Finite Group Databasebeta

← #11922852

Subgroups — #11922852

Order 1152.

822 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 87 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×#64259C211yesmaximal, characteristic, complemented
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4#10593647C211yesmaximal, characteristic
5#10593647C211yesmaximal, characteristic
6#10593645C211yesmaximal, characteristic, complemented
7#10593645C211yesmaximal, characteristic, complemented
8#85311—33nomaximal
9#82173—31nomaximal
10C22×C3⋊S4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3×C4×A4C2211yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
16C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
17C4×C3⋊S4—22no—
18#64259—22no—
19#64105—41nomaximal
20C2×C4×S4—33no—
21C2×A4⋊Q8—33no—
22C22×C3×C4⋊C4D311yescharacteristic, complemented
23C2×C4×#67—31no—
24C2×#924—31no—
25C2×#896—31no—
26A4×C4⋊C4D313yescomplemented
27(C2×S4)⋊C4—33no—
28(C2×S4)⋊C4—33no—
29SL(2,ℤ/4)⋊C4—33no—
30SL(2,ℤ/4)⋊C4—33no—
31#5213—31no—
32#5213—31no—
33#5210—31no—
34#5210—31no—
35#4994—31no—
36#4993—31no—
37#4989—31no—
38#4983—31no—
39#4980—31no—
40#4979—31no—
41#4976—31no—
42C22×C3×A4C2311yescharacteristic
43C2×C3⋊S4—21no—
44C2×C3⋊S4—21no—
45C2×C3⋊S4Q812yescomplemented
46C2×C4×C3⋊D3—41no—
47C2×#634—41no—
48C3×C4×A4—21no—
49C3×C4×A4—21no—
50C3×C4×A4—21no—
51C3⋊SL(2,ℤ/4)—21no—
52C3⋊SL(2,ℤ/4)—21no—
53C3⋊SL(2,ℤ/4)D412yes—
54C32×C4⋊C4—41no—
55#3752—41no—
56#3752—41no—
57#3750—41no—
58#3750—41no—
59#2449—91noSylow
60C22×S4—33no—
61C23×C3×C4C2×D311yescharacteristic
62C23×C3×C4C2×D311yescharacteristic
63C23×C3×C4C2×D311yescharacteristic
64C22×#67—31no—
65C22×C4×D3—31no—
66C22×Dic6—31no—
67C2×C4×A4C2×D313yes—
68C2×C4×A4C2×D313yes—
69C2×C4×A4C2×D313yes—
70C2×SL(2,ℤ/4)—33no—
71C2×SL(2,ℤ/4)—33no—
72C2×SL(2,ℤ/4)—33no—
73C4×S4—66no—
74A4⋊Q8—66no—
75C2×C3×C4⋊C4—31no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C4⋊C4—31no—
78C2×C3×C4⋊C4—31no—
79C2×#217—31no—
80C2×#217—31no—
81C2×#217—31no—
82C4×#67—62no—
83C4×#67—62no—
84C2×(C2×D3)⋊C4—31no—
85C2×(C2×D3)⋊C4—31no—
86C2×(C2×D3)⋊C4—31no—
87C2×(C2×D3)⋊C4—31no—
88C2×(C2×D3)⋊C4—31no—
89C2×C4⋊Dic3—31no—
90#924—62no—
91#924—62no—
92C2×Dic3⋊C4—31no—
93C2×Dic3⋊C4—31no—
94C2×Dic3⋊C4—31no—
95C2×Dic3⋊C4—31no—
96C2×Dic3⋊C4—31no—
97C2×Dic3⋊C4—31no—
98C2×Dic3⋊C4—31no—
99C2×C4×Dic3—31no—
100#896—61no—
101#896—61no—
102#896—61no—
103#896—61no—
104#896—123no—
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×A4C2×D411yescharacteristic
115C2×C3×A4C2×Q811yescharacteristic
116C2×C3×A4#4111yescharacteristic
117C3⋊S4—22no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C3⋊Dic3—41no—
122C2×C3⋊Dic3—41no—
123C2×C3⋊Dic3—41no—
124C4×C3⋊D3—82no—
125#634—82no—
126C2×C22⋊Q8—91no—
127C2×C22⋊Q8—91no—
128C2×C4×D4—91no—
129C22×C4⋊C4C3⋊D311yescharacteristic, complemented
130#399—91no—
131#398—91no—
132#396—91no—
133#395—91no—
134#389—91no—
135#387—91no—
136#386—91no—
137#386—91no—
138#385—91no—
139#385—91no—
140#385—91no—
141C24×C3C22×D311yescharacteristic
142C23×D3—31no—
143C22×A4C22×D313yes—
144C2×S4—36no—
145C2×S4—63no—
146C2×S4—63no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—31no—
156C22×C3×C4—61no—
157C22×C3×C4—61no—
158C22×C3×C4—61no—
159C2×#67—32no—
160C2×#67—32no—
161C2×#67—61no—
162C2×#67—61no—
163C2×#67—61no—
164C2×#67—61no—
165C22×Dic3—31no—
166C22×Dic3—31no—
167C22×Dic3—31no—
168C22×Dic3—31no—
169C22×Dic3—31no—
170C22×Dic3—31no—
171C22×Dic3—31no—
172C2×C4×D3—62no—
173C2×C4×D3—61no—
174C2×C4×D3—61no—
175C2×C4×D3—123no—
176C2×C4×D3—121no—
177C2×Dic6—62no—
178C2×Dic6—61no—
179C2×Dic6—61no—
180C2×Dic6—123no—
181C2×Dic6—121no—
182C4×A4—23no—
183C4×A4—23no—
184C4×A4—23no—
185SL(2,ℤ/4)—36no—
186SL(2,ℤ/4)—63no—
187SL(2,ℤ/4)—63no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—43no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4S411yescharacteristic, complemented
194#217—32no—
195#217—32no—
196#217—61no—
197#217—61no—
198#217—61no—
199#217—61no—
200(C2×D3)⋊C4—62no—
201(C2×D3)⋊C4—61no—
202(C2×D3)⋊C4—61no—
203(C2×D3)⋊C4—62no—
204(C2×D3)⋊C4—61no—
205(C2×D3)⋊C4—61no—
206(C2×D3)⋊C4—123no—
207(C2×D3)⋊C4—123no—
208(C2×D3)⋊C4—121no—
209C4⋊Dic3—62no—
210Dic3⋊C4—62no—
211Dic3⋊C4—62no—
212Dic3⋊C4—62no—
213Dic3⋊C4—61no—
214Dic3⋊C4—61no—
215Dic3⋊C4—62no—
216Dic3⋊C4—61no—
217Dic3⋊C4—61no—
218Dic3⋊C4—123no—
219Dic3⋊C4—123no—
220Dic3⋊C4—121no—
221C4×Dic3—62no—
222C22×C32—41no—
223C2×C3⋊D3—42no—
224C2×C3⋊D3—81no—
225C2×C3⋊D3—81no—
226C3×A4C22⋊Q811yescharacteristic, complemented
227C32×C4—81no—
228C32×C4—81no—
229C32×C4—81no—
230C3⋊Dic3—42no—
231C3⋊Dic3—81no—
232C3⋊Dic3—81no—
233C22×Q8—91no—
234C22×D4—91no—
235C23×C4—91no—
236C23×C4C2×C3⋊D311yescharacteristic
237C23×C4C2×C3⋊D311yescharacteristic
238C23×C4C2×C3⋊D311yescharacteristic
239C22⋊Q8—182no—
240C22⋊Q8—182no—
241C22⋊Q8—181no—
242C22⋊Q8—181no—
243C22⋊Q8—181no—
244C22⋊Q8—181no—
245C4×D4—182no—
246C4×D4—182no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
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×C4⋊C4—91no—
259C2×C22⋊C4—91no—
260C2×C22⋊C4—91no—
261C2×C22⋊C4—91no—
262C2×C22⋊C4—91no—
263C2×C22⋊C4—91no—
264C2×C22⋊C4—91no—
265C2×C22⋊C4—91no—
266C2×C22⋊C4—91no—
267C2×C42—91no—
268#95—91no—
269#95—91no—
270#95—91no—
271#95—91no—
272#95—91no—
273#95—91no—
274#95—91no—
275#95—91no—
276C23×C3—31no—
277C23×C3—31no—
278C23×C3—31no—
279C23×C3—31no—
280C23×C3#23511yescharacteristic
281C23×C3D3×D411yescharacteristic
282C23×C3D3×Q811yescharacteristic
283C22×D3—32no—
284C22×D3—61no—
285C22×D3—61no—
286C22×D3—61no—
287C22×D3—61no—
288C22×D3—61no—
289C22×D3—61no—
290C22×D3—123no—
291C2×A4#23513yes—
292C2×A4D3×D413yes—
293C2×A4D3×Q813yes—
294S4—66no—
295C2×C3×C4—32no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—43no—
300C2×C3×C4—43no—
301C2×C3×C4—43no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
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×C4C2×S411yescharacteristic
314C2×C3×C4C2×S411yescharacteristic
315C2×C3×C4C2×S411yescharacteristic
316#67—62no—
317#67—62no—
318#67—62no—
319#67—62no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—61no—
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—123no—
342C2×Dic3—123no—
343C2×Dic3—123no—
344C4×D3—122no—
345C4×D3—122no—
346C4×D3—246no—
347Dic6—122no—
348Dic6—122no—
349Dic6—246no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C3⋊D3—82no—
354C24—91no—
355C24C22×C3⋊D311yescharacteristic
356C2×Q8—181no—
357C2×Q8—182no—
358C2×Q8—181no—
359C2×Q8—361no—
360C2×D4—92no—
361C2×D4—92no—
362C2×D4—181no—
363C2×D4—181no—
364C2×D4—181no—
365C2×D4—181no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—181no—
386C22×C4—182no—
387C22×C4—181no—
388C22×C4—361no—
389C4⋊C4—31no—
390C4⋊C4—31no—
391C4⋊C4—31no—
392C4⋊C4—61no—
393C4⋊C4—181no—
394C4⋊C4—181no—
395C4⋊C4—182no—
396C4⋊C4—182no—
397C4⋊C4—182no—
398C4⋊C4—182no—
399C4⋊C4—182no—
400C4⋊C4—181no—
401C4⋊C4—181no—
402C4⋊C4—361no—
403C4⋊C4C3⋊S411yescharacteristic, complemented
404C22⋊C4—92no—
405C22⋊C4—92no—
406C22⋊C4—181no—
407C22⋊C4—181no—
408C22⋊C4—182no—
409C22⋊C4—182no—
410C22⋊C4—181no—
411C22⋊C4—181no—
412C22⋊C4—181no—
413C22⋊C4—181no—
414C22⋊C4—181no—
415C22⋊C4—181no—
416C22⋊C4—361no—
417C42—182no—
418C22×C3—31no—
419C22×C3—31no—
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—43no—
428C22×C3—61no—
429C22×C3#89611yescharacteristic, complemented
430C22×C3C22×S411yescharacteristic
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—126no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—121no—
438C2×D3—121no—
439C2×D3—243no—
440C2×D3—243no—
441A4#89613yescomplemented
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—83no—
449C3×C4—83no—
450C3×C4—83no—
451Dic3—62no—
452Dic3—62no—
453Dic3—62no—
454Dic3—126no—
455Dic3—121no—
456Dic3—121no—
457Dic3—121no—
458Dic3—121no—
459Dic3—243no—
460Dic3—243no—
461C32—41noSylow
462C23—31no—
463C23—31no—
464C23—31no—
465C23—31no—
466C23—92no—
467C23—181no—
468C23—181no—
469C23—181no—
470C23—181no—
471C23—181no—
472C23—181no—
473C23#382811yescharacteristic
474C23D4×C3⋊D311yescharacteristic
475C23Q8×C3⋊D311yescharacteristic
476Q8—362no—
477Q8—362no—
478D4—182no—
479D4—182no—
480D4—182no—
481D4—182no—
482C2×C4—32no—
483C2×C4—31no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—61no—
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—92no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—181no—
501C2×C4—181no—
502C2×C4—181no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4—181no—
509C2×C4—181no—
510C2×C4—181no—
511C2×C4—181no—
512C2×C4—181no—
513C2×C4—181no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C4—362no—
519C2×C4—362no—
520C2×C4C2×C3⋊S411yescharacteristic
521C2×C4C2×C3⋊S411yescharacteristic
522C2×C4C2×C3⋊S411yescharacteristic
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—43no—
528C2×C3—43no—
529C2×C3—43no—
530C2×C3#591211yescharacteristic
531C2×C3D4×S411yescharacteristic
532C2×C3Q8×S411yescharacteristic
533D3—122no—
534D3—246no—
535C22—31no—
536C22—31no—
537C22—31no—
538C22—31no—
539C22—31no—
540C22—31no—
541C22—31no—
542C22—31no—
543C22—31no—
544C22—61no—
545C22—182no—
546C22—182no—
547C22—182no—
548C22—361no—
549C22—361no—
550C22—361no—
551C22—361no—
552C22#6410511yescharacteristic, complemented
553C22C22×C3⋊S411yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—61no—
558C4—61no—
559C4—61no—
560C4—182no—
561C4—182no—
562C4—182no—
563C4—361no—
564C4—361no—
565C4—361no—
566C4—361no—
567C3—43no—
568C3#8531111yescharacteristic, complemented
569C2—31no—
570C2—31no—
571C2—31no—
572C2—31no—
573C2—362no—
574C2#1059657511yescharacteristic
575C2D4×C3⋊S411yescharacteristic
576C2Q8×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3⋊S4C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4C22⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#64259C2semidirect
3. C3×A4×C4⋊C4C2semidirect
6. #10593645C2semidirect
7. #10593645C2semidirect
22. C22×C3×C4⋊C4D3semidirect
26. A4×C4⋊C4D3semidirect
45. C2×C3⋊S4Q8semidirect
129. C22×C4⋊C4C3⋊D3semidirect
193. C3×C4⋊C4S4semidirect
226. C3×A4C22⋊Q8semidirect
403. C4⋊C4C3⋊S4semidirect
429. C22×C3#896semidirect
441. A4#896semidirect
552. C22#64105semidirect
568. C3#85311semidirect

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×#642592
C3×A4×C4⋊C42
#105936472
#105936452
#853113
#821733
#641054