Finite Group Databasebeta

← #11923035

Subgroups — #11923035

Order 1152.

1182 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 115 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×C4×A4C211yesmaximal, characteristic, complemented
2C2×#64269C212yesmaximal, complemented
3C2×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
4#10593647C212yesmaximal, complemented
5C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
6#85486—33nomaximal
7#84032—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C22×C3⋊S4C2212yes—
10C2×C3×C4×A4—22no—
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13#64269—24no—
14C2×C3⋊SL(2,ℤ/4)C2212yes—
15C4⋊(C3⋊S4)—22no—
16#64139—41nomaximal
17C24×C3×C4D311yescharacteristic, complemented
18C22×C4×A4D313yescomplemented
19C2×GL(2,ℤ/4)—36no—
20C2×C4⋊S4—33no—
21C2×#953—32no—
22C2×#930—31no—
23C2×#930—31no—
24C2×#930—31no—
25(C2×S4)⋊C4—36no—
26C4⋊SL(2,ℤ/4)—33no—
27#5213—32no—
28#5213—32no—
29#5213—32no—
30#5212—31no—
31#5212—31no—
32#5212—31no—
33C22×C3×A4—21no—
34C22×C3×A4—21no—
35C22×C3×A4D412yescomplemented
36C22×C3×A4C2311yescharacteristic
37C2×C3⋊S4—22no—
38C2×C3⋊S4—22no—
39C2×C3⋊S4—22no—
40C22×C32×C4—41no—
41C2×#638—42no—
42C2×C4⋊(C3⋊D3)—41no—
43C3×C4×A4—21no—
44C3×C4×A4D412yescomplemented
45C3⋊SL(2,ℤ/4)—22no—
46#3752—42no—
47C4⋊(C3⋊Dic3)—41no—
48#2847—91noSylow
49C25×C3C2×D311yescharacteristic
50C23×A4C2×D313yes—
51C22×S4—36no—
52C23×C3×C4—22no—
53C23×C3×C4—31no—
54C23×C3×C4—31no—
55C23×C3×C4—31no—
56C23×C3×C4—31no—
57C23×C3×C4—62no—
58C23×C3×C4C2×D311yescharacteristic
59C23×C3×C4C2×D311yescharacteristic
60C22×#67—32no—
61C22×#67—32no—
62C22×#67—32no—
63C22×D12—31no—
64C2×C4×A4—26no—
65C2×C4×A4C2×D313yes—
66C2×C4×A4C2×D313yes—
67GL(2,ℤ/4)—612no—
68C2×SL(2,ℤ/4)—36no—
69C4⋊S4—66no—
70#953—64no—
71#953—64no—
72C2×#217—32no—
73C2×#217—32no—
74C2×#217—32no—
75#930—62no—
76#930—64no—
77#930—61no—
78#930—61no—
79#930—61no—
80#930—62no—
81#930—61no—
82#930—123no—
83C2×(C2×D3)⋊C4—32no—
84C2×(C2×D3)⋊C4—32no—
85C2×(C2×D3)⋊C4—32no—
86C2×C4⋊Dic3—31no—
87C2×C4⋊Dic3—31no—
88C2×C4⋊Dic3—31no—
89C3⋊#95—32no—
90C3⋊#95—32no—
91C3⋊#95—32no—
92C23×C32—41no—
93C22×C3⋊D3—42no—
94C2×C3×A4—22no—
95C2×C3×A4C2×D411yescharacteristic
96C2×C3×A4C2×D411yescharacteristic
97C2×C3×A4#4111yescharacteristic
98C3⋊S4—42no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—82no—
102#638—84no—
103C2×C3⋊Dic3—42no—
104C4⋊(C3⋊D3)—82no—
105C24×C4C3⋊D311yescharacteristic, complemented
106C2×#121—91no—
107C2×#121—91no—
108C2×#121—91no—
109C2×#120—92no—
110#386—92no—
111#386—92no—
112#386—92no—
113#380—91no—
114#380—91no—
115#380—91no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—31no—
119C24×C3—31no—
120C24×C3—32no—
121C24×C3—61no—
122C24×C3—61no—
123C24×C3D1212yescomplemented
124C24×C3C22×D311yescharacteristic
125C23×D3—32no—
126C22×A4—23no—
127C22×A4—23no—
128C22×A4D1216yescomplemented
129C22×A4C22×D313yes—
130C2×S4—66no—
131C2×S4—66no—
132C2×S4—66no—
133C22×C3×C4—21no—
134C22×C3×C4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—32no—
138C22×C3×C4—34no—
139C22×C3×C4—31no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—43no—
144C22×C3×C4—62no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147C22×C3×C4—61no—
148C22×C3×C4—62no—
149C22×C3×C4—62no—
150C22×C3×C4—62no—
151C22×C3×C4—62no—
152C22×C3×C4—62no—
153C22×C3×C4#6712yescomplemented
154C22×C3×C4S411yescharacteristic, complemented
155C2×#67—62no—
156C2×#67—64no—
157C2×#67—62no—
158C2×#67—64no—
159C2×#67—62no—
160C2×#67—64no—
161C2×#67—62no—
162C2×#67—64no—
163C2×#67—64no—
164C2×#67—62no—
165C2×#67—62no—
166C2×#67—62no—
167C2×#67—62no—
168C2×#67—126no—
169C22×Dic3—32no—
170C22×Dic3—32no—
171C22×Dic3—32no—
172C2×D12—61no—
173C2×D12—62no—
174C2×D12—62no—
175C2×D12—61no—
176C2×D12—123no—
177C4×A4—23no—
178C4×A4#6716yescomplemented
179SL(2,ℤ/4)—66no—
180#217—64no—
181#217—62no—
182#217—64no—
183#217—62no—
184(C2×D3)⋊C4—64no—
185(C2×D3)⋊C4—64no—
186(C2×D3)⋊C4—62no—
187(C2×D3)⋊C4—62no—
188(C2×D3)⋊C4—126no—
189C4⋊Dic3—62no—
190C4⋊Dic3—62no—
191C4⋊Dic3—61no—
192C4⋊Dic3—61no—
193C4⋊Dic3—123no—
194C22×C32—42no—
195C22×C32—41no—
196C22×C32—81no—
197C22×C32—81no—
198C2×C3⋊D3—82no—
199C2×C3⋊D3—82no—
200C2×C3⋊D3—82no—
201C3×A4#12111yescharacteristic, complemented
202C32×C4—42no—
203C32×C4—81no—
204C3⋊Dic3—82no—
205C25C2×C3⋊D311yescharacteristic
206C22×D4—91no—
207C22×D4—92no—
208C22×D4—92no—
209C22×D4—92no—
210C23×C4—22no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—62no—
216C23×C4C2×C3⋊D311yescharacteristic
217C23×C4C2×C3⋊D311yescharacteristic
218#121—182no—
219#121—184no—
220#121—181no—
221#121—181no—
222#121—181no—
223#121—182no—
224#121—181no—
225#120—184no—
226#120—184no—
227C2×C4⋊C4—91no—
228C2×C4⋊C4—91no—
229C2×C4⋊C4—91no—
230C2×C22⋊C4—92no—
231C2×C22⋊C4—92no—
232C2×C22⋊C4—92no—
233C2×C22⋊C4—92no—
234C2×C22⋊C4—92no—
235C2×C22⋊C4—92no—
236#95—92no—
237#95—92no—
238#95—92no—
239C23×C3—22no—
240C23×C3—31no—
241C23×C3—32no—
242C23×C3—31no—
243C23×C3—31no—
244C23×C3—31no—
245C23×C3—32no—
246C23×C3—32no—
247C23×C3—32no—
248C23×C3—32no—
249C23×C3—31no—
250C23×C3—43no—
251C23×C3—61no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—62no—
256C23×C3—61no—
257C23×C3—61no—
258C23×C3—62no—
259C23×C3—62no—
260C23×C3—61no—
261C23×C3—64no—
262C23×C3C2×D1211yescharacteristic
263C23×C3#23511yescharacteristic
264C23×C3C2×#6711yescharacteristic
265C23×C3C2×S411yescharacteristic
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—126no—
274C2×A4—26no—
275C2×A4C2×D1213yes—
276C2×A4#23513yes—
277C2×A4C2×#6713yes—
278S4—126no—
279C2×C3×C4—22no—
280C2×C3×C4—31no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—32no—
285C2×C3×C4—31no—
286C2×C3×C4—43no—
287C2×C3×C4—43no—
288C2×C3×C4—62no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—62no—
293C2×C3×C4—64no—
294C2×C3×C4—62no—
295C2×C3×C4—61no—
296C2×C3×C4—86no—
297C2×C3×C4C2×S411yescharacteristic
298C2×C3×C4C2×S411yescharacteristic
299#67—122no—
300#67—122no—
301#67—122no—
302#67—128no—
303#67—124no—
304#67—122no—
305#67—124no—
306#67—2412no—
307C2×Dic3—62no—
308C2×Dic3—62no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—62no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—126no—
317D12—122no—
318D12—122no—
319D12—246no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—82no—
324C3⋊D3—162no—
325C24—21no—
326C24—21no—
327C24—31no—
328C24—31no—
329C24—32no—
330C24—61no—
331C24—61no—
332C24—92no—
333C24C4⋊(C3⋊D3)12yescomplemented
334C24C22×C3⋊D311yescharacteristic
335C2×D4—181no—
336C2×D4—182no—
337C2×D4—182no—
338C2×D4—184no—
339C2×D4—182no—
340C2×D4—184no—
341C2×D4—182no—
342C2×D4—184no—
343C2×D4—182no—
344C2×D4—182no—
345C2×D4—184no—
346C2×D4—184no—
347C2×D4—182no—
348C2×D4—182no—
349C2×D4—182no—
350C2×D4—182no—
351C2×D4—181no—
352C22×C4—21no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—32no—
357C22×C4—34no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—92no—
372C22×C4—92no—
373C22×C4—92no—
374C22×C4#63812yescomplemented
375C22×C4C3⋊S411yescharacteristic, complemented
376C4⋊C4—181no—
377C4⋊C4—182no—
378C4⋊C4—182no—
379C4⋊C4—181no—
380C22⋊C4—184no—
381C22⋊C4—184no—
382C22⋊C4—182no—
383C22⋊C4—184no—
384C22⋊C4—182no—
385C22⋊C4—184no—
386C22⋊C4—182no—
387C22⋊C4—182no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—31no—
396C22×C3—31no—
397C22×C3—31no—
398C22×C3—31no—
399C22×C3—34no—
400C22×C3—32no—
401C22×C3—46no—
402C22×C3—43no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—61no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3—61no—
412C22×C3—61no—
413C22×C3—83no—
414C22×C3—83no—
415C22×C3#93011yescharacteristic, complemented
416C22×C3C4⋊S412yescomplemented
417C22×C3C22×S411yescharacteristic
418C2×D3—122no—
419C2×D3—122no—
420C2×D3—122no—
421C2×D3—122no—
422C2×D3—122no—
423C2×D3—122no—
424C2×D3—122no—
425C2×D3—246no—
426C2×D3—246no—
427C2×D3—246no—
428A4#93013yescomplemented
429C3×C4—21no—
430C3×C4—32no—
431C3×C4—46no—
432C3×C4—61no—
433C3×C4—83no—
434C3×C4GL(2,ℤ/4)12yescomplemented
435Dic3—122no—
436Dic3—122no—
437Dic3—122no—
438Dic3—246no—
439C32—41noSylow
440C23—22no—
441C23—31no—
442C23—32no—
443C23—31no—
444C23—31no—
445C23—31no—
446C23—32no—
447C23—32no—
448C23—32no—
449C23—32no—
450C23—31no—
451C23—61no—
452C23—61no—
453C23—61no—
454C23—61no—
455C23—62no—
456C23—61no—
457C23—61no—
458C23—62no—
459C23—62no—
460C23—61no—
461C23—64no—
462C23—182no—
463C23—182no—
464C23—182no—
465C23—182no—
466C23—182no—
467C23—182no—
468C23—182no—
469C23C2×C4⋊(C3⋊D3)11yescharacteristic
470C23#382811yescharacteristic
471C23C2×#63811yescharacteristic
472C23C2×C3⋊S411yescharacteristic
473D4—362no—
474D4—362no—
475D4—362no—
476D4—362no—
477D4—368no—
478D4—364no—
479D4—364no—
480D4—362no—
481D4—362no—
482C2×C4—22no—
483C2×C4—31no—
484C2×C4—32no—
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—32no—
488C2×C4—31no—
489C2×C4—62no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—62no—
494C2×C4—64no—
495C2×C4—62no—
496C2×C4—61no—
497C2×C4—182no—
498C2×C4—182no—
499C2×C4—182no—
500C2×C4—182no—
501C2×C4—182no—
502C2×C4—182no—
503C2×C4—182no—
504C2×C4—182no—
505C2×C4—182no—
506C2×C4C2×C3⋊S411yescharacteristic
507C2×C4C2×C3⋊S411yescharacteristic
508C2×C3—22no—
509C2×C3—31no—
510C2×C3—31no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—43no—
514C2×C3—43no—
515C2×C3—43no—
516C2×C3—62no—
517C2×C3—86no—
518C2×C3C2×C4⋊S411yescharacteristic
519C2×C3#591211yescharacteristic
520C2×C3C2×GL(2,ℤ/4)11yescharacteristic
521D3—242no—
522D3—486no—
523C22—21no—
524C22—21no—
525C22—31no—
526C22—31no—
527C22—31no—
528C22—31no—
529C22—31no—
530C22—31no—
531C22—31no—
532C22—31no—
533C22—31no—
534C22—34no—
535C22—32no—
536C22—62no—
537C22—62no—
538C22—62no—
539C22—62no—
540C22—61no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—61no—
545C22—61no—
546C22—362no—
547C22—362no—
548C22—362no—
549C22—362no—
550C22—362no—
551C22—362no—
552C22—362no—
553C22#6413911yescharacteristic, complemented
554C22C4⋊(C3⋊S4)12yescomplemented
555C22C22×C3⋊S411yescharacteristic
556C4—21no—
557C4—32no—
558C4—61no—
559C4—362no—
560C4—362no—
561C4—362no—
562C4#6426912yescomplemented
563C3—43no—
564C3#8548611yescharacteristic, complemented
565C2—22no—
566C2—31no—
567C2—31no—
568C2—31no—
569C2—31no—
570C2—62no—
571C2—722no—
572C2C2×C4⋊(C3⋊S4)11yescharacteristic
573C2#1059657511yescharacteristic
574C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC24×C3×C4D3
socleC24×C3C22×D3
hypercenterC22×C4C3⋊S4
hypocenterC3×A4#121
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×A4C2semidirect
2. C2×#64269C2semidirect
3. C2×C4⋊(C3⋊S4)C2semidirect
4. #10593647C2semidirect
5. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
17. C24×C3×C4D3semidirect
18. C22×C4×A4D3semidirect
35. C22×C3×A4D4semidirect
44. C3×C4×A4D4semidirect
105. C24×C4C3⋊D3semidirect
123. C24×C3D12semidirect
128. C22×A4D12semidirect
153. C22×C3×C4#67semidirect
154. C22×C3×C4S4semidirect
178. C4×A4#67semidirect
201. C3×A4#121semidirect
333. C24C4⋊(C3⋊D3)semidirect
374. C22×C4#638semidirect
375. C22×C4C3⋊S4semidirect
415. C22×C3#930semidirect
416. C22×C3C4⋊S4semidirect
428. A4#930semidirect
434. C3×C4GL(2,ℤ/4)semidirect
553. C22#64139semidirect
554. C22C4⋊(C3⋊S4)semidirect
562. C4#64269semidirect
564. C3#85486semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×A42
C2×#642692
C2×C4⋊(C3⋊S4)2
#105936472
C4⋊(C3⋊SL(2,ℤ/4))2
#854863
#840323
#641394