Finite Group Databasebeta

← #11922857

Subgroups — #11922857

Order 1152.

842 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
3#10593647C212yesmaximal
4C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
5C4×C3⋊SL(2,ℤ/4)C212yesmaximal, complemented
6#85307—33nomaximal
7#82161—31nomaximal
8C22×C3⋊S4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
13C4×C3⋊S4C414yescomplemented
14#64101—41nomaximal
15C2×C4×S4—33no—
16C22×C3×C4⋊C4D311yescharacteristic, complemented
17C2×C4×#67—31no—
18C2×#892—31no—
19A4×C4⋊C4D313yescomplemented
20(C2×S4)⋊C4—36no—
21C4⋊SL(2,ℤ/4)—33no—
22C4×SL(2,ℤ/4)—36no—
23#5213—32no—
24#5212—31no—
25C4×#217—32no—
26#4990—32no—
27#4987—31no—
28#4978—32no—
29#4977—31no—
30Dic3×C4⋊C4—31no—
31C22×C3×A4C2311yescharacteristic
32C2×C3⋊S4—21no—
33C2×C3⋊S4—21no—
34C2×C3⋊S4C2×C412yes—
35C2×C4×C3⋊D3—41no—
36C3×C4×A4—22no—
37C3×C4×A4C2×C412yes—
38C3⋊SL(2,ℤ/4)—22no—
39C3⋊SL(2,ℤ/4)C2×C412yes—
40C32×C4⋊C4—41no—
41#3752—42no—
42C4⋊(C3⋊Dic3)—41no—
43C4×C3⋊Dic3—42no—
44#2352—91noSylow
45C22×S4—33no—
46C23×C3×C4C2×D312yes—
47C23×C3×C4C2×D311yescharacteristic
48C22×#67—31no—
49C22×C4×D3—31no—
50C2×C4×A4C2×D316yes—
51C2×C4×A4C2×D313yes—
52C2×SL(2,ℤ/4)—33no—
53C2×SL(2,ℤ/4)—36no—
54C4×S4—312no—
55C2×C3×C4⋊C4—31no—
56C2×C3×C4⋊C4—31no—
57C2×C3×C4⋊C4—32no—
58C2×#217—31no—
59C2×#217—32no—
60C4×#67—34no—
61C4×#67—34no—
62C2×(C2×D3)⋊C4—31no—
63C2×(C2×D3)⋊C4—32no—
64C2×(C2×D3)⋊C4—32no—
65C2×C4⋊Dic3—31no—
66C2×C4⋊Dic3—31no—
67C2×Dic3⋊C4—31no—
68C2×C4×Dic3—31no—
69C2×C4×Dic3—32no—
70C2×C4×Dic3—32no—
71#892—61no—
72#892—62no—
73#892—61no—
74#892—123no—
75C3⋊#95—32no—
76C3⋊#95—32no—
77C3⋊#95—32no—
78C3⋊#95—32no—
79C3⋊#95—32no—
80C22×C3⋊D3—41no—
81C2×C3×A4C22×C411yescharacteristic
82C2×C3×A4#4111yescharacteristic
83C2×C3×A4#4111yescharacteristic
84C3⋊S4—22no—
85C2×C32×C4—41no—
86C2×C32×C4—42no—
87C2×C3⋊Dic3—41no—
88C2×C3⋊Dic3—42no—
89C4×C3⋊D3—44no—
90C2×C4×D4—91no—
91C2×#117—91no—
92C22×C4⋊C4C3⋊D311yescharacteristic, complemented
93#388—92no—
94#387—92no—
95#386—92no—
96#382—91no—
97#380—91no—
98#380—91no—
99C4×C4⋊C4—91no—
100C4×C22⋊C4—92no—
101C24×C3C22×D311yescharacteristic
102C23×D3—31no—
103C22×A4C22×D313yes—
104C2×S4—36no—
105C2×S4—63no—
106C2×S4—63no—
107C22×C3×C4—22no—
108C22×C3×C4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—32no—
111C22×C3×C4—32no—
112C22×C3×C4—32no—
113C22×C3×C4—62no—
114C22×C3×C4C4×D312yes—
115C2×#67—32no—
116C2×#67—32no—
117C2×#67—61no—
118C2×#67—61no—
119C2×#67—61no—
120C2×#67—61no—
121C22×Dic3—31no—
122C22×Dic3—31no—
123C22×Dic3—31no—
124C22×Dic3—32no—
125C22×Dic3—32no—
126C2×C4×D3—34no—
127C2×C4×D3—61no—
128C2×C4×D3—62no—
129C2×C4×D3—61no—
130C2×C4×D3—123no—
131C4×A4—26no—
132C4×A4C4×D316yes—
133SL(2,ℤ/4)—36no—
134SL(2,ℤ/4)—66no—
135C3×C4⋊C4—32no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—43no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4S411yescharacteristic, complemented
140#217—32no—
141#217—32no—
142#217—62no—
143#217—62no—
144(C2×D3)⋊C4—34no—
145(C2×D3)⋊C4—62no—
146(C2×D3)⋊C4—62no—
147(C2×D3)⋊C4—126no—
148(C2×D3)⋊C4—122no—
149C4⋊Dic3—61no—
150C4⋊Dic3—62no—
151C4⋊Dic3—61no—
152C4⋊Dic3—123no—
153Dic3⋊C4—34no—
154C4×Dic3—34no—
155C4×Dic3—62no—
156C4×Dic3—62no—
157C4×Dic3—126no—
158C4×Dic3—122no—
159C22×C32—41no—
160C2×C3⋊D3—42no—
161C2×C3⋊D3—81no—
162C2×C3⋊D3—81no—
163C3×A4#11711yescharacteristic, complemented
164C32×C4—42no—
165C32×C4—82no—
166C3⋊Dic3—42no—
167C3⋊Dic3—82no—
168C22×D4—91no—
169C23×C4—91no—
170C23×C4C2×C3⋊D312yes—
171C23×C4C2×C3⋊D311yescharacteristic
172C4×D4—94no—
173C4×D4—94no—
174#117—181no—
175#117—182no—
176#117—181no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—91no—
181C2×C4⋊C4—91no—
182C2×C4⋊C4—91no—
183C2×C22⋊C4—91no—
184C2×C22⋊C4—91no—
185C2×C22⋊C4—92no—
186C2×C22⋊C4—92no—
187C2×C22⋊C4—92no—
188C2×C42—91no—
189C2×C42—92no—
190C2×C42—92no—
191#95—92no—
192#95—92no—
193#95—92no—
194#95—92no—
195#95—92no—
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—31no—
199C23×C3—31no—
200C23×C3C2×C4×D311yescharacteristic
201C23×C3#23711yescharacteristic
202C23×C3#23911yescharacteristic
203C22×D3—32no—
204C22×D3—61no—
205C22×D3—61no—
206C22×D3—61no—
207C22×D3—61no—
208C22×D3—61no—
209C22×D3—61no—
210C22×D3—123no—
211C2×A4C2×C4×D313yes—
212C2×A4#23713yes—
213C2×A4#23913yes—
214S4—66no—
215C2×C3×C4—31no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—43no—
221C2×C3×C4—46no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4C2×S412yes—
228C2×C3×C4C2×S411yescharacteristic
229#67—62no—
230#67—62no—
231#67—62no—
232#67—62no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—123no—
249C2×Dic3—126no—
250C4×D3—64no—
251C4×D3—64no—
252C4×D3—1212no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C3⋊D3—82no—
257C24—91no—
258C24C22×C3⋊D311yescharacteristic
259C2×D4—92no—
260C2×D4—92no—
261C2×D4—181no—
262C2×D4—181no—
263C2×D4—181no—
264C2×D4—181no—
265C22×C4—22no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—32no—
269C22×C4—32no—
270C22×C4—32no—
271C22×C4—62no—
272C22×C4—94no—
273C22×C4—91no—
274C22×C4—91no—
275C22×C4—91no—
276C22×C4—92no—
277C22×C4—92no—
278C22×C4—181no—
279C22×C4—182no—
280C22×C4—181no—
281C22×C4C4×C3⋊D312yes—
282C4⋊C4—32no—
283C4⋊C4—31no—
284C4⋊C4—61no—
285C4⋊C4—94no—
286C4⋊C4—181no—
287C4⋊C4—182no—
288C4⋊C4—181no—
289C4⋊C4C3⋊S411yescharacteristic, complemented
290C22⋊C4—94no—
291C22⋊C4—92no—
292C22⋊C4—92no—
293C22⋊C4—182no—
294C22⋊C4—182no—
295C22⋊C4—182no—
296C22⋊C4—182no—
297C22⋊C4—362no—
298C42—94no—
299C42—182no—
300C42—182no—
301C42—362no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—43no—
312C22×C3—61no—
313C22×C3#89211yescharacteristic, complemented
314C22×C3C22×S411yescharacteristic
315C2×D3—62no—
316C2×D3—62no—
317C2×D3—62no—
318C2×D3—126no—
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—243no—
324C2×D3—243no—
325A4#89213yescomplemented
326C3×C4—22no—
327C3×C4—32no—
328C3×C4—46no—
329C3×C4—62no—
330C3×C4—86no—
331C3×C4C4×S412yes—
332Dic3—62no—
333Dic3—62no—
334Dic3—62no—
335Dic3—126no—
336Dic3—122no—
337Dic3—122no—
338Dic3—246no—
339C32—41noSylow
340C23—31no—
341C23—31no—
342C23—31no—
343C23—31no—
344C23—92no—
345C23—181no—
346C23—181no—
347C23—181no—
348C23—181no—
349C23—181no—
350C23—181no—
351C23C2×C4×C3⋊D311yescharacteristic
352C23#383011yescharacteristic
353C23#383211yescharacteristic
354D4—182no—
355D4—182no—
356D4—182no—
357D4—182no—
358C2×C4—31no—
359C2×C4—32no—
360C2×C4—32no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—61no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—92no—
369C2×C4—92no—
370C2×C4—92no—
371C2×C4—184no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4—184no—
379C2×C4—182no—
380C2×C4—182no—
381C2×C4—182no—
382C2×C4—182no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4C2×C3⋊S412yes—
386C2×C4C2×C3⋊S411yescharacteristic
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—43no—
392C2×C3—43no—
393C2×C3—43no—
394C2×C3C2×C4×S411yescharacteristic
395C2×C3#591411yescharacteristic
396C2×C3#591911yescharacteristic
397D3—122no—
398D3—246no—
399C22—31no—
400C22—31no—
401C22—31no—
402C22—31no—
403C22—31no—
404C22—31no—
405C22—31no—
406C22—31no—
407C22—31no—
408C22—61no—
409C22—182no—
410C22—182no—
411C22—182no—
412C22—361no—
413C22—361no—
414C22—361no—
415C22—361no—
416C22#6410111yescharacteristic, complemented
417C22C22×C3⋊S411yescharacteristic
418C4—22no—
419C4—32no—
420C4—62no—
421C4—182no—
422C4—182no—
423C4—182no—
424C4—362no—
425C4—362no—
426C4C4×C3⋊S412yes—
427C3—43no—
428C3#8530711yescharacteristic, complemented
429C2—31no—
430C2—31no—
431C2—31no—
432C2—31no—
433C2—362no—
434C2C2×C4×C3⋊S411yescharacteristic
435C2#1059657711yescharacteristic
436C2#1059658211yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutatorC2×C3×A4C22×C4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3⋊S4C22
cosocleC22×C3×A4C23
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4#117
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×A4×C4⋊C4C2semidirect
4. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
5. C4×C3⋊SL(2,ℤ/4)C2semidirect
13. C4×C3⋊S4C4semidirect
16. C22×C3×C4⋊C4D3semidirect
19. A4×C4⋊C4D3semidirect
92. C22×C4⋊C4C3⋊D3semidirect
139. C3×C4⋊C4S4semidirect
163. C3×A4#117semidirect
289. C4⋊C4C3⋊S4semidirect
313. C22×C3#892semidirect
325. A4#892semidirect
416. C22#64101semidirect
428. C3#85307semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊S42
C3×A4×C4⋊C42
#105936472
C4⋊(C3⋊SL(2,ℤ/4))2
C4×C3⋊SL(2,ℤ/4)2
#853073
#821613
#641014