Finite Group Databasebeta

← Dic6×Dic3⋊C4 #11805563

Subgroups — Dic6×Dic3⋊C4 #11805563

Order 1152.

1221 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Dic6C211yesmaximal, characteristic
2C2×Dic3×Dic6C211yesmaximal, characteristic
3C2×Dic3×Dic6C211yesmaximal, characteristic
4#10592627C211yesmaximal, characteristic
5C3×C4×Dic3⋊C4C211yesmaximal, characteristic
6#10591349C211yesmaximal, characteristic
7#10591327C212yesmaximal
8#10591244C212yesmaximal
9Dic3×Dic3⋊C4C212yesmaximal
10#10591234C212yesmaximal
11#10591212C211yesmaximal, characteristic
12Q8×Dic3⋊C4—31nomaximal
13C4⋊C4×Dic6—31nomaximal
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×C42C2211yescharacteristic
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×#3750C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4C2212yes—
21C2×C3×Dic3⋊C4C2211yescharacteristic
22C2×C3×Dic3⋊C4C2211yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2212yes—
25C2×C3×C4×Dic3C2211yescharacteristic
26C3×C4×Dic6—24no—
27C2×#3724C2212yes—
28C2×Dic3⋊Dic3C2212yes—
29C2×Dic3⋊Dic3C2212yes—
30C2×Dic3⋊Dic3C2212yes—
31C2×Dic3⋊Dic3C2212yes—
32C2×Dic32C2212yes—
33C2×Dic32C2212yes—
34Dic3×Dic6—24no—
35Dic3×Dic6C418yescomplemented
36#63658C2212yes—
37C3×C3⋊#95C2211yescharacteristic
38C3×C3⋊#95C2211yescharacteristic
39#63579C2212yes—
40C2×C3×C4×Q8—31no—
41C2×Q8×Dic3—31no—
42C2×Q8×Dic3—31no—
43C2×C4×Dic6—31no—
44C2×C4×Dic6—31no—
45C2×C4×Dic6D311yescharacteristic
46C3×C4×C4⋊C4—31no—
47#4975—32no—
48#4975—32no—
49#4975—31no—
50Dic3×C4⋊C4—32no—
51#4972—31no—
52#4972—32no—
53#4935—31no—
54C4×Dic3⋊C4—32no—
55C4×Dic3⋊C4D311yescharacteristic
56#4928—31no—
57#4928—31no—
58#4928—31no—
59#4928—32no—
60#4928—61no—
61#4647—32no—
62#4647—32no—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C3⋊Dic3C2312yes—
67C22×C3⋊Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2311yescharacteristic
74C2×C3×Dic6C2×C414yes—
75C2×C3×Dic6D414yes—
76C2×C3×Dic6Q814yes—
77C32×C42—22no—
78C4⋊(C3⋊Dic3)—22no—
79C4⋊(C3⋊Dic3)C2×C414yes—
80#3750—42no—
81C3×C4⋊Dic3—22no—
82C3×Dic3⋊C4—24no—
83C3×Dic3⋊C4—22no—
84C3×Dic3⋊C4Q814yescomplemented
85C3×C4×Dic3—22no—
86C3×C4×Dic3—24no—
87C3×C4×Dic3C2×C414yes—
88#3724—42no—
89Dic3⋊Dic3—24no—
90Dic3⋊Dic3—24no—
91Dic3⋊Dic3—42no—
92Dic3⋊Dic3C2×C418yes—
93Dic32—24no—
94Dic32C2×C418yes—
95Q8×C4⋊C4—91noSylow
96C22×C3×Q8—31no—
97C22×Dic6C2×D311yescharacteristic
98C3×C4×Q8—64no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—32no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C42—21no—
104C2×C3×C42—31no—
105C2×C3×C42—32no—
106C2×C3×C42—31no—
107C2×C3×C42C2×D311yescharacteristic
108C2×C3×C42C2×D311yescharacteristic
109Q8×Dic3—38no—
110Q8×Dic3—64no—
111C2×C4⋊Dic3—31no—
112C2×C4⋊Dic3—31no—
113C2×C4⋊Dic3—32no—
114C2×C4⋊Dic3—31no—
115C2×C4⋊Dic3—31no—
116C2×C4⋊Dic3—32no—
117C2×C4⋊Dic3—32no—
118C2×C4⋊Dic3—32no—
119C2×C4⋊Dic3—61no—
120C2×C4⋊Dic3—61no—
121C2×C4⋊Dic3C2×D311yescharacteristic
122C2×Dic3⋊C4—32no—
123C2×Dic3⋊C4—32no—
124C2×Dic3⋊C4—32no—
125C2×Dic3⋊C4—32no—
126C2×Dic3⋊C4—32no—
127C2×Dic3⋊C4—32no—
128C2×Dic3⋊C4—32no—
129C2×Dic3⋊C4—32no—
130C2×Dic3⋊C4—62no—
131C2×Dic3⋊C4C2×D312yes—
132C2×Dic3⋊C4C2×D311yescharacteristic
133C2×Dic3⋊C4C2×D311yescharacteristic
134C2×C4×Dic3—32no—
135C2×C4×Dic3—32no—
136C2×C4×Dic3—32no—
137C2×C4×Dic3—32no—
138C2×C4×Dic3C2×D312yes—
139C2×C4×Dic3C2×D311yescharacteristic
140C2×C4×Dic3C2×D311yescharacteristic
141C4×Dic6—24no—
142C4×Dic6—38no—
143C4×Dic6—64no—
144C3×#95—31no—
145C3×#95—31no—
146C3⋊#95—32no—
147C3⋊#95—32no—
148C3⋊#95—32no—
149C3⋊#95—32no—
150C3⋊#95—62no—
151C3⋊#95C2×D311yescharacteristic
152C3⋊#95C2×D311yescharacteristic
153C23×C32C2411yescharacteristic
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4C22×C412yes—
160C2×C32×C4C2×D412yes—
161C2×C32×C4C2×Q812yes—
162C2×C32×C4C2×Q812yes—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3C22×C414yes—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3C22×C412yes—
180C2×C3×Dic3C22×C414yes—
181C2×C3×Dic3C22×C412yes—
182C2×C3×Dic3C2×D414yes—
183C2×C3×Dic3C2×Q812yes—
184C2×C3×Dic3C2×Q814yes—
185C2×C3×Dic3C2×Q812yes—
186C2×C3×Dic3#4112yes—
187C3×Dic6C4⋊C4116yescomplemented
188C2×C4×Q8—31no—
189C2×C4×Q8—91no—
190C2×C4×Q8—91no—
191#389—92no—
192#389—92no—
193#389—91no—
194#389—91no—
195#384—91no—
196#384—92no—
197C4×C4⋊C4—31no—
198C4×C4⋊C4—92no—
199C2×C3×Q8—34no—
200C2×C3×Q8—34no—
201C2×C3×Q8—34no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—31no—
206C22×C3×C4—32no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—32no—
210C22×C3×C4—31no—
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D311yescharacteristic
217C22×Dic3—32no—
218C22×Dic3—32no—
219C22×Dic3—32no—
220C22×Dic3—32no—
221C22×Dic3—62no—
222C22×Dic3—62no—
223C22×Dic3C22×D312yes—
224C22×Dic3C22×D311yescharacteristic
225C22×Dic3C22×D311yescharacteristic
226C22×Dic3C22×D311yescharacteristic
227C22×Dic3C22×D311yescharacteristic
228C22×Dic3C22×D312yes—
229C2×Dic6Dic614yes—
230C2×Dic6C4×D314yes—
231C2×Dic6#6714yes—
232C3×C4⋊C4—34no—
233C3×C4⋊C4—62no—
234C3×C4⋊C4—64no—
235C3×C4⋊C4—62no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—34no—
239C3×C42—42no—
240C3×C42—64no—
241C3×C42—62no—
242C4⋊Dic3—22no—
243C4⋊Dic3—34no—
244C4⋊Dic3—34no—
245C4⋊Dic3—38no—
246C4⋊Dic3—64no—
247C4⋊Dic3—64no—
248C4⋊Dic3—64no—
249C4⋊Dic3—62no—
250C4⋊Dic3—62no—
251C4⋊Dic3—122no—
252C4⋊Dic3—122no—
253Dic3⋊C4—22no—
254Dic3⋊C4—24no—
255Dic3⋊C4—38no—
256Dic3⋊C4—64no—
257Dic3⋊C4—64no—
258Dic3⋊C4—122no—
259Dic3⋊C4—122no—
260Dic3⋊C4—122no—
261Dic3⋊C4—122no—
262Dic3⋊C4—122no—
263Dic3⋊C4—124no—
264Dic3⋊C4Dic614yesdirect product factor
265C4×Dic3—22no—
266C4×Dic3—24no—
267C4×Dic3—38no—
268C4×Dic3—38no—
269C4×Dic3—64no—
270C4×Dic3—64no—
271C4×Dic3C4×D314yes—
272C22×C32C23×C411yescharacteristic
273C22×C32C22×D411yescharacteristic
274C22×C32C22×Q811yescharacteristic
275C22×C32C22×Q811yescharacteristic
276C22×C32C2×#4111yescharacteristic
277C22×C32H(5,𝔽2)11yescharacteristic
278C22×C322−1+411yescharacteristic
279C32×C4—22no—
280C32×C4—41no—
281C32×C4C2×C4⋊C414yes—
282C3⋊Dic3—24no—
283C3⋊Dic3—42no—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—41no—
287C3×Dic3—42no—
288C3×Dic3C2×C4⋊C418yes—
289C3×Dic3C4×Q814yescomplemented
290C22×Q8—31no—
291C4×Q8—64no—
292C4×Q8—98no—
293C4×Q8—184no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—92no—
299C2×C4⋊C4—91no—
300C2×C4⋊C4—92no—
301C2×C4⋊C4—92no—
302C2×C4⋊C4—91no—
303C2×C4⋊C4—92no—
304C2×C4⋊C4—92no—
305C2×C4⋊C4—92no—
306C2×C42—31no—
307C2×C42—32no—
308C2×C42—31no—
309C2×C42—92no—
310C2×C42—92no—
311C2×C42D3211yescharacteristic
312#95—31no—
313#95—31no—
314#95—92no—
315#95—92no—
316C23×C3—21no—
317C23×C3C23×D311yescharacteristic
318C23×C3C23×D311yescharacteristic
319C3×Q8—316no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—32no—
336C2×C3×C4—34no—
337C2×C3×C4—34no—
338C2×C3×C4—32no—
339C2×C3×C4—34no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4C2×Dic612yes—
358C2×C3×C4C2×Dic612yes—
359C2×C3×C4C2×C4×D312yes—
360C2×C3×C4C2×C4×D312yes—
361C2×C3×C4D3×D412yes—
362C2×C3×C4D3×Q812yes—
363C2×C3×C4D3×Q812yes—
364C2×C3×C4C2×#6712yes—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—34no—
376C2×Dic3—34no—
377C2×Dic3—64no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—62no—
381C2×Dic3—62no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—122no—
389C2×Dic3—122no—
390C2×Dic3—122no—
391C2×Dic3—122no—
392C2×Dic3—122no—
393C2×Dic3C2×Dic612yes—
394C2×Dic3C2×Dic612yes—
395C2×Dic3C2×Dic614yes—
396C2×Dic3C2×C4×D312yes—
397C2×Dic3C2×C4×D312yes—
398C2×Dic3C2×C4×D314yes—
399C2×Dic3#23512yes—
400C2×Dic3C2×#6714yes—
401Dic6Dic3⋊C4116yesdirect product factor
402C2×C32C22×C4⋊C411yescharacteristic
403C2×C32C2×C4×Q811yescharacteristic
404C2×C32#52011yescharacteristic
405C2×C32D4×Q811yescharacteristic
406C2×C32#55011yescharacteristic
407C2×C32#55711yescharacteristic
408C2×C32Q8211yescharacteristic
409C2×Q8—34no—
410C2×Q8—34no—
411C2×Q8—34no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—32no—
418C22×C4—92no—
419C22×C4—92no—
420C22×C4C2×D3211yescharacteristic
421C22×C4C2×D3211yescharacteristic
422C22×C4C2×D3211yescharacteristic
423C4⋊C4—34no—
424C4⋊C4—64no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—98no—
428C4⋊C4—94no—
429C4⋊C4—184no—
430C4⋊C4—184no—
431C4⋊C4—182no—
432C4⋊C4—362no—
433C4⋊C4—362no—
434C4⋊C4—362no—
435C42—22no—
436C42—34no—
437C42—64no—
438C42—62no—
439C42—98no—
440C42—184no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3C22×Dic611yescharacteristic
449C22×C3C22×Dic611yescharacteristic
450C22×C3C22×C4×D311yescharacteristic
451C22×C3C22×C4×D311yescharacteristic
452C22×C3C2×#23511yescharacteristic
453C22×C3C2×D3×D411yescharacteristic
454C22×C3C2×D3×Q811yescharacteristic
455C22×C3C2×D3×Q811yescharacteristic
456C22×C3#100711yescharacteristic
457C22×C3D3×#4111yescharacteristic
458C22×C3#100911yescharacteristic
459C22×C3#100911yescharacteristic
460C22×C3#101011yescharacteristic
461C22×C3C22×#6711yescharacteristic
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—24no—
465C3×C4—38no—
466C3×C4—34no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—62no—
472C3×C4—62no—
473C3×C4—122no—
474C3×C4—121no—
475C3×C4D3×C4⋊C414yes—
476C3×C4C2×Dic3⋊C414yes—
477Dic3—22no—
478Dic3—22no—
479Dic3—41no—
480Dic3—42no—
481Dic3—64no—
482Dic3—64no—
483Dic3—124no—
484Dic3—122no—
485Dic3—122no—
486Dic3—242no—
487Dic3C4×Dic614yescomplemented
488Dic3C2×Dic3⋊C418yes—
489C32Q8×C4⋊C411yescharacteristic, complemented, Sylow
490C23C22×D3211yescharacteristic
491Q8—316no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—32no—
498C2×C4—34no—
499C2×C4—32no—
500C2×C4—34no—
501C2×C4—32no—
502C2×C4—34no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—61no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—94no—
516C2×C4—182no—
517C2×C4—182no—
518C2×C4—182no—
519C2×C4—182no—
520C2×C4—182no—
521C2×C4D3×Dic612yes—
522C2×C4D3×Dic612yes—
523C2×C4C4×D3212yes—
524C2×C4D3×#6712yes—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3C2×C4×Dic611yescharacteristic
533C2×C3C2×D3×C4⋊C411yescharacteristic
534C2×C3#552311yescharacteristic
535C2×C3D4×Dic611yescharacteristic
536C2×C3#555311yescharacteristic
537C2×C3Q8×Dic611yescharacteristic
538C2×C3Q8×Dic611yescharacteristic
539C2×C3#556711yescharacteristic
540C2×C3#556911yescharacteristic
541C2×C3C4×D3×Q811yescharacteristic
542C2×C3#557411yescharacteristic
543C2×C3C22×Dic3⋊C411yescharacteristic
544C2×C3Q8×#6711yescharacteristic
545C2×C3#583411yescharacteristic
546C22C2×D3×Dic611yescharacteristic
547C22C2×D3×Dic611yescharacteristic
548C22#6429711yescharacteristic
549C22C2×C4×D3211yescharacteristic
550C22D3×#23511yescharacteristic
551C22#6430811yescharacteristic
552C22C2×D3×#6711yescharacteristic
553C4—22no—
554C4—38no—
555C4—34no—
556C4—41no—
557C4—62no—
558C4—62no—
559C4—121no—
560C4—122no—
561C4—184no—
562C4—362no—
563C4D3×Dic3⋊C414yes—
564C3—21no—
565C3C4⋊C4×Dic611yescharacteristic, complemented
566C3Q8×Dic3⋊C411yescharacteristic, complemented
567C2#1059385111yescharacteristic
568C2Dic6211yescharacteristic
569C2C4×D3×Dic611yescharacteristic
570C2#1059389011yescharacteristic
571C2C2×D3×Dic3⋊C411yescharacteristic
572C2Dic6×#6711yescharacteristic
573C2#1059500711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C42C22
socle, prime radical, cosocleC23×C32C24
hypercenterC2×C42D32
hypocenterC32Q8×C4⋊C4
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
35. Dic3×Dic6C4semidirect
84. C3×Dic3⋊C4Q8semidirect
187. C3×Dic6C4⋊C4semidirect
264. Dic3⋊C4Dic6direct (N × H)
289. C3×Dic3C4×Q8semidirect
401. Dic6Dic3⋊C4direct (N × H)
487. Dic3C4×Dic6semidirect
489. C32Q8×C4⋊C4semidirect
565. C3C4⋊C4×Dic6semidirect
566. C3Q8×Dic3⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Dic62
C2×Dic3×Dic62
#105926272
C3×C4×Dic3⋊C42
#105913492
#105913272
#105912442
Dic3×Dic3⋊C42
#105912342
#105912122
Q8×Dic3⋊C43
C4⋊C4×Dic63