Finite Group Databasebeta

← C3×#14186 #10632793

Subgroups — C3×#14186 #10632793

Order 768.

1786 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic
3C2×C3×#389C212yesmaximal
4C2×C3×#387C211yesmaximal, characteristic
5C3×#1446C211yesmaximal, characteristic
6C3×#1444C211yesmaximal, characteristic
7C3×#1441C211yesmaximal, characteristic
8C3×#1436C212yesmaximal
9C3×#1436C212yesmaximal
10C3×#1436C212yesmaximal
11C3×C4×#95C211yesmaximal, characteristic
12#14186C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×C4×Q8—28no—
15C2×C3×C4×Q8—24no—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C4⋊C4C2212yes—
22C22×C3×C42C2212yes—
23C22×C3×C42C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#395C418yescomplemented
26C3×#389—28no—
27C3×#387—24no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2211yescharacteristic
39C2×C3×#95C2212yes—
40C2×C3×#95C2212yes—
41C22×C4×Q8C2×C311yescharacteristic
42C2×#395C2×C311yescharacteristic
43C2×#389C2×C312yes—
44C2×#387C2×C311yescharacteristic
45#1446C2×C311yescharacteristic
46#1444C2×C311yescharacteristic
47#1441C2×C311yescharacteristic
48#1436C2×C312yes—
49#1436C2×C312yes—
50#1436C2×C312yes—
51C4×#95C2×C311yescharacteristic
52C22×C3×Q8—24no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—24no—
56C22×C3×Q8C2×C414yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C3×C4×Q8—232no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4—42no—
82C2×C3×C4⋊C4C2×C414yes—
83C2×C3×C4⋊C4C2×C418yes—
84C2×C3×C4⋊C4D418yes—
85C2×C3×C42—24no—
86C2×C3×C42—24no—
87C2×C3×C42—24no—
88C2×C3×C42—24no—
89C2×C3×C42—24no—
90C2×C3×C42—24no—
91C2×C3×C42—42no—
92C2×C3×C42D414yes—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—22no—
100C3×#95—24no—
101C3×#95—24no—
102C3×#95—42no—
103C3×#95C2×C418yes—
104C3×#95C2×C414yes—
105C3×#95Q818yescomplemented
106C23×Q8C22×C311yescharacteristic
107C2×C4×Q8—28no—
108C2×C4×Q8—24no—
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C4⋊C4C22×C312yes—
112C22×C4⋊C4C22×C312yes—
113C22×C4⋊C4C22×C311yescharacteristic
114C22×C4⋊C4C22×C312yes—
115C22×C42C22×C312yes—
116C22×C42C22×C312yes—
117C22×C42C22×C311yescharacteristic
118#395C3×C418yescomplemented
119#389—28no—
120#387—24no—
121C2×#95C22×C311yescharacteristic
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C311yescharacteristic
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C311yescharacteristic
127C2×#95C22×C312yes—
128C2×#95C22×C312yes—
129C2×#95C22×C311yescharacteristic
130C2×#95C22×C312yes—
131C2×#95C22×C311yescharacteristic
132C2×#95C22×C312yes—
133C2×#95C22×C312yes—
134C24×C3C2411yescharacteristic
135C2×C3×Q8—28no—
136C2×C3×Q8—28no—
137C2×C3×Q8—216no—
138C2×C3×Q8—28no—
139C2×C3×Q8—216no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C22×C412yes—
189C22×C3×C4C22×C412yes—
190C22×C3×C4C22×C414yes—
191C22×C3×C4C22×C412yes—
192C22×C3×C4C22×C414yes—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×Q814yes—
200C22×C3×C4C2×Q814yes—
201C22×C3×C4C2×Q814yes—
202C22×C3×C4#4112yes—
203C22×C3×C4#4114yes—
204C22×C3×C4#4114yes—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—216no—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—216no—
209C3×C4⋊C4—48no—
210C3×C4⋊C4—44no—
211C3×C42—216no—
212C3×C42—28no—
213C3×C42—48no—
214C22×Q8—24no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—24no—
218C22×Q8C2×C3×C414yes—
219C23×C4C23×C311yescharacteristic
220C23×C4C23×C311yescharacteristic
221C23×C4C23×C311yescharacteristic
222C23×C4C23×C311yescharacteristic
223C23×C4C23×C312yes—
224C23×C4C23×C312yes—
225C23×C4C23×C312yes—
226C23×C4C23×C311yescharacteristic
227C23×C4C23×C312yes—
228C23×C4C23×C311yescharacteristic
229C23×C4C23×C311yescharacteristic
230C4×Q8—232no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C2×C3×C414yes—
245C2×C4⋊C4C2×C3×C418yes—
246C2×C4⋊C4C3×D418yes—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42C3×D414yes—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—24no—
263#95—24no—
264#95—42no—
265#95C2×C3×C418yes—
266#95C2×C3×C414yes—
267#95C3×Q818yescomplemented
268C23×C3C23×C411yescharacteristic
269C23×C3C22×D411yescharacteristic
270C23×C3C22×D412yes—
271C23×C3C22×Q812yes—
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4112yes—
274C23×C3C2×#4112yes—
275C23×C3H(5,𝔽2)11yescharacteristic
276C23×C32−1+411yescharacteristic
277C23×C32−1+411yescharacteristic
278C23×C32−1+411yescharacteristic
279C3×Q8—232no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4C4×D414yes—
341C2×C3×C4C4×D418yes—
342C2×C3×C4C4×Q818yescomplemented
343C2×C3×C4#12014yes—
344C2×C3×C4C4⋊Q814yes—
345C24C24×C311yescharacteristic
346C2×Q8—28no—
347C2×Q8—216no—
348C2×Q8—28no—
349C2×Q8—216no—
350C2×Q8—28no—
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—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4C22×C3×C412yes—
400C22×C4C22×C3×C412yes—
401C22×C4C22×C3×C412yes—
402C22×C4C22×C3×C414yes—
403C22×C4C22×C3×C414yes—
404C22×C4C2×C3×D414yes—
405C22×C4C2×C3×D414yes—
406C22×C4C2×C3×D412yes—
407C22×C4C2×C3×D412yes—
408C22×C4C2×C3×D414yes—
409C22×C4C2×C3×D412yes—
410C22×C4C2×C3×Q814yes—
411C22×C4C2×C3×Q814yes—
412C22×C4C2×C3×Q814yes—
413C22×C4C3×#4112yes—
414C22×C4C3×#4114yes—
415C22×C4C3×#4114yes—
416C4⋊C4—28no—
417C4⋊C4—216no—
418C4⋊C4—28no—
419C4⋊C4—216no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C42—216no—
423C42—28no—
424C42—48no—
425C22×C3C2×C4×D411yescharacteristic
426C22×C3C2×C4×D412yes—
427C22×C3C2×C4×Q812yes—
428C22×C3#51911yescharacteristic
429C22×C3#52011yescharacteristic
430C22×C3C2×#12011yescharacteristic
431C22×C3#52511yescharacteristic
432C22×C3#52512yes—
433C22×C3C2×C4⋊Q811yescharacteristic
434C22×C3#53211yescharacteristic
435C22×C3#53312yes—
436C22×C3#53611yescharacteristic
437C22×C3#53711yescharacteristic
438C22×C3#54111yescharacteristic
439C22×C3#54211yescharacteristic
440C22×C3#54612yes—
441C22×C3#54712yes—
442C22×C3D4×Q812yes—
443C22×C3D4×Q812yes—
444C22×C3#55212yes—
445C22×C3#55412yes—
446C22×C3#55612yes—
447C22×C3#55712yes—
448C3×C4—28no—
449C3×C4—24no—
450C3×C4—24no—
451C3×C4—28no—
452C3×C4—24no—
453C3×C4—42no—
454C3×C4—44no—
455C3×C4—44no—
456C3×C4—42no—
457C3×C4—81no—
458C3×C4—81no—
459C23C23×C3×C411yescharacteristic
460C23C22×C3×D411yescharacteristic
461C23C22×C3×D412yes—
462C23C22×C3×Q812yes—
463C23C2×C3×#4111yescharacteristic
464C23C2×C3×#4112yes—
465C23C2×C3×#4112yes—
466C23C3×H(5,𝔽2)11yescharacteristic
467C23C3×2−1+411yescharacteristic
468C23C3×2−1+411yescharacteristic
469C23C3×2−1+411yescharacteristic
470Q8—232no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4C3×C4×D418yes—
532C2×C4C3×C4×D414yes—
533C2×C4C3×C4×Q818yescomplemented
534C2×C4C3×#12014yes—
535C2×C4C3×C4⋊Q814yes—
536C2×C3C4×#12011yescharacteristic
537C2×C3C4×C4⋊Q811yescharacteristic
538C2×C3#232311yescharacteristic
539C2×C3#236012yes—
540C2×C3#236512yes—
541C2×C3#243011yescharacteristic
542C2×C3#243411yescharacteristic
543C2×C3#246212yes—
544C2×C3#246512yes—
545C2×C3#258411yescharacteristic
546C2×C3#260411yescharacteristic
547C22C2×C3×C4×D412yes—
548C22C2×C3×C4×D411yescharacteristic
549C22C2×C3×C4×Q812yes—
550C22C3×#51911yescharacteristic
551C22C3×#52011yescharacteristic
552C22C2×C3×#12011yescharacteristic
553C22C3×#52512yes—
554C22C3×#52511yescharacteristic
555C22C2×C3×C4⋊Q811yescharacteristic
556C22C3×#53211yescharacteristic
557C22C3×#53312yes—
558C22C3×#53611yescharacteristic
559C22C3×#53711yescharacteristic
560C22C3×#54111yescharacteristic
561C22C3×#54211yescharacteristic
562C22C3×#54612yes—
563C22C3×#54712yes—
564C22C3×D4×Q812yes—
565C22C3×D4×Q812yes—
566C22C3×#55212yes—
567C22C3×#55412yes—
568C22C3×#55612yes—
569C22C3×#55712yes—
570C4—28no—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—28no—
575C4—44no—
576C4—44no—
577C4—42no—
578C4—42no—
579C4—81no—
580C4—81no—
581C3#1418611yescharacteristic, direct product factor, Sylow
582C2C3×C4×#12011yescharacteristic
583C2C3×C4×C4⋊Q811yescharacteristic
584C2C3×#232311yescharacteristic
585C2C3×#236012yes—
586C2C3×#236512yes—
587C2C3×#243011yescharacteristic
588C2C3×#243411yescharacteristic
589C2C3×#246212yes—
590C2C3×#246512yes—
591C2C3×#258411yescharacteristic
592C2C3×#260411yescharacteristic
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. #14186C3direct (N × H)
25. C3×#395C4semidirect
105. C3×#95Q8semidirect
118. #395C3×C4semidirect
267. #95C3×Q8semidirect
342. C2×C3×C4C4×Q8semidirect
533. C2×C4C3×C4×Q8semidirect
581. C3#14186direct (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
C2×C3×#3952
C2×C3×#3892
C2×C3×#3872
C3×#14462
C3×#14442
C3×#14412
C3×#14362
C3×C4×#952
#141863