Finite Group Databasebeta

← #11921957

Subgroups — #11921957

Order 1152.

700 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C2×#64204C211yesmaximal, characteristic, complemented
3A4×Dic3⋊C4C211yesmaximal, characteristic, complemented
4#10593645C211yesmaximal, characteristic, complemented
5(C2×S4)⋊Dic3C211yesmaximal, characteristic
6(C2×S4)⋊Dic3C211yesmaximal, characteristic
7SL(2,ℤ/4)⋊Dic3C211yesmaximal, characteristic, complemented
8#85311—31nomaximal
9#82360—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
17C3×C4×S4—22no—
18#64204—22no—
19#63851—41nomaximal
20C2×C4×S4D311yescharacteristic
21C2×A4⋊Q8—31no—
22C2×C3×C4×D4—31no—
23C2×#924—31no—
24C22×Dic3⋊C4D311yescharacteristic, complemented
25C2×#878—31no—
26A4×C4⋊C4—31no—
27(C2×S4)⋊C4—31no—
28(C2×S4)⋊C4—31no—
29SL(2,ℤ/4)⋊C4—31no—
30SL(2,ℤ/4)⋊C4—31no—
31SL(2,ℤ/4)⋊C4—61no—
32#5220—31no—
33#5220—31no—
34#5210—31no—
35#5210—31no—
36#4982—31no—
37#4979—31no—
38#4950—31no—
39#4949—31no—
40#4948—31no—
41#4933—31no—
42#4928—31no—
43C22×C3×A4C2311yescharacteristic
44C2×C3×S4—21no—
45C2×C3×S4—21no—
46C2×C3×S4Q812yescomplemented
47C2×C3×C4×D3—41no—
48C3×C4×A4—21no—
49C2×#627—41no—
50Dic3×A4—21no—
51Dic3×A4—21no—
52C3⋊SL(2,ℤ/4)—21no—
53C3⋊SL(2,ℤ/4)—21no—
54C3×SL(2,ℤ/4)D412yes—
55#3750—41no—
56C3×Dic3⋊C4—41no—
57Dic3⋊Dic3—41no—
58(C2×D3)⋊Dic3—41no—
59(C2×D3)⋊Dic3—41no—
60#2449—91noSylow
61C22×S4C2×D311yescharacteristic
62C22×C3×D4—31no—
63C23×C3×C4—31no—
64C23×C3×C4C2×D311yescharacteristic
65C23×Dic3C2×D311yescharacteristic
66C23×Dic3C2×D311yescharacteristic
67C22×Dic6—31no—
68C2×C4×A4—21no—
69C2×C4×A4—31no—
70C2×C4×A4—31no—
71C2×C4×A4C2×D311yescharacteristic
72C2×SL(2,ℤ/4)—31no—
73C2×SL(2,ℤ/4)—31no—
74C2×SL(2,ℤ/4)—61no—
75C2×SL(2,ℤ/4)—61no—
76C2×SL(2,ℤ/4)C2×D311yescharacteristic
77C4×S4—22no—
78A4⋊Q8—62no—
79C3×C4×D4—62no—
80C3×C4×D4—62no—
81C2×C3×C4⋊C4—31no—
82C2×C3×C22⋊C4—31no—
83C2×C3×C22⋊C4—31no—
84C2×C3×C42—31no—
85C2×#217—31no—
86C2×#217—31no—
87C2×#217—31no—
88C2×#217—31no—
89C2×#217—31no—
90C2×#217—31no—
91C2×C4⋊Dic3—31no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—31no—
94#924—61no—
95#924—61no—
96#924—61no—
97#924—61no—
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×Dic3⋊C4—31no—
105C2×Dic3⋊C4—31no—
106#896—121no—
107#878—62no—
108#878—62no—
109C3⋊#95—31no—
110C3⋊#95—31no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C3⋊#95—31no—
114C3⋊#95—31no—
115C3⋊#95—31no—
116C3⋊#95—31no—
117C22×C3×D3—41no—
118C2×C3×A4C2×D411yescharacteristic
119C2×C3×A4C2×Q811yescharacteristic
120C2×C3×A4#4111yescharacteristic
121C3×S4—22no—
122C2×C32×C4—41no—
123C2×C3⋊Dic3—41no—
124C2×C3⋊Dic3—41no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—41no—
127C2×C3×Dic3—41no—
128C3×C4×D3—82no—
129#627—82no—
130C2×C22⋊Q8—91no—
131C2×C22⋊Q8—91no—
132C2×C4×D4—31no—
133C22×C4⋊C4—31no—
134#399—91no—
135#398—91no—
136#396—91no—
137#395—91no—
138#389—91no—
139#387—91no—
140#386—91no—
141#386—91no—
142#385—91no—
143#385—91no—
144#385—91no—
145C24×C3—31no—
146C24×C3C22×D311yescharacteristic
147C22×A4—21no—
148C22×A4C22×D311yescharacteristic
149C2×S4—21no—
150C2×S4—21no—
151C2×S4Dic612yescomplemented
152C2×C3×D4—32no—
153C2×C3×D4—32no—
154C2×C3×D4—61no—
155C2×C3×D4—61no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C22×C3×C4—21no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—61no—
165C22×C3×C4—62no—
166C22×C3×C4—61no—
167C22×C3×C4—61no—
168C22×C3×C4—121no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—31no—
172C22×Dic3—31no—
173C22×Dic3—31no—
174C22×Dic3—31no—
175C22×Dic3—31no—
176C22×Dic3—31no—
177C22×Dic3—31no—
178C22×Dic3—31no—
179C22×Dic3—61no—
180C22×Dic3—61no—
181C2×C4×D3—41no—
182C2×Dic6—61no—
183C2×Dic6—62no—
184C2×Dic6—61no—
185C2×Dic6—121no—
186C2×Dic6—121no—
187C4×A4—21no—
188C4×A4—41no—
189C4×A4—61no—
190C4×A4—61no—
191SL(2,ℤ/4)—62no—
192SL(2,ℤ/4)—61no—
193SL(2,ℤ/4)—61no—
194SL(2,ℤ/4)—121no—
195SL(2,ℤ/4)#6712yes—
196C3×C4⋊C4—62no—
197C3×C4⋊C4—121no—
198C3×C22⋊C4—32no—
199C3×C22⋊C4—32no—
200C3×C22⋊C4—62no—
201C3×C42—62no—
202#217—61no—
203#217—61no—
204#217—62no—
205#217—61no—
206#217—61no—
207#217—61no—
208#217—61no—
209#217—61no—
210#217—61no—
211#217—121no—
212(C2×D3)⋊C4—121no—
213(C2×D3)⋊C4—121no—
214C4⋊Dic3—61no—
215C4⋊Dic3—62no—
216C4⋊Dic3—62no—
217C4⋊Dic3—61no—
218Dic3⋊C4—31no—
219Dic3⋊C4—31no—
220Dic3⋊C4—31no—
221Dic3⋊C4—61no—
222Dic3⋊C4—62no—
223Dic3⋊C4—62no—
224Dic3⋊C4—61no—
225Dic3⋊C4—61no—
226Dic3⋊C4—121no—
227Dic3⋊C4—121no—
228Dic3⋊C4—121no—
229Dic3⋊C4—241no—
230Dic3⋊C4S411yescharacteristic, complemented
231C22×C32—41no—
232C2×C3×D3—42no—
233C2×C3×D3—81no—
234C2×C3×D3—81no—
235C3×A4C22⋊Q811yescharacteristic, complemented
236C32×C4—81no—
237C3⋊Dic3—81no—
238C3⋊Dic3—81no—
239C3×Dic3—42no—
240C3×Dic3—81no—
241C3×Dic3—81no—
242C22×Q8—91no—
243C22×D4—31no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4D3211yescharacteristic
248C22⋊Q8—182no—
249C22⋊Q8—182no—
250C22⋊Q8—181no—
251C22⋊Q8—181no—
252C22⋊Q8—181no—
253C22⋊Q8—181no—
254C4×D4—62no—
255C4×D4—62no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—91no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—91no—
260C2×C4⋊C4—91no—
261C2×C4⋊C4—91no—
262C2×C4⋊C4—91no—
263C2×C4⋊C4—91no—
264C2×C4⋊C4—91no—
265C2×C4⋊C4—91no—
266C2×C4⋊C4—91no—
267C2×C4⋊C4—91no—
268C2×C22⋊C4—31no—
269C2×C22⋊C4—31no—
270C2×C22⋊C4—91no—
271C2×C22⋊C4—91no—
272C2×C22⋊C4—91no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—91no—
275C2×C22⋊C4—91no—
276C2×C42—31no—
277#95—91no—
278#95—91no—
279#95—91no—
280#95—91no—
281#95—91no—
282#95—91no—
283#95—91no—
284#95—91no—
285C23×C3—31no—
286C23×C3—31no—
287C23×C3—31no—
288C23×C3—31no—
289C23×C3—32no—
290C23×C3—61no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3#23511yescharacteristic
297C23×C3D3×D411yescharacteristic
298C23×C3D3×Q811yescharacteristic
299C22×D3—41no—
300C2×A4—21no—
301C2×A4—21no—
302C2×A4—21no—
303C2×A4C2×Dic611yescharacteristic
304C2×A4#23511yescharacteristic
305C2×A4C2×#6711yescharacteristic
306S4—22no—
307C3×D4—62no—
308C3×D4—62no—
309C3×D4—62no—
310C3×D4—62no—
311C2×C3×C4—32no—
312C2×C3×C4—32no—
313C2×C3×C4—32no—
314C2×C3×C4—31no—
315C2×C3×C4—41no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—81no—
327C2×C3×C4—121no—
328C2×C3×C4—121no—
329C2×C3×C4—122no—
330C2×C3×C4—122no—
331C2×C3×C4C2×S411yescharacteristic
332C2×Dic3—32no—
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—41no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—121no—
356C2×Dic3—121no—
357C2×Dic3—241no—
358C2×Dic3—241no—
359C2×Dic3C2×S411yescharacteristic
360C2×Dic3C2×S411yescharacteristic
361C4×D3—82no—
362Dic6—122no—
363Dic6—122no—
364Dic6—242no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C3×D3—82no—
369C24—31no—
370C24C2×D3211yescharacteristic
371C2×Q8—181no—
372C2×Q8—182no—
373C2×Q8—181no—
374C2×Q8—361no—
375C2×D4—32no—
376C2×D4—32no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C22×C4—21no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—31no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—91no—
398C22×C4—91no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—121no—
402C22×C4—181no—
403C22×C4—181no—
404C4⋊C4—31no—
405C4⋊C4—62no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—91no—
409C4⋊C4—181no—
410C4⋊C4—181no—
411C4⋊C4—182no—
412C4⋊C4—182no—
413C4⋊C4—181no—
414C4⋊C4—182no—
415C4⋊C4—181no—
416C4⋊C4—182no—
417C4⋊C4—181no—
418C4⋊C4—361no—
419C22⋊C4—32no—
420C22⋊C4—32no—
421C22⋊C4—62no—
422C22⋊C4—181no—
423C22⋊C4—181no—
424C22⋊C4—181no—
425C22⋊C4—181no—
426C22⋊C4—182no—
427C22⋊C4—181no—
428C22⋊C4—181no—
429C22⋊C4—181no—
430C22⋊C4—181no—
431C22⋊C4—361no—
432C42—62no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—41no—
443C22×C3—61no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—62no—
447C22×C3—81no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3—121no—
452C22×C3#89611yescharacteristic, complemented
453C22×C3C22×S411yescharacteristic
454C2×D3—42no—
455C2×D3—81no—
456C2×D3—81no—
457A4—21no—
458A4#92411yescharacteristic, complemented
459C3×C4—21no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—61no—
464C3×C4—81no—
465C3×C4—161no—
466C3×C4—241no—
467C3×C4—241no—
468Dic3—21no—
469Dic3—21no—
470Dic3—42no—
471Dic3—61no—
472Dic3—61no—
473Dic3—121no—
474Dic3—121no—
475Dic3—121no—
476Dic3—121no—
477Dic3—241no—
478Dic3—241no—
479Dic3—242no—
480Dic3—481no—
481C32—41noSylow
482C23—31no—
483C23—31no—
484C23—31no—
485C23—31no—
486C23—32no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23D3×Dic611yescharacteristic
494C23#379811yescharacteristic
495C23D3×#6711yescharacteristic
496Q8—362no—
497Q8—362no—
498D4—62no—
499D4—62no—
500D4—62no—
501D4—62no—
502C2×C4—32no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—92no—
519C2×C4—91no—
520C2×C4—91no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—181no—
524C2×C4—181no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4—181no—
528C2×C4—181no—
529C2×C4—181no—
530C2×C4—181no—
531C2×C4—181no—
532C2×C4—181no—
533C2×C4—181no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4—181no—
542C2×C4D3×S411yescharacteristic
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—81no—
551C2×C3—81no—
552C2×C3—81no—
553C2×C3—122no—
554C2×C3#591211yescharacteristic
555C2×C3D4×S411yescharacteristic
556C2×C3Q8×S411yescharacteristic
557D3—82no—
558C22—31no—
559C22—31no—
560C22—31no—
561C22—31no—
562C22—31no—
563C22—31no—
564C22—31no—
565C22—31no—
566C22—31no—
567C22—61no—
568C22—62no—
569C22—62no—
570C22—62no—
571C22—121no—
572C22—121no—
573C22—121no—
574C22—121no—
575C22#6385111yescharacteristic, complemented
576C22C2×D3×S411yescharacteristic
577C4—21no—
578C4—62no—
579C4—62no—
580C4—61no—
581C4—61no—
582C4—62no—
583C4—61no—
584C4—181no—
585C4—181no—
586C4—361no—
587C4—361no—
588C4—361no—
589C4—361no—
590C3—41no—
591C3—81no—
592C3#8531111yescharacteristic, complemented
593C2—31no—
594C2—31no—
595C2—31no—
596C2—31no—
597C2—122no—
598C2Dic6×S411yescharacteristic
599C2#1059642611yescharacteristic
600C2#67×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC2×C4D3×S4
hypocenterC3×A4C22⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#64204C2semidirect
3. A4×Dic3⋊C4C2semidirect
4. #10593645C2semidirect
7. SL(2,ℤ/4)⋊Dic3C2semidirect
24. C22×Dic3⋊C4D3semidirect
46. C2×C3×S4Q8semidirect
151. C2×S4Dic6semidirect
230. Dic3⋊C4S4semidirect
235. C3×A4C22⋊Q8semidirect
452. C22×C3#896semidirect
458. A4#924semidirect
575. C22#63851semidirect
592. C3#85311semidirect

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×#642042
A4×Dic3⋊C42
#105936452
(C2×S4)⋊Dic32
SL(2,ℤ/4)⋊Dic32
#853113
#823603
#638514