C414yes—
57C22×C3×Q8C2×C414yes—
58C22×C3×Q8D414yes—
59C22×C3×Q8D414yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C3×C4⋊Q8—232no—
70C3×C4×Q8—232no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—28no—
77C2×C3×C4⋊C4—28no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2×C4116yes—
82C2×C3×C4⋊C4D414yes—
83C2×C3×C42—24no—
84C2×C3×C42—24no—
85C2×C3×C42—24no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42—22no—
91C2×C3×C42—22no—
92C2×C3×C42—22no—
93C2×C3×C42C2×C414yes—
94C2×C3×C42D414yes—
95C3×#95—28no—
96C3×#95—28no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—28no—
101C3×#95Q818yescomplemented
102C23×Q8C22×C311yescharacteristic
103C23×Q8C22×C311yescharacteristic
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8C3×C418yescomplemented
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C22×C4⋊C4C22×C312yes—
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C4⋊C4C22×C314yes—
113C22×C4⋊C4C22×C311yescharacteristic
114C22×C42C22×C312yes—
115C22×C42C22×C311yescharacteristic
116C22×C42C22×C311yescharacteristic
117C22×C42C22×C311yescharacteristic
118#391—24no—
119#391—24no—
120#391—24no—
121C2×#95C22×C312yes—
122C2×#95C22×C314yes—
123C2×#95C22×C312yes—
124C2×#95C22×C314yes—
125C2×#95C22×C314yes—
126C2×#95C22×C312yes—
127C2×#95C22×C312yes—
128C24×C3C2411yescharacteristic
129C2×C3×Q8—28no—
130C2×C3×Q8—28no—
131C2×C3×Q8—28no—
132C2×C3×Q8—28no—
133C2×C3×Q8—28no—
134C2×C3×Q8—28no—
135C2×C3×Q8—28no—
136C2×C3×Q8—28no—
137C2×C3×Q8—28no—
138C2×C3×Q8—44no—
139C2×C3×Q8—44no—
140C2×C3×Q8—44no—
141C2×C3×Q8—44no—
142C2×C3×Q8C22⋊C4116yes—
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—21no—
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—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4C22×C414yes—
180C22×C3×C4C22×C414yes—
181C22×C3×C4C22×C414yes—
182C22×C3×C4C22×C412yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×Q814yes—
193C22×C3×C4C2×Q814yes—
194C22×C3×C4C2×Q814yes—
195C22×C3×C4#4114yes—
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C3×C4⋊C4—216no—
199C3×C4⋊C4—232no—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—44no—
202C3×C42—216no—
203C3×C42—28no—
204C3×C42—28no—
205C3×C42—44no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8C2×C3×C414yes—
217C22×Q8C2×C3×C414yes—
218C22×Q8C3×D414yes—
219C22×Q8C3×D414yes—
220C23×C4C23×C312yes—
221C23×C4C23×C312yes—
222C23×C4C23×C312yes—
223C23×C4C23×C311yescharacteristic
224C23×C4C23×C312yes—
225C23×C4C23×C312yes—
226C23×C4C23×C312yes—
227C23×C4C23×C311yescharacteristic
228C23×C4C23×C311yescharacteristic
229C4⋊Q8—232no—
230C4×Q8—232no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C2×C3×C4116yes—
242C2×C4⋊C4C3×D414yes—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42C2×C3×C414yes—
254C2×C42C3×D414yes—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95C3×Q818yescomplemented
262C23×C3C23×C411yescharacteristic
263C23×C3C22×D411yescharacteristic
264C23×C3C22×D411yescharacteristic
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D411yescharacteristic
267C23×C3C22×Q812yes—
268C23×C3C2×#4111yescharacteristic
269C23×C3C2×#4111yescharacteristic
270C23×C3C2×#4112yes—
271C23×C32−1+412yes—
272C23×C32−1+412yes—
273C3×Q8—232no—
274C3×Q8—416no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—28no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
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—44no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4C2×C22⋊C418yes—
321C2×C3×C4C2×C22⋊C414yes—
322C2×C3×C4C4×D414yes—
323C2×C3×C4C4×D414yes—
324C2×C3×C4C4×Q818yescomplemented
325C2×C3×C4#12014yes—
326C2×C3×C4#12114yes—
327C2×C3×C4#12114yes—
328C2×C3×C4#12314yes—
329C2×C3×C4C4⋊Q814yescomplemented
330C24C24×C311yescharacteristic
331C2×Q8—28no—
332C2×Q8—28no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8C3×C22⋊C4116yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4C22×C3×C414yes—
382C22×C4C22×C3×C414yes—
383C22×C4C22×C3×C414yes—
384C22×C4C22×C3×C412yes—
385C22×C4C2×C3×D414yes—
386C22×C4C2×C3×D412yes—
387C22×C4C2×C3×D414yes—
388C22×C4C2×C3×D414yes—
389C22×C4C2×C3×D412yes—
390C22×C4C2×C3×D412yes—
391C22×C4C2×C3×D412yes—
392C22×C4C2×C3×D412yes—
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×Q814yes—
395C22×C4C2×C3×Q814yes—
396C22×C4C2×C3×Q814yes—
397C22×C4C3×#4114yes—
398C22×C4C3×#4112yes—
399C22×C4C3×#4112yes—
400C4⋊C4—216no—
401C4⋊C4—232no—
402C4⋊C4—28no—
403C4⋊C4—44no—
404C42—216no—
405C42—28no—
406C42—28no—
407C42—44no—
408C22×C3C22×C22⋊C411yescharacteristic
409C22×C3C2×C4×D411yescharacteristic
410C22×C3C2×C4×D411yescharacteristic
411C22×C3C2×C4×Q812yes—
412C22×C3#51912yes—
413C22×C3C2×#12011yescharacteristic
414C22×C3C2×#12111yescharacteristic
415C22×C3C2×#12111yescharacteristic
416C22×C3C2×#12311yescharacteristic
417C22×C3C2×C4⋊Q811yescharacteristic
418C22×C3#53211yescharacteristic
419C22×C3#53312yes—
420C22×C3#53611yescharacteristic
421C22×C3#53611yescharacteristic
422C22×C3#54112yes—
423C22×C3#54712yes—
424C22×C3#54812yes—
425C22×C3#54812yes—
426C22×C3D4×Q812yes—
427C22×C3D4×Q812yes—
428C22×C3D4×Q812yes—
429C22×C3#55612yes—
430C22×C3#55712yes—
431C3×C4—28no—
432C3×C4—24no—
433C3×C4—28no—
434C3×C4—44no—
435C3×C4—42no—
436C3×C4—44no—
437C3×C4—44no—
438C3×C4—82no—
439C3×C4#38618yes—
440C23C23×C3×C411yescharacteristic
441C23C22×C3×D411yescharacteristic
442C23C22×C3×D411yescharacteristic
443C23C22×C3×D411yescharacteristic
444C23C22×C3×D411yescharacteristic
445C23C22×C3×Q812yes—
446C23C2×C3×#4111yescharacteristic
447C23C2×C3×#4111yescharacteristic
448C23C2×C3×#4112yes—
449C23C3×2−1+412yes—
450C23C3×2−1+412yes—
451Q8—232no—
452Q8—416no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4C2×C3×C22⋊C418yes—
499C2×C4C2×C3×C22⋊C414yes—
500C2×C4C3×C4×D414yes—
501C2×C4C3×C4×D414yes—
502C2×C4C3×C4×Q818yescomplemented
503C2×C4C3×#12014yes—
504C2×C4C3×#12114yes—
505C2×C4C3×#12114yes—
506C2×C4C3×#12314yes—
507C2×C4C3×C4⋊Q814yescomplemented
508C2×C3C2×#38611yescharacteristic
509C2×C3C4×C4⋊Q811yescharacteristic
510C2×C3#232711yescharacteristic
511C2×C3Q8×C22⋊C412yes—
512C2×C3#236512yes—
513C2×C3#242912yes—
514C2×C3#243211yescharacteristic
515C2×C3#243411yescharacteristic
516C2×C3#244612yes—
517C2×C3#254511yescharacteristic
518C2×C3#254611yescharacteristic
519C22C22×C3×C22⋊C411yescharacteristic
520C22C2×C3×C4×D411yescharacteristic
521C22C2×C3×C4×D411yescharacteristic
522C22C2×C3×C4×Q812yes—
523C22C3×#51912yes—
524C22C2×C3×#12011yescharacteristic
525C22C2×C3×#12111yescharacteristic
526C22C2×C3×#12111yescharacteristic
527C22C2×C3×#12311yescharacteristic
528C22C2×C3×C4⋊Q811yescharacteristic
529C22C3×#53211yescharacteristic
530C22C3×#53312yes—
531C22C3×#53611yescharacteristic
532C22C3×#53611yescharacteristic
533C22C3×#54112yes—
534C22C3×#54712yes—
535C22C3×#54812yes—
536C22C3×#54812yes—
537C22C3×D4×Q812yes—
538C22C3×D4×Q812yes—
539C22C3×D4×Q812yes—
540C22C3×#55612yes—
541C22C3×#55712yes—
542C4—24no—
543C4—28no—
544C4—28no—
545C4—44no—
546C4—42no—
547C4—44no—
548C4—44no—
549C4—82no—
550C4C3×#38618yes—
551C3#1402411yescharacteristic, direct product factor, Sylow
552C2C2×C3×#38611yescharacteristic
553C2C3×C4×C4⋊Q811yescharacteristic
554C2C3×#232711yescharacteristic
555C2C3×Q8×C22⋊C412yes—
556C2C3×#236512yes—
557C2C3×#242912yes—
558C2C3×#243211yescharacteristic
559C2C3×#243411yescharacteristic
560C2C3×#244612yes—
561C2C3×#254511yescharacteristic
562C2C3×#254611yescharacteristic
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
10. #14024C3direct (N × H)
15. C2×C3×C4⋊Q8C4semidirect
101. C3×#95Q8semidirect
106. C2×C4⋊Q8C3×C4semidirect
261. #95C3×Q8semidirect
324. C2×C3×C4C4×Q8semidirect
329. C2×C3×C4C4⋊Q8semidirect
502. C2×C4C3×C4×Q8semidirect
507. C2×C4C3×C4⋊Q8semidirect
551. C3#14024direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C22×C3×C4×Q82
C2×C3×#3912
C3×#14412
C3×#14362
C3×C4⋊#952
C3×C4×#952
#140243