Finite Group Databasebeta

← #11922842

Subgroups — #11922842

Order 1152.

874 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
2C2×#64259C211yesmaximal, characteristic, complemented
3C3×A4×C22⋊C4C211yesmaximal, characteristic
4#10593666C211yesmaximal, characteristic
5C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
6#10593645C211yesmaximal, characteristic, complemented
7#10593645C211yesmaximal, characteristic, complemented
8#85293—33nomaximal
9#82110—31nomaximal
10C23×C3×A4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3⋊SL(2,ℤ/4)—22no—
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
16C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
17C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
18#64259—22no—
19#64087—41nomaximal
20C22×SL(2,ℤ/4)—33no—
21C2×A4⋊Q8—33no—
22C22×C3×C22⋊C4D311yescharacteristic
23C22×#217—31no—
24C2×#924—31no—
25C2×#878—31no—
26A4×C22⋊C4D313yes—
27#5432—33no—
28C4⋊SL(2,ℤ/4)—33no—
29SL(2,ℤ/4)⋊C4—33no—
30SL(2,ℤ/4)⋊C4—33no—
31#5247—31no—
32#5212—31no—
33#5210—31no—
34#5210—31no—
35#4950—31no—
36#4949—31no—
37#4948—31no—
38#4948—31no—
39#4947—31no—
40#4946—31no—
41#4942—31no—
42C22×C3×A4—21no—
43C22×C3×A4—21no—
44C22×C3×A4Q812yescomplemented
45C22×C3×A4C2311yescharacteristic
46C22×C3⋊Dic3—41no—
47C2×#634—41no—
48C3×C4×A4—21no—
49C3×C4×A4—21no—
50C3⋊SL(2,ℤ/4)—21no—
51C3⋊SL(2,ℤ/4)—21no—
52C3⋊SL(2,ℤ/4)—21no—
53C3⋊SL(2,ℤ/4)D412yes—
54C32×C22⋊C4—41no—
55#3757—41no—
56C4⋊(C3⋊Dic3)—41no—
57#3750—41no—
58#3750—41no—
59#2437—91noSylow
60C25×C3C2×D311yescharacteristic
61C23×A4C2×D313yes—
62C23×C3×C4C2×D311yescharacteristic
63C23×C3×C4C2×D311yescharacteristic
64C23×Dic3—31no—
65C23×Dic3—31no—
66C22×Dic6—31no—
67C2×C4×A4C2×D313yes—
68C2×C4×A4C2×D313yes—
69C2×SL(2,ℤ/4)—33no—
70C2×SL(2,ℤ/4)—33no—
71C2×SL(2,ℤ/4)—33no—
72C2×SL(2,ℤ/4)—33no—
73C2×SL(2,ℤ/4)—66no—
74A4⋊Q8—66no—
75C2×C3×C22⋊C4—31no—
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—
84C2×#217—31no—
85C2×#217—31no—
86C2×#217—31no—
87C2×#217—31no—
88C2×#217—31no—
89C2×#217—62no—
90C2×#217—62no—
91C2×C4⋊Dic3—31no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—31no—
94#924—62no—
95#924—62no—
96C2×Dic3⋊C4—31no—
97C2×Dic3⋊C4—31no—
98C2×Dic3⋊C4—31no—
99C2×Dic3⋊C4—31no—
100C2×Dic3⋊C4—31no—
101C2×Dic3⋊C4—31no—
102#878—61no—
103#878—61no—
104#878—61no—
105#878—61no—
106#878—123no—
107C3⋊#95—31no—
108C3⋊#95—31no—
109C3⋊#95—31no—
110C3⋊#95—31no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C23×C32—41no—
114C2×C3×A4—22no—
115C2×C3×A4C2×D411yescharacteristic
116C2×C3×A4C2×Q811yescharacteristic
117C2×C3×A4#4111yescharacteristic
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C3⋊Dic3—41no—
121C2×C3⋊Dic3—41no—
122C2×C3⋊Dic3—41no—
123C2×C3⋊Dic3—41no—
124C2×C3⋊Dic3—82no—
125#634—82no—
126C2×C22⋊Q8—91no—
127C2×C22⋊Q8—91no—
128C22×C22⋊C4—91no—
129C22×C22⋊C4C3⋊D311yescharacteristic
130#399—91no—
131#396—91no—
132#396—91no—
133#393—91no—
134#385—91no—
135#385—91no—
136#385—91no—
137#385—91no—
138#380—91no—
139#380—91no—
140#379—91no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—31no—
144C24×C3—31no—
145C24×C3—32no—
146C24×C3—61no—
147C24×C3—61no—
148C24×C3Dic612yescomplemented
149C24×C3C22×D311yescharacteristic
150C22×A4—23no—
151C22×A4—23no—
152C22×A4Dic616yescomplemented
153C22×A4C22×D313yes—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—61no—
161C22×C3×C4—61no—
162C22×Dic3—31no—
163C22×Dic3—31no—
164C22×Dic3—31no—
165C22×Dic3—31no—
166C22×Dic3—31no—
167C22×Dic3—31no—
168C22×Dic3—31no—
169C22×Dic3—31no—
170C22×Dic3—62no—
171C22×Dic3—62no—
172C22×Dic3—61no—
173C22×Dic3—62no—
174C22×Dic3—61no—
175C22×Dic3—123no—
176C22×Dic3—121no—
177C22×Dic3—121no—
178C2×Dic6—62no—
179C2×Dic6—61no—
180C2×Dic6—61no—
181C2×Dic6—123no—
182C2×Dic6—121no—
183C4×A4—23no—
184C4×A4—23no—
185SL(2,ℤ/4)—36no—
186SL(2,ℤ/4)—63no—
187SL(2,ℤ/4)—63no—
188SL(2,ℤ/4)—63no—
189C3×C22⋊C4—31no—
190C3×C22⋊C4—31no—
191C3×C22⋊C4—31no—
192C3×C22⋊C4—43no—
193C3×C22⋊C4—61no—
194C3×C22⋊C4S411yescharacteristic
195#217—32no—
196#217—32no—
197#217—62no—
198#217—62no—
199#217—61no—
200#217—61no—
201#217—61no—
202#217—62no—
203#217—61no—
204#217—61no—
205#217—61no—
206#217—61no—
207#217—61no—
208#217—123no—
209#217—121no—
210#217—121no—
211C4⋊Dic3—62no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—123no—
215C4⋊Dic3—121no—
216Dic3⋊C4—62no—
217Dic3⋊C4—62no—
218Dic3⋊C4—61no—
219Dic3⋊C4—61no—
220Dic3⋊C4—61no—
221Dic3⋊C4—61no—
222Dic3⋊C4—123no—
223Dic3⋊C4—123no—
224Dic3⋊C4—121no—
225Dic3⋊C4—121no—
226C22×C32—42no—
227C22×C32—41no—
228C22×C32—81no—
229C22×C32—81no—
230C3×A4C22⋊Q811yescharacteristic, complemented
231C32×C4—81no—
232C32×C4—81no—
233C3⋊Dic3—42no—
234C3⋊Dic3—81no—
235C3⋊Dic3—81no—
236C3⋊Dic3—81no—
237C25C2×C3⋊D311yescharacteristic
238C22×Q8—91no—
239C23×C4—91no—
240C23×C4—91no—
241C23×C4C2×C3⋊D311yescharacteristic
242C23×C4C2×C3⋊D311yescharacteristic
243C22⋊Q8—181no—
244C22⋊Q8—181no—
245C22⋊Q8—181no—
246C22⋊Q8—182no—
247C22⋊Q8—182no—
248C22⋊Q8—181no—
249C2×C4⋊C4—91no—
250C2×C4⋊C4—91no—
251C2×C4⋊C4—91no—
252C2×C4⋊C4—91no—
253C2×C4⋊C4—91no—
254C2×C4⋊C4—91no—
255C2×C4⋊C4—91no—
256C2×C4⋊C4—91no—
257C2×C4⋊C4—91no—
258C2×C22⋊C4—31no—
259C2×C22⋊C4—31no—
260C2×C22⋊C4—31no—
261C2×C22⋊C4—31no—
262C2×C22⋊C4—91no—
263C2×C22⋊C4—91no—
264C2×C22⋊C4—91no—
265C2×C22⋊C4—91no—
266C2×C22⋊C4—91no—
267C2×C22⋊C4—91no—
268C2×C22⋊C4—91no—
269C2×C22⋊C4—91no—
270C2×C22⋊C4—91no—
271C2×C22⋊C4—91no—
272C2×C22⋊C4—182no—
273C2×C22⋊C4—182no—
274#95—91no—
275#95—91no—
276#95—91no—
277#95—91no—
278#95—91no—
279#95—91no—
280C23×C3—22no—
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—31no—
284C23×C3—31no—
285C23×C3—32no—
286C23×C3—32no—
287C23×C3—32no—
288C23×C3—31no—
289C23×C3—43no—
290C23×C3—61no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—62no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—61no—
299C23×C3—62no—
300C23×C3—62no—
301C23×C3—61no—
302C23×C3—122no—
303C23×C3C2×Dic611yescharacteristic
304C23×C3D3×D411yescharacteristic
305C23×C3#23711yescharacteristic
306C23×C3C2×S411yescharacteristic
307C2×A4—26no—
308C2×A4C2×Dic613yes—
309C2×A4D3×D413yes—
310C2×A4#23713yes—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—43no—
314C2×C3×C4—43no—
315C2×C3×C4—61no—
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×C4C2×S411yescharacteristic
324C2×C3×C4C2×S411yescharacteristic
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
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—123no—
350C2×Dic3—123no—
351C2×Dic3—123no—
352C2×Dic3—123no—
353C2×Dic3—122no—
354C2×Dic3—124no—
355C2×Dic3—122no—
356C2×Dic3—246no—
357Dic6—122no—
358Dic6—122no—
359Dic6—246no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—82no—
364C24—21no—
365C24—21no—
366C24—31no—
367C24—31no—
368C24—32no—
369C24—61no—
370C24—61no—
371C24#63412yescomplemented
372C24C22×C3⋊D311yescharacteristic
373C2×Q8—181no—
374C2×Q8—182no—
375C2×Q8—181no—
376C2×Q8—361no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—91no—
386C22×C4—91no—
387C22×C4—91no—
388C22×C4—91no—
389C22×C4—91no—
390C22×C4—91no—
391C22×C4—91no—
392C22×C4—91no—
393C22×C4—182no—
394C22×C4—182no—
395C22×C4—181no—
396C22×C4—182no—
397C22×C4—181no—
398C22×C4—361no—
399C22×C4—361no—
400C4⋊C4—181no—
401C4⋊C4—181no—
402C4⋊C4—181no—
403C4⋊C4—182no—
404C4⋊C4—182no—
405C4⋊C4—182no—
406C4⋊C4—181no—
407C4⋊C4—181no—
408C4⋊C4—181no—
409C4⋊C4—361no—
410C4⋊C4—361no—
411C4⋊C4—361no—
412C22⋊C4—31no—
413C22⋊C4—31no—
414C22⋊C4—31no—
415C22⋊C4—61no—
416C22⋊C4—92no—
417C22⋊C4—92no—
418C22⋊C4—182no—
419C22⋊C4—182no—
420C22⋊C4—181no—
421C22⋊C4—181no—
422C22⋊C4—181no—
423C22⋊C4—181no—
424C22⋊C4—182no—
425C22⋊C4—181no—
426C22⋊C4—181no—
427C22⋊C4—181no—
428C22⋊C4—181no—
429C22⋊C4—361no—
430C22⋊C4—361no—
431C22⋊C4C3⋊S411yescharacteristic
432C22×C3—21no—
433C22×C3—21no—
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—31no—
443C22×C3—32no—
444C22×C3—46no—
445C22×C3—43no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—62no—
449C22×C3—62no—
450C22×C3—61no—
451C22×C3—62no—
452C22×C3—62no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—83no—
456C22×C3—83no—
457C22×C3—121no—
458C22×C3—121no—
459C22×C3#87811yescharacteristic, complemented
460C22×C3A4⋊Q812yescomplemented
461C22×C3C22×S411yescharacteristic
462A4#87813yescomplemented
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—61no—
466C3×C4—61no—
467C3×C4—83no—
468C3×C4—83no—
469Dic3—62no—
470Dic3—62no—
471Dic3—126no—
472Dic3—121no—
473Dic3—121no—
474Dic3—121no—
475Dic3—121no—
476Dic3—121no—
477Dic3—121no—
478Dic3—243no—
479Dic3—243no—
480Dic3—243no—
481C32—41noSylow
482C23—22no—
483C23—31no—
484C23—31no—
485C23—31no—
486C23—31no—
487C23—32no—
488C23—32no—
489C23—32no—
490C23—31no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23—62no—
497C23—61no—
498C23—61no—
499C23—61no—
500C23—62no—
501C23—62no—
502C23—61no—
503C23—122no—
504C23C2×#63411yescharacteristic
505C23D4×C3⋊D311yescharacteristic
506C23#383011yescharacteristic
507C23C2×C3⋊S411yescharacteristic
508Q8—362no—
509Q8—362no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—181no—
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×C4—181no—
543C2×C4—181no—
544C2×C4—362no—
545C2×C4—364no—
546C2×C4—362no—
547C2×C4C2×C3⋊S411yescharacteristic
548C2×C4C2×C3⋊S411yescharacteristic
549C2×C3—22no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—31no—
553C2×C3—31no—
554C2×C3—43no—
555C2×C3—43no—
556C2×C3—43no—
557C2×C3—62no—
558C2×C3—86no—
559C2×C3C2×A4⋊Q811yescharacteristic
560C2×C3D4×S411yescharacteristic
561C2×C3#591411yescharacteristic
562C22—21no—
563C22—21no—
564C22—31no—
565C22—31no—
566C22—31no—
567C22—31no—
568C22—31no—
569C22—31no—
570C22—31no—
571C22—31no—
572C22—31no—
573C22—32no—
574C22—62no—
575C22—62no—
576C22—62no—
577C22—62no—
578C22—61no—
579C22—62no—
580C22—62no—
581C22—61no—
582C22—61no—
583C22—121no—
584C22—121no—
585C22#6408711yescharacteristic, complemented
586C22#6425912yescomplemented
587C22C22×C3⋊S411yescharacteristic
588C4—21no—
589C4—21no—
590C4—61no—
591C4—61no—
592C4—182no—
593C4—182no—
594C4—361no—
595C4—361no—
596C4—361no—
597C4—361no—
598C4—361no—
599C4—361no—
600C3—43no—
601C3#8529311yescharacteristic, complemented
602C2—22no—
603C2—31no—
604C2—31no—
605C2—31no—
606C2—31no—
607C2—62no—
608C2C2×#6425911yescharacteristic
609C2D4×C3⋊S411yescharacteristic
610C2#1059657711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC22×C3×C22⋊C4D3
socleC24×C3C22×D3
prime radicalC23×C3×A4C22
hypercenterC22⋊C4C3⋊S4
hypocenterC3×A4C22⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#64259C2semidirect
5. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
6. #10593645C2semidirect
7. #10593645C2semidirect
44. C22×C3×A4Q8semidirect
148. C24×C3Dic6semidirect
152. C22×A4Dic6semidirect
230. C3×A4C22⋊Q8semidirect
371. C24#634semidirect
459. C22×C3#878semidirect
460. C22×C3A4⋊Q8semidirect
462. A4#878semidirect
585. C22#64087semidirect
586. C22#64259semidirect
601. C3#85293semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊SL(2,ℤ/4)2
C2×#642592
C3×A4×C22⋊C42
#105936662
C4⋊(C3⋊SL(2,ℤ/4))2
#105936452
#852933
#821103
#640874