Finite Group Databasebeta

← #11921917

Subgroups — #11921917

Order 1152.

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

normal subgroups only

This group has 732 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×C3×C4×S4C211yesmaximal, characteristic
2C4×Dic3×A4C211yesmaximal, characteristic, complemented
3C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4(C2×S4)⋊Dic3C212yesmaximal
5SL(2,ℤ/4)⋊Dic3C212yesmaximal, complemented
6#85287—31nomaximal
7#82318—31nomaximal
8C22×C3×S4C2211yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C3×C4×S4C414yescomplemented
14#63855—41nomaximal
15C2×C4×S4D311yescharacteristic
16C2×C3×C4×D4—31no—
17C2×#926—31no—
18C22×C4×Dic3D311yescharacteristic, complemented
19C42×A4—31no—
20(C2×S4)⋊C4—32no—
21SL(2,ℤ/4)⋊C4—32no—
22C4×SL(2,ℤ/4)—31no—
23C4×SL(2,ℤ/4)—61no—
24#5220—32no—
25C4×#217—31no—
26C4×#217—31no—
27#4980—32no—
28#4951—32no—
29#4950—32no—
30C4×C4⋊Dic3—31no—
31#4932—31no—
32C22×C3×A4C2311yescharacteristic
33C2×C3×S4—22no—
34C2×C3×S4C2×C412yes—
35C2×C3×C4×D3—41no—
36C3×C4×A4C2×C412yes—
37Dic3×A4—22no—
38C3⋊SL(2,ℤ/4)—22no—
39C3×SL(2,ℤ/4)C2×C412yes—
40C4×C3⋊Dic3—41no—
41C3×C4×Dic3—41no—
42Dic3⋊Dic3—42no—
43(C2×D3)⋊Dic3—42no—
44#2290—91noSylow
45C22×S4C2×D311yescharacteristic
46C22×C3×D4—31no—
47C23×C3×C4—31no—
48C23×C3×C4C2×D311yescharacteristic
49C23×Dic3C2×D312yes—
50C2×C4×A4—21no—
51C2×C4×A4—32no—
52C2×C4×A4C2×D311yescharacteristic
53C2×SL(2,ℤ/4)—32no—
54C2×SL(2,ℤ/4)—62no—
55C2×SL(2,ℤ/4)C2×D311yescharacteristic
56C4×S4Dic314yescomplemented
57C3×C4×D4—34no—
58C3×C4×D4—34no—
59C2×C3×C4⋊C4—31no—
60C2×C3×C22⋊C4—31no—
61C2×C3×C22⋊C4—31no—
62C2×C3×C42—31no—
63C2×#217—32no—
64C2×#217—32no—
65C2×#217—32no—
66#926—61no—
67#926—62no—
68#926—61no—
69C2×C4⋊Dic3—32no—
70C2×C4⋊Dic3—32no—
71C2×C4×Dic3—31no—
72C2×C4×Dic3—31no—
73C2×C4×Dic3—32no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—31no—
76#872—121no—
77C3⋊#95—32no—
78C3⋊#95—32no—
79C3⋊#95—32no—
80C3⋊#95—32no—
81C3⋊#95—32no—
82C22×C3×D3—41no—
83C2×C3×A4C22×C411yescharacteristic
84C2×C3×A4#4112yes—
85C3×S4—22no—
86C2×C32×C4—41no—
87C2×C3⋊Dic3—42no—
88C2×C3×Dic3—41no—
89C2×C3×Dic3—42no—
90C3×C4×D3—44no—
91C2×C4×D4—31no—
92C2×#117—91no—
93C22×C42—31no—
94#388—92no—
95#387—92no—
96#386—92no—
97#385—92no—
98C4×C4⋊C4—91no—
99C4×C22⋊C4—91no—
100C4×C22⋊C4—91no—
101#376—91no—
102C24×C3—31no—
103C24×C3C22×D311yescharacteristic
104C22×A4—21no—
105C22×A4C22×D311yescharacteristic
106C2×S4—22no—
107C2×S4C2×Dic312yes—
108C2×C3×D4—32no—
109C2×C3×D4—32no—
110C2×C3×D4—62no—
111C2×C3×D4—62no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—34no—
119C22×C3×C4—61no—
120C22×C3×C4—62no—
121C22×C3×C4—61no—
122C22×C3×C4C4×D312yes—
123C22×Dic3—22no—
124C22×Dic3—32no—
125C22×Dic3—32no—
126C22×Dic3—32no—
127C22×Dic3—32no—
128C22×Dic3—62no—
129C2×C4×D3—41no—
130C4×A4—22no—
131C4×A4—62no—
132C4×A4C2×Dic312yes—
133SL(2,ℤ/4)—64no—
134SL(2,ℤ/4)—62no—
135SL(2,ℤ/4)C2×Dic312yes—
136C3×C4⋊C4—34no—
137C3×C22⋊C4—34no—
138C3×C22⋊C4—32no—
139C3×C22⋊C4—32no—
140C3×C42—34no—
141C3×C42—121no—
142#217—62no—
143#217—64no—
144#217—62no—
145#217—62no—
146#217—62no—
147(C2×D3)⋊C4—122no—
148C4⋊Dic3—64no—
149C4⋊Dic3—62no—
150C4⋊Dic3—62no—
151Dic3⋊C4—122no—
152C4×Dic3—31no—
153C4×Dic3—32no—
154C4×Dic3—61no—
155C4×Dic3—62no—
156C4×Dic3—61no—
157C4×Dic3—61no—
158C4×Dic3—121no—
159C4×Dic3—241no—
160C4×Dic3S411yescharacteristic, complemented
161C22×C32—41no—
162C2×C3×D3—42no—
163C2×C3×D3—82no—
164C3×A4#11711yescharacteristic, complemented
165C32×C4—42no—
166C3⋊Dic3—82no—
167C3×Dic3—42no—
168C3×Dic3—82no—
169C22×D4—31no—
170C23×C4—31no—
171C23×C4—32no—
172C23×C4D3211yescharacteristic
173C4×D4—34no—
174C4×D4—34no—
175#117—181no—
176#117—182no—
177#117—181no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—92no—
180C2×C4⋊C4—92no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—92no—
184C2×C22⋊C4—92no—
185C2×C22⋊C4—92no—
186C2×C42—31no—
187C2×C42—91no—
188C2×C42—91no—
189C2×C42—92no—
190C2×C42—91no—
191C2×C42—91no—
192#95—92no—
193#95—92no—
194#95—92no—
195#95—92no—
196#95—92no—
197C23×C3—31no—
198C23×C3—32no—
199C23×C3—31no—
200C23×C3—32no—
201C23×C3—62no—
202C23×C3—61no—
203C23×C3—62no—
204C23×C3—61no—
205C23×C3C2×C4×D311yescharacteristic
206C23×C3#23512yes—
207C22×D3—41no—
208C2×A4—22no—
209C2×A4—21no—
210C2×A4#23512yes—
211C2×A4C22×Dic311yescharacteristic
212S4—22no—
213C3×D4—62no—
214C3×D4—64no—
215C3×D4—62no—
216C2×C3×C4—31no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—41no—
224C2×C3×C4—61no—
225C2×C3×C4—64no—
226C2×C3×C4—62no—
227C2×C3×C4—61no—
228C2×C3×C4—62no—
229C2×C3×C4—61no—
230C2×C3×C4—64no—
231C2×C3×C4—81no—
232C2×C3×C4—122no—
233C2×C3×C4C2×S411yescharacteristic
234C2×Dic3—34no—
235C2×Dic3—32no—
236C2×Dic3—41no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C2×Dic3—122no—
247C2×Dic3—242no—
248C2×Dic3C2×S412yes—
249C4×D3—44no—
250C2×C32—41no—
251C2×C32—42no—
252C3×D3—82no—
253C24—31no—
254C24C2×D3211yescharacteristic
255C2×D4—32no—
256C2×D4—32no—
257C2×D4—62no—
258C2×D4—62no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—32no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—34no—
266C22×C4—61no—
267C22×C4—62no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4—92no—
271C22×C4—92no—
272C22×C4—92no—
273C22×C4—92no—
274C22×C4—182no—
275C22×C4D3×Dic312yes—
276C4⋊C4—34no—
277C4⋊C4—182no—
278C4⋊C4—182no—
279C4⋊C4—184no—
280C22⋊C4—34no—
281C22⋊C4—32no—
282C22⋊C4—32no—
283C22⋊C4—182no—
284C22⋊C4—182no—
285C22⋊C4—184no—
286C22⋊C4—182no—
287C22⋊C4—182no—
288C42—34no—
289C42—31no—
290C42—92no—
291C42—91no—
292C42—181no—
293C42—181no—
294C42—182no—
295C42—181no—
296C22×C3—31no—
297C22×C3—32no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—41no—
303C22×C3—61no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—62no—
307C22×C3—81no—
308C22×C3—122no—
309C22×C3—122no—
310C22×C3#87211yescharacteristic, complemented
311C22×C3C22×S411yescharacteristic
312C2×D3—42no—
313C2×D3—82no—
314A4—21no—
315A4#92611yescharacteristic, complemented
316C3×C4—32no—
317C3×C4—42no—
318C3×C4—62no—
319C3×C4—62no—
320C3×C4—62no—
321C3×C4—82no—
322C3×C4—242no—
323C3×C4C4×S412yes—
324Dic3—22no—
325Dic3—42no—
326Dic3—62no—
327Dic3—122no—
328Dic3—122no—
329Dic3—242no—
330Dic3—244no—
331C32—41noSylow
332C23—31no—
333C23—31no—
334C23—32no—
335C23—32no—
336C23—62no—
337C23—61no—
338C23—62no—
339C23—61no—
340C23#379812yes—
341C23C2×D3×Dic311yescharacteristic
342D4—64no—
343D4—62no—
344D4—62no—
345C2×C4—31no—
346C2×C4—32no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—32no—
353C2×C4—61no—
354C2×C4—64no—
355C2×C4—62no—
356C2×C4—61no—
357C2×C4—62no—
358C2×C4—61no—
359C2×C4—64no—
360C2×C4—94no—
361C2×C4—92no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—182no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—182no—
370C2×C4—182no—
371C2×C4D3×S411yescharacteristic
372C2×C3—32no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—42no—
376C2×C3—41no—
377C2×C3—82no—
378C2×C3—81no—
379C2×C3—122no—
380C2×C3C2×C4×S411yescharacteristic
381C2×C3#591212yes—
382D3—82no—
383C22—32no—
384C22—31no—
385C22—32no—
386C22—32no—
387C22—31no—
388C22—31no—
389C22—61no—
390C22—62no—
391C22—62no—
392C22—62no—
393C22—122no—
394C22—122no—
395C22#6385511yescharacteristic, complemented
396C22C2×D3×S411yescharacteristic
397C4—32no—
398C4—62no—
399C4—62no—
400C4—62no—
401C4—62no—
402C4—182no—
403C4—362no—
404C4—362no—
405C4Dic3×S412yes—
406C3—41no—
407C3—81no—
408C3#8528711yescharacteristic, complemented
409C2—31no—
410C2—31no—
411C2—32no—
412C2—122no—
413C2#1059642612yes—
414C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×S4
commutatorC2×C3×A4C22×C4
FrattiniC22C2×D3×S4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
cosocleC22×C3×A4C23
hypocenterC3×A4#117
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×Dic3×A4C2semidirect
3. C4×C3⋊SL(2,ℤ/4)C2semidirect
5. SL(2,ℤ/4)⋊Dic3C2semidirect
13. C3×C4×S4C4semidirect
18. C22×C4×Dic3D3semidirect
56. C4×S4Dic3semidirect
160. C4×Dic3S4semidirect
164. C3×A4#117semidirect
310. C22×C3#872semidirect
315. A4#926semidirect
395. C22#63855semidirect
408. C3#85287semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
C4×Dic3×A42
C4×C3⋊SL(2,ℤ/4)2
(C2×S4)⋊Dic32
SL(2,ℤ/4)⋊Dic32
#852873
#823183
#638554