Finite Group Databasebeta

← #11921964

Subgroups — #11921964

Order 1152.

728 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×S4C211yesmaximal, characteristic
2C4×Dic3×A4C211yesmaximal, characteristic, complemented
3#10593645C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C211yesmaximal, characteristic
5(C2×S4)⋊Dic3C211yesmaximal, characteristic
6#10593238C211yesmaximal, characteristic, complemented
7Dic3×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8#85287—31nomaximal
9#74657—31nomaximal
10C22×C3×S4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×Dic3×A4C2211yescharacteristic
13C2×Dic3×A4C2211yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
16C2×C3×SL(2,ℤ/4)C2211yescharacteristic
17Dic3×S4C414yescomplemented
18#63877—41nomaximal
19C2×C4×S4—31no—
20C2×D4×Dic3—31no—
21C22×C4×Dic3D311yescharacteristic, complemented
22C2×#877—31no—
23C42×A4—31no—
24(C2×S4)⋊C4—31no—
25(C2×S4)⋊C4D311yescharacteristic
26SL(2,ℤ/4)⋊C4—31no—
27SL(2,ℤ/4)⋊C4—31no—
28SL(2,ℤ/4)⋊C4—61no—
29C4×SL(2,ℤ/4)—31no—
30C3×#386—31no—
31#5220—31no—
32#5210—31no—
33#4946—31no—
34#4945—31no—
35#4944—31no—
36Dic3×C22⋊C4—31no—
37Dic3×C22⋊C4—31no—
38#4654—31no—
39#4653—31no—
40#4649—31no—
41#4648—31no—
42C22×C3×A4C2311yescharacteristic
43C2×C3×S4—21no—
44C2×C3×S4—21no—
45C2×C3×S4C2×C412yes—
46C3×C4×A4—21no—
47C2×D3×Dic3—41no—
48Dic3×A4—21no—
49Dic3×A4C2×C412yes—
50C3⋊SL(2,ℤ/4)—21no—
51C3⋊SL(2,ℤ/4)C2×C412yes—
52C3×SL(2,ℤ/4)—21no—
53#3750—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×C4×Dic3—41no—
56#3724—41no—
57(C2×D3)⋊Dic3—41no—
58Dic32—41no—
59#2290—91noSylow
60C22×S4C2×D311yescharacteristic
61C22×C3×D4—31no—
62C23×C3×C4C2×D311yescharacteristic
63C23×Dic3—31no—
64C23×Dic3C2×D311yescharacteristic
65C23×Dic3C2×D311yescharacteristic
66C2×C4×A4—21no—
67C2×C4×A4—31no—
68C2×C4×A4—31no—
69C2×C4×A4C2×D311yescharacteristic
70C2×SL(2,ℤ/4)—31no—
71C2×SL(2,ℤ/4)—31no—
72C2×SL(2,ℤ/4)—61no—
73C2×SL(2,ℤ/4)—61no—
74C2×SL(2,ℤ/4)C2×D311yescharacteristic
75C4×S4—34no—
76C2×C3×C22⋊C4—31no—
77C2×C3×C22⋊C4—31no—
78C2×C3×C22⋊C4—31no—
79C2×#217—31no—
80C2×#217—31no—
81C2×#217—31no—
82C2×#217—31no—
83C2×#217—31no—
84D4×Dic3—34no—
85D4×Dic3—34no—
86C2×C4⋊Dic3—31no—
87C2×Dic3⋊C4—31no—
88C2×Dic3⋊C4—31no—
89C2×Dic3⋊C4—31no—
90C2×Dic3⋊C4—31no—
91C2×C4×Dic3—31no—
92C2×C4×Dic3—31no—
93C2×C4×Dic3—31no—
94C2×C4×Dic3—31no—
95C2×C4×Dic3—31no—
96C2×C4×Dic3—31no—
97C2×C4×Dic3—31no—
98#877—61no—
99#877—61no—
100#877—61no—
101#877—61no—
102#872—121no—
103C3×#95—31no—
104C3×#95—31no—
105C3⋊#95—31no—
106C3⋊#95—31no—
107C3⋊#95—31no—
108C3⋊#95—31no—
109C3⋊#95—31no—
110C3⋊#95—31no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C22×C3×D3—41no—
114C2×C3×A4C22×C411yescharacteristic
115C2×C3×A4#4111yescharacteristic
116C2×C3×A4#4111yescharacteristic
117C3×S4—22no—
118C2×C32×C4—41no—
119C2×C3⋊Dic3—41no—
120C2×C3⋊Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—41no—
124D3×Dic3—44no—
125C2×C4×D4—91no—
126C2×#117—91no—
127C22×C42—31no—
128#388—91no—
129#388—91no—
130#387—91no—
131#387—91no—
132#386—31no—
133#386—91no—
134#385—91no—
135#385—91no—
136C4×C4⋊C4—91no—
137C4×C22⋊C4—91no—
138C4×C22⋊C4—91no—
139#376—91no—
140C24×C3—31no—
141C24×C3C22×D311yescharacteristic
142C22×A4—21no—
143C22×A4C22×D311yescharacteristic
144C2×S4—21no—
145C2×S4—21no—
146C2×S4C4×D312yes—
147C2×C3×D4—32no—
148C2×C3×D4—32no—
149C2×C3×D4—61no—
150C2×C3×D4—61no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C22×C3×C4—21no—
154C22×C3×C4—31no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—61no—
160C22×Dic3—21no—
161C22×Dic3—31no—
162C22×Dic3—31no—
163C22×Dic3—31no—
164C22×Dic3—32no—
165C22×Dic3—31no—
166C22×Dic3—31no—
167C22×Dic3—34no—
168C22×Dic3—31no—
169C22×Dic3—31no—
170C22×Dic3—31no—
171C22×Dic3—61no—
172C22×Dic3—61no—
173C22×Dic3—62no—
174C22×Dic3—61no—
175C22×Dic3C4×D312yes—
176C2×C4×D3—121no—
177C4×A4—21no—
178C4×A4—32no—
179C4×A4—41no—
180C4×A4—61no—
181SL(2,ℤ/4)—21no—
182SL(2,ℤ/4)—32no—
183SL(2,ℤ/4)—62no—
184SL(2,ℤ/4)—61no—
185SL(2,ℤ/4)—121no—
186C3×C22⋊C4—61no—
187C3×C22⋊C4—62no—
188C3×C22⋊C4—61no—
189C3×C22⋊C4—61no—
190C3×C22⋊C4—61no—
191C3×C42—121no—
192#217—34no—
193#217—32no—
194#217—32no—
195#217—61no—
196#217—62no—
197#217—61no—
198#217—61no—
199#217—61no—
200(C2×D3)⋊C4—41no—
201(C2×D3)⋊C4—121no—
202C4⋊Dic3—34no—
203Dic3⋊C4—61no—
204Dic3⋊C4—62no—
205Dic3⋊C4—61no—
206Dic3⋊C4—62no—
207Dic3⋊C4—61no—
208Dic3⋊C4—61no—
209Dic3⋊C4—121no—
210Dic3⋊C4—121no—
211Dic3⋊C4—241no—
212C4×Dic3—34no—
213C4×Dic3—31no—
214C4×Dic3—31no—
215C4×Dic3—31no—
216C4×Dic3—61no—
217C4×Dic3—61no—
218C4×Dic3—62no—
219C4×Dic3—61no—
220C4×Dic3—121no—
221C4×Dic3S411yescharacteristic, complemented
222C22×C32—41no—
223C2×C3×D3—42no—
224C2×C3×D3—81no—
225C2×C3×D3—81no—
226C3×A4#11711yescharacteristic, complemented
227C32×C4—81no—
228C3⋊Dic3—42no—
229C3⋊Dic3—81no—
230C3×Dic3—42no—
231C3×Dic3—81no—
232C3×Dic3—81no—
233C22×D4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—91no—
237C23×C4D3211yescharacteristic
238C4×D4—94no—
239C4×D4—94no—
240#117—181no—
241#117—181no—
242#117—181no—
243#117—181no—
244C2×C4⋊C4—91no—
245C2×C4⋊C4—91no—
246C2×C4⋊C4—91no—
247C2×C4⋊C4—91no—
248C2×C4⋊C4—91no—
249C2×C22⋊C4—31no—
250C2×C22⋊C4—31no—
251C2×C22⋊C4—31no—
252C2×C22⋊C4—91no—
253C2×C22⋊C4—91no—
254C2×C22⋊C4—91no—
255C2×C22⋊C4—91no—
256C2×C22⋊C4—91no—
257C2×C42—91no—
258C2×C42—91no—
259C2×C42—91no—
260C2×C42—91no—
261C2×C42—91no—
262C2×C42—91no—
263C2×C42—91no—
264#95—31no—
265#95—31no—
266#95—91no—
267#95—91no—
268#95—91no—
269#95—91no—
270#95—91no—
271#95—91no—
272#95—91no—
273#95—91no—
274C23×C3—31no—
275C23×C3—31no—
276C23×C3—31no—
277C23×C3—31no—
278C23×C3—32no—
279C23×C3—61no—
280C23×C3—61no—
281C23×C3—61no—
282C23×C3—61no—
283C23×C3—61no—
284C23×C3—61no—
285C23×C3C2×C4×D311yescharacteristic
286C23×C3#23511yescharacteristic
287C23×C3#23511yescharacteristic
288C22×D3—41no—
289C2×A4—21no—
290C2×A4—21no—
291C2×A4—21no—
292C2×A4C2×C4×D311yescharacteristic
293C2×A4#23711yescharacteristic
294C2×A4#23711yescharacteristic
295S4—22no—
296C3×D4—62no—
297C3×D4—62no—
298C3×D4—62no—
299C3×D4—62no—
300C2×C3×C4—32no—
301C2×C3×C4—32no—
302C2×C3×C4—31no—
303C2×C3×C4—41no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—81no—
316C2×C3×C4—121no—
317C2×C3×C4—121no—
318C2×C3×C4C2×S411yescharacteristic
319C2×Dic3—31no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—31no—
327C2×Dic3—41no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—64no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—64no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—121no—
345C2×Dic3—121no—
346C2×Dic3—241no—
347C2×Dic3—241no—
348C2×Dic3C2×S411yescharacteristic
349C2×Dic3C2×S411yescharacteristic
350C4×D3—124no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—41no—
354C3×D3—82no—
355C24—31no—
356C24C2×D3211yescharacteristic
357C2×D4—32no—
358C2×D4—32no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C22×C4—21no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—32no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—91no—
375C22×C4—91no—
376C22×C4—92no—
377C22×C4—91no—
378C22×C4—91no—
379C22×C4—94no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4—181no—
383C22×C4—181no—
384C22×C4—181no—
385C22×C4—182no—
386C4⋊C4—94no—
387C4⋊C4—181no—
388C4⋊C4—181no—
389C4⋊C4—182no—
390C4⋊C4—181no—
391C4⋊C4—182no—
392C4⋊C4—181no—
393C22⋊C4—61no—
394C22⋊C4—62no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C22⋊C4—61no—
398C22⋊C4—94no—
399C22⋊C4—92no—
400C22⋊C4—92no—
401C22⋊C4—181no—
402C22⋊C4—181no—
403C22⋊C4—182no—
404C22⋊C4—181no—
405C22⋊C4—181no—
406C42—31no—
407C42—91no—
408C42—91no—
409C42—94no—
410C42—91no—
411C42—181no—
412C42—181no—
413C42—181no—
414C42—182no—
415C22×C3—31no—
416C22×C3—31no—
417C22×C3—31no—
418C22×C3—31no—
419C22×C3—31no—
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—41no—
425C22×C3—61no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—81no—
430C22×C3—121no—
431C22×C3—121no—
432C22×C3—121no—
433C22×C3—121no—
434C22×C3#87211yescharacteristic, complemented
435C22×C3C22×S411yescharacteristic
436C2×D3—42no—
437C2×D3—81no—
438C2×D3—81no—
439A4—21no—
440A4#87711yescharacteristic, complemented
441C3×C4—21no—
442C3×C4—62no—
443C3×C4—61no—
444C3×C4—81no—
445C3×C4—122no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—161no—
449C3×C4—241no—
450Dic3—21no—
451Dic3—32no—
452Dic3—62no—
453Dic3—62no—
454Dic3—61no—
455Dic3—81no—
456Dic3—122no—
457Dic3—121no—
458Dic3—121no—
459Dic3—242no—
460Dic3—241no—
461Dic3—481no—
462Dic3C4×S412yes—
463C32—41noSylow
464C23—31no—
465C23—31no—
466C23—31no—
467C23—31no—
468C23—32no—
469C23—61no—
470C23—61no—
471C23—61no—
472C23—61no—
473C23—61no—
474C23—61no—
475C23#379511yescharacteristic
476C23C4×D3211yescharacteristic
477C23#380411yescharacteristic
478D4—62no—
479D4—62no—
480D4—62no—
481D4—62no—
482C2×C4—31no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—31no—
486C2×C4—31no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—91no—
501C2×C4—92no—
502C2×C4—92no—
503C2×C4—92no—
504C2×C4—92no—
505C2×C4—91no—
506C2×C4—184no—
507C2×C4—181no—
508C2×C4—181no—
509C2×C4—181no—
510C2×C4—184no—
511C2×C4—181no—
512C2×C4—181no—
513C2×C4—181no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C4—181no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4D3×S411yescharacteristic
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—81no—
531C2×C3—81no—
532C2×C3—81no—
533C2×C3—122no—
534C2×C3C2×C4×S411yescharacteristic
535C2×C3#591211yescharacteristic
536C2×C3#591211yescharacteristic
537D3—82no—
538C22—31no—
539C22—31no—
540C22—31no—
541C22—31no—
542C22—31no—
543C22—31no—
544C22—31no—
545C22—31no—
546C22—31no—
547C22—61no—
548C22—62no—
549C22—62no—
550C22—62no—
551C22—121no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22#6387711yescharacteristic, complemented
556C22C2×D3×S411yescharacteristic
557C4—21no—
558C4—32no—
559C4—62no—
560C4—61no—
561C4—61no—
562C4—92no—
563C4—121no—
564C4—121no—
565C4—182no—
566C4—181no—
567C4—182no—
568C4—361no—
569C4—361no—
570C3—41no—
571C3—81no—
572C3#8528711yescharacteristic, complemented
573C2—31no—
574C2—31no—
575C2—31no—
576C2—31no—
577C2—122no—
578C2#1059642711yescharacteristic
579C2C4×D3×S411yescharacteristic
580C2#1059646311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
cosocleC22×C3×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4#117
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×Dic3×A4C2semidirect
3. #10593645C2semidirect
6. #10593238C2semidirect
7. Dic3×SL(2,ℤ/4)C2semidirect
17. Dic3×S4C4semidirect
21. C22×C4×Dic3D3semidirect
221. C4×Dic3S4semidirect
226. C3×A4#117semidirect
434. C22×C3#872semidirect
440. A4#877semidirect
555. C22#63877semidirect
572. C3#85287semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×S42
C4×Dic3×A42
#105936452
C3×(C2×S4)⋊C42
(C2×S4)⋊Dic32
#105932382
Dic3×SL(2,ℤ/4)2
#852873
#746573
#638774