Finite Group Databasebeta

← #11920742

Subgroups — #11920742

Order 1152.

534 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 47 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×GL(2,ℤ/4)C211yesmaximal, characteristic
2A4×#217C211yesmaximal, characteristic, complemented
3#10593666C211yesmaximal, characteristic, complemented
4#73049—31nomaximal
5#71670—31nomaximal
6C23×C3×A4C2211yescharacteristic
7C22×C3×S4C411yescharacteristic, complemented
8C2×Dic3×A4—21no—
9C2×C3⋊SL(2,ℤ/4)—21no—
10C3×GL(2,ℤ/4)—21no—
11C3×GL(2,ℤ/4)—21no—
12C2×C3×SL(2,ℤ/4)C411yescharacteristic, complemented
13#63580—41nomaximal
14C2×GL(2,ℤ/4)D311yescharacteristic
15C2×C3×#120—31no—
16C22×#217D311yescharacteristic, complemented
17A4×C22⋊C4—31no—
18#5432—31no—
19#5432—61no—
20#5247—31no—
21C2×#834—31no—
22C2×#834—31no—
23#4526—31no—
24#4526—31no—
25C22×C3×A4—41no—
26C22×C3×A4C2×C411yescharacteristic
27C22×C3×A4D411yescharacteristic
28C22×C3×A4D411yescharacteristic
29C2×C3×S4—21no—
30C2×C3×S4—41no—
31C2×C3×#67—41no—
32Dic3×A4—41no—
33C3⋊SL(2,ℤ/4)—41no—
34C3×SL(2,ℤ/4)—21no—
35#3757—41no—
36C3×#217—41no—
37#1898—91noSylow
38C25×C3C2×D311yescharacteristic
39C23×A4—21no—
40C23×A4C2×D311yescharacteristic
41C22×S4Dic311yescharacteristic, complemented
42C22×C3×D4—31no—
43C22×C3×D4—31no—
44C22×C3×D4—31no—
45C23×Dic3—21no—
46C2×C4×A4—61no—
47GL(2,ℤ/4)—21no—
48GL(2,ℤ/4)—21no—
49C2×SL(2,ℤ/4)—62no—
50C2×SL(2,ℤ/4)—61no—
51C2×SL(2,ℤ/4)Dic311yescharacteristic, complemented
52C3×#120—61no—
53C3×#120—61no—
54C3×#120—61no—
55C3×#120—61no—
56C2×C3×C22⋊C4—31no—
57C2×C3×C22⋊C4—31no—
58C2×C3×C22⋊C4—31no—
59C2×#217—31no—
60C2×#217—31no—
61C2×#217—31no—
62C2×#217—31no—
63C2×#217—61no—
64C2×#217—61no—
65C2×#217—61no—
66#834—62no—
67#834—61no—
68#834—61no—
69#806—121no—
70C23×C32—41no—
71C22×C3×D3—41no—
72C2×C3×A4—21no—
73C2×C3×A4—21no—
74C2×C3×A4—21no—
75C2×C3×A4C22⋊C411yescharacteristic
76C3×S4—41no—
77C2×C3⋊Dic3—81no—
78C3×#67—81no—
79C3×#67—81no—
80C2×C3×Dic3—41no—
81C2×C3×Dic3—81no—
82C2×#120—31no—
83C22×C22⋊C4—31no—
84C2×#99—91no—
85C2×#99—91no—
86#379—91no—
87#342—91no—
88#342—91no—
89C24×C3—31no—
90C24×C3—31no—
91C24×C3—31no—
92C24×C3—31no—
93C24×C3—31no—
94C24×C3—41no—
95C24×C3—121no—
96C24×C3C4×D311yescharacteristic
97C24×C3D1211yescharacteristic
98C24×C3#6711yescharacteristic
99C22×A4—21no—
100C22×A4—21no—
101C22×A4—21no—
102C22×A4—41no—
103C22×A4—42no—
104C22×A4C2×Dic311yescharacteristic
105C22×A4#6711yescharacteristic
106C22×A4#6711yescharacteristic
107C2×S4—21no—
108C2×S4—41no—
109C2×C3×D4—61no—
110C2×C3×D4—61no—
111C2×C3×D4—61no—
112C2×C3×D4—61no—
113C2×C3×D4—61no—
114C2×C3×D4—61no—
115C2×C3×D4—61no—
116C2×C3×D4—61no—
117C2×C3×D4—61no—
118C2×C3×D4—61no—
119C2×C3×D4—121no—
120C2×C3×D4—121no—
121C2×C3×D4—122no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C2×#67—41no—
126C22×Dic3—41no—
127C22×Dic3—61no—
128C22×Dic3—61no—
129C22×Dic3—61no—
130C22×Dic3—61no—
131C22×Dic3—121no—
132C4×A4—121no—
133SL(2,ℤ/4)—21no—
134SL(2,ℤ/4)—122no—
135SL(2,ℤ/4)—121no—
136C3×C22⋊C4—61no—
137C3×C22⋊C4—61no—
138C3×C22⋊C4—61no—
139C3×C22⋊C4—61no—
140C3×C22⋊C4—61no—
141C3×C22⋊C4—61no—
142C3×C22⋊C4—121no—
143#217—32no—
144#217—31no—
145#217—61no—
146#217—61no—
147#217—61no—
148#217—61no—
149#217—61no—
150#217—121no—
151#217—121no—
152#217—121no—
153#217—122no—
154#217—241no—
155#217S411yescharacteristic, complemented
156C22×C32—41no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32—161no—
160C2×C3×D3—81no—
161C2×C3×D3—161no—
162C3×A4#9911yescharacteristic, complemented
163C3⋊Dic3—161no—
164C3×Dic3—81no—
165C3×Dic3—161no—
166C25D3211yescharacteristic
167C22×D4—31no—
168C22×D4—31no—
169C22×D4—31no—
170C23×C4—61no—
171#120—61no—
172#120—61no—
173#120—61no—
174#120—61no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—91no—
179C2×C22⋊C4—91no—
180C2×C22⋊C4—91no—
181C2×C22⋊C4—91no—
182C2×C22⋊C4—181no—
183C2×C22⋊C4—181no—
184C2×C22⋊C4—181no—
185#99—181no—
186#99—182no—
187#99—181no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—31no—
195C23×C3—31no—
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—31no—
199C23×C3—31no—
200C23×C3—41no—
201C23×C3—61no—
202C23×C3—61no—
203C23×C3—61no—
204C23×C3—61no—
205C23×C3—62no—
206C23×C3—62no—
207C23×C3—61no—
208C23×C3—61no—
209C23×C3—61no—
210C23×C3—81no—
211C23×C3—121no—
212C23×C3—121no—
213C23×C3—121no—
214C23×C3—121no—
215C23×C3—121no—
216C23×C3—121no—
217C23×C3—121no—
218C23×C3—121no—
219C23×C3(C2×D3)⋊C411yescharacteristic
220C23×C3C2×S411yescharacteristic
221C22×D3—41no—
222C2×A4—22no—
223C2×A4—21no—
224C2×A4—21no—
225C2×A4—21no—
226C2×A4—21no—
227C2×A4—41no—
228C2×A4—41no—
229C2×A4#21711yescharacteristic
230S4—41no—
231C3×D4—121no—
232C3×D4—121no—
233C3×D4—121no—
234C3×D4—122no—
235C3×D4—122no—
236C3×D4—122no—
237C3×D4—121no—
238C3×D4—121no—
239C3×D4—121no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—121no—
246C2×C3×C4—121no—
247C2×C3×C4—241no—
248#67—81no—
249#67—81no—
250C2×Dic3—21no—
251C2×Dic3—41no—
252C2×Dic3—61no—
253C2×Dic3—121no—
254C2×Dic3—121no—
255C2×Dic3—121no—
256C2×Dic3—121no—
257C2×Dic3—121no—
258C2×Dic3—121no—
259C2×Dic3—121no—
260C2×Dic3—121no—
261C2×Dic3—121no—
262C2×Dic3—121no—
263C2×Dic3—241no—
264C2×Dic3—242no—
265C2×C32—41no—
266C2×C32—81no—
267C2×C32—81no—
268C2×C32—81no—
269C3×D3—161no—
270C24—31no—
271C24—31no—
272C24—31no—
273C24—31no—
274C24—31no—
275C24—41no—
276C24—121no—
277C24D3×Dic311yescharacteristic
278C24#62511yescharacteristic
279C24#62611yescharacteristic
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—122no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—121no—
297C22×C4—181no—
298C22×C4—181no—
299C22×C4—181no—
300C22×C4—181no—
301C22×C4—361no—
302C22⋊C4—31no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C22⋊C4—61no—
308C22⋊C4—61no—
309C22⋊C4—92no—
310C22⋊C4—91no—
311C22⋊C4—181no—
312C22⋊C4—181no—
313C22⋊C4—181no—
314C22⋊C4—181no—
315C22⋊C4—181no—
316C22⋊C4—361no—
317C22⋊C4—361no—
318C22⋊C4—362no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—32no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—61no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—122no—
349C22×C3—121no—
350C22×C3—161no—
351C22×C3—162no—
352C22×C3—241no—
353C22×C3—241no—
354C22×C3#80611yescharacteristic, complemented
355C22×C3C4×S411yescharacteristic
356C22×C3C4⋊S411yescharacteristic
357C22×C3GL(2,ℤ/4)11yescharacteristic
358C2×D3—81no—
359C2×D3—161no—
360A4—21no—
361A4#83411yescharacteristic, complemented
362C3×C4—121no—
363C3×C4—121no—
364C3×C4—121no—
365C3×C4—481no—
366Dic3—41no—
367Dic3—81no—
368Dic3—121no—
369Dic3—241no—
370Dic3—241no—
371Dic3—481no—
372Dic3—482no—
373C32—41noSylow
374C23—21no—
375C23—21no—
376C23—21no—
377C23—31no—
378C23—31no—
379C23—31no—
380C23—31no—
381C23—31no—
382C23—31no—
383C23—31no—
384C23—31no—
385C23—31no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—62no—
391C23—62no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23—121no—
399C23—121no—
400C23—121no—
401C23—121no—
402C23—121no—
403C23(C2×D3)⋊Dic311yescharacteristic
404C23D3×S411yescharacteristic
405D4—121no—
406D4—121no—
407D4—121no—
408D4—122no—
409D4—122no—
410D4—122no—
411D4—121no—
412D4—121no—
413D4—121no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—181no—
423C2×C4—361no—
424C2×C4—361no—
425C2×C4—361no—
426C2×C4—361no—
427C2×C4—361no—
428C2×C4—361no—
429C2×C4—361no—
430C2×C4—361no—
431C2×C4—361no—
432C2×C4—361no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—41no—
439C2×C3—61no—
440C2×C3—61no—
441C2×C3—61no—
442C2×C3—81no—
443C2×C3—81no—
444C2×C3—81no—
445C2×C3—82no—
446C2×C3—81no—
447C2×C3—161no—
448C2×C3—161no—
449C2×C3—241no—
450C2×C3(C2×S4)⋊C411yescharacteristic
451D3—161no—
452C22—31no—
453C22—31no—
454C22—32no—
455C22—31no—
456C22—31no—
457C22—31no—
458C22—41no—
459C22—61no—
460C22—61no—
461C22—61no—
462C22—61no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—121no—
471C22—121no—
472C22—121no—
473C22—121no—
474C22—121no—
475C22—122no—
476C22—121no—
477C22—241no—
478C22—241no—
479C22#6358011yescharacteristic, complemented
480C22Dic3×S411yescharacteristic
481C22#6420711yescharacteristic
482C22#6420911yescharacteristic
483C4—121no—
484C4—121no—
485C4—121no—
486C4—121no—
487C4—361no—
488C4—721no—
489C4—721no—
490C3—41no—
491C3—81no—
492C3#7304911yescharacteristic, complemented
493C2—21no—
494C2—21no—
495C2—21no—
496C2—31no—
497C2—31no—
498C2—61no—
499C2—61no—
500C2—61no—
501C2—241no—
502C2(C2×S4)⋊Dic311yescharacteristic
subgroupG/H
centerC2(C2×S4)⋊Dic3
commutatorC22×C3×A4C2×C4
Frattini, hypercenterC23D3×S4
FittingC25×C3C2×D3
socleC23×C3(C2×D3)⋊C4
prime radicalC2×C3×GL(2,ℤ/4)C2
cosocleC23×C3×A4C22
hypocenterC3×A4#99
LCSG ▷ C22×C3×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. A4×#217C2semidirect
3. #10593666C2semidirect
7. C22×C3×S4C4semidirect
12. C2×C3×SL(2,ℤ/4)C4semidirect
16. C22×#217D3semidirect
41. C22×S4Dic3semidirect
51. C2×SL(2,ℤ/4)Dic3semidirect
155. #217S4semidirect
162. C3×A4#99semidirect
354. C22×C3#806semidirect
361. A4#834semidirect
479. C22#63580semidirect
492. C3#73049semidirect

Maximal subgroups

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

subgroupindex
C2×C3×GL(2,ℤ/4)2
A4×#2172
#105936662
#730493
#716703
#635804