Finite Group Databasebeta

← #11921993

Subgroups — #11921993

Order 1152.

888 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 79 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×C3×C4×S4C211yesmaximal, characteristic, complemented
2C2×#64207C211yesmaximal, characteristic
3C2×(C3⋊S4)⋊C4C211yesmaximal, characteristic, complemented
4A4×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#10593647C211yesmaximal, characteristic, complemented
6(C2×S4)⋊Dic3C211yesmaximal, characteristic, complemented
7Dic3×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8#85308—31nomaximal
9#82340—31nomaximal
10C22×C3⋊S4C2211yescharacteristic
11C22×C3×S4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13C2×Dic3×A4C2211yescharacteristic
14C2×Dic3×A4C2211yescharacteristic
15C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
16C2×C3×SL(2,ℤ/4)C2211yescharacteristic
17C3×C4×S4—22no—
18#64207C414yescomplemented
19(C3⋊S4)⋊C4—22no—
20#63904—41nomaximal
21C2×C4⋊S4—31no—
22C2×C4×S4—31no—
23C2×C4×S4D311yescharacteristic, complemented
24C2×C3×C4×D4—31no—
25C2×#939—31no—
26C2×C4×#67—31no—
27C22×Dic3⋊C4D311yescharacteristic, complemented
28C2×#67⋊C4—31no—
29A4×C4⋊C4—31no—
30(C2×S4)⋊C4—31no—
31(C2×S4)⋊C4—31no—
32(C2×S4)⋊C4—61no—
33C4×SL(2,ℤ/4)—31no—
34#5220—31no—
35#5213—31no—
36#5210—31no—
37#4989—31no—
38#4976—31no—
39#4957—31no—
40#4944—31no—
41Dic3×C22⋊C4—31no—
42#4940—31no—
43C4×Dic3⋊C4—31no—
44C22×C3×A4C2311yescharacteristic
45C2×C3⋊S4—21no—
46C2×C3⋊S4—21no—
47C2×C3⋊S4C2×C412yescomplemented
48C2×C3×S4—21no—
49C2×C3×S4—21no—
50C2×C3×S4C2×C412yescomplemented
51C2×C3×C4×D3—41no—
52C3×C4×A4—21no—
53C2×#626—41no—
54C2×#624—41no—
55Dic3×A4—21no—
56Dic3×A4C2×C412yescomplemented
57C3⋊SL(2,ℤ/4)—21no—
58C3×SL(2,ℤ/4)D412yescomplemented
59#3752—41no—
60C3×Dic3⋊C4—41no—
61(C2×D3)⋊Dic3—41no—
62Dic32—41no—
63#2361—91noSylow
64C22×S4—31no—
65C22×S4—61no—
66C22×S4C2×D311yescharacteristic
67C22×C3×D4—31no—
68C23×C3×C4—31no—
69C23×C3×C4C2×D311yescharacteristic
70C22×#67—31no—
71C22×#67—31no—
72C23×Dic3C2×D311yescharacteristic
73C23×Dic3C2×D311yescharacteristic
74C22×C4×D3—31no—
75C2×C4×A4—21no—
76C2×C4×A4—31no—
77C2×C4×A4—31no—
78C2×C4×A4C2×D311yescharacteristic
79C2×SL(2,ℤ/4)—31no—
80C2×SL(2,ℤ/4)—61no—
81C2×SL(2,ℤ/4)C2×D311yescharacteristic
82C4⋊S4—34no—
83C4×S4—22no—
84C4×S4—62no—
85C3×C4×D4—62no—
86C3×C4×D4—62no—
87C2×C3×C4⋊C4—31no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—31no—
90C2×C3×C42—31no—
91C2×#217—31no—
92C2×#217—31no—
93C2×#217—31no—
94C2×#217—31no—
95#939—34no—
96#939—34no—
97C4×#67—61no—
98C4×#67—61no—
99C4×#67—61no—
100C4×#67—61no—
101C2×(C2×D3)⋊C4—31no—
102C2×(C2×D3)⋊C4—31no—
103C2×(C2×D3)⋊C4—31no—
104C2×(C2×D3)⋊C4—31no—
105C2×Dic3⋊C4—31no—
106C2×Dic3⋊C4—31no—
107C2×Dic3⋊C4—31no—
108C2×Dic3⋊C4—31no—
109C2×Dic3⋊C4—31no—
110C2×Dic3⋊C4—31no—
111C2×C4×Dic3—31no—
112C2×C4×Dic3—31no—
113C2×C4×Dic3—31no—
114#893—121no—
115#67⋊C4—62no—
116#67⋊C4—62no—
117C3⋊#95—31no—
118C3⋊#95—31no—
119C3⋊#95—31no—
120C3⋊#95—31no—
121C3⋊#95—31no—
122C3⋊#95—31no—
123C22×C3⋊D3—41no—
124C22×C3×D3—41no—
125C2×C3×A4C22×C411yescharacteristic
126C2×C3×A4C2×D411yescharacteristic
127C2×C3×A4#4111yescharacteristic
128C3⋊S4—22no—
129C3×S4—22no—
130C2×C32×C4—41no—
131C2×C3⋊Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C3×C4×D3—82no—
136#626—44no—
137#624—82no—
138C2×#121—91no—
139C2×C4×D4—31no—
140C2×C4×D4—91no—
141C2×C4×D4—91no—
142C22×C4⋊C4—31no—
143#390—91no—
144#389—91no—
145#388—91no—
146#388—91no—
147#386—91no—
148#386—91no—
149#385—91no—
150#385—91no—
151C4×C4⋊C4—91no—
152C4×C22⋊C4—91no—
153C24×C3—31no—
154C24×C3C22×D311yescharacteristic
155C23×D3—31no—
156C22×A4—21no—
157C22×A4C22×D311yescharacteristic
158C2×S4—21no—
159C2×S4—21no—
160C2×S4—32no—
161C2×S4—62no—
162C2×S4—61no—
163C2×S4—61no—
164C2×S4—121no—
165C2×S4—121no—
166C2×S4C4×D312yescomplemented
167C2×C3×D4—32no—
168C2×C3×D4—32no—
169C2×C3×D4—61no—
170C2×C3×D4—61no—
171C2×C3×D4—61no—
172C2×C3×D4—61no—
173C22×C3×C4—21no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—61no—
180C22×C3×C4—62no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4—121no—
184C2×#67—34no—
185C2×#67—32no—
186C2×#67—32no—
187C2×#67—61no—
188C2×#67—62no—
189C2×#67—61no—
190C2×#67—61no—
191C2×#67—61no—
192C2×#67—61no—
193C2×#67—61no—
194C22×Dic3—21no—
195C22×Dic3—31no—
196C22×Dic3—31no—
197C22×Dic3—32no—
198C22×Dic3—31no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—31no—
203C22×Dic3—61no—
204C22×Dic3C4×D312yescomplemented
205C2×D12—121no—
206C2×C4×D3—41no—
207C2×C4×D3—61no—
208C2×C4×D3—62no—
209C2×C4×D3—61no—
210C2×C4×D3—121no—
211C2×C4×D3—121no—
212C4×A4—21no—
213C4×A4—32no—
214C4×A4—41no—
215C4×A4—61no—
216SL(2,ℤ/4)—61no—
217SL(2,ℤ/4)—121no—
218SL(2,ℤ/4)#6712yescomplemented
219C3×C4⋊C4—62no—
220C3×C4⋊C4—121no—
221C3×C22⋊C4—32no—
222C3×C22⋊C4—32no—
223C3×C22⋊C4—62no—
224C3×C42—62no—
225#217—34no—
226#217—61no—
227#217—61no—
228#217—61no—
229#217—61no—
230#217—121no—
231(C2×D3)⋊C4—34no—
232(C2×D3)⋊C4—62no—
233(C2×D3)⋊C4—61no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—121no—
236(C2×D3)⋊C4—121no—
237(C2×D3)⋊C4—121no—
238(C2×D3)⋊C4—241no—
239Dic3⋊C4—34no—
240Dic3⋊C4—31no—
241Dic3⋊C4—31no—
242Dic3⋊C4—31no—
243Dic3⋊C4—61no—
244Dic3⋊C4—62no—
245Dic3⋊C4S411yescharacteristic, complemented
246C4×Dic3—62no—
247C4×Dic3—61no—
248C4×Dic3—62no—
249C4×Dic3—61no—
250C4×Dic3—121no—
251C22×C32—41no—
252C2×C3⋊D3—42no—
253C2×C3⋊D3—81no—
254C2×C3⋊D3—81no—
255C2×C3×D3—42no—
256C2×C3×D3—81no—
257C2×C3×D3—81no—
258C3×A4C4×D411yescharacteristic, complemented
259C32×C4—81no—
260C3⋊Dic3—81no—
261C3×Dic3—42no—
262C3×Dic3—42no—
263C3×Dic3—81no—
264C22×D4—31no—
265C22×D4—91no—
266C22×D4—91no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—91no—
271C23×C4D3211yescharacteristic
272#121—94no—
273#121—94no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—182no—
277C4×D4—182no—
278C4×D4—181no—
279C4×D4—181no—
280C4×D4—181no—
281C4×D4—181no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—91no—
284C2×C4⋊C4—91no—
285C2×C4⋊C4—91no—
286C2×C4⋊C4—91no—
287C2×C4⋊C4—91no—
288C2×C4⋊C4—91no—
289C2×C22⋊C4—31no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—91no—
293C2×C22⋊C4—91no—
294C2×C22⋊C4—91no—
295C2×C22⋊C4—91no—
296C2×C22⋊C4—91no—
297C2×C22⋊C4—91no—
298C2×C22⋊C4—91no—
299C2×C42—31no—
300C2×C42—91no—
301C2×C42—91no—
302C2×C42—91no—
303#95—91no—
304#95—91no—
305#95—91no—
306#95—91no—
307#95—91no—
308#95—91no—
309C23×C3—31no—
310C23×C3—31no—
311C23×C3—31no—
312C23×C3—31no—
313C23×C3—32no—
314C23×C3—61no—
315C23×C3—61no—
316C23×C3—61no—
317C23×C3—61no—
318C23×C3—61no—
319C23×C3—61no—
320C23×C3C2×C4×D311yescharacteristic
321C23×C3D3×D411yescharacteristic
322C23×C3#23911yescharacteristic
323C22×D3—32no—
324C22×D3—41no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—61no—
329C22×D3—61no—
330C22×D3—61no—
331C22×D3—121no—
332C22×D3—241no—
333C2×A4—21no—
334C2×A4—21no—
335C2×A4—21no—
336C2×A4C2×C4×D311yescharacteristic
337C2×A4#23511yescharacteristic
338C2×A4C2×#6711yescharacteristic
339S4—22no—
340S4—62no—
341S4—122no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—62no—
345C3×D4—62no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—31no—
350C2×C3×C4—41no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—81no—
362C2×C3×C4—121no—
363C2×C3×C4—121no—
364C2×C3×C4—122no—
365C2×C3×C4—122no—
366C2×C3×C4C2×S411yescharacteristic
367#67—64no—
368#67—62no—
369#67—62no—
370#67—62no—
371#67—62no—
372#67—64no—
373C2×Dic3—31no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—31no—
379C2×Dic3—41no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—121no—
394C2×Dic3—241no—
395C2×Dic3C2×S411yescharacteristic
396C2×Dic3C2×S411yescharacteristic
397D12—124no—
398C4×D3—82no—
399C4×D3—122no—
400C4×D3—122no—
401C4×D3—242no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—41no—
405C3⋊D3—82no—
406C3×D3—82no—
407C24—31no—
408C24—91no—
409C24C2×D3211yescharacteristic
410C2×D4—32no—
411C2×D4—32no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—94no—
417C2×D4—92no—
418C2×D4—92no—
419C2×D4—181no—
420C2×D4—181no—
421C2×D4—182no—
422C2×D4—181no—
423C2×D4—181no—
424C2×D4—181no—
425C2×D4—181no—
426C22×C4—21no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—32no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—92no—
442C22×C4—91no—
443C22×C4—91no—
444C22×C4—91no—
445C22×C4—91no—
446C22×C4—121no—
447C22×C4—181no—
448C22×C4—182no—
449C22×C4—181no—
450C22×C4—181no—
451C22×C4—361no—
452C4⋊C4—31no—
453C4⋊C4—62no—
454C4⋊C4—91no—
455C4⋊C4—91no—
456C4⋊C4—94no—
457C4⋊C4—91no—
458C4⋊C4—181no—
459C4⋊C4—182no—
460C22⋊C4—32no—
461C22⋊C4—32no—
462C22⋊C4—62no—
463C22⋊C4—94no—
464C22⋊C4—94no—
465C22⋊C4—181no—
466C22⋊C4—182no—
467C22⋊C4—181no—
468C22⋊C4—181no—
469C22⋊C4—181no—
470C22⋊C4—181no—
471C22⋊C4—181no—
472C22⋊C4—361no—
473C22⋊C4—361no—
474C42—62no—
475C42—181no—
476C42—182no—
477C42—181no—
478C42—182no—
479C22×C3—31no—
480C22×C3—31no—
481C22×C3—31no—
482C22×C3—31no—
483C22×C3—31no—
484C22×C3—31no—
485C22×C3—31no—
486C22×C3—31no—
487C22×C3—31no—
488C22×C3—41no—
489C22×C3—61no—
490C22×C3—62no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—81no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3#89311yescharacteristic, complemented
499C22×C3C22×S411yescharacteristic
500C2×D3—42no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—122no—
507C2×D3—121no—
508C2×D3—121no—
509C2×D3—121no—
510C2×D3—121no—
511C2×D3—242no—
512C2×D3—241no—
513C2×D3—241no—
514C2×D3—481no—
515C2×D3—481no—
516A4—21no—
517A4C4×#6711yescharacteristic, complemented
518C3×C4—21no—
519C3×C4—62no—
520C3×C4—62no—
521C3×C4—62no—
522C3×C4—61no—
523C3×C4—81no—
524C3×C4—122no—
525C3×C4—161no—
526C3×C4—241no—
527Dic3—21no—
528Dic3—32no—
529Dic3—42no—
530Dic3—62no—
531Dic3—61no—
532Dic3—121no—
533Dic3—121no—
534Dic3—241no—
535Dic3—481no—
536Dic3C4×S412yescomplemented
537C32—41noSylow
538C23—31no—
539C23—31no—
540C23—31no—
541C23—31no—
542C23—32no—
543C23—61no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—92no—
550C23—181no—
551C23—181no—
552C23—181no—
553C23—181no—
554C23—181no—
555C23—181no—
556C23#379911yescharacteristic
557C23C4×D3211yescharacteristic
558C23D3×#6711yescharacteristic
559D4—62no—
560D4—62no—
561D4—62no—
562D4—62no—
563D4—184no—
564D4—184no—
565D4—182no—
566D4—182no—
567D4—182no—
568D4—182no—
569C2×C4—31no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—31no—
574C2×C4—31no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—92no—
586C2×C4—92no—
587C2×C4—91no—
588C2×C4—92no—
589C2×C4—92no—
590C2×C4—91no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—181no—
594C2×C4—181no—
595C2×C4—181no—
596C2×C4—181no—
597C2×C4—181no—
598C2×C4—181no—
599C2×C4—181no—
600C2×C4—181no—
601C2×C4—181no—
602C2×C4—181no—
603C2×C4—181no—
604C2×C4—181no—
605C2×C4—181no—
606C2×C4—362no—
607C2×C4—362no—
608C2×C4D3×S411yescharacteristic
609C2×C3—31no—
610C2×C3—31no—
611C2×C3—31no—
612C2×C3—31no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3—41no—
616C2×C3—81no—
617C2×C3—81no—
618C2×C3—81no—
619C2×C3—122no—
620C2×C3C2×C4×S411yescharacteristic
621C2×C3D4×S411yescharacteristic
622C2×C3#591911yescharacteristic
623D3—82no—
624D3—122no—
625D3—242no—
626D3—482no—
627C22—31no—
628C22—31no—
629C22—31no—
630C22—31no—
631C22—31no—
632C22—31no—
633C22—31no—
634C22—31no—
635C22—31no—
636C22—61no—
637C22—62no—
638C22—62no—
639C22—62no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22—182no—
645C22—182no—
646C22—182no—
647C22—361no—
648C22—361no—
649C22—361no—
650C22—361no—
651C22#6390411yescharacteristic, complemented
652C22C2×D3×S411yescharacteristic
653C4—21no—
654C4—32no—
655C4—62no—
656C4—62no—
657C4—61no—
658C4—62no—
659C4—61no—
660C4—92no—
661C4—182no—
662C4—181no—
663C4—361no—
664C4—361no—
665C3—41no—
666C3—81no—
667C3#8530811yescharacteristic, complemented
668C2—31no—
669C2—31no—
670C2—31no—
671C2—31no—
672C2—122no—
673C2—362no—
674C2#1059642111yescharacteristic
675C2C4×D3×S411yescharacteristic
676C2#67×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC2×#64207C2
cosocleC22×C3×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4C4×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×S4C2semidirect
3. C2×(C3⋊S4)⋊C4C2semidirect
4. A4×Dic3⋊C4C2semidirect
5. #10593647C2semidirect
6. (C2×S4)⋊Dic3C2semidirect
7. Dic3×SL(2,ℤ/4)C2semidirect
18. #64207C4semidirect
23. C2×C4×S4D3semidirect
27. C22×Dic3⋊C4D3semidirect
47. C2×C3⋊S4C2×C4semidirect
50. C2×C3×S4C2×C4semidirect
56. Dic3×A4C2×C4semidirect
58. C3×SL(2,ℤ/4)D4semidirect
166. C2×S4C4×D3semidirect
204. C22×Dic3C4×D3semidirect
218. SL(2,ℤ/4)#67semidirect
245. Dic3⋊C4S4semidirect
258. C3×A4C4×D4semidirect
498. C22×C3#893semidirect
517. A4C4×#67semidirect
536. Dic3C4×S4semidirect
651. C22#63904semidirect
667. C3#85308semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
C2×#642072
C2×(C3⋊S4)⋊C42
A4×Dic3⋊C42
#105936472
(C2×S4)⋊Dic32
Dic3×SL(2,ℤ/4)2
#853083
#823403
#639044