Finite Group Databasebeta

← #11921970

Subgroups — #11921970

Order 1152.

892 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4×C3⋊S4C211yesmaximal, characteristic
2C2×(C3⋊S4)⋊C4C211yesmaximal, characteristic
3C2×(C3⋊S4)⋊C4C211yesmaximal, characteristic
4A4×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C3×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
6Dic3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
7SL(2,ℤ/4)⋊Dic3C211yesmaximal, characteristic, complemented
8C4⋊C4×S4—31nomaximal
9#74393—31nomaximal
10C22×C3⋊S4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×Dic3×A4C2211yescharacteristic
13C2×Dic3×A4C2211yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C4×C3⋊S4—22no—
16C2×C3×SL(2,ℤ/4)C2211yescharacteristic
17C2×C3×SL(2,ℤ/4)C2211yescharacteristic
18(C3⋊S4)⋊C4—22no—
19(C3⋊S4)⋊C4C414yescomplemented
20#63883—41nomaximal
21C2×C4×S4—31no—
22C2×C4×S4—31no—
23C2×C4×S4—31no—
24C2×C4×S4—61no—
25C2×C4×#67—31no—
26C22×Dic3⋊C4D311yescharacteristic, complemented
27C2×D3×C4⋊C4—31no—
28C2×#67⋊C4—31no—
29C2×#67⋊C4—31no—
30A4×C4⋊C4—31no—
31C4⋊SL(2,ℤ/4)—31no—
32SL(2,ℤ/4)⋊C4—31no—
33SL(2,ℤ/4)⋊C4D311yescharacteristic, complemented
34C3×#385—31no—
35#4989—31no—
36#4976—31no—
37#4975—31no—
38#4950—31no—
39#4942—31no—
40#4668—31no—
41#4667—31no—
42#4649—31no—
43#4647—31no—
44C22×C3×A4C2311yescharacteristic
45C2×C3⋊S4C2×C412yes—
46C2×C3⋊S4D412yes—
47C2×C3⋊S4Q812yes—
48C2×C4×C3⋊D3—41no—
49C3×C4×A4—21no—
50C2×#624—41no—
51C2×#624—41no—
52Dic3×A4—21no—
53Dic3×A4C2×C412yescomplemented
54C3⋊SL(2,ℤ/4)—21no—
55C3×SL(2,ℤ/4)—21no—
56C3×SL(2,ℤ/4)C2×C412yescomplemented
57C3×Dic3⋊C4—41no—
58C3×Dic3⋊C4—41no—
59Dic3⋊Dic3—41no—
60Dic3⋊Dic3—41no—
61D4×C4⋊C4—91noSylow
62C22×S4—31no—
63C22×S4—61no—
64C23×C3×C4C2×D311yescharacteristic
65C22×#67—31no—
66C23×Dic3C2×D311yescharacteristic
67C23×Dic3C2×D311yescharacteristic
68C22×C4×D3—31no—
69C22×C4×D3—31no—
70C22×C4×D3—31no—
71C2×C4×A4—21no—
72C2×C4×A4—31no—
73C2×C4×A4—31no—
74C2×C4×A4C2×D311yescharacteristic
75C2×SL(2,ℤ/4)—31no—
76C2×SL(2,ℤ/4)—61no—
77C2×SL(2,ℤ/4)C2×D311yescharacteristic
78C2×SL(2,ℤ/4)C2×D311yescharacteristic
79C4×S4—34no—
80C4×S4—62no—
81C4×S4—62no—
82C4×S4—64no—
83C2×C3×C4⋊C4—31no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C22⋊C4—31no—
87C2×#217—31no—
88C4×#67—62no—
89C4×#67—62no—
90C2×(C2×D3)⋊C4—31no—
91C2×(C2×D3)⋊C4—31no—
92C2×(C2×D3)⋊C4—31no—
93C2×C4⋊Dic3—31no—
94C2×C4⋊Dic3—31no—
95C2×Dic3⋊C4—31no—
96C2×Dic3⋊C4—31no—
97C2×Dic3⋊C4—31no—
98C2×Dic3⋊C4—31no—
99C2×Dic3⋊C4—31no—
100C2×Dic3⋊C4—31no—
101C2×Dic3⋊C4—31no—
102C2×Dic3⋊C4—31no—
103C2×Dic3⋊C4—31no—
104C2×C4×Dic3—31no—
105C2×C4×Dic3—31no—
106C2×C4×Dic3—31no—
107D3×C4⋊C4—61no—
108D3×C4⋊C4—61no—
109D3×C4⋊C4—61no—
110D3×C4⋊C4—61no—
111D3×C4⋊C4—121no—
112#67⋊C4—34no—
113#67⋊C4—34no—
114#67⋊C4—62no—
115#67⋊C4—62no—
116C3×#95—31no—
117C3×#95—31no—
118C3⋊#95—31no—
119C3⋊#95—31no—
120C3⋊#95—31no—
121C3⋊#95—31no—
122C22×C3⋊D3—41no—
123C2×C3×A4C22×C411yescharacteristic
124C2×C3×A4C2×D411yescharacteristic
125C2×C3×A4C2×Q811yescharacteristic
126C3⋊S4C4⋊C414yescomplemented
127C2×C32×C4—41no—
128C2×C3⋊Dic3—41no—
129C4×C3⋊D3—82no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—41no—
134#624—44no—
135#624—82no—
136C2×C4×D4—91no—
137C2×C4×D4—91no—
138C2×C4×D4—91no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—91no—
141#389—91no—
142#389—91no—
143#385—31no—
144#385—91no—
145#385—91no—
146#385—91no—
147#384—91no—
148#380—91no—
149#380—91no—
150C4×C4⋊C4—91no—
151C24×C3C22×D311yescharacteristic
152C23×D3—31no—
153C22×A4—21no—
154C22×A4C22×D311yescharacteristic
155C2×S4—32no—
156C2×S4—32no—
157C2×S4—32no—
158C2×S4—62no—
159C2×S4—62no—
160C2×S4—62no—
161C22×C3×C4—21no—
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4—61no—
169C2×#67—32no—
170C2×#67—32no—
171C2×#67—32no—
172C2×#67—32no—
173C2×#67—32no—
174C2×#67—32no—
175C22×Dic3—21no—
176C22×Dic3—31no—
177C22×Dic3—31no—
178C22×Dic3—31no—
179C22×Dic3—32no—
180C22×Dic3—31no—
181C22×Dic3—31no—
182C22×Dic3—31no—
183C22×Dic3—31no—
184C22×Dic3—61no—
185C22×Dic3C4×D312yescomplemented
186C2×C4×D3—34no—
187C2×C4×D3—61no—
188C2×C4×D3—62no—
189C2×C4×D3—61no—
190C2×C4×D3—61no—
191C2×C4×D3—61no—
192C2×C4×D3—62no—
193C2×C4×D3—62no—
194C2×C4×D3—61no—
195C2×C4×D3—61no—
196C2×C4×D3—121no—
197C2×C4×D3—121no—
198C2×C4×D3—121no—
199C2×C4×D3—121no—
200C2×C4×D3—121no—
201C2×C4×D3—241no—
202C4×A4—22no—
203C4×A4—21no—
204C4×A4—32no—
205C4×A4—61no—
206SL(2,ℤ/4)—21no—
207SL(2,ℤ/4)—62no—
208SL(2,ℤ/4)—61no—
209SL(2,ℤ/4)C4×D312yescomplemented
210C3×C4⋊C4—61no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—121no—
213C3×C4⋊C4—121no—
214C3×C22⋊C4—32no—
215C3×C22⋊C4—32no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4—61no—
218#217—61no—
219#217—61no—
220(C2×D3)⋊C4—34no—
221(C2×D3)⋊C4—62no—
222(C2×D3)⋊C4—62no—
223C4⋊Dic3—61no—
224C4⋊Dic3—61no—
225C4⋊Dic3—121no—
226C4⋊Dic3—121no—
227Dic3⋊C4—34no—
228Dic3⋊C4—31no—
229Dic3⋊C4—31no—
230Dic3⋊C4—31no—
231Dic3⋊C4—41no—
232Dic3⋊C4—61no—
233Dic3⋊C4—62no—
234Dic3⋊C4—61no—
235Dic3⋊C4—62no—
236Dic3⋊C4—62no—
237Dic3⋊C4—61no—
238Dic3⋊C4—121no—
239Dic3⋊C4S411yescharacteristic, complemented
240C4×Dic3—34no—
241C4×Dic3—62no—
242C4×Dic3—62no—
243C22×C32—41no—
244C2×C3⋊D3—42no—
245C2×C3⋊D3—42no—
246C2×C3⋊D3—42no—
247C3×A4C2×C4⋊C411yescharacteristic, complemented
248C32×C4—81no—
249C3⋊Dic3—81no—
250C3×Dic3—42no—
251C3×Dic3—42no—
252C3×Dic3—81no—
253C3×Dic3—81no—
254C22×D4—91no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—91no—
258C23×C4—91no—
259C23×C4—91no—
260C23×C4D3211yescharacteristic
261C4×D4—94no—
262C4×D4—94no—
263C4×D4—182no—
264C4×D4—182no—
265C4×D4—182no—
266C4×D4—182no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—91no—
270C2×C4⋊C4—91no—
271C2×C4⋊C4—91no—
272C2×C4⋊C4—91no—
273C2×C4⋊C4—91no—
274C2×C4⋊C4—91no—
275C2×C4⋊C4—91no—
276C2×C4⋊C4—91no—
277C2×C4⋊C4—91no—
278C2×C4⋊C4—91no—
279C2×C4⋊C4—91no—
280C2×C4⋊C4—181no—
281C2×C4⋊C4—181no—
282C2×C4⋊C4—181no—
283C2×C4⋊C4—181no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—91no—
287C2×C22⋊C4—91no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—91no—
290C2×C42—91no—
291C2×C42—91no—
292C2×C42—91no—
293#95—31no—
294#95—31no—
295#95—91no—
296#95—91no—
297#95—91no—
298#95—91no—
299C23×C3—31no—
300C23×C3—31no—
301C23×C3—31no—
302C23×C3—31no—
303C23×C3C2×C4×D311yescharacteristic
304C23×C3D3×D411yescharacteristic
305C23×C3D3×Q811yescharacteristic
306C22×D3—32no—
307C22×D3—32no—
308C22×D3—32no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—121no—
314C22×D3—241no—
315C2×A4—21no—
316C2×A4—21no—
317C2×A4—21no—
318C2×A4C2×C4×D311yescharacteristic
319C2×A4D3×D411yescharacteristic
320C2×A4D3×Q811yescharacteristic
321S4—34no—
322S4—64no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—31no—
326C2×C3×C4—41no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—81no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4C2×S411yescharacteristic
345#67—34no—
346#67—34no—
347#67—34no—
348#67—34no—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—31no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—31no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—121no—
371C2×Dic3—241no—
372C2×Dic3C2×S411yescharacteristic
373C2×Dic3C2×S411yescharacteristic
374C4×D3—64no—
375C4×D3—64no—
376C4×D3—124no—
377C4×D3—122no—
378C4×D3—122no—
379C4×D3—122no—
380C4×D3—122no—
381C4×D3—242no—
382C4×D3—242no—
383C4×D3—244no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C3⋊D3—44no—
388C24—91no—
389C24C2×D3211yescharacteristic
390C2×D4—92no—
391C2×D4—92no—
392C2×D4—92no—
393C2×D4—92no—
394C2×D4—92no—
395C2×D4—92no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—32no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—91no—
407C22×C4—91no—
408C22×C4—94no—
409C22×C4—91no—
410C22×C4—91no—
411C22×C4—92no—
412C22×C4—91no—
413C22×C4—91no—
414C22×C4—91no—
415C22×C4—181no—
416C22×C4—181no—
417C22×C4—181no—
418C22×C4—181no—
419C22×C4—182no—
420C22×C4—181no—
421C22×C4—182no—
422C22×C4—182no—
423C22×C4—181no—
424C22×C4—181no—
425C22×C4—361no—
426C22×C4—361no—
427C4⋊C4—31no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—91no—
431C4⋊C4—91no—
432C4⋊C4—94no—
433C4⋊C4—91no—
434C4⋊C4—121no—
435C4⋊C4—181no—
436C4⋊C4—181no—
437C4⋊C4—181no—
438C4⋊C4—181no—
439C4⋊C4—181no—
440C4⋊C4—182no—
441C4⋊C4—182no—
442C4⋊C4—182no—
443C4⋊C4—361no—
444C22⋊C4—32no—
445C22⋊C4—32no—
446C22⋊C4—61no—
447C22⋊C4—61no—
448C22⋊C4—94no—
449C22⋊C4—182no—
450C22⋊C4—182no—
451C22⋊C4—181no—
452C22⋊C4—181no—
453C42—94no—
454C42—182no—
455C42—182no—
456C22×C3—31no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—31no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—31no—
463C22×C3—31no—
464C22×C3—31no—
465C22×C3—41no—
466C22×C3—61no—
467C22×C3—81no—
468C22×C3D3×C4⋊C411yescharacteristic, complemented
469C22×C3C22×S411yescharacteristic
470C2×D3—34no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—122no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—242no—
481C2×D3—242no—
482C2×D3—242no—
483A4—21no—
484A4D3×C4⋊C411yescharacteristic, complemented
485C3×C4—21no—
486C3×C4—62no—
487C3×C4—62no—
488C3×C4—61no—
489C3×C4—81no—
490C3×C4—82no—
491C3×C4—122no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—241no—
495Dic3—21no—
496Dic3—34no—
497Dic3—32no—
498Dic3—42no—
499Dic3—61no—
500Dic3—81no—
501Dic3—121no—
502Dic3—121no—
503Dic3—241no—
504Dic3—242no—
505Dic3C4×S412yescomplemented
506C32—41noSylow
507C23—31no—
508C23—31no—
509C23—31no—
510C23—31no—
511C23—92no—
512C23—92no—
513C23—92no—
514C23—181no—
515C23—181no—
516C23—181no—
517C23—181no—
518C23#379711yescharacteristic
519C23C4×D3211yescharacteristic
520C23#381111yescharacteristic
521D4—94no—
522D4—94no—
523D4—94no—
524D4—94no—
525C2×C4—31no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—92no—
545C2×C4—92no—
546C2×C4—92no—
547C2×C4—92no—
548C2×C4—92no—
549C2×C4—91no—
550C2×C4—92no—
551C2×C4—91no—
552C2×C4—184no—
553C2×C4—184no—
554C2×C4—181no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4—181no—
562C2×C4—181no—
563C2×C4—181no—
564C2×C4—181no—
565C2×C4—362no—
566C2×C4—362no—
567C2×C4—362no—
568C2×C4—362no—
569C2×C4D3×S411yescharacteristic
570C2×C3—31no—
571C2×C3—31no—
572C2×C3—31no—
573C2×C3—31no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—81no—
578C2×C3—81no—
579C2×C3—81no—
580C2×C3C2×C4×S411yescharacteristic
581C2×C3D4×S411yescharacteristic
582C2×C3Q8×S411yescharacteristic
583D3—64no—
584D3—124no—
585D3—244no—
586C22—31no—
587C22—31no—
588C22—31no—
589C22—31no—
590C22—31no—
591C22—31no—
592C22—31no—
593C22—31no—
594C22—31no—
595C22—61no—
596C22—94no—
597C22—182no—
598C22—182no—
599C22—182no—
600C22—182no—
601C22—182no—
602C22—182no—
603C22#6388311yescharacteristic, complemented
604C22C2×D3×S411yescharacteristic
605C4—21no—
606C4—32no—
607C4—62no—
608C4—62no—
609C4—61no—
610C4—61no—
611C4—94no—
612C4—92no—
613C4—121no—
614C4—121no—
615C4—181no—
616C4—361no—
617C4—361no—
618C3—41no—
619C3—81no—
620C3C4⋊C4×S411yescharacteristic, complemented
621C2—31no—
622C2—31no—
623C2—31no—
624C2—31no—
625C2—184no—
626C2(C3⋊S4)⋊Q811yescharacteristic
627C2C4×D3×S411yescharacteristic
628C2#1059647311yescharacteristic
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×A4C2×C4⋊C4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. A4×Dic3⋊C4C2semidirect
5. C3×SL(2,ℤ/4)⋊C4C2semidirect
6. Dic3⋊SL(2,ℤ/4)C2semidirect
7. SL(2,ℤ/4)⋊Dic3C2semidirect
19. (C3⋊S4)⋊C4C4semidirect
26. C22×Dic3⋊C4D3semidirect
33. SL(2,ℤ/4)⋊C4D3semidirect
53. Dic3×A4C2×C4semidirect
56. C3×SL(2,ℤ/4)C2×C4semidirect
126. C3⋊S4C4⋊C4semidirect
185. C22×Dic3C4×D3semidirect
209. SL(2,ℤ/4)C4×D3semidirect
239. Dic3⋊C4S4semidirect
247. C3×A4C2×C4⋊C4semidirect
468. C22×C3D3×C4⋊C4semidirect
484. A4D3×C4⋊C4semidirect
505. Dic3C4×S4semidirect
603. C22#63883semidirect
620. C3C4⋊C4×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊S42
C2×(C3⋊S4)⋊C42
A4×Dic3⋊C42
C3×SL(2,ℤ/4)⋊C42
Dic3⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊Dic32
C4⋊C4×S43
#743933
#638834